# Philosophy of Logic

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On Logic, Aristotle and Rand

Normativity of Logic – Kant v. Rand

Normativity of Logic – Robert Hanna

Between False, Invalid, and Meaningless

Meaningless Tautology

Analytic-Synthetic Distinction
Part 1 – Quine
Part 2 – White and Rand-Peikoff
Part 3 – Objective Analyticity

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From First Section of "Induction on Identity"
(1990 – Objectivity V1N2)

"Existence is Identity, Consciousness is Identification. . . . Logic is the art of noncontradicory identification" (AS 1016). I think this view is correct, fresh, and important.

"He whose subject is existing things qua existing must be able to state the most certain principle of all things. This is the philosopher, and the most certain principle of all is that regarding which it is impossible to be mistaken. . . . Which principle this is, let us proceed to say. It is that the same attribute cannot at the same time belong and not belong to the same subject and in the same respect" (Metaph. 1005b8–20). There are no contradictions in reality. Contradiction is the fundamental indicator of discordance with reality. Contradiction is the fundamental fallacy of deductive inference.

Noncontradiction is the fundamental rule of valid deductive inference because identity is the fundamental law of reality. The law of identity is commonly stated as "A is A" or "a thing is itself." An existent is itself and not something else. Then an inference that something both is a certain thing and not a certain thing is faulty. A thing cannot both be and not be a certain thing. The ground of noncontradiciton, the ground of validity in deductive inference, is the law of identity.

In moving from identity to noncontradiction, we move from "a thing is itself" to "a thing is itself and not something else." This maneuver draws to the fore our knowledge that any thing is a certain, specific thing. A thing is something. A thing is what it is. "To exist is to be something. . . . It is to be an entity of a specific nature made of specific attributes (AS 1016). This is the law of identity come of age.

Gottfried Wilhelm von Leibniz was cognizant of the intimate connection between the principle of (non)contradiction and the principle of identity. (Nicolaus of Autrecourt cognized much of the connection already in the fourteenth century; see Weinberg 1967, 9–30.) "The first of the truths of reason is the principle of contradiction or, what comes to the same thing, that of identity (Leibniz 1969, 385). By truths of reason he meant necessary propositions, and by necessary propositions he meant propositions whose contradictories cannot be true. By truth he meant correspondence of a proposition with reality, possible or actual (Leibniz 1981; see also Rescher 1979, 130–34).

Primitive truths of reason, Leibniz called identicals. Among affirmative identicals are "the equilateral rectangle is a rectangle" and "A is A" and "each thing is what it is" and "at any given time a thing is as it is." As negative identicals, we have "what is A cannot be non-A." There are also negative identicals that are called disparates. An example would be "warmth is not the same thing as color" (Leibniz 1981, 4.2.361–63; 1966, 306).

The principle of noncontradiction (or identity), according to Leibniz, tells us only what is possible, not what is actual. Its truth is founded in the essence of things. It is an innate truth, and it applies to ideas that are innate (such as geometrical ideas), which is to say ideas not derived from the senses, but ideas found ready in the mind. The principle of noncontradiction also applies to sensible truths, as when we say "sweet is not bitter." The actual is among the possible, and Leibniz might admit that the principle of noncontradiction has application to sensible truths without going on to sink the principle into the bedrock of the actual.

As Leibniz saw it, truths of reason and their necessities do come from outside us "since all that we do consists in recognizing them, in spite of ourselves and in a constant manner." To demonstrate the existence of these necessities,
"I have taken for granted that we thing and that we have sensations. So there are two absolute general truths: truths that is, which tell of the actual existence of things. One is that we think; the other, that there is a great variety in our thoughts. From the former it follows that we are; from the latter, that there is something other than us, that is to say, something other than that which thinks, which is the cause of the great variety of our experiences. Now the one of these is just as incontestable and as independent as the other . . . ." (1966, 307)

In contrast to truths of reason, in Leibniz's analysis, are truths of fact. Truths of fact are known by observation and induction, not by deduction. Their truth "is founded not in the essence (of things) but in their existence; and they are true as though by chance" (Leibniz 1963, 274). Truths of existence are contingent. Truths of existence are true. They have hypothetical and physical necessity, if not absolute and logical necessity (Mates 1986, 116–19; Ishiguro 1982; Wilson 1976). But denial of existential truths does not result in contradiction, at least not in the finite mind of man. Really?

—Aristotle 1941 [c. 348–322 B.C.]. The Basic Works of Aristotle. R. McKeon, editor. Random House.
—Ishiguro, H. 1982. Leibniz on Hypothetical Truths. In Leibniz: Critical and Intepretive Essays. M. Hooker, editor. University of Minnesota.
—Leibniz, G.W. 1963 [1666]. Dissertation on the Art of Combinations. Quoted in Copleston, F. 1963. A History of Philosophy, volume 4. Image Books.
——. 1966 [1675]. Letter to Simon Foucher. In The Philosophy of the 16th and 17th Centuries. R.H. Popkin, editor. The Free Press.
——. 1969 [1692]. Critical Thoughts on the General Part of the Principles of Descartes. In G. W. Leibniz: Philosophical Papers and Letters. 2nd ed. L.E. Loemker, translator.
——. 1981 [1704]. New Essays on Human Understanding. P. Remnant and J. Bennett, translators. Cambridge.
—Mates, B. 1986. The Philosophy of Leibniz: Metaphysics and Language. Oxford.
—Rand, A. 1957. Atlas Shrugged. Random House.
—Rescher, N. 1979. Leibniz: An Introduction to His Philosophy. Rowman and Littlefield.
—Weinberg, J.R. 1969 [1948]. Nicolaus of Autrecourt. Greenwood Press.
—Wilson, M.D. 1976 [1972]. On Leibniz's Explication of "Necessary Truth." Reprinted in Leibniz: A Collection of Critical Essays. H.G. Frankfurt, editor. Notre Dame.

Remark

The preceding contentions of Aristotle, Leibniz, and Rand are about logic, they are not presentations of the discipline that is logic. I wrote, in step with Leibniz and Rand, "the ground of validity in deductive inference is the law of identity." I should note that validity in deductive logic is more than consistency. The logical formula "p and not-q" is called consistent because it will be sometimes true, specifically, if the proposition p is true and the proposition q is false. The formula "p and not-p" is called inconsistent because it will be always false, whether p is true or false. The formula "if p, then p" is called valid because it will be always true, whether p is true or false (Methods of Logic [Quine 1982, 40–41]). There are many formulas that are valid, though, unlike the preceding example, they are not seen to be valid so quickly.

To claim that logic is the art of noncontradictory identification is to set logic in its relations to metaphysics and to epistemology. That claim and my claim that the ground of validity in deductive inference is the law of identity are claims about logic. They belong to the philosophy of logic.

Philosophy of logic is something in which philosophers engaged before this class of inquiry came to have its present name and scope. Kant was engaged in it when he conceived of transcendental logic as distinct from general logic. The latter was simply the discipline of logic itself, as it had been developed (mainly by Aristotle) to Kant's time, whereas the former was dealing with questions in philosophy of logic. Similarly, when Kant conceives of general logic as "the science of the necessary laws of the understanding," he is engaged in philosophy of logic.

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At a recent meeting of the American Philosophical Association, Gila Sher delivered a paper titled “Is Logic in the Mind or in the World?” According to her abstract, she presented an outline in this paper of a unified answer to these questions among others: Is logic in the mind or in the world? Does logic need a foundation? What is the main obstacle to a foundation for logic? Can it be overcome?

Part III of Penelope Maddy’s Second Philosophy (2007 Oxford) is a contemporary, naturalist philosophy of logic:

III.1 Naturalistic Options: Psychologism / Empiricism / Conventionalism / Analyticity

“I hope this brief survey creates or reinforces existing discomfort with the available options, and thus provides an incentive to search for an alternative. My own interest in finding another way springs partly from the shortcomings of the views just rehearsed and partly from an embarrassingly hazy impression from elementary arithmetic: 2 + 2 = 4 seems to me to report something about the world, and that something seems closely connected with logic. But oddball motivations aside, my goal in what follows is to sketch an . . . account of the ground of logical truth that differs sharply from those sketched above.” (206)

III.2 Kant on Logic: Analytic a priori / The Discursive Intellect / Analyticity Revisited

III.3 Undoing the Copernican Revolution

III.4 The Logical Structure of the World: Objects / Properties and Relations / Dependencies / Indeterminacy

III.5 The Logical Structure of Cognition: Objects / Properties, Relations, and Dependencies / From the World to Cognition

III.6 The Status of Rudimentary Logic

III.7 From Rudimentary to Classical Logic

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Our standard modern logic has come to be called classical logic. This logic expanded and revised the logic of Aristotle as it had been developed up to the time of Kant. Classical logic, as taught in texts such as R. L. Simpson’s Essentials of Symbolic Logic and W. V. O. Quine’s Methods of Logic, is the culmination of innovations by Boole, De Morgan, Jevons, Peirce, and above all, Frege (1879).

Standard modern logic is called classical to distinguish it from extensions of it in modal logics and from rivals of it, such as intuitionist logic, many-valued logics, paraconsistent logics, fuzzy logics, quantum logics, and relevance logics. This last and modal logic, as well as the ways in which classical logic improves on Aristotelian logic (e.g., existential fallacy), stand out as promising productive integration with Rand’s metaphysics and conception of logic.

Predications are conceptual identifications. Edward Zalta takes the discipline of logic to be “the study of the forms and consequences of predication” (2004, 433).* That conception of logic fits well with Rand’s conception of logic as “the art of non-contradictory identification.” The ramifications of Rand’s idea that “logic rests on the axiom that existence exists,” combined with her E-, I-, and C-axioms, need to be charted through the terrain of classical logic, modal logic, and relevance logic.

* “In Defense of the Law of Non-Contradiction” in The Law of Non-Contradiction, Priest, Beall, and Armour-Garb, editors (Oxford). Two beginning works have addressed how predication can be taken under Rand’s thesis “existence is identity.” These are the final section (IX) of my 1991 Objectivity essay “Induction on Identity” (V1N3) and David Kelley’s paper “Concepts and Propositions” read at the 1996 summer seminar of the Institute of Objectivist Studies. See also.

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Remark

Rand’s metaphysics and epistemology may favor certain developments in two nonclassical logics: modal logic and relevance logic. But within our classical logic itself, it is unlikely that Rand’s 1957 metaphysics and the conception of logic she situates in it have any definite ramifications for second-order logic, only for elements up to first-order predicate logic with quantification and with identity.

Said of any existent, “A is A” can mean either “A is being A, specifically, A is predicatively being A the way it is and not in other ways” (Metaph. 1041a10–26) or it can mean “A is the same as A”. The latter can be divided into the merely verbal, as when we say “a belly is a tummy” or it can be more than merely verbal interchangeability, as when we say “a triangle is a trilateral” or “the morning star is the evening star.”

It is because identity has various bearings in the real that it has various bearings in logic. These would include the license of substituting like for like and the proscription of equivocation. Truth is preserved under the former, spoiled under the latter.

Another bearing of identity in logic is the logical relation of identity, which is usually denoted by the equals-sign in the texts (Copi’s Symbolic Logic, 158–68; Quine’s Methods of Logic, 268–73). Logic assimilates this relation by adding two axioms to those sufficient for the logic of (logical) quantification. One of those additional axioms is: for any a, a = a.

Within the syllogistic, we have the identity formula “Every A is A”. This was being used by logicians at least by the time of Albert the Great (13th cent.). They used it, for example, to prove the convertibility of "No B is A" to "No A is B". They added "Every A is A" to "No B is A" to infer "No A is B", relying on one of Aristotle’s forms of syllogism (first mood of the second figure):

No L is M

Every S is M

No S is L

No B is A

Every A is A

No A is B

(See Kneale and Kneale’s The Development of Logic, 235–36; also.)

The workings of identity in logic can sometimes look like a barren exercise. But these workings are for true thinking about the way the world is.

Edited by Stephen Boydstun

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[...]

~~~~~~~~~~~~~~~~

Remark

Rand’s metaphysics and epistemology may favor certain developments in two nonclassical logics: modal logic and relevance logic. But within our classical logic itself, it is unlikely that Rand’s 1957 metaphysics and the conception of logic she situates in it have any definite ramifications for second-order logic, only for elements up to first-order predicate logic with quantification and with identity.

Said of any existent, “A is A” can mean either “A is being A, specifically, A is predicatively being A the way it is and not in other ways” (Metaph. 1041a10–26) or it can mean “A is the same as A”. The latter can be divided into the merely verbal, as when we say “a belly is a tummy” or it can be more than merely verbal interchangeability, as when we say “a triangle is a trilateral” or “the morning star is the evening star.”

It is because identity has various bearings in the real that it has various bearings in logic. These would include the license of substituting like for like and the proscription of equivocation. Truth is preserved under the former, spoiled under the latter.

Another bearing of identity in logic is the logical relation of identity, which is usually denoted by the equals-sign in the texts (Copi’s Symbolic Logic, 158–68; Quine’s Methods of Logic, 268–73). Logic assimilates this relation by adding two axioms to those sufficient for the logic of (logical) quantification. One of those additional axioms is: for any a, a = a.

[...]

David Kelley in his 2003 lecture "Concepts and Categories" commented that modal logic rests on the notion of "possible worlds" (or "possible interpretations" or "possible substitutions"). This notion steals the concept "possibility" from the conditions that make something possible. If this interpretation is correct, then modal logic is fundamentally flawed. And since relevance logic depends on modal logic, it too is flawed on this view.

One issue a philosophy of logic has to address is the regimentation of basic axioms. I take the axiom of identity, in the form "A is A," to imply three things:

• Each "A" is a concept, not a concrete.
• The "is" stands for an "identification" instance of "integration."
• The form "a = a" is not the same as "A is A" because "a" is a concrete (e.g., "Hesperus is Phosphorus").

My take on the meanings of "is" is here.

Edited by Thom T G

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Philosophy of Modal Logic

From the thread “The Analytic-Synthetic Dichotomy”

Note A

Rand's definition of logic is given on page 1016 of Atlas. "Logic is the art of non-contradictory identification." She remarks also on that page "logic rests on the axiom that existence exists."

It follows, I notice, that it is not logically possible that nothing exists. It is not logically possible that existence might not have been. One logician, David Bostock, remarks to the contrary in his Intermediate Logic (1997 Oxford). He maintains that "it is a possibility that nothing should have existed at all" (354). The two opposing views, Rand's and Bostock's, commend different ways of developing modal logic. I won't pursue that just now.

Let me indicate, instead, a little more of Rand's conception of logic. Her definition of logic as the art of non-contradictory identification is made in the context of having cast consciousness as identification. In this conception, logic is embedded in wider processes of identification, all of them relying on Rand's thesis that no existents are without identity.

Note B

I was too hasty and overstepping when I said that Rand’s position on metaphysics and logic commends ways of developing modal logic different from what Bostock would or should commend given their divergence on the following: Rand’s conception of logic as resting on the axiom existence exists entails, I maintain, that it is not logically possible that existence might not have been. Whereas Professor Bostock maintains that it is a logical possibility that nothing should have existed at all.

Perhaps the two perspectives should indeed lead to different developments of modal logic, but I have not followed that through to see if it is so. Here is what difference is evident from Bostock's Chapter 8, "Existence and Identity."

Sticking to Rand’s idea that logic rests on the axiom that existence exists, I would say that all logically possible worlds are relatable to the actual world and that all logically possible worlds are relatable to each other via the actual world. That is, the appropriate modal logic for broadly logical necessity is some variety of S5, which is a normal modal logic.

Bostock uses his contention that it is a logical possibility that nothing should have existed at all to motivate a dilemma: Either one must plunk for a non-normal modal logic as the logic appropriate for logical necessity or one must adopt a certain sort of non-orthodox classical logic (where classical here means non-modal). He favors the latter alternative, and for all I know, it may be that that is a classical logic square with Rand’s conception of metaphysics and logic. But from an Objectivist perspective, neither the dilemma nor the non-orthodox classical logic can be motivated by an alleged logical possibility that nothing should have existed at all.

~~~~~~~~~~~~~~~~

Thom,

Modal logic begins with this much logic already in hand: first-order predicate logic, outfitted with quantification and identity. In our modern logic texts, we first learn propositional logic, then predicate logic, then add to the latter, quantification (all and some) and identity (the). Quine’s text mentioned above is superb for logic to that point. In modal logic, we study the usual aspects of logic—consistency, validity, soundness, completeness—with the added expressions necessarily (or it is necessary that) and possibly (or it is possible that).

For history of modal logic, I would turn to that entry in the Index of Kneale and Kneale’s

The Development of Logic and to the ninth chapter “Modal Logic from Kant to Possible World Semantics” in The Development of Modern Logic. A good text on modal logic today is Modal Logic for Philosophers (2006 Cambridge) by James Garson.

The use of the phrase “possible world” in modal logic is as a marker for a context of possibility. (See also situation semantics.) Professor Garson writes:

“A pervasive feature of natural languages is that sentences depend for their truth value on the context or situation in which they are evaluated. For example, sentences like ‘It is raining’ and ‘I am glad’ cannot be assigned truth values unless the time, place of utterance, and the identity of the speaker is known. The same sentence may be true in one situation and false in another. In modal language, where we consider how things might have been, sentences may be evaluated in different possible worlds.” (57)

One might try “to repair ordinary language by translating each of its context-dependent sentences into a complex one that makes the context of its evaluation explicit” (57). Consider all one would need to specify in order to do that for it is raining. “There is a more satisfactory alternative. . . . [Let] the account of truth assignment [be] adjusted to reflect the fact that the truth value depends on the context. The central idea of intensional semantics is to include contexts in our description of the truth conditions of sentences in this way. / To do this [for modal logic], let us introduce the set W, which contains the relevant contexts of evaluation. Since logics for necessity and possibility are the paradigm modal logics, W will be called the set of (possible) worlds. But in the same way that necessarily is a generic operator, W should be understood as a generic set including whatever contexts are relevant for the understanding of necessarily at issue.” (57–58)

The discipline of logic is for discovery and teaching of the most general forms of truth-preservation in conceptual thought. In modal logic, “no attempt is made in intensional semantics to fix the ‘true nature’ of W, and there is no need to do so. When one wishes to apply modal logic to the analysis of a particular expression of language, then more details about what the members of W are like will be apparent” (58).

Edited by Stephen Boydstun

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The notion "truth condition" is the lynchpin of representationalism, i.e., the ontology of cognition that mental contents are true only if they correspond to things in the external world. Within representationalism, as reported by David Kelley, "truth conditions" are the internal propositional contents of a belief, specified as conditions from the proposition's intentional meaning, and against which external facts are compared so as to assess truth. (DK "WIK" 6a-b )

I take Professor James Garson to say, therefore, that a sentence, e.g., "It is raining," is not an instance of a judgment reached by some conscious mind by a perceptual observation or a process of reason, but is instead an amalgamated bundle of representational internal contents that constitutes the "context of evaluation" (e.g., Wn) to be compared somehow to the external facts of the actual world. This view entails the mind to be some colossal consciousness (supreme or collective) in order to represent all logically permutable contents that may relate truth functionally to the actual world.

From this view of logic, I can see why logician David Bostock may imagine that his modal logic can dispense with the actual world, that it is entirely possible that nothing should have existed. The internal contents are rich enough without the need for external comparison. After all, the underlying ontology of representationalism presupposes the total independence of the contents of consciousness from the external facts of the world.

On the subject of predicate logic, don't you think, Stephen, that it is a system that entirely leaves out "identification"? If predication is conceptual identification, predicate logic does not help anyone to predicate. Consider the proposition "It is raining." In predicate logic, it is symbolized as Ra, where R is a pure abstraction, standing for the predicate "is raining" in the world; and a is a constant, standing for a bare particular. Somehow, the juxtaposition of R and a is supposed to yield a belief with truth conditions to an atomic fact? There is something lacking about this ontology. Adding functional connectives and quantifications to the system only compounds this lack. Without identification, predicate logic is at best a second-person tool for symbolic manipulation, not a first-person art. Metalogically, it fails Ayn Rand's definition of logic.

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Correction to #4:

Where I wrote “and identity (the)” I should have written “and identity (the very one).”

~~~~~~~~~~~~~~~

Thom,

“Socrates is human” is a singular proposition. The logical subject is Socrates; the logical predicate—that which is asserted of the subject—is is human.

“The Washington Monument is partly mineral” is a singular proposition. The logical subject is The Washington Monument; the logical predicate is is partly mineral.

Qa is a compact way of denoting any singular proposition, where the a stands for the needed singular subject, any such one, and the Q stands for the needed predicate, any such one, asserted of the subject. To introduce abstract notation for compact reference to certain classes of things is to aid comprehension, not to deny the particular and specific identities of things in the class.

All humans are mortal.

Socrates is human.

Therefore Socrates is mortal.

To capture under a compact notation the class of arguments to which that one belongs is only to facilitate comprehension of truth-preserving forms in the part of logic we call predicate logic. The abstractness of logic is no saying against its setting within processes of identification.

Good point you make about the perspective of Bostock. I would rather you found another name than representationalism for the faulty perspective at which you are driving. Concepts and propositions are indeed representations (in a way that percepts are not). I could not figure out more specifically where Bostock was coming from on the specific point in contention. He was so opaque about it. I wondered if he were repeating a finding in logic itself, of which I am ignorant. Then too, I wondered if he were bringing in an extra-logical thesis such as is affirmed in concepts of God in the accounts by Aquinas, Leibniz, or Whitehead.

Edited by Stephen Boydstun

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Stephen,

I take the name representationalism directly from DK's The Evidence of the Senses (p. 10). While there is in the cognitive sciences a conventional theory of perception by the same name, that that which we perceive are our percepts, which are the sense data, ideas, images, etc.; the sense for which I use this name is much broader and reaches toward the ontology of cognition. From this perspective, there are three basic models: realism, representationalism, and phenomenalism. (TEOTS 7)

In modern logic, only names of concretes can be translated into the logical subjects of a standard logical form. All abstractions must be moved over to the logical predicates. So, "All humans are mortal" seemingly has no explicit logical subjects, only logical predicates, namely, "is human" and "is mortal." Why is this so? What is the philosophy underlying this bifurcation?

Traditional logic by contrast takes the term "humans" in the above sentence to be a logical subject, not a "disguised predicate." ("CAP" 12) But it also takes concretes to be classical; i.e., "Socrates" is an an abstract classical of one member. What is the philosophy underlying the allowance of abstractions but only abstractions to be both subjects and predicates?

The sentence "The man holding a glass of vinegar and standing in the middle of the crowd is loquacious"--this sentence is understood as a singular proposition. Yet when translated into logical formulas, it takes the compound form in modern logic, not the expected form Qa; and it takes the universal statement form in traditional logic. Errors in translation like these should signal a deeper error underlying the logics.

I contend that the philosophy under modern logic is representationalism ("WIK" 5), and that the philosophy under traditional logic is phenomenalism or more specifically neo-Platonic idealism ("CAP" 18). Moreover, I find one sentence in "Concepts and Propositions" to be significant: "As far as I can see, the epistemological considerations I have discussed [concerning the Objectivist theory of propositions] do not support either logical theory, or favor one above the other." (Ibid. 12) I take this to mean that the existing logics, modern and traditional, must be set aside, for they cannot rest on a realist, Aristotelian foundation.

The discipline of logic needs to be reformed anew if realism is the proper ontology of cognition. In which case, this is an area Objectivists surely should go forth to stake out and explore.

Edited by Thom T G

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The paper “Concepts and Propositions” to which Thom referred is by David Kelley (1996). It is available online here. The statement Thom quotes is on page 12. The interpretation Thom gives to that statement is incorrect. Dr. Kelley was only saying that the theory of propositions he offers—a theory of how concept and propositions fit together and relate to reality—does not appear to favor traditional term logic over modern predicate logic nor favor the latter over the former. He takes his Objectivist theory to be consonant in large measure with both. Ditto for my 1991. (Cf.)

~~~~~~~~~~~~~~~~

I mentioned above that standard modern logic (propositional logic plus predicate logic with identity) is called classical logic for two reasons. One is to distinguish it from its extensions into modal logic. The other is to distinguish it from logics that are partly at odds with it, such as intuitionist logic, many-valued logics, paraconsistent logics, fuzzy logics, quantum logics, and relevance logics.

I have written the following note on Quantum Logic.

In her journal The Objectivist, immediately after the issues of the journal containing her "Introduction to Objectivist Epistemology," Rand published Leonard Peikoff's "The Analytic-Synthetic Dichotomy" (1967). Peikoff, speaking also for Rand, added to the dissents that had been raised against validity of the distinction in the preceding couple of decades.

It was Quine's essay "Two Dogmas of Empiricism," published in 1951, that had brought Quine’s debate with Carnap over the analytic-synthetic distinction to widespread attention among philosophers. In this essay, Quine argued against the validity of the distinction. Carnap wanted to maintain a sharp distinction between analytic statements depending entirely on the meanings being used and synthetic statements making assertions about the empirical world. Quine's alternative view had it that all statements face the world as part of a corporate body of statements. On this view, experience bears the same kind of evidential relation to the theoretical parts of natural science as it does to mathematics and logic. (See also Quine's 1960 essay "Carnap and Logical Truth," which is in the collection The Ways of Paradox and Other Essays. "Two Dogmas" is in the collection From a Logical Point of View.)

During the 1950s, Hilary Putnam was also writing about the analyticity of various statements, such as the statement of Rand's in 1957 that a leaf "cannot be all red and all green at the same time." Other philosophers, too, such as Arthur Pap and Morton White, were writing on the analytic-synthetic controversy during the 50s.

In 1968 Putnam published the essay "Is Logic Empirical?" which was renamed "The Logic of Quantum Mechanics" in a later collection of his papers. He proposed a quantum-logical interpretation of quantum mechanics to give an example of how we might have an empirical reason for revising logic.

Hans Reichenbach had attempted an interpretation of quantum mechanics in 1944 using a three-valued truth-functional logic. (Our standard logic is a two-valued truth-functional logic.) The Reichenbach quantum-logical interpretation of quantum mechanics has no adherents today.

Peter Gibbons writes in Particles and Paradoxes: The Limits of Quantum Logic (CUP 1987):

"Far more important than artificial many-valued logics are the quantum logics that arise naturally in the Hilbert space formalism. The strongest of these logics---strongest in the sense of most closely approximating to classical logic---is the logic which mirrors the structure of the set of closed subspaces of Hilbert space. This quantum logic was first studied by the mathematicians Garrett Birkoff and John von Neumann in a classic paper of 1936. By quantum logic we mean Birkoff and von Neumann's quantum logic and the various ways in which it may be formulated as a logic.

"A lattice of propositions is of course not quite a logic, whatever a logic is. But a lattice of propositions has a structure which is at least very much like the structure of a logic. So if quantum logic really is a logic, we should first make it look like a logic. Then we will be in a position to discuss whether it really is a logic.

"Quantum logic can in fact be fairly easily transcribed as a logic in the usual logical styles---as an axiomatic system, as a sequent calculus, and as a natural deduction system. . ." (127-28).

Quantum logics utilize algebraic accounts of quantum theory. They make use of Boolean algebras, partial Boolean algebras, and orthomodular lattices. These structures can be found embedded in Hilbert spaces, those complex topological vector spaces appropriate to quantum mechanics. [see Chapter 7 of The Structure and Interpretation of Quantum Mechanics R.I.G. Hughes (HUP 1989).]

The basic idea is to supersede the classical logical constants of AND, OR, and NOT with the lattice-theoretic (algebraic) operations of MEET, JOIN, and ORTHOCOMPLEMENT. Recall the distribution equivalences for AND and OR that we learn in elementary logic, such as {[p AND (q OR r)] Equiv [(p AND q) OR (p AND r)]}. Corresponding equivalences of distributions do not obtain for MEET and JOIN in the algebra appropriate to quantum mechanics.

It seems to me as to most philosophers of physics today that the word supersedes in the first sentence of the preceding paragraph gets it wrong. The very Hilbert spaces and algebraic structures embedded in them from which some would draw a new logic are themselves deductively certified as sound mathematics using simply classical logic.

It is possible, for all I know at this time, that the mathematics required for quantum theory can be rederived using the quantum logic I described. If that proved not possible, then the quantum logic would not do as a revised general logic. If such a rederivation of the necessary mathematics were executed, there would be the further question of whether other interpretations of quantum mechanics---say the transactional interpretation or sum-over-histories interpretations---were superior to an interpretation requiring a revision of general logic.

So I don't mean to give the impression that revisions and improvements in the discipline that is our modern, classical logic are not possible. It should be noticed, however, that adoption of the quantum logic as more fundamental than our classical logic would not require any loosening of the law of non-contradiction. Rand's conception of logic as the art of non-contradictory identification would still stand.

~~~~~~~~~~~~~~~~

Recent work on quantum logics can be found here.

I delve into Quine’s famous essay here.

Edited by Stephen Boydstun

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“Existence exists—and the act of grasping that statement implies two corollary axioms: that something exists which one perceives and that one exists possessing consciousness, consciousness being the faculty of perceiving that which exists” (AS 1015). Axiomatic status is being conveyed in the preceding statement from a general proposition “Existence exists” to the particular corollary propositions “Something exists” and “One exists having consciousness of that which exists” via a particular act of grasping the meaning and truth of the general assertion “Existence exists.”

“Something exists” is a proposition that on its own would leave open whether more than one thing exists. Affirmation of this proposition does not leave it on its own, of course. And anyway, it is clear from the context of the statement in Rand’s text, which speaks of individual acts of grasping by individual minds, that “Something exists” is to be understood as affirming that multiple individual existents exist.

John Duns Scotus (1265–1308) would say that it would not be a contradiction to deny that individual existents exist, yet affirm the general proposition “Existence exists.” Likewise, according to Scotus, it would not be a contradiction to affirm that stoneness or combustion or redness exist, yet to deny that there are any particular stones or particular occasions of things burning or things being red.

Rand says to the contrary: it is a contradiction to affirm the existence of stoneness, yet deny the existence of individual stones. Fundamentally, “that which exists is concrete” (P-E of Art 23), and stoneness as a nature does not itself amount to a concrete individual. I venture that the conveyance of axiomatic status to Rand’s two implied corollaries is the same as her logical join of stoneness to particular stones. The act of grasping the proposition that there is a character of stoneness implies that one grasps that there are individual stones. The act of grasping that there is space implies that there are regions of space.

It would not be a contradiction to affirm “Existence exists,” yet deny that there are stones or that there is space. It would be a contradiction to affirm that “Existence exists,” yet deny that any particular individual things exist. Such individuals known are particular stones and regions of space, but knowledge of these existents is not axiomatic in Rand’s system because they are not entailed by comprehension of the meaning and truth of “Existence exists.”

A philosophic axiom is a statement identifying the base of knowledge such that the axiom statement is “necessarily contained in all others” (AS 1040). The containment of “There are particular regions of space” within “There is space” has the same logical necessity as that in the containment of “Something exists” within “Existence exists.” But that same logical necessity of containment does not hold “There is space” within “Existence exists.” Nor does that same logical necessity of containment hold “There are regions of space” within “Something exists.” True, “There are regions of space” is contained under some stripe of logical necessity in “There are stones,” but the latter is not contained under any stripe of logical necessity in “Something exists.”

It may well be—as I think it be—that not only is every existent concrete, every existent is physical and therefore spatially situated. These are further general propositions of metaphysics not implied as axioms by Rand’s method.

~~~~~~~~~~~~~~~~

On the Validation of Philosophical Axioms

The theme of the 2005 meeting of the Ayn Rand Society was “Ayn Rand as an Aristotelian.” James Lennox presented a paper “Axioms and Their Validation,” which I have summarized as follows:

Professor Lennox first presented Rand’s views on axiomatic concepts, then Aristotle’s view on axioms and their validation.* He then briefly compared their views in this area.

Lennox stressed that Rand draws her philosophic axioms so as to be highly abstract, yet to be based on concretes given in perception. As is well-known to readers here, Rand’s axiomatic concepts are existence, identity, and consciousness. It was incumbent on Rand to explain “how these concepts, the most abstract of all concepts, are related to the perceptually given” (AV 4). Rand’s answer: The axiomatic status of these concepts derives from the character of their referents. The facts identified by these concepts are directly perceived, and they are fundamental givens implicit in any knowledge or proof procedure. As readers here know, the truth of these identifying concepts cannot be proven, but their axiomatic status can be shown by showing that they are presupposed in any attempt to deny them.

On Rand’s view, these axioms are implicit in every state of awareness of any sentient animal. For humans the axiomatic facts of existence, identity, and consciousness are “perceived or experienced directly, but grasped conceptually.” Explicit conceptual identification of these axiomatic facts provides an ever-present widest conceptual context for all one’s conceptual constructions concerning reality.

Aristotle’s indemonstrable starting points for knowledge are the principle of non-contradiction and the principle of excluded middle. These principles are presuppositions of all demonstration. Aristotle says “we come to know the primaries [of a special discipline such as geometry] by induction; for that is in fact how perception produces the universal in us.” He seems to think the same faculty of reason enables us to know both the starting points of a special discipline and the fundamental principles of demonstration, which are non-contradiction and excluded middle. [Note from SB: For more on this, see Leonard Peikoff’s (1985) “Aristotle’s Intuitive Induction” in The New Scholasticism 59(2):185-99.]

For Aristotle these two principles identify fundamental facts of reality. Because these are the most fundamental facts about reality, cognizance of them is required to comprehend anything of the more special characters of things. Aristotle knows one cannot prove the truth of non-contradiction or excluded middle, but because these truths hold in every area of knowledge, one can always show their merit by showing the dissolution of thought that follows on their denial.

Both Rand and Aristotle see philosophical axioms as explicit identifications of fundamental facts of reality. Lennox goes on to observe that, also for both these philosophers, “anyone who knows anything and anyone seeking to prove anything has grasped, at some tacit level, these fundamental facts. Both insist that any attempt to deny such fundamental truths is self-refuting” (AV 13).

An obvious difference between Aristotle and Rand on philosophic axioms is in their selection of which facts are the axiomatic ones. Rand does not select non-contradiction and excluded middle as the most basic facts of reality. Beneath both of these, Rand sees the fact of identity: to be is to be something specific. That difference between Aristotle and Rand is a harmonious one. Lastly, Lennox sees Rand’s axiom of consciousness—that anyone who makes claims implicitly affirms their own consciousness of existence—as propounded also by Aristotle.

*In Rand’s parlance, to say that an idea has been validated is to say that its truth has been established by perceptual evidence or by induction (either the kind known as abstractive, or as intuitive; or the kind known as enumerative, or as ampliative) or by deduction upon such evidence.

~~~~~~~~~~~~~~~~

Earlier works on Rand and Aristotle concerning philosophic axioms and their validation are these:

Douglas Rasmussen (1973). Aristotle and the Defense of the Law of Contradiction. The Personalist (Spring).

Douglas Den Uyl and Douglas Rasmussen (1984). Ayn Rand’s Realism in The Philosophic Thought of Ayn Rand. University of Illinois.

Leonard Peikoff (1991). Chapter 1 of Objectivism: The Philosophy of Ayn Rand. Dutton.

Tibor Machan (1992). “Evidence of Necessary Existence” Objectivity 1(4):31–62.

———. (1999). Chapter 2 of Ayn Rand.

Fred Seddon (2005). Implied Axioms

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Normativity of Logic – Kant v. Rand

In the perspective of Immanuel Kant, reasoning in accordance with logic can falter due to various empirical circumstances of the reasoning mind. Knowing those pitfalls and how to avoid them is what Kant would call applied logic. Principles of applied logic are partly from empirical principles. As for the principles of pure logic itself, logic apart from such applications, “it has no empirical principles” (B78 A54). The principles of logic are not principles of empirical psychology, and their ultimate authority stems from something deeper than empirical necessities of thought.

Logic for Kant was Aristotelian logic. He thought this discipline to have been set out completely by Aristotle, and he thought such finality of the discipline was due to the distinctive character of the discipline that is logic. “Logic is a science that provides nothing but a comprehensive exposition and strict proof of the formal rules of all thought” (Bxiii). The office of logic is “to abstract from all objects of cognition and their differences; hence in logic the understanding deals with nothing more than itself and its form” (Bix; B170 A131). Logic is “vestibule” of the sciences in which we acquire knowledge. Logic is presupposed in all judgments constituting knowledge (Bix).

Knowledge requires the joint operation of a receptivity of the mind and a spontaneity of the mind. In our receptivity, sensible objects are given to us. In our spontaneity of conceptualization and judgment, those objects are thought (B29 A15). Sensory presentations are givens. The spontaneity of cognition is the ability to produce presentations ourselves. Kant calls understanding the faculty for bringing given sensible objects under concepts and therewith thinking those objects (B74–75 A50–51). Logic is “the science of the rules of the understanding as such” (B76 A52). These are “the absolutely necessary rules of thought without which the understanding cannot be used at all” (B76 A52).

Kant distinguishes the faculty of understanding from its superintendent, the faculty of reason. The understanding can arrive at universal propositions by induction. Correct syllogistic inferences among propositions are from reason (B169 A130; B359–60 A303–4). By its formal principles, reason provides unity to the rules of understanding (B359 A302). I should mention that it is not the role of reason (or of understanding) in logic that Kant tries to curb in his Critique of Pure Reason (B=1787 A=1781). This role Kant takes as within the proper jurisdiction of reason.

Kant regards logic as “a canon of understanding and of reason” (B77 A53; B170 A131). A canon is a standard or rule to be followed. How can rules of logic be rules to be followed by the understanding if they are the rules that characterize what is the form of all thought? How can the rules prescribe for X if they are descriptive of what X is? Let X be alternatively the faculty of understanding or the faculty of reason, the question arises.

Kant calls such logic general logic, and this he takes as abstracting away “from all reference of cognition to its object” (B79 A55). This conception of logic is significantly different from that of Rand: Logic is an art of identification, regimented by and towards the fact of existence and the fact that existence is identity.

Over a period of forty years, Kant taught logic at least thirty-two times. Syllogistic inference and non-contradiction were the rules for formal logic. Kant took these rules to concern some of the requirements for truth. They do not amount to all of the requirements for truth, “for even if a cognition accorded completely with logical form, i.e., even if it did not contradict itself, it could still contradict its object” (B84 A59). That much is correct, and Kant is correct too in saying that “whatever contradicts these rules is false” (B84 A59). Why? “Because the understanding is then in conflict with its own universal rules of thought, and hence with itself” (B84 A59).

How can the normativity of logic be accounted for if its principles are taken for correct independently of any relations they might have to existence and any of the most general structure of existence? Kant needs to explain how general logical norms for our thinking can be norms without taking their standard from the world and how such norms can be rules restricting what is possibly true in the world.

Might the source of norms for the construction of concepts be the source of norms for inferences when concepts are working in judgments? Can the normativity of forms of inference among judgments be tied to normativity in forms of judgments and normativity in the general forms of concepts composing those forms of judgment?

What requirements must concepts meet if they are to be concepts comprehending particulars in true ways? From the side of the understanding itself, the fundamental forms concepts may take are required to be systematically interconnected to satisfy the circumstance that the understanding “is an absolute unity” (B92 A67). Considered apart from their content, concepts rest on functions. “By function I mean the unity of the act of arranging various presentations under one common presentation” (B93 A68). So far, so good, but then Kant’s account stumbles badly.

Concepts are employed in the understanding to make judgments. In judgments, according to Kant, “a concept is never referred directly to an object” (B93 A68). Concepts, when not referring to other concepts, refer to sensory or otherwise given presentations (B177 A138–42). This is part of Kant’s systematic rejection of what he called intellectual intuition. That rejection is not entirely wrongheaded, but this facet of the rejection is one of Kant’s really bad errors. I say as follows: the fact that concepts relate perceptually given particulars does not mean that concepts do not refer directly to the particulars of which we have perceptual experience. It simply does not square with the phenomenology of thought to say that when we are using a concept we are not referring directly to the existents (or the possibility of them) falling under the concept.

Kant will have cut himself off from an existential source of normativity in judgment through concepts, thence a possible source of normativity for inferences among judgments, unless that normativity can be gotten through his indirect reference for concepts to existents through given presentations of existents. For Kant, as for most every epistemologist, concepts are unities we contrive among diverse things according to their common characteristics (B39 A25, B377 A320). The problem for Kant is that the diverse things unified are diverse given presentations in consciousness that become objects of consciousness only at the moments of conceptualization and judgment themselves (A103–6, A113–14, A119–23, B519–25 A491–97, B141–46). (Kant’s empirical realism, in A367–77, B274–79, and B232–47 A189–202, is subordinate to his transcendental idealism, which is idealism, not realism. I need yet to consider more closely the mitigating arguments of Abela 2002 and Westphal 2004.)

The concept body can be used as a logical subject or in the predicate of a judgment. As subject in “Bodies are divisible,” body refers directly to certain given presentations of objects, but body does not refer to those objects unless in use in a judgment. In use for predicate in “Metals are bodies,” body refers to the subject concept metal, which in turn refers to certain given presentations of objects (B94 A69).

Compare that casting of predication to the 1991 casting I made for Rand’s philosophy in which existence is identity and consciousness is identification. “The compositions between subject and predicate in a true proposition capture objective identity relations. There are at least three such relations affirmed simultaneously in every true proposition.” In the assertion “Metals are bodies,” the copula are executes a triple coordination. It wires the subject concept metals and the predicate concept bodies to the same existents. This is a relation of particular identity. Secondly, it carries the specification of metals in respect of their physical species: they are bodies, not colors. This is a relation of specific identity. Thirdly, it affirms the relation of particular identity between proposition and reality: this particular identity relation and this specific identity relation of this proposition are affirmed as identity structures in reality, in the existents. (I hope yet to further cast predication with its measurement traits and to assimilate identity-casting of predication into the developments in logic and theory of judgment since the era of Kant [Martin 2006]).

“The only use that the understanding can make of . . . concepts is to judge by means of them” (B93 A68). According to Kant, we cannot begin to understand the concept body otherwise than as in judgments. Right understanding of body means only knowledge of its particular right uses as the logical subject or in the logical predicate.

Kant observes that judgments, like concepts, are unities. It is the faculty of understanding that supplies those unities by its acts. The logical forms of judgment are not conformed to identity structures in the world or in given sensory presentations. Kant conceived those presentations as having their limitations set by relations of part to whole. He thought they could not also, in their state as givens, have relations of class inclusion (B39 A25, B377 A320). This is a facet of his overly sharp divide between sensibility and understanding. I have long held that relations of class inclusion are not concrete relations, unlike the relations of part-whole, containment, proximity, or perceptual similarity. That does not conflict, however, with the idea that what should be placed in which classes should be actively conformed to particular concrete relations found in the world.

Kant thought that our receptivity of given sensory presentation is not cognitive and requires conceptualization in order to become experience (B74–75 A50–51). “All experience, besides containing the senses’ intuition through which something is given, does also contain a concept of an object that is given in intuition, or that appears. Accordingly, concepts of objects as such presumably underlie all experiential cognition as its a priori conditions” (B126 A93). The sensory given presentation contains particular and specific information about the object that can be thought in concepts and judgments concerning the object. But the most general and necessary forms of objects in experience is not information supplied by the sensory given presentations (sensory intuitions), but by the understanding itself for agreement with itself (B114–16, B133n).

Without the general form of objects supplied by the understanding, there is no cognitive experience of an object. “Understanding is required for all experience and for its possibility. And the first thing that understanding does for these is not that of making the presentation of objects distinct, but that of making the presentation of an object possible at all” (B244 A199).

Kant is concerned to show that there are general patterns of necessity found in experience that are seamless with logical necessities. He errs in supposing that that seamlessness comes about because the general forms for any possible experience of objects logically precedes any actual experience of objects. That a percipient subject must have organization capable of perception if it is to perceive is surely so. Consider, however, that a river needs channels in order to flow, yet that does not rule out the possibility (and actual truth) that the compatibility of a valley and a river was the result of the flow of water.

According to Kant, we could have no experience of objects without invoking concepts bearing, independently of experience, certain of the general forms had by any object whatsoever. The unity-act of the understanding that is the conceptual act, which gives a unified content, an object, to given sensory presentations is also the very unity-act that unifies the various concepts in a judgment (B104–5 A78–79).

An additional power Kant gives to the understanding is the power of immediate inference. From a single premise, certain conclusions can be rightly drawn. “The proposition All human beings are mortal already contains the propositions that human beings are mortal, that some mortals are human beings, and that nothing that is immortal is a human being” (B360 A303). In these inferences, all of the material concepts, human being and mortal, appearing in conclusions were in the premise. Such inferences can be made out to be the mediate inferences of a syllogism, but only by adding a premise that is a tautology such as Some mortals are mortal (D-W Logic 769; J Logic 115).

Mediate inferences require addition of a second judgment, a second premise, in order to bring about the conclusion from a given premise. The proposition All scholars are mortal is not contained in the basic judgment All men are mortal since the concept scholar does not appear in the latter. The intermediate judgment All scholars are men must be introduced to draw the conclusion (B360 A304).

The basic judgment—the major premise of the syllogism—is thought by the understanding. This is the thinking of a rule. Under condition of that rule, the minor premise of the syllogism is subsumed, by the power of judgment. Lastly, reason makes determinate cognition by the predicate of the basic rule the new judgment, which is the conclusion (B360–61 A304).

What usually happens is that the conclusion has been assigned as a judgment in order to see whether it does not issue from judgments already given, viz., judgments through which a quite different object is thought. When this is the task set for me, then I locate the assertion of this conclusion in the understanding, in order to see whether it does not occur in it under certain conditions according to a universal rule. If I then find such a condition, and if the object of the conclusion can be subsumed under the given condition, then the conclusion is inferred from the rule which holds also for other objects of cognition. We see from this that reason in making inferences seeks to reduce the great manifoldness of understanding’s cognition to the smallest number of principles (universal conditions) and thereby to bring about the highest unity of this cognition. (B361 A304–5)

The faculty of reason, in contradistinction from understanding, does not deal with given sensory presentations, but with concepts and judgments. “Just as the understanding brings the manifold of intuition under concepts and thereby brings the intuition into connection,” so does reason “bring the understanding into thoroughgoing coherence with itself” (B362 A305–6). Reason provides cognition with logical form a priori, independently of experience. The principles of the understanding may be said to be immanent “because they have as their subject only the possibility of experience” (B365 A309). The principles of reason may be said to be transcendent in regard to all empirical givens.

The spontaneity of thought is unifying activity, whether in conceiving, judging, or inferring. Readers here will have probably noticed in Kant the themes of integration and economy, which are major in Rand’s analyses of cognition. However, for Kant the unifying activity of the understanding and of reason is not “an insight into anything like the ‘intelligible’ structure of the world” (Pippin 1982, 93).

Kant represents understanding and reason as working together as a purposive system. I maintain, in step with Rand, that all purposive systems are living systems or artifacts of those living systems. We hold that only life is an ultimate end in itself; life is the ultimate setter of all needs (cf.). The purposive system that is the human mind is the information-and-control system having its own dynamic needs derivative to serving the needs of the human individual and species for continued existence. Life has rules set by its needs for further life.

Life requires not only coherent work among its subsystems, but fitness with its environment. Rules of life pertain to both. Rules of mind pertain to both (cf. Peikoff 1991, 117-19, 147-48). Rules of logic do indeed enable coherent work of the mind, but they also yield effective comprehension of the world. Identity and unity are structure in the world, and, in their organic elaboration, they are structure of the viable organism (cf. Peikoff 1991, 125–26). The normativity of logic arises from the need of the human being for life in the world as it is.

References

Abela, P. 2002. Kant’s Empirical Realism. Oxford.

Kant, I. 1992. Lectures on Logic. J. M. Young, translator and editor. Cambridge.

——. 1996. Critique of Pure Reason. W. S. Pluhar, translator. Hackett.

Martin, W. M. 2006. Theories of Judgment. Cambridge.

Peikoff, L. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.

Pippin, R. B. 1982. Kant’s Theory of Form. Yale.

Westphal, K. R. 2004. Kant’s Transcendental Proof of Realism. Cambridge.

Edited by Stephen Boydstun

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Normativity of Logic – Robert Hanna

Robert Hanna does not accept Kant’s idealism. His account of the normativity of logic in Rationality and Logic (RL) is nonetheless sensibly characterized as a quasi-Kantian account. Professor Hanna proposes that we have a faculty of logic in which dwells a protologic—a set of schematic logical structures—upon which any formal logic, classical or nonclassical, is constructed. This protologic Hanna argues to be presupposed, constructively and epistemologically, by any formal logical system.

I mentioned above that standard modern logic, which is an enlargement (and some correction) of logic beyond its development by Aristotle, is called classical to distinguish it from its further extensions and from its rivals. Logics that only extend classical logic will have modified some of “the classical logical constants, interpretation rules, axioms, or inference rules such that all the tautologies, theorems, valid inferences and laws of elementary logic still hold, along with some additional ones” (RL 40–41). Such are modern modal logics (Garson 2006; Priest 2001). The rivals of classical logic make modifications to “logical operators [constants], interpretation rules, axioms, or inference rules such that not all the tautologies, theorems, valid inferences, and laws of classical or elementary logic still hold” (RL 41). Such is relevance logic (Mares 2004; Priest 2001) or paraconsistent logic (Priest 2006).

The logics in rivalry with classical logic are contestants on the field of specifying which inferences, among our inferences in the vernacular, are valid. One inference certified in classical logic that many students find repugnant is the taking as valid any inference from false premises to a true conclusion. Paraconsistent logics and relevance logics are systematic formulations of logic in which such inferences come out as invalid. I should mention, too, that various informal fallacies of classical logic come to be formal fallacies within various nonclassical logics (RL 218).

All systems of logic are systematic formulations of “the necessary relation of consequence” (RL 43). Examples from classical logic would be “if p and q, then p” and “if both (i) if p then q and (ii) p, then q” and “if both (i) if p then q and (ii) not-q, then not-p.” Hanna is proposing that behind all logical systems, whether classical or not, there must be a single set of schematic logical structures which determine what will count as a possible logical system; there is a single protologic epistemologically presupposed by every logical system. This protologic is used in justifying assertions about any classical or nonclassical logic; the protologic is constructively presupposed by every logical system (RL 44).

In justifying claims about logic, we are invoking conscious logical beliefs about protologic. Having such a role, there is no way the protologic set of schematic structures could be revised. Moreover, they must be a priori.

Kant took a priori to mean necessarily so (B3–4, B119–24, A87–92); true independently of all experience, not having its source in experience (B2, B117–19 A84–87, B163); but true of the experienced world and needed for any empirical cognition (B5–6, B121–27 A89–94, B163–65, B196–97 A157–58). (Further, see Robinson 1969 and Tait 1992.)

For his own concept a priori, Hanna defines its cognitive facet as cognition not entirely determined by “inner, proprioceptive or outer sensory experiences even though it is always actually accompanied by such sensory experiences” (RL 273n25). He defines the semantic facet of the a priori as sentence meaning wherein the truth conditions of the sentence are not entirely determined by its verification conditions. He defines the epistemic facet of the a priori as belief wherein its justification is not entirely determined by sensory evidence.

No system of logic rejects all of classical logic. Hanna conjectures that the metalogical principles in the protologic might consist of weak versions of four basic principles in classical logic. One would be that “an argument is valid if it is impossible for all of its premises to be true and its conclusion false” (RL 45). Another would be that “not every sentence is both true and false” (RL 45). Cognitive psychology can contribute to the further specification of principles such as these, principles a priori and not revisable, in our repertoire. Our faculty of language epistemologically presupposes our faculty of protologic (cf. Macnamara in Note 36 here).

The protologic faculty of logic is sensitive to external experiential stimuli, but not entirely determined by such stimuli. It is an essential aspect of the mind of a rational animal, and it is an a priori aspect of such a mind. “It is not modally controlled by the empirical world, although it inevitably tracks the empirical world” (RL 83). “The logic faculty is a central and informationally promiscuous faculty of the human mind (and apparently the only one), whose role it is to mediate between the peripheral faculties and the central processes of theory-formation, judgment, belief, desire, and volition” (RL 109). That we have innate logical powers does not entail that we have any innate ideas (RL 135).

Hanna is a realist about logic and logical necessity. Any explanation and justification of logic must presuppose logic. No explanatory reduction of logic to other things is possible. The thesis that we are endowed with a logical faculty offers not an explanatory reduction, but a connection for logic of a nonreductive, yet realist nature: “(i) logic is cognitively constructed by rational animals, and (ii) logic is objectively real via language, and consequently logical necessity is an objectively real property or fact in a world that objectively and really contains linguistic structures” (RL 158; also 80–81).

Hanna argues that the innate logical faculty includes a capability for logical intuition. This is an act in which we grasp logical rules, and grasp them as justified and necessary, in a noninferential, a priori, yet fallible way (RL 167–82).

To say that formal logic is normative is to say “humans ought to reason soundly or validly (more generally, cogently). Otherwise put, the normativity of logic consists in the fact . . . that the justification of human beliefs or intentional actions depends on our ability to reason cogently” (RL 203).

Hanna maintains that logic is categorically normative, not hypothetically normative. Logic enjoins one to hold a certain belief or take a certain action under all circumstances and primarily because of logic alone. A view of logic as hypothetically normative would say that logic enjoins one to hold a certain belief or take a certain action because of logic, but only in certain circumstances and primarily because of something extralogical (RL 203).

Mill held that logic is intrinsically normative (necessarily normative) and that logic is explanatorily reducible to empirical psychology. This implies that logic is intrinsically but hypothetically normative, not categorically normative. Then conditions from particular human interests or conditions from natural facts could constrain the scope of the applicability of logical obligation. Hanna maintains to the contrary that logical norms apply in all possible contexts.

Hanna constructs a logical argument, which leans on elements of modal logic, to refute the thesis that logic is reducible to empirical psychology (RL 20–21, 27). That is not to say that there is no essential connection between the logical and the psychological. Hanna argues for such an essential connection: logic is cognitively constructed by rational animals who are essentially logical animals (RL 25). The logical in the phrase “essentially logical animals” is to be understood as primarily normative (RL 215–16). Although we obey the protologic perfectly whenever we reason, we shall adhere only imperfectly to normative mental principles that we construct from the protologic, and only imperfectly to formal logical norms we construct from the protologic (RL 149–53).

Hanna elects the path of Kant, Boole, and Frege. “Logic is the universal, topic-neutral, a priori science of the necessary laws of truth, and also a pure normative science based directly on rationality itself” (RL 204). He calls this the moral science conception of logic. Logic is a moral science (RL 205). It is “an integral part of human morality, namely the part that consists in justifying moral judgments and decisions, including direct moral arguments and reflective equilibrium” (RL 206).

Hanna rejects the idea, put forth by Otto Weininger of virtually identifying logic with ethics (RL 205–6). To the contrary, “moral wrongdoing is not necessarily or even usually connected with wrong logical reasoning; and on the other hand, wrong logical reasoning is not necessarily or even usually sinful” (RL 217). Weininger’s idea was that “logic and ethics are fundamentally the same, they are no more than duty to oneself” (1903). (In an appendix, I shall display some of Weininger’s elaboration of this idea.)

Hanna rejects the idea that morality is entirely a system of hypothetical imperatives. Kant’s categorical imperative of ethics is not “an all-purpose practical decision procedure or algorithm. . . . Negatively described, the categorical imperative is a filter for screening out bad maxims; positively described, it is a constructive protocol for correctly generating maxims, given the multifarious array of concrete input-materials to practical reasoning . . .” (RL 212). In parallel with the role of an ethical categorical imperative, Hanna alleges a logical categorical imperative. Specifically, that imperative would be: “Think only according to those processes of reasoning that satisfy the protologic” (RL 213).

From this perspective, Hanna would have us see through the errors of radically conventionalist theories of logic (RL 210–11) and skeptical, even nihilistic, attacks on the objectivity of the norms of logic (RL 206, 223–30). Good for Robert Hanna.

I think Hanna’s philosophy of logic is largely compatible with Rand’s thought on logic and epistemology. To Hanna’s logical imperative, we should add this prior one: think. And we should supplement Hanna’s theory with the circumstance that the choice to think is the choice to live, that the categorical demands of logic are vested by the categorical structure of existence and the hypothetical standing of human life.*

*See also the review by Gila Sher, especially her criticisms 2 and 5.

References

Garson, J. W. 2006. Modal Logic for Philosophers. Cambridge.

Hanna, R. 2006. Rationality and Logic. MIT.

Kant, I. 1996 [1787]. Critique of Pure Reason. W. S. Pluhar, translator. Hackett.

Mares, E. D. 2004. Relevant Logic: A Philosophic Interpretation. Oxford.

Priest, G. 2001. An Introduction to Non-Classical Logic. Cambridge.

——. 2006. Doubt Truth to Be a Liar. Oxford.

Robinson, R. 1969 [1958]. Necessary Propositions. In The First Critique. T. Penelhum and J. J. MacIntosh, editors. Wadsworth.

Tait, W. W. 1992. Reflections on the Concept of A Priori Truth and Its Corruption by Kant. In Proof and Knowledge in Mathematics. Routledge.

Appendix

Otto Weininger’s is a quasi-Kantian view of logic as categorically normative. This bright young man took his own life at age 23, shortly after the publication of his book Sex and Character in 1903. Weininger was Viennese, but his philosophical positions were not of the patterns then favored in Vienna, rather in Greater Germany: patterns from Kant, Fichte, Schopenhauer, and Nietzsche.

Weininger conceived the obligation of logic to belong to the moral obligation we have to ourselves. To have a real relation to truth there must be an active permanency that conveys the identity of items forward in time. That active, responsible permanency is none other than one’s own personality (Chap. VI).

“Logic deals with the true significance of the principle of identity (also with that of contradiction . . .). The proposition A=A is axiomatic and self-evident. It is the primitive measure of truth for all other propositions. . . . A=A, the principle of all truth, cannot itself be a special truth. . . . [it] cannot be a source of positive knowledge. . . . [it is] the common standard for all acts of thought. . . . The proposition of identity does not add to our knowledge; it does not increase but rather founds a kingdom. / Logic . . . is the supreme standard by which the individual can test his own psychological ideas [general, processional ideas] and those of others . . . .

“When I enunciate the proposition A=A, the meaning of the proposition is not that a special individual A of experience or of thought is like itself. The judgment of identity does not depend on the existence of an A. It means only that if an A exists, or even if it does not exist, then A=A. Something is posited, the existence of A=A whether or no A itself exists. It cannot be the result of experience, as Mill supposed, for it is independent of the existence of A. But an existence has been posited; it is not the existence of the object; it must be the existence of the subject. The reality of the existence is not the first A or the second A, but in the simultaneous identity of the two. And so the proposition A=A is no other than the proposition ‘I am’.

“Man realizes himself only insofar as he is logical. He finds himself in cognition. / Duty is only duty to oneself, duty of the empirical ego to the intelligible ego. . . . Logic and ethics are fundamentally the same, they are no more than duty to oneself. They celebrate their union by the highest service of truth. . . . All ethics are possible only by the laws of logic, and logic is no more than the ethical side of law. / Logic proves the absolute actual existence of the ego; ethics controls the form the activity assumes. . . . Ethics makes it possible for the intelligible ego to act free from the shackles of empiricism.” (Chap. VII)

I should object to the idea that A=A posits an existent relationship where A does not exist. The identities of things that do not exist are wholly parasitic on the identities of existents, and the guide A=A is only a guide to bring one or keep one among the latter.

Edited by Stephen Boydstun

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Between False, Invalid, and Meaningless

Rand defined truth as “the recognition (i.e., identification) of the facts of reality” (ITOE 48; also Rand 1957, 1017). She maintained: “Man identifies and integrates the facts of reality by means of concepts. He retains concepts in his mind by means of definitions. He organizes concepts into propositions—and the truth or falsehood of his propositions rests, not only on their relation to the facts he asserts, but also on the truth or falsehood of the definitions of the concepts he uses to assert them” (ITOE 48; see also Peikoff 1991, 137–39).

Rand is here speaking of definitions of concepts not in whatever stipulative character they have, but in the assertion a definition makes for a concept’s relation to reality, in the full context of one’s knowledge. The hallmark of falsity for Rand is the contradictory. “Logic is the art of non-contradictory identification. . . . No concept man forms is valid unless he integrates it without contradiction into the sum total of his knowledge” (Rand 1957, 1016).

I noted in #2 that predications are conceptual identifications, that Edward Zalta takes the discipline of logic to be “the study of the forms and consequences of predication,” and that this fits well with Rand’s conception. Comportment between Rand’s conception of logic and prominent contemporary senses of logic,* such as Zalta’s, is comportment in which we incorporate theirs into Rand’s with its distinctive general situation with existence. Contemporary logicians and philosophers of logic should see Rand’s conception of logic as including more of what they would call rationality beyond their formal logic.

Rand holds that “logic rests on the axiom that existence exists” (1957, 1016; further, Peikoff 1967, 112–13). She holds that widest existence with its principle existence is identity is the widest frame of knowledge. Nothing real is beyond existence with its fundamental principle. The elementary function of logic is to provide general constraints on concepts, definitions, propositions, and inferences for attaining and keeping: integrated truth, integrated in the widest context, ever in the widest frame, which is existence with its identity.

Rand holds also that all values (hence all problems, questions, solutions, games, wars, or victories) arise only for living existence, within wider existence. Consonant with that existential situation, we have that “It is value” logically presupposes “It is.” Widest existence with its fundamental principle frames the realm of value like any other. I say then for Rand, as for me, truth as recognition of reality is epistemologically more basic than functional correctness. (This does not entail that a sense of correctness is acquired later in development than a sense of truth.) In the functional system that is knowledge, the meaningless is epistemologically rooted in falsehoods.

The sense of truth of which I am speaking as epistemologically prior to a sense of correctness is only the sense of truth manifest in “It is.” The further, normative sense of truth manifest in “It is true that” presupposes not only “It is” but “Truth is to be pursued and can be missed.”

Truth is a form of organic functional correctness, and the latter is metaphysically prior to the former in that conceptual consciousness arises from non-conscious biological systems.* Correctness is accordance with function. Meaningfulness is one success, one correctness in the functional system that is human thought.

To the extent that a concept’s definition is not assessable for contradiction, neither with other propositions warranted to be true, including other concepts’ definitions, nor more directly with reality,* it will be most natural in Rand’s system to call the concept objectively meaningless. (Note that philosophic axioms are assessable.) A concept that has no definition—not ostensively, not expressly, not implicitly in a set of propositions using the concept—would also be meaningless. One important example of this latter sort of objectively meaningless concept would be one reifying the concept nothing into anything more than not anything. The idea that I might have been conceived by parents other than the ones who actually conceived me is of that genre. Such a concept of I relies on a reification of nothing. My tissue, in which development my mind arose, does not in any sense exist and bear possibilities before it existed. It is objectively meaningless to say I might have been conceived by parents wealthier, better educated, and so forth. The dense thicket of contingencies in such alternative personal histories leaves talk of such possibilities meaningless (cf. Boydstun 1996, 183–87). The I of such possibilities has no identity, no definition.

In the functional system that is knowledge, another important example of the meaningless by way of having no definition would be concepts whose entire ambit of propositions in which they are logical subjects are propositions stating what the referents of the concept are not. No positive definition, no identification of existence, is thereby attained. The subject that is center and base for negative-way, beyond-being theology* is objectively meaningless (cf. Rand 1957* and Kant KrV B149; B308–9; A254–56 B310–12; A780–82 B808–10; Westfal 2004, 64–66).

The subject of a purely negative theology has not even the limits of Pegasus or of a partial mathematical function such as 5 minus 7 among the natural numbers (positive, counting numbers). There is logic appropriate to characters like Pegasus and to 5 minus 7 among the naturals, which logic has its relations to logic for the concrete world (Epstein 2006, 413–36). There is no such logic for the absolute non-identity of the subject of purely negative theology. On the positive side, concerning the putative personality that is God, in the full context of present human knowledge, we may put personality-God in the same region as Pegasus (cf. ITOE Appendix, 148).

What is objectively meaningless in the context of knowledge attained in a culture is not necessarily objectively meaningless also in the more restricted context of knowledge of a less advanced culture. The meaninglessness of purely negative theology can be seen at our culture’s stage of knowledge. It could also have been seen in the eras of the Middle Platonists and the Neo-Platonists who embraced negative theology. Was my first example of the meaningless by way of having no definition—the example of me before my tissue began—objectively meaningless in the culture of Plato and Aristotle? I do not know. My judgments, in present study, of what is objectively meaningless are with respect to the higher knowledge-stage of our world today. And they are for individuals old enough to be reading books such as Atlas Shrugged.

Shortly, I shall craft a niche for cases of the objectively meaningless in defined concepts and in propositions, looking to implications of Rand’s epistemology. That is, I shall expose a way, in Rand’s framework, in which propositions, including definitions, can fail to be assessable for contradiction. The objectivity of which I speak in the objectively meaningful is, all together, Rand’s recognition sense of the objective with its metaphysical and epistemological faces as well as her sense of the objective in her partition intrinsic-subjective-objective.*

Rand was cognizant of Aristotle’s principles of definition according to genus and species and according to essentials.* In Introduction to Objectivist Epistemology (ITOE), she integrates the former of those Aristotelian principles with her analysis of concepts in terms of measurement omission. She recasts the Aristotelian virtue of definitions according to essentials by dropping his conception of essence as intellectual, universal form joined to and ontologically separable from matter it informs and by dropping the Aristotelian information of our minds by such forms in perceptual, memorial experience. Rather, she upholds definitions of concepts reflecting reality by taking for essence of the class under the concept the shared distinguishing characteristic that explains, within present extent of knowledge, the greatest number of other characteristics distinctive of members of the class. Essential characteristics are constructively discerned by actively thinking over the differences, similarities, thematic relations,* and causal relationships discerned in particulars.

Notice that what was properly the essential characteristic of a concept at an earlier stage of human knowledge could become improper with the advance of knowledge. Holding onto the old form of the concept could become irrational, a defiance of reality, an obfuscating embrace of false for true (e.g.). The old concept could be sub-optimum in truth and sub-optimum in the strength of objective meaning possible in the more advanced context of knowledge.

Rand was also aware of the controversy among contemporary philosophers concerning the soundness of the modern division of propositions into analytic or synthetic (a, b, c).This controversy had been prominent in academic print* and in semi-popular works such as A. J. Ayers’ Language, Truth, and Logic (1952; quoted in Peikoff 1967, 94) and the writings on philosophy of science by Hans Reichenbach and Philipp Frank (who was quoted in Basic Principles of Objectivism lectures in the nineteen-sixties; Branden 2009, 22–23) in the four decades before Rand’s ITOE. Immediately after completing the ITOE-series in The Objectivist, she issued therein Leonard Peikoff’s “The Synthetic-Analytic Dichotomy,” which came down on the side of those who had argued variously against the distinction. In ITOE Rand rejected the distinction in the course of laying out her theory of right definition (cf. White 1952, 318–30; Peikoff 1967, 94–97, 100–106, 115; Browne 2007, starting here). “The nominalists of modern philosophy, particularly the logical positivists and linguistic analysts, claim that the alternative of true or false is not applicable to definitions, only to ‘factual’ propositions. Since words, they claim, represent arbitrary human (social) conventions, and concepts have no objective referents in reality, a definition can be neither true nor false” (ITOE 47–48).

Rand’s theory of concepts is not nominalist, but objectivist (and mensural). Proper concepts are aimed to reflect realities and those realities’ places in and relationships to wider realities. Proper definition of a concept enables us to keep concepts straight in mind and ever-tuned to reality in our ever-growing grasp of it.

Writing, I should mention, is a tremendous tool for stabilizing and sharpening expressly defined concepts, for better remembering them, and for drawing them together to discern contradictions and implications. I should mention, too, as I frequently do mention, with reason: when a mathematician or scientist speaks out the equation she is writing on the board for class, she is speaking the concepts and relationships in purely natural language. Neither she nor the class will need to have facility in concepts of artificial, formal languages, meta-language, and logics within formal languages to solve a partial differential equation or to prove that every group is a subgroup of the permutation group on some set.

Proper concepts and their definitions have objective bases in the world and in the nature of consciousness, including the nature of thought. The concepts triangle, seven, hand, and justice are not arbitrary, purely conventional constructs. They have meaning, truth, and utility, whatever the natural language in which they are expressed. Objective meaning is structured after properly integrated truth. Objective meaning, like an objective concept and its definition, has validity by reality as it stands in the fullest present human knowledge of reality. Meaning in the system that is knowledge is not an arbitrary free invention. For meaning, I say, as Rand says for definition, “objective validity is determined by reference to the facts of reality” (ITOE 46).

Rand reserves the term arbitrary for cases of the arbitrary entailing dissonance with reality rationally differentiated and integrated. In the negative sense of the subjective, as spoiling objectivity, we can say more briefly that Rand reserves the term arbitrary for cases of subjective arbitrariness (e.g. ITOE 42–43). Such is Rand’s use when she writes against arbitrariness in modern empiricism, rationalism, and philosophy of science: “Its exponents dismissed philosophical problems by declaring that fundamental concepts—such as existence, entity, identity, reality—are meaningless; they declared that concepts are arbitrary social conventions . . .” (Rand 1970, 83). Notice Rand concurs that if concepts were arbitrary conventions they would be meaningless.

Where we might say something was arbitrary in a sense not subversive of objectivity, Rand uses the term optional. “After the first stage of learning certain fundamentals, there is no particular order in which a child learns new concepts; there is for a while, a broad area of the optional, where he may learn simple, primary concepts and complex, derivative ones almost concurrently, depending on . . .” (ITOE 20, also 70, 72, and Appendix, 180). Where we might say, to indicate universality, “consider an arbitrary scalene triangle” or “consider an arbitrary body having angular velocity omega,” we mean nothing more than any when we say arbitrary, and Rand has no complaint about the arbitrary in that sense. Further, she should have no issue with objectivity-comporting arbitrariness in choices of coordinate system, number base, or natural language. But when she uses arbitrary, she means the subjectively arbitrary, the one necessarily objectively meaningless.

We saw earlier Rand’s awareness of logical positivism with its attachments to nominalism and to sharp division of propositions into either synthetic or analytic. Logical positivism is also known as logical empiricism.* Rudolf Carnap was a principal in that philosophical movement. He held that metaphysical statements such as Heidegger’s “the Nothing nothings” are meaningless (Carnap 1959, 69–73). Carnap defended a divide between synthetic and analytic statements. Furthermore, statements that are neither one of those are meaningless. We have seen above the different reason that Rand too would regard the famous statement of Heidegger as objectively meaningless.

Validity within propositional and predicate logic is generally taken to mean: that merit of argument in which the conclusion cannot be false if the premises are true.* We speak also of validity in property titles and in contracts. Kant had much work for a sense of validity in epistemology joining those two senses. He announces in the Preface to the first edition of the Critique of Pure Reason that a central component of that work “refers to the objects of pure understanding and is intended to make comprehensible the objective validity of understanding’s a priori concepts” (xvi; see Pippin 1982, 154–58).

That general epistemological sense of objective validity in concepts is useful in application to concepts and propositions in philosophical systems besides Kant’s. In his mature, pragmatic philosophy, Dewey writes: “According to experimental inquiry, the validity of the object of thought depends upon the consequences of the operations which define the object of thought” (1929, 103). Speaking of experiment, Dewey refers to the process “by which the conclusion is reached that such and such a judgment of an object is valid” (ibid., 230). Logical positivist Ayer writes: “In saying that we propose to show ‘how propositions are validated’, we do not of course mean to suggest that all propositions are validated in the same way. On the contrary we lay stress on the fact that the criterion by which we determine the validity of an a priori or analytic proposition is not sufficient to determine the validity of an empirical or synthetic proposition. For it is characteristic of empirical propositions that their validity is not purely formal” (Ayer 1952, 90). For Ayer one can validate a proposition either by finding it to be analytic or by finding it to be empirically verified.

In Rand’s philosophy, Peikoff takes validation to be “any process of establishing an idea’s relationship to reality, whether deductive reasoning, inductive reasoning, or perceptual self-evidence” (1991, 8).* As Peikoff had expressed it in his 1976 lecture series The Philosophy of Objectivism,validation, in the broad sense includes any process of relating mental contents to the facts of reality. Direct perception . . . is one such process. Proof designates another type of validation. Proof is the process of deriving a conclusion logically from antecedent knowledge” (see further, Peikoff 1991, 118–20, 137–38).

Rand observes “there are such things as invalid concepts, i.e., words that represent attempts to integrate errors, contradictions or false propositions, . . . or words without specific definitions, without referents, which can mean anything to anyone . . . . An invalid concept invalidates every proposition or process of thought in which it is used as a cognitive assertion” (ITOE 49).

That is a big umbrella. I think of objectively meaningless concepts as a type of invalid concept. They are a subdivision of that type of error; not every invalid concept is objectively meaningless. Rand constructs an invalid concept man in which his ability to run is taken for his essential characteristic and running entities is taken as genus of the concept (ITOE 71). To better absorb this example, one should imagine that one does not have the other conception and definition of man rational animal, not even implicitly. Try to imagine that for basic adult definition of man one has only the definition running animal, alongside running water and so forth as species of running entities.

That would be a misidentification of the essential trait of man. It would be an objectively invalid concept, but not an objectively meaningless concept. Such a concept is assessable for validity. For one thing, it is assessable for contradiction with reality, including contradiction with other concepts warranted by reality in the present context of knowledge. That is to say, such a concept is objectively meaningful though it is objectively invalid.

Earlier I examined invalid concepts objectively meaningless by way of having no definition. There are defined concepts empty of distinctive positive identity. Such a concept will yield propositions incorporating specifically it: not assessable for contradiction. Such a concept and such propositions composed distinctively by them are objectively meaningless. Consider a concept super-knowledge (which is what some concepts of faith or intuition come to) that is stipulated to be nothing other than imageless and wordless apprehension higher than knowledge resulting from perceptual and logical means. This would be an invalid concept that is objectively meaningless. Keeping to the stipulated sense of the subject term, in saying “Super-knowledge is beautiful,” the subject is without positive identity beyond its genus apprehension. There is nothing objectively meaningful being said beyond “Some apprehension is beautiful.” No assessments of this proposition for contradiction by particular or specific identities are by test of the proposed species super-knowledge. Similarly it goes with susceptibility to assessment for contradictions in “Beauty is super-knowable.” That beauty is apprehensible is all the objective meaning.*

References

Ayer, A. J. 1952 [1936, 1946]. Language, Truth and Logic. 2nd ed. Dover.

Boydstun, S. 1996. Volitional Synapses. Objectivity 2(4):183–204.*

Branden, N. 2009. The Vision of Ayn Rand. Cobden.

Carnap, R. 1959 [1932]. The Elimination of Metaphysics through Logical Analysis of Language. A. Pap, translator. In Logical Positivism. A. J. Ayer, editor. Free Press.

Dewey, J. 1929. The Quest for Certainty. J. A. Boydston, editor. 1984. Southern Illinois.

Epstein, R. L. 2006. Classical Mathematical Logic. Princeton.

Kant, I. 1781. Critique of Pure Reason. W. J. Pluhar, translator. 1996. Hackett.

Peikoff, L. The Analytic-Synthetic Dichotomy. In Rand 1966–67.

——. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.

Pippin, R. B. 1982. Kant’s Theory of Form. Yale.

Rand, A. 1957. Atlas Shrugged. Random House.

——. 1966–67. Introduction to Objectivist Epistemology. 2nd ed. H. Binswanger and L. Peikoff, editors. 1990. Meridian.

——. 1970. Kant versus Sullivan. In Philosophy: Who Needs It. 1982. Signet.

Westfal, K. R. 2004. Kant’s Transcendental Proof of Realism. Cambridge.

White, M. G. 1952. The Analytic and the Synthetic: An Untenable Dualism. In Semantics and the Philosophy of Language. L. Linsky, editor. Illinois.

*A different response to Rand’s epistemology in areas treated in the present study can be found here.

Sequel

~~~~~~~~~~~~~~~~

With Measurement

Measurement is the identification of a relationship in numerical terms—and the complexity of the science of measurement indicates the complexity of the relationships which exist in the universe and which man has barely begun to investigate. They exist, even if the appropriate standards and methods of measurement are not always . . . easily available . . . . If anything were actually “immeasurable,” it would bear no relationship of any kind to the rest of the universe, it would not affect nor be affected by anything else in any manner whatever, it would enact no causes and bear no consequences.[cf. Note 3]

The motive of the anti-measurement attitude is obvious: it is . . . the desire, epistemologically, to escape from the responsibility of cognitive precision and wide-scale integration; and metaphysically, the desire to escape from the absolutism of existence, of facts, of reality and, above all, of identity. (ITOE 39)

I have proposed that in Rand’s philosophy objective meaningfulness requires the setting of identity by definition. Objective meaning is given an additional level of strength by Rand’s epistemology and its metaphysical presuppositions, by the principles: (i) All concretes, whether physical or mental, stand in measurable relations to other concretes. (ii) Cognitive systems are measurement systems.* Best and most meaningful concepts are those given mathematical, measure-theoretic analysis. After one has defined a concept, one can further amplify meaning by specifying the concept’s dimensions of the subsumed items and the measurement scale(s) those dimensions afford. This specification is an objective issue, a matter of discerning what is the magnitude character of the dimensions shared by the items covered by the species in the definition (see Note 27).

In my dictionary, solid is defined as “a substance that is neither liquid nor gaseous.” Yes, that much is necessary for understanding the meaning of solid. A little reflection yields “a substance that is neither liquid nor gaseous, that is, a substance that does not flow” (presuming a context of common durations and precisions of observation). With advances in our network of mechanical concepts, including their mathematical characterization, we can say further

. . .

Solids are distinguished from fluids in virtue of the fact that they have moduli of rigidity that are not (too close to) zero; any solid is capable of withstanding shearing forces up to some particular measure. The particular modulus of rigidity of a particular solid object is part of what constitutes its individual identity, and the fact that its particular modulus is not zero is what qualifies it for the class solid.

. . .

Meaning that is strengthened by measure-theoretic analysis expands the possibilities for uncovering contradictions with reality together with contradictions among one’s concepts as cohorts reflecting the unity that is existence with identity. Consider the ways in which explicitly specifying quantitative dimensions of mechanical concepts perfected their meaning and helped to weed out their contradictions in the progression from Descartes to Leibniz to Newton. Made quantitative and free of contradiction, those concepts have powered invention and engineering in the modern world. Consider also:

. . .

From those percepts, general causal principles (from “Pushed balls roll” to “Applied torque causes onset of rolling”) are formed after the general pattern of how universal concepts are formed from percepts. Harriman’s book is an attempt to spell out more specifically the abstraction process from elementary causal principles such as “pushed balls roll” to general scientific principles—the tremendous abstraction process that is ampliative induction—illustrated by episodes in the history of science. . . .

I remarked previously, taking issue with Rand, as follows:

. . .

There are indeed some indispensable concepts we should not expect to be susceptible to being cast under a measurement-omission form of concepts. Among these would be the logical constants such as negation, conjunction, or disjunction. The different occasions of these concepts are substitution units under them, but the occasions under these concepts are not with any measure values along dimensions, not with any measure values on any measure scale having the structure of ordinal scale or above. Similarly, it would seem that logical concepts on which the fundamental concepts of set theory and mathematical category theory rely have substitution units, but not measure-value units at ordinal or above. The membership concept, back of substitution units and sets, hence back of concepts, is also a concept whose units are only substitution units. Indeed, all of the logical concepts required as presupposition of arithmetic and measurement have only substitution units. Still, to claim that all concretes can be subsumed under some concept(s) other than those, said concept(s) having not only substitution units, but measure values at ordinal or above, is a very substantial claim about all concrete particulars.

. . .

I said earlier in this addendum that in Rand’s philosophy objective meaningfulness requires the setting of identity by definition. I say further: Some logical and set-theoretic concepts—not, or, and, all, some, set—are defined by implicit definitions, a specification of their roles supporting meaning and truth of propositions, displayed most essentially in the propositions of logic and mathematics themselves. To be sure, these concepts are rooted in structures of action and situation learned in child development. (Notice also that some functions of these concepts can be implemented in machines.) Later they are rarified for use in language and abstract thought. (On action origins, see a, b. On Piaget’s perspective, see the contributions of Smith, Boom, and Campbell here. On acquisition of logical notions in language acquisition, see a, b, c, .)

The concept collection is not the very same as the concept set as the latter is used in logic or mathematics. Then too, logical class membership, which is used in Rand’s explication of conceptual class, is not the very same as natural-species membership such as Silver’s belonging to the horse species. Validity of the concept natural-species membership is not the full warrant for the concept class membership.

Perfecting the meaning and warrant of the concept class membership will not rely on a measure-value omission. Substitution units are not to be analyzed in terms of measure-value units. The concepts from logic and mathematics that are required for an analysis of measurement, thence measurement omission, are not to be analyzed in terms of the latter. That is why I have held, contrary to Rand, that certain logical and set-theoretic concepts are not to be analyzed in terms of measure-value omission, which is the explanatory structure distinctive of Rand’s analysis of concepts. It remains that the proposal to analyze all other concepts in terms of measurement omission (at least to the level of ordinal measurement) is a substantial, definite, meaningful, and meaning-giving proposal.

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At a recent meeting of the American Philosophical Association, Gila Sher delivered a paper titled "Is Logic in the Mind or in the World?" According to her abstract, she presented an outline in this paper of a unified answer to these questions among others...

The workings of identity in logic can sometimes look like a barren exercise. But these workings are for true thinking about the way the world is.

Stephen, myself I think that the difference between smart people and stupid people is that stupid people don't know how ignorant they are, and they think that everyone else is at their level; whereas a smart person knows when someone else is a whole head taller than they are. And you are. So, you will excuse me if I am being obvious here, but by "necessary facts" or "necessary factual truths" would logic not be in the mind because it is in the world? I know the difference between inside-me and outside-of-me, but, ultimately, unlike leprechauns and unicorns or centralized economic planning, logic actually works. So, that validates it.

You are an architect. Could you imagine a conference held in any building wherein the artists debated the validity of structural engineering? (It is easier to do that the other way around. I work in a Beaux Art structure and from the roof, I can see a skyscraper with a medieval cathedral on top. What were they thinking?) To me, the brilliance in Rand's work was her re-ignition of objectivism, an Enlightenment philosophy that was eclipsed first by positivism, and then all the rest in succession.

I am now reading The Rational Optimist by Matt Ridley. He talks of Japan in the late medieval era. They retreated from mechanization and gave over their work and energy to human efforts at the lowest levels. If you had popped in to say, "Would you like some machines?" you would have gotten a long list of arguments - including a sword... So, too, with modern philosophy. I see Objectivism as a machinery for analysis and investigation. Obviously, though, most other people interested in such topics do not share that with us.

Be that as it may, thanks for posting. I always benefit.

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Michael (and Jim),

Mathematicians, unlike the rest of us, have retained something of the original Enlightenment spirit, thought the novelist Robert Musil;* they provide examples of spiritual daring that has otherwise fallen by the wayside. “We others,” Musil regretted, “have let our courage drop since the time of the Enlightenment. Some small bungle was enough to get us off the track of reason, and we now let every soft-headed visionary denounce the project of a d’Alembert or a Diderot as misguided rationalism.” We are apt to plead the cause of feeling against the intellect, forgetting that we inhabit an intellect-constructed world (Musil 1913 – “Der mathematische Mensch”).

So begins the Preface to A. W. Carus’ Carnap and Twentieth-Century Thought – Explication as Enlightenment (Cambridge 2007). As you know, across the nineteenth-century, there had raged an intellectual conflict between, on the one side, literary and philosophic romantics who extolled the subordination of scientific reason to intuitive life and, on the other side, philosophic rationalists (small r) who extolled unfettered scientific reason, seen as the potential savior of humanity from disease, hunger, and war. The former included Goethe, Schelling, and Nietzsche; the latter included Mill and the positivists Comte and Mach.

Michael, I should question casting positivism and logical positivism as outside the Enlightenment quarter. In central Europe, by the time of Musil’s remark and beyond World War I, neo-Romantic, anti-Enlightenment views predominated.

. . . The atmosphere is well captured, and pitilessly satirised, in Musil’s great novel The Man without Qualities.

The Vienna Circle is impossible to understand outside this very specific cultural context. The Circle reasserted Enlightenment values against this comprehensive Romantic fervour. It countered with an equally comprehensive programme of re-Enlightenment. Unlike previous German movements that had taken the Enlightenment partially on board—especially the venerable tradition of German classicism deriving from Goethe, Schiller, and Humboldt, within Kant’s philosophical framework—the Vienna Circle resolved to accept no compromises. Everything was to be rethought from the bottom up. To begin with, the basis of scientific knowledge itself—the backbone of the cosmopolitan ideal—had to be reconstructed. The older Enlightenment philosophies of Mill, Comte, or Mach had been glaringly unable to cope with recent advances in the sciences. Instead, the Vienna Circle turned to Bertrand Russell, to Russell’s student Wittgenstein, and to scientific thinkers such as Helmholtz and Poincaré. . . . (Carus 2007, 3–4)

I highly recommend Carus 2007.

Related vis-à-vis theoretical philosophy:

Friedman, M. 1999. Reconsidering Logical Positivism. Cambridge.

Juhl, C., and E. Loomis 2010. Analyticity. Routledge.

Ben-Menahem, Y. Conventionalism. Cambridge.

Russell, G. 2008. Truth in Virtue of Meaning – A Defence of the Analytic/Synthetic Distinction. Oxford.

Zahar, E. 2001. Poincaré’s Philosophy – From Conventionalism to Phenomenology. Open Court.

Related vis-à-vis practical philosophy:

Wolin, R. 2004. The Seduction of Unreason: The Intellectual Romance with Fascism from Nietzsche to Postmodernism. Princeton.

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Stephen, thanks! I will remember Carus when I qualifiy for more library privileges here. In the mean time, I do like taxonomies. There are several ways to arrange the Periodic Table, some visually stunning. As you know many Objectivists identify Nietzsche as an "individualist" and Comte as a (well, the) "collectivist." So, that axis is another way to arrange the thinkers. I do agree with you, though, that the "classical" versus "romantic" is another method. I point out, also, that both Comte and Spencer looked for "social physics" before using "positivism" to seek for "sociology." So, collectivism is not inherent in positivism. Likewise, Karl Popper was famous for his plea for an Open Society, though, I feel that his epistemology would leave many Objectivists cold. I tried Wittgenstein's Tracticus twice and was not impressed, though I was, indeed impressed - if not blown away, as were some of his comrades - that these were the speculations of a solider in the trenches between battles. War is truly destructive. But, I mention the work because of my interest in languages. I have here a phrase book for Central Asia: Uzbek, Kazhak, Uighur, a little Pashto... When Wittgenstein and Popper argued (without hot pokers) about natural language, they probably meant German, likely included English, and just stopped there. I just found it silly to speculate on the "meaning of meaning" when you limit your guesses to your own tribe. What do other people mean by meaning? I wonder...

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Meaningless Tautology

I exhibited a defined concept that was objectively meaningless in “Between False, Invalid, and Meaningless.”* That was the concept super-knowledge, stipulated to be nothing other than imageless and wordless apprehension higher than knowledge resulting from perceptual and logical means. That concept I said was objectively meaningless since it contained no positive identity beyond its genus apprehension; its objective meaning is only apprehension; the putative species super-knowledge adds nothing.

Ordinary parlance takes tautology to mean a needless repetition, a redundancy. Repetition is needed for learning. That sort of repetition would not be a tautology. Tautology is a repetition needless to the cognitive purpose at hand. There is, for this particular purpose, no gain in the repetition.

The failing to cognitive purpose that is tautology, in ordinary parlance, is not the utterance or thought that is simply A, A, A, . . . . Rather, it is paradigmatically the utterance or thought having the form A is A that is alleged to be needless repetition to the cognitive purpose. Statements of the form A is A, if tautologies in the present sense, could well be thought to be objectively meaningless in the manner of the concept super-knowledge.

A is A is called tautology in logic and among philosophers, without necessarily prejudging the repetition to be needless to their cognitive purposes. Indeed, to the purpose of valid deduction, tautologous repetition is the seal (Carnap 1934, 44). With logic and philosophy, whether and in what ways A is A is meaningless are further questions, unlike ordinary parlance of tautology (e.g. Wittgenstein 1918, §§ 4.46, 6.1; Ayer 1946, 41, 81, 87, 135; Pap 1958, 438).

Logic texts inform us that A is A is always true, even if A is a false statement. That tautology (A is A) could ever be meaningless is in some tension, to say the least, with holding it is always true. The sense, if any, under which A is A is meaningless must be a sense in which A is A is not true.

From the circumstance that A is A is always true, Wittgenstein and Carnap, in their different ways, concluded that A is A has no factual content. In the view of Moritz Schlick, affirmations of present observation and affirmations of A is A

have in common with them that in both cases the process of understanding is at the same time the process of verification—I grasp the truth at the same time as the sense. In the case of an affirmation [in an observational proposition] it would have just as little sense to ask whether I could perhaps be mistaken about its truth as it would in the case of tautology. [Cf.] Both are absolutely valid. Only the analytic proposition or tautology is at the same time empty of content, while the observational proposition provides us the satisfaction of genuine cognition of reality. (1934, quoted in Friedman 1999, 149)

Super-knowledge is nothing beyond apprehension, and “Super-knowledge is super-knowledge” is nothing beyond “Apprehension is apprehension.” Do statements such as “Apprehension is apprehension” or “Man is man” ever amount to a cognitive gain? When, if ever, are statements of the form A is A objectively meaningless?

Aristotle observed that saying “Man is man” or “The musical is the musical” and so forth are all occasions of affirming that a thing is itself. Someone who had said “Each thing is itself” might have meant “Each thing is inseparable from itself; and its being one just meant this” (Met. 1041a1). This meaning in A is A states a truth of all existing things—an existing thing is one, one with itself—and that is sufficient for some tare weight of objective meaningfulness in the statement. That much of A is A is background assumption in all other meaningful statements. Saying A is non-A is objectively meaningless and false. As I said in “Between False, Invalid, and Meaningless,” the objectively meaningless always stems from falsehood. In the case of affirming a manifest contradiction, the root falsehood is self-same with the meaningless statement.

In mathematics if we can show that an equation can be reduced to the equation 1=1 or 2=2 or sin(theta)=sin(theta), and so forth, we have proven that the initial equation is true. That is a usefulness of “A thing is itself.” In science one aim (and thrill) is discovery of A is A sleepers. This has been accomplished through the join of observation, mathematics, and induction. Examples are the discovery that the evening star is the morning star and that light is electromagnetic radiation (within a certain range of radiation frequency).

One objectively meaningful way in which one might say “Apprehension is apprehension” is in the context of having concluded that the concept super-knowledge adds nothing to the concept apprehension. Stating that “Apprehension is apprehension” could be, in this conversational context, an acknowledgment that a kind of reduction from two to one has been seen. Less constructively, yet still with objective meaningfulness, the statement could also be sarcasm implicitly stating that the definition of super-knowledge merely states the truth everyone already knew, which is that apprehension is itself and has the character it has.

There are occasions in which one could say “Man is man” by way of stressing that some stunning actions of man are among human capabilities, parts of human nature. Seeing the first man walk on the moon, one might say “Man is man.” Seeing Romeo take his life, one might say “Man is man.” The reader of Atlas Shrugged finds Ayn Rand proclaiming “Man is man” with a meaning along these lines. Her proclamation was to stress that, notwithstanding his freedom of mind, man has a definite nature, that he is nothing but man, and that he is one. This was important for the purpose of setting forth her vision of the nature of man.

Rand’s statement “Existence exists” is a repetition. The statement had been considered before Rand. Bradley thought it a meaningless tautology; Russell thought otherwise. Tibor Machan has explained the inner cognitive contour of Rand’s statement; he has explained how it means its truth about its object (1992, 40–43; 1999, Chapter 2). Rand was using the statement for first philosophical axiom, and it serves the functions that axioms serve in her philosophy (Machan 1992, 32–40).

Logicians of the thirteenth century found a use for A is A in syllogistic logic. As previously noted,* the form it took there was Every A is A, and it was used in the first mood of the second-figure syllogism to finally convert No B is A to No A is B.

We find Leibniz, four centuries later, illustrating the utility of Some A is A for concluding Some A is B from All A is B via a syllogism of the first figure. “I offer these examples . . . to show that identities do indeed have a use and that no truth, however slight it may seem, is completely barren; on the contrary, . . . these identities contain the foundations for all the rest” (Leibniz 1679, 226; see also 1705, 362–63).

For Leibniz the necessity of the conclusion of a demonstration consists in the circumstance that
its contrary implies a contradiction; this is the true and unique mark of impossibility. Just as necessity corresponds to impossibility, furthermore, so an identity corresponds to a proposition which implies a contradiction. For the primary impossibility is this: A is not A; just as the primary necessity in propositions is this: A is A. (Leibniz 1678, 187; see also 1689, 267; 1789?, 96)

The identity of which Leibniz speaks as a basis of logical necessity is the identity of sameness. A demonstration establishes a sameness between the subject and predicate in the conclusion. It shows that the conclusion’s predicate is contained in the conclusion’s subject. Among the premises could be observations or intellectual truths (e.g. “Nothing is greater or less than itself.”). The demonstration proceeds by recognizing definitions and by substitution. For effective use in proofs, definitions must not contain contradictions, manifest or concealed. It is not enough that we understand what we say in a definition, for it can still be the case that our definition is of something impossible (Leibniz 1684, 293). Natures of things are implicated in proofs by the observations, intellectual truths, and definitions employed in the demonstration.

Loemker writes of Leibniz: “Contradiction, . . . or the principle of impossibility, is implied in identity, and the two are opposite aspects of the same law, which Leibniz sometimes calls the basic law of being” (1989, 24). In Leibniz’ view, the law of identity entails that predicates of affirmative propositions are contained in their subjects. As with Aristotle, with Leibniz the primary form of being is substance. Identity-containments by subjects of their predicates record the relation of substance to its modifications.

Rand did not continue with Aristotle’s central concept of substance, rather, she made existence most fundamental and made natured entity the primary form of existence. Entity, not substance, takes the role of bearing attributes and actions. Rand’s conception that “logic is the art of non-contradictory identification” and that “logic rests on the axiom existence exists” embed logic in her fundamental metaphysics: Existence exists, existence is identity; consciousness is identification (1957, 1016). In amplification of her compact statement “Existence is identity,” Rand goes on to say that the law of identity (and lack of contradiction) applies to objects, to attributes, to actions, and to their compositions into larger wholes (1957, 1015–16).* In Rand’s metaphysics, identity as to nature is tied at the most basic level to identity as self-sameness. For Leibniz identity as self-sameness (of reals) is the deeper reality of the two. Not so for Rand.

Rand shared with Leibniz the view that the principle of non-contradiction rests on the law of identity. In the 1960’s lectures The Basic Principles of Objectivism, Nathaniel Branden held forth and explained Rand’s idea that the law of identity is the basic principle of metaphysics and of epistemology. The law of identity
is the link between the two sciences, the bridge between existence and consciousness, between reality and knowledge.

As a principle of metaphysics, the Law of Identity tells us that everything which is, is what it is. As a principle of epistemology, it tells us that contradictions cannot exist, that a thing cannot be A and not-A, and, therefore . . . we have made an error. Our thinking is wrong. Our thinking does not correspond to reality, and we must check our conclusions, our reasons for them, our premises. (Branden 2009, 66–67)

Those relations on identity and non-contradiction were also presented by Leonard Peikoff in his 1972 lectures on the history of philosophy, where he indicated historical philosophic puzzles resolved by these Randian conceptions. In Objectivism: The Philosophy of Ayn Rand, Peikoff writes:
The law of identity acts as a bridge linking existence and consciousness, or metaphysics and epistemology. The law acts as a bridge in a second respect also. The law defines the basic rule of method required for a conceptual consciousness to achieve its task. In this regard, the law tells man: identifications must be noncontradictory.

. . . Aristotle’s law of contradiction states . . . nothing can be A and non-A at the same time and in the same respect. This is not a different fact from the law of identity. It is a corollary of the latter, a restatement of it for the purpose of guiding human cognition. (1991, 118–19)

Rand’s sense of identity basing noncontradiction goes beyond Leibniz to include natures at the most fundamental level of identity. Moreover, identities of nature or character are not modifications of existence. Existence is identity, not only identity of sameness, but identity of character. Existence is both identities. Existence is identity of character.

In modern logic, we could certify the inference from No B is A to No A is B in the following way: It is not the case that there is an individual thing x that is of the class B and an individual thing y that is of the class A where x is the very same as y in the way the evening star is the morning star. The way in which the evening star is the morning star is the very way in which the morning star is the evening star; the way in which x is y is the very way in which y is x. Therefore, it is not the case that there is an individual thing y that is of the class A and an individual thing x that is of the class B where y is the very same as x in the way the morning star is the evening star.

Where x, which is of the class B, is the very same individual thing as y, which is of the class A, in the way the evening star is the very same as the morning star, we have shown that if there is no x=y member of B that is x=y member of A, then there is no y=x member of A that is y=x member of B. That is, we have shown that if No B is A, then No A is B. The inference can be certified under the same pattern where instead of “in the same way that the evening star is the morning star,” we use “in the same way that a belly is a tummy.” The conversion of No B is A to No A is B is certified for both of those types of sameness of individuals.

Those logical certifications presume that for any individual thing a, it is itself. The morning star is the morning star, Venus is Venus, and a belly is a belly. Also, a class A to which a belongs is itself. A is A. (See also Feingo and May 2002.)

“B and non-B is B and non-B” affirms a sameness, an identity between thoughts. But the thought is only of nothing. “B and non-B is B and non-B” has the form A is A. This affirmation having the form A is A fails totally in objective meaningfulness, for it affirms not that B and non-B is one—there is not one there to affirm—and it affirms nothing of positive character-identity. Without the former, it is without even the tare weight of objective meaningfulness; without the latter, it is absent any weight of meaningfulness on the scale. Similarly, A is A, where A is stipulated as meaning “a characterless thing” is grossly defective in objective meaningfulness. The usual source of objective meaninglessness in the repetition that is tautology in the logical sense is neglect of the circumstance that existence is character-identity. No character-identity, no objective meaning on the scale. (See also Peikoff 1967, 100–101.)

In Tractatus Wittgenstein maintained that tautology allows each and every state of affairs, whereas contradiction voids possible states of affairs. Tautologies are unconditionally true; they present no picture of reality, no specific state of affairs. Tautologies are without sense, though they are not nonsensical (§4.46). I have allowed that even the tautology “B and non-B is B and non-B” is not entirely nonsense, though it is devoid of objective meaningfulness. However, “No possible state of affairs is no possible state of affairs” is not an allowance of a state of affairs. Here tautology allows non-states-of-affairs. Then tautologies allow not only all states-of-affairs, but non-states-of-affairs. Yet B and non-B is a voided state of affairs by the principle of contradiction. Tautology then allows what contradiction voids. Then the principle of identity contradicts the principle of non-contradiction. This predicament suggests that Wittgenstein was off-track in thinking that states of affairs allowed by tautology were entirely without restrictions (§4.46) and in not fully assimilating (beyond §6.124) that all possible states of affairs have some specific character or other.

“The method of logic,” non-contradictory identification, “does reflect the nature and needs of man’s consciousness. It also reflects the other factor essential to a proper method: the facts of external reality. The principle which logic provides to guide men’s mental steps is the fundamental law of reality” (Peikoff 1991, 120–21). A is A, where A is objectively meaningful.

References

Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton.

Ayer, A. J. 1946. Language, Truth and Logic. Dover.

Branden, N. 2009. The Vision of Ayn Rand. Cobden.

Carnap, R. 1934. The Logical Syntax of Language. A. Smeaton, translator. 1937 (reprinted 2002). Open Court.

Fiengo, R., and R. May 2002. Identity Statements. In Logical Form and Language. G. Preyer and G. Peter, editors. Oxford.

Friedman, M. 1999. Reconsidering Logical Positivism. Cambridge.

Leibniz, G. W. 1678. Letter to Herman Conring – March 19. In Loemker (L) 1989.
——. 1679. On the General Characteristic (L).
——. 1684. Meditations on Knowledge, Truth, and Ideas (L).
——. 1689. First Truths (L).
——. 1689? On Freedom. In G. W. Leibniz: Philosophical Essays. R. Ariew and D. Garber, translators. Hackett.
——. 1705. New Essays on Human Understanding. P. Remnant and J. Bennett, translators. 1996. Cambridge.

Loemker, L. E., editor, 1989 [1952]. Gottfried Wilhelm Leibniz: Philosophical Papers and Letters. L. E. Loemker, translator. Kluwer.

Machan, T. 1992. Evidence of Neceassary Existence. Objectivity 1(4):31–62.
——. 1999. Ayn Rand. Peter Lang.

Pap, A. 1958. Semantics and Necessary Truth. Yale.

Peikoff, L. 1967. The Analytic-Synthetic Dichotomy. In Rand 1990.
——. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.

Rand, A. 1957. Atlas Shrugged. Random House.
——. 1990 [1966–67]. Introduction to Objectivist Epistemology. Expanded second edition. H. Binswanger and L. Peikoff, editors. Meridian.

Schlick, M. 1934. On the Foundation of Knowledge. In Moritz Schlick: Philosophical Papers. H. L. Mulder and B. F. B. Van de Velde-Schlick, editors. 1978–79. Reidel.

Wittgenstein, L. 1918. Tractatus Logico-Philosophicus. C. K. Ogden, translator. 1922. Routledge.

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Analytic-Synthetic Distinction

Part 1 – Quine

In the 1940’s and ’50’s, some eminent philosophers mounted challenges to the sharp distinction between analytic and synthetic statements, propositions, and judgments. The sharp distinction had been important in the modern philosophy of empiricism. In the present paper, I set out the relation of Willard Quine’s opposition to the distinction in the ’50’s to the Rand-Peikoff opposition in the following decade.* In a sequel, I shall address the relation of Morton White’s opposition to the distinction in the ’50’s to the Rand-Peikoff opposition in the following decade; the overlap of issues raised by White in the controversy with those raised by the Objectivists is considerable. The issues bearing on the analytic-synthetic controversy in Ayn Rand’s epistemology treatise (ITOE) will partially overlap Quine’s battery of issues in the controversy. Leonard Peikoff’s “The Analytic-Synthetic Dichotomy” will be the main Objectivist treatment of the issues raised in White’s “The Analytic and the Synthetic: An Untenable Dualism.”

It was Quine's essay "Two Dogmas of Empiricism" (TD), published in 1951, that brought his debate with Carnap over the analytic-synthetic distinction to widespread attention among philosophers. In this essay, Quine argues against the validity of the distinction. Carnap wanted to maintain a sharp distinction between analytic statements depending entirely on the meanings being used and synthetic statements making assertions about the empirical world. Quine's alternative view had it that all statements face the world as part of a corporate body of statements. On this view, experience bears the same kind of evidential relation to the theoretical parts of natural science as it does to mathematics and logic (see also Quine 1954).

One of Quine’s aims in “Two Dogmas” is to argue that no sharp distinction can be drawn between analytic statements and synthetic statements. Analytic statements are ones alleged to be “true by virtue of meaning and independently of fact” (TD 21). Truths grounded in fact are known as synthetic truths; statements of such truths are called synthetic statements.

Quine reminds the reader that meaning “is not to be identified with naming” (TD 21). My height and my stature (under one of its definitions) name the same thing and mean the same thing. Likewise for three and drei. In general, however, concepts with different meanings can name the same thing, as my right hand and my writing hand.

Quine writes of terms, but these are terms working in a certain way, terms employed in statements admitting of truth or falsity. Quine’s “terms in statements” would seem not far from “concepts in propositions” which is the technical vocabulary adopted by Rand.

Turning to general terms like right hand, Quine observes that we must distinguish between the meaning of the term and the extension (the referents) of the term. Think of the essence of right hands. Instead of thinking of the essence in the Aristotelian way—as inhering in those hands (actual and possible)—let it inhere in the term. That thought is the meaning of a general term, and, like Aristotelian essence, it is not one and the same as the thing signified (TD 21–22).

I better hit the gavel for Rand at this point. We speak of the meanings of words, but words are only markers for concepts, “and the meaning of a concept consists of its units.” We define concepts “by specifying their referents.” Concepts and definitions are certain ways of specifying referents (ITOE 44).

I look up the word derelict in my dictionary and find one of its meanings: abandoned property; especially, a ship abandoned at sea. Knowing how to apply the latter term (grammatically, a phrase), I might now use this sense of the word derelict. The word being defined and its definition have the same meaning. They are cognitively synonymous.

Quine thought that the useful conceptions of meanings come down to (i) giving synonyms or (ii) making significant utterances (1948, 11). Rand held that when we make significant utterances that engage concepts, or general terms, those concepts have definitions specifying their referents. (Prior to being able to state propositions in which concepts figure, a concept like ball is [marked by a word and] nested in image and action schemata [ITOE 13, 20, 43; further Boydstun 2004, 290]. Presumably, this rudimentary mentation, alternative to explicit propositions and definitions, informs them.)

Quine notices that there are definitions of a sort that are not simply the giving of synonyms, and he calls these sorts of definitions explications.

In explication the purpose is not merely to paraphrase the definiendum [the term being defined] into an outright synonym, but actually to improve upon the definiendum by refining or supplementing its meaning. But even explication, though not merely reporting a preexisting synonymy between definiendum and definiens [the definition], does rest nevertheless on other preexisting synonymies. . . . Any word worth explicating has some contexts which, as wholes, are clear and precise enough to be useful; and the purpose of explication is to preserve the usage of these favored contexts while sharpening the usage of other contexts. In order that a given definition be suitable for purposes of explication, therefore, what is required is not that the definiendum in its antecedent usage be synonymous with the definiens, but just that each of these favored contexts of the definiendum, taken as a whole in its antecedent usage, be synonymous with the corresponding context of the definiens.

Two alternative definientia may be equally appropriate for the purposes of a given task of explication and yet not be synonymous with each other; for they may serve interchangeably within the favored contexts but diverge elsewhere. By cleaving to one of these definientia rather than the other, a definition of explicative kind generates, by fiat, a relation of synonymy between definiendum and definiens which did not hold before. But such a definition still owes its explicative function, as seen, to preexisting synonymies. (TD 25)

When philosophers lay out theories of good definition, they are theories of an explicative kind of definition (see Kelley 1988, chapter 3). Consider Rand’s definition of reason as the faculty that identifies and integrates the evidence of the senses. In my dictionary, I find reason defined as the capacity for rational thought, rational inference, or rational discrimination. The terms rational and thought go to already familiar synonymies with reason. The differentia within the rational, in this dictionary definition, are the discriminatory and the inferential.

Rand’s definition stays close to the common usage reflected by the dictionary, but it replaces discrimination and inference by their kin identification and integration, it eliminates the non-explicative rational, and it adds a base for the activities of reason, specifically, deliverances of the senses. Rand’s definition is explanatory of the common usage found in the dictionary, and it is tailored to tie neatly to a particular wider philosophical view.

Quine could say this is a fine explicative type of definition. Rand has given the term reason a new synonymy. The various contexts in which reason under the dictionary definition is properly used remain contexts in which reason under the new, explicative definition is properly used. The new definition covers the processes of drawing distinctions and making inferences. The new definition also applies to the wider processes of identification and integration of sensory evidence, processes in which the narrower processes are embedded. Quine would stress that, nonetheless, “such a definition still owes its explicative function . . . to preexisting synonymies” (TD 25). Quine is being too short here.

Quine’s argument against the idea that there are clearly statements true purely by virtue of meanings, and true independently of fact, hangs on his conception of meaning. We have joined Quine in saying that meaning is distinct from reference. We have not allowed that meaning can be independent of reference. The meanings of my right hand and my writing hand differ, but both meanings are specifications of a referent. Similarly, the meanings of right hand and writing hand differ, but both are specifications of the extensions (the referents) under those concepts.

Quine’s conception of meaning is shriveled into “synonymy of linguistic forms” (TD 22). He allows that a logical truth such as “Every tall man is a man” has a guarantee of truth that rests on more than one’s experiences of facts about men. He realizes that logical truths are sometimes called analytic, but his target is other statements taken for analytic: statements reducible to logical truths by synonymies, statements reducible to logical truths by meaning.

Quine shares with defenders of analyticities the conviction that logical truths are true, and true regardless of particular facts to which they are applied. If there are statements reducible to logical truths by virtue of meaning and independently of fact, then their truth would be guaranteed by virtue of meaning and independently of fact. A candidate analytic statement from Kant would be “Bodies have location.” (A contrasting synthetic statement would be “Orbiting bodies are weightless.”) One who has the concept body knows that having location is part of the meaning of the concept. Substituting “Things having location (and . . .)” for body yields the logical truth “Things having location (and . . .) have location.”

Now, we know “Necessarily, things having location (and . . .) have location.” Does only that sense of necessity attach when we claim “Necessarily, bodies have location”? Quine disputes the idea that purported analyticity of a statement can be adequately explained by cognitive synonymies and logical truth (TD 29–31). Analyticity cannot be explained by a sensible conception of meaning joined with logical truth. An adequate way of distinguishing analytic from synthetic statements has not been produced.

Quine uncovered a narrow, but serious, problem for the analytic-synthetic distinction. It would seem that there are wider problems for the distinction that he passes over because of his cramped conceptions of meaning, definition, and essential characteristics.

References

Kelley, D. 1988. The Art of Reasoning. W. W. Norton.

Peikoff, L. 1967. The Analytic-Synthetic Dichotomy. In Rand 1990.

Rand, A. 1990 [1966–67]. Introduction to Objectivist Epistemology. Expanded 2nd edition. Meridian.

Quine, W. V. O. 1948. On What There Is. In Quine 1953.

——. 1951. Two Dogmas of Empiricism. In Quine 1953.

——. 1953. From a Logical Point of View. Harvard.

——. 1954. Carnap and Logical Truth. In The Ways of Paradox and Other Essays. 2nd edition. 1976. Harvard.

White, M. G. 1952 [1950]. The Analytic and the Synthetic: An Untenable Dualism. In Semantics and the Philosophy of Language. L. Linsky, editor. Illinois.

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Analytic-Synthetic Distinction

Part 2 – White and Rand-Peikoff

Near the close of “Two Dogmas of Empiricism,” after saying once more that he rejects the distinction between the analytic and the synthetic, Quine enters a footnote directing the reader to a paper by Morton White “for an effective expression of further misgivings over this distinction” (1953, 46). That paper is “The Analytic and the Synthetic: An Untenable Dualism” (UD).[1] It appeared originally in Hook 1950,[2] then again, in Linsky 1952. My page references are from the latter.

Morton White noted two kinds of statements that had lately been regarded as analytic. The first are purely formal logical truths such as “A is A” and “A or not-A.” The second are cases of “what is traditionally known as essential predication” (UD 318). He ponders especially the example “All men are rational animals.” That statement is logically the same as “Any man is a rational animal” or “A man is a rational animal.” This last expression of the proposition is one of Leonard Peikoff’s examples of a purportedly analytic statement in “The Analytic-Synthetic Dichotomy” ([A-S] 1967, 90).

White did not pursue in this paper whether it is correct to characterize logical truths as analytic (UD 318–19). It will be recalled that Peikoff held forth Rand’s conception of logical truth against that of A. J. Ayer, who had maintained: “The principles of logic and mathematics are true universally simply because we never allow them to be anything else. . . . In other words, the truths of logic and mathematics are analytic propositions or tautologies” (1946, 77; Branden 1963, 7; A-S 94, 101, 111–18).

As with Quine’s “Two Dogmas,” White undermined the distinction between the analytic and the synthetic by finding fault with various explications of what analyticity amount to. They concluded there is no durable articulate way of classifying propositions and truths as analytic in sharp contrast to synthetic.

One way of conceiving an analytic statement is as expressing a proposition deducible from a logical truth by substitution of a synonym of one of its terms. (i) Every A is A. Therefore, (ii) Every man is a man. With rational animal as synonym for man, we obtain (iii) Every man is a rational animal (UD 319).

Thence analyticity is explicated in terms of logical truth and synonymy. White rejects the view that whether man and rational animal are synonymous is a matter of arbitrarily selected convention. Similarly, that man and featherless biped are not synonymous is not a matter of arbitrarily selected convention. Natural language is not like an artificial logical language in which meanings of terms are set entirely by stipulation (UD 321–24).

White allows we certainly have some sort of working distinction between propositions such as, on the one hand, “Man is an animal” and “Man is a rational animal” and, on the other hand, “Man is a featherless biped” and “Man has two eyes” (Peikoff’s example, A-S 90). White concludes that distinction between those two classes of statement is not that statements in the first class are analytic, the latter not, where the analytic is defined as consequence of a logical truth under substitution of a synonym and we are given no objective criterion for synonymy (UD 318–24).

Could analytic statements be defined instead as those whose denials are self-contradictory? White did not relate this criterion for analyticity to Kant, but I should do so. One of Kant’s characterizations of analytic judgments is that in them the predicate is “thought through identity” with thought of the subject. Synthetic judgments connect predicate to subject, but not in the relation of identity (KrV A6–8 B10–12). The law of noncontradiction is the principle of analytic judgments. According to Kant, all judgments must conform to that law, but only judgments certifiable by that law alone, apart from their truth in experience, are analytic (A151–53 B190–93; 1783, 4:266–70; 1790, 8:228–30, 244–45; Allison 2004, 89–93; Garrett 2008, 204–6).

White queries how it is that “Man is not a rational animal” leads to “Man is not man,” yet “Man is a quadruped” does not lead to “Man is not man.” He again notes that appealing to synonymies in the language is not illuminating in the absence of objective criteria for synonymy (UD 324). If it is said that one’s sense of wrongness in “Man is not a rational animal” differs from one’s sense of wrongness in “Man is a quadruped,” White replies that that is surely only a matter of degree, not a sharp difference in kind. Between one’s response to contradiction of “Man is a rational animal” and contradiction of “Man is a biped,” there is not a sharp difference in kind. If self-contradiction upon denial of a proposition is the criterion for analyticity of the proposition, then there is no sharp divide between the analytic and the synthetic (UD 325–26).

Suppose we adopt the following criterion for analyticity. Were we to come across an animal we determine to be not a rational animal, we would dismiss it instantly as being a man. By contrast, were we to come across an animal we see is not a featherless biped (it is, say, a quadruped), but whose rationality is not yet confirmed or disconfirmed, we hesitate over whether this animal is a man. We know that we might give up the proposition “All men are featherless bipeds” if we learn this animal is rational (UD 326–28). White responds: “Now I suspect that this criterion will be workable but it will not allow us to distinguish what we think in advance are the analytic equivalences. It will result in our finding that many firmly believed ‘synthetic’ equivalences are analytic on this criterion” (UD 328).

White gives no example, but I think his point is illustrated by an analytic-synthetic pair of judgments, favorites with Kant: “All bodies are extended” (analytic) and “All bodies have weight” (synthetic). By the latter, given his knowledge of Newtonian physics, I think Kant rightly understands “All bodies not in gravitational orbit have weight.” Be that as it may, Kant and his contemporaneous intellectuals would dismiss as body just as quickly an entity lacking weight (in the appropriate setting) as they would dismiss as body an entity lacking extension.[3] The criterion of speed of dismissal upon counterfactual encounter fails to always sort what is taken for analytic from what is taken for synthetic.

White observes that the obscurity of proposed criteria for distinguishing analytic from synthetic statements, propositions, and judgments, is not fixed by incorporating the sound Millian point that what is synonymous with man, for example, varies with discursive context. In a biological discourse, “mammiferous animal having two hands” (Mill’s example) might be synonym for man. It remains that analyticity is not illuminated by proposing logical truth and synonymy as its base, not illuminated so as to yield a sharp divide, rather than a gradual divide, between the analytic and the synthetic. The arguments run against such an explication of the analyticity of “Man is a rational animal” will rerun for “Man is a mammiferous animal with two hands” (UD 329–30).

White saw the myth of a sharp divide between the analytic and the synthetic as affiliate of an older mythically sharp division: the Aristotelian division between essential and accidental predication (UD 319, 325, 330). This kinship was also recognized in Peikoff 1967 (A-S 95).

In Cratylus Plato has Socrates uphold the principle that contrary attributes never belong to a fully real thing simultaneously and the principle that “things have some fixed being or essence of their own. They are not in relation to us and are not made to fluctuate by how they appear to us. They are by themselves, in relation to their own being or essence, which is theirs by nature” (386d–e; see also Euthyphro 6d–e; Phaedo 65d, 75c–d, 78d, 100c; Republic 475e–76d, 479–80). Each thing has attributes such as shape, sound, or color; but in addition, each thing has a being or essence. Indeed, “color or sound each have a being or essence, just like every other thing that we say ‘is’” (Cra. 423d–e). Plato maintained moreover that what each thing essentially is, such as Man, Good, Size, or Strength is not discovered by sight or hearing, but by reason when it is most free from bodily, sensory distractions (Phd. 65, 74–75, 78c–79d, 83, 86, 96–105; Theaetetus 184b–87a).

The character of each thing that is always the same is a kind—call it a form—that is “a being itself by itself” (Parmenides 135a–c). Sensory perceptions are as shadows and reflections of these intelligible forms, these intrinsic natures, these essences and being of things (Rep. 509d–e). Plato has no notion of ideas or concepts encompassing both visible forms (such as shapes, sounds, or colors) and intelligible forms (cf. Metaphysics 987b1–13; Notomi 2005, 193–201). Modern notions of concepts or ideas are, in Plato’s frame, only our thoughts grasping intelligible forms. (See further, Kraut 1992, 7–12; White 1992.)

Peikoff put forth the proposition that every version of the analytic-synthetic distinction is based on the idea that some of the characteristics of the existents referred to by a concept are included in the concept, while some other characteristics of those existents are not included in the concept (A-S 94–95). Rand’s theory of concepts has them referring to all the existents of the kind sorted by the concept, including all of the characteristics of those existents (ITOE 27, 65–69). Therefore, Rand’s theory of concepts precludes a dichotomy between the analytic and the synthetic.

Peikoff took Plato’s partition between intelligible forms and their visible shadows or reflections—Plato’s partition between intelligible forms and visible forms—to be the primary historical source of the idea that a concept means only some of the characteristics of the existents it designates. That is incorrect.

It is true that since what we call concepts are for Plato thoughts of intelligible forms and since those forms are the kinds, essences, and being of things themselves, the perceptual characteristics of things are not part of our thoughts of intelligible forms, not part of our concepts. Lines drawn in a geometric proof only approximate lines that are the subject of the proof; the perceived lines are not those geometric lines. But Peikoff went further and referred to Plato’s perceptual characteristics of things as accidents, in contrast to the intelligible characteristics that are essences (A-S 95). Peikoff noted parenthetically that there is a naturalistic distinction (where the home of essences is not in the mind of a supernatural deity as later Platonists would have it) between essential and accidental characteristics of things in Aristotle and his intellectual descendants. That acknowledgment is a big understatement, and Peikoff's additional remark that the essence-accident dichotomy in the theory of concepts endorsed by Aristotelians (he was probably thinking of moderate realists such as Aquinas) “reflects a strong Platonic influence” is unsustainable.[4]

Aristotelians’ and Platonists’ placement of Aristotle’s active intellect, which is part of our means to universals, in a divine mind with which we participate reflects a Platonic influence on the interpretation and development of Aristotle. But the distinction between essential and accidental characteristics is the making of Aristotle. The distinction is within nature, and it is this distinction, Aristotle’s distinction, that constrained Scholastic theories of universals, concepts, and predication. Insofar as the analytic-synthetic distinction rests on the idea that some characteristics of the existents referred to by a concept are included in the concept, while other characteristics are not included in the concept, the analytic-synthetic distinction finds paternity among the ancients, not substantially with Plato, but with Aristotle in his division of essential and accidental characteristics.

(To be continued.)

Notes

1. Nelson Goodman writes in a 1953 footnote: “Perhaps I should explain for the sake of some unusually sheltered reader that the notion of a necessary connection of ideas, or of an absolutely analytic statement, is no longer sacrosanct. Some, like Quine and White, have forthrightly attacked the notion; others, like myself, have simply discarded it; and still others have begun to feel acutely uncomfortable about it” (60). PS: I have learned that White’s paper has been included also in the collection From a Philosophical Point of View (2004).

2. Sidney Hook was Leonard Peikoff’s dissertation advisor.

3. Notice also, looking beyond the present criterion of analyticity, that in modern physics of elementary particles, we take electrons and the other leptons to be bodies (matter) because they have weight (because of nonzero rest mass), yet they have no extension. The feature Kant took for analytic, we eventually took as dispensable, whereas the feature he took for synthetic, we have retained.

4. Concerning Aquinas’ correct understanding that with Platonists such as Pseudo-Dionysius abstract principles, natures, and essences are not part of concrete individuals, but are forms simple, immaterial, and separate from the concrete individuals participating in them, see Aertsen 2010, 82–83.

On Aristotle’s theory of essential and accidental predication radically departing from Plato and (i) turning Plato’s theory of forms upside down, a traditional understanding of Aristotle, or in the alternative, (ii) holding in effect that existence is specific identity to the result that particulars and their essential characteristics are equalized, see Riik 2012.

References

Aertsen, J. A. 2010. Platonism. In The Cambridge History of Medieval Philosophy. R. Pasnau, editor. Cambridge.

Allison, H. 2004 [1983]. Kant’s Trascendental Idealism. Revised and enlarged edition. Yale.

Allison, H., and P. Heath, editors, 2002. Immanuel Kant: Theoretical Philosophy after 1781. Cambridge.

Aristotle c. 348–322 B.C. Metaphysics. In The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton.

Ayer, A. J. 1946. Language, Truth and Logic. Dover.

Branden, N. 1963. Review of Brand Blanshard’s Reason and Analysis. The Objectivist Newsletter 2(2):7–8.

Garrett, D. 2008. Should Hume Have Been a Transcendental Idealist? In Kant and the Early Moderns. D. Garber and B. Longuenesse, editors. Princeton.

Goodman, N. 1953. The New Riddle of Induction. In Fact, Fiction, and Forecast. 4th edition. 1983. Harvard.

Hook, S., editor, 1950. John Dewey: Philosopher of Science and Freedom. Dial.

Kant, I. 1781(A), 1787(B). Critique of Pure Reason. W. Pluhar, translator. 1996. Hackett.

——. 1783. Prolegomena to Any Future Metaphysics That Will Be Able to Come Forward as Science. G. Hatfield, translator. In Allison and Heath 2002.

——. 1790. On a Discovery Whereby Any New Critique of Pure Reason Is to Be Made Superfluous by an Older One. H. Allison, translator. In Allison and Heath 2002.

Kraut, R. 1992. The Cambridge Companion to Plato. Cambridge.

Linsky, L., editor, 1952. Semantics and the Philosophy of Language. Illinois.

Notomi, N. 2005. Plato’s Metaphysics and Dialectic. In A Companion to Ancient Philosophy. M. L. Gill and P. Pellegrin, editors. Wiley-Blackwell.

Peikoff, L. 1967. The Analytic-Synthetic Dichotomy. In Rand 1990.

Plato c. 428–348 B.C. Plato – Complete Works. J. M. Cooper, editor. 1997. Hackett.

Quine, W. V. O. 1951. Two Dogmas of Empiricism. In From a Logical Point of View. 1953. Harvard.

Rand, A. 1990 [1966–67]. Introduction to Objectivist Epistemology. Expanded 2nd edition. Meridian.

White, M. G. 1952 [1950]. The Analytic and the Synthetic: An Untenable Dualism. In Linsky 1952.

White, N. 1992. Plato’s Metaphysical Epistemology. In Kraut 1992.

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Part 3 shown here is an expanded version

of the original, which appeared 27 June 2012.

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Analytic-Synthetic Distinction

Part 3 – Objective Analyticity

The most basic element in the conception of analytic propositions, in the many varieties of the conception, is that in such propositions the proposition can be known to be necessarily true if only one understands the concepts that are its subject and predicate. In this study, I shall formulate a new analytic-synthetic distinction.

My analytic-synthetic distinction would not have been possible before the development of modern science and its methods, including the way it crafts concepts into a system and characterizes their relationships mathematically in every way it can discover. Rand’s mensural-objective general analysis of concepts also would not have been possible without that development, and my work builds on hers.

The individual being that is Socrates is a man. The man who is Socrates is not something inhering in Socrates; he simply is Socrates. Individual beings are what Aristotle calls primary substance and what Rand calls entities (Categories 2a13–16; Barnes 1995, 98–101; Rand 1957, 1016, 1036–37; 1966–67, 5-6, 15; Branden c. 1968, 38, 80–81; Peikoff 1976, 146; 1991, 12–13).

The individual, primary substance belongs to species and genera, which are utterly dependent for reality on the existence of the individual, primary substance. Man as an animal species rightly characterizes Socrates and sorts him into a specific class, though man in this sense is not in Socrates. Man as species in characterization and sorting is what Aristotle calls secondary substance (Cat. 2a15–18, 2b8–15; Metaphysics 1038b8–34) and what Rand called the concept man, as a concept grouping and characterizing individual entities (1966–67, 15–17)[5]. In this sense, for Rand as for Aristotle, man is rightly said likewise of both Socrates and Protagoras (On Interpretation 17a38–39; Met. 1038b10–12). Porphyry identified as universals Aristotle’s things “said of a subject,” in contrast to things “in a subject” (Gerson 2005, 83–84; Karamandis 2006, 314–18).

Everything predicable of man as species in characterization can be rightly predicated also of the individual man (Cat. 1b10–15). The definition of man as species in characterization can be rightly predicated of Socrates or Protagoras. We say properly “Socrates is a rational animal.” The genus animal in the definition of man, like man as species in characterization, can be rightly said in specification of Socrates, though it is not in him (Cat. 2a19–26). The genus, like the species, is secondary substance. The genus animal, though not in these primary substances, is rightly said likewise of Socrates, Alexander, and Bucephalus.

There are many things that are in the individual being. Some that are in the individual being can be predicated of it, yet the definition of the predicate is not rightly predicated of the individual being. “For example, white, which is in a subject; for a body is called white. But the definition of white will never be predicated of the body” (Cat. 2a32–34). To say what white is we must refer to body, for white has no being except as in a substance, more specifically in a body (Met. 1028a10–b4). On closer consideration, we see white cannot be defined strictly speaking, for we can say of its being only that it is the color of such-and-such body. Similarly with the definition of biped: we can say only that it is a certain means of locomotion of an animal (Met. 1030b14–1031a13; Gill 2010, 113–16).

The substance animal must be brought into the account of what is biped, whereas animal body need not be brought into the account of white. Any body can be brought into an account of color. Though body in general must be brought into its account, it need not be specifically animal body. In Rand’s terms, we would say the concept locomotion presupposes the concept animal. We rightly say as well that locomotion presupposes motion, which presupposes body. But for capturing (part of) what is animal specifically, it is locomotion, not motion, and not color, that needs to be understood.

Aristotle calls accidental those things in substance, but such that they are neither necessary to the substance being what it is nor specific to the substance (Topics 102a18–b5). An accident is “something which may either belong or not belong to any one and the self-same thing. Likewise also whiteness; for there is nothing to prevent the same thing being at one time white and at another not white” (Top. 102b6–10; also Top. 109a34–b12, 116a31–b6, 155a27–35; Met. 1007a1–b18, 1013b34–1014a6, 1015b17–34, 1017a8–23, 1025a14–33, 1026b3–12, 1031b22–28).

Notice that white is an accidental trait of man not only because some men are white, others not. Aristotle takes white to be also an accidental trait of snow, for it does not explain what snow is (Top. 120b21–29). Accident hangs on the possibility of self-sameness of its subject under alteration of the trait. Snow would be snow even were it black (Top. 120b30–35; further complexity, Cat. 12b32–13a2).

Although manners of locomotion are non-accidental traits of animals, the actual occasions of locomotion are states of doing. Saying that a man is running no more says what the man is than saying that he is white. Among the things that can be truly predicated of a primary substance, predicated of the individual being, only the secondary substances, the species in classification, “reveal the primary substance. For if one says of the individual man what he is, it will be in place to give the species or the genus (though more informative to give man than animal); but to give any of the other things would be out of place—for example to say white or runs or anything like that” (Cat. 2b30–35; also Top. 139a24–30). More generally, in Aristotle’s view, neither a quality nor an action will tell what a substance is when predicated of it (Top. 120b25–29).

A quality that rightly divides a genus into species is the differentia in the proper definition of a class of beings such as man (Top. 142b20–145b20). Rationality, not whiteness and not being biped, is the marker within the genus animal that indicates why a man is what he is, that is, it indicates the essence of that distinctive class of animals. The differentia must not be a dispensable attribute. “The differentia is never an accidental attribute, any more than the genus is” (Top. 144a25–26; see Smith 1995, 53–54; further, Code 2010, 93–95).

“By what is ‘in a subject’ I mean what is in something, not as a part, and cannot exist separately from what it is in” (Cat. 1a24–25; further, Lloyd 1981, 41–43, 67–68). Aristotle’s ontological and epistemological focus is on structures of kind, not structures of parts, in the texts I am drawing upon. His quest for account of what is is heavily weighted toward account by specific identity, as opposed to account by particular identity.[6]

We have seen that in Aristotle’s understanding, there are things not said of anything and not in anything; they are individual beings, primary substance (Rand’s entities). We have seen his things said of individual beings and not in them; these are secondary substances (Rand’s concepts of entities). We have seen his things that are said of and are in individual being (Rand’s attributes and actions of entities). Into a fourth bin, Aristotle puts things that are in but not said of primary substance. One of his examples is the individual whiteness that is in a subject (Cat. 1a23–28).

The whiteness in Socrates’ body is a particular whiteness. Aristotle seems correct in thinking we cannot say that particular whiteness of Socrates. For an example known to us today by our own experience, think of the tenor Pavarotti. Aristotle has it that tenor voice is in Pavarotti and tenor can be said of him. However, the particular tenor voice, the one by which we recognize Pavarotti singing (in contrast to Domingo or anyone else singing), though it is in Pavarotti, we cannot make it into a predicate of him.

Agreeing with Rand’s measurement-omission analysis of concepts in their relations to particulars, I say Pavarotti’s tenor voice is in principle susceptible to mathematical analysis as a set of particular values along multiple psychophysical dimensions. Algebraic variables along those dimensions are not in the particular voice; particular values of the variables (the magnitudes reflected by the measure values) are in the particular voice (as heard). There remains some sense, however, in Aristotle’s position that we cannot say of Socrates merely his particular whiteness (cf. Gotthelf 2000, 43n14). We cannot express the particular values of measure without appealing to variability of values along a dimension, which is to say without appealing to whiteness as general or to tenor as general.

We find yet some sense also in Aristotle’s view that white even as a class is in Socrates. The particular value of whiteness in Socrates has its objective relations by common dimension(s) and common variable(s) to the whiteness of Protagoras (assuming that he too is a white man).

With these parts of Aristotle’s thought as background and with Rand’s metaphysics and her theory of concepts and definition as background that I accept, I attempt a new division of statements into analytic and synthetic, capturing some of what earlier analytic-synthetic divisions were after. The analyticity is not nominalist.

Let objective analytic statements be those validly stating necessary specific identity, formal or existential. Their predicates are necessary to their subjects being specifically what they are. Let objective synthetic statements be existential ones validly stating non-necessary specific identity or stating particular identity. Existential synthetic statements of particular identity can be further divided according to whether they are necessary for necessary specific identity.

Objective existential statements of necessary specific identity include all and only existential statements in which the subject presupposes specifically, essentially, or generically that which is predicated or vice versa. Such would be all objective existential definitions, but more generally, all objective existential statements of species or genus. Objective analytic statements in my sense, whether they are formal or existential, are objectively meaningful* statements standing in logical, conceptual relations of presupposition, often implicitly (in the logical sense). Given appropriate understanding of the concepts, such statements can be validated by analysis and proof, apart from facts outside those in the presupposition structure. By formal objective analytic statements, I mean of course the truths of logic and mathematics.

EXISTENTIAL ANALYTIC

“Man is a rational animal.”

—Definition

“Man is body”

Body is a genus for, necessary for, and a presupposition of man.

“Walking is locomotion.”

Locomotion is a genus for, . . . walking. (Italics here are to indicate the term written is as concept in addition to its referents; though italics are appropriate for that reason, I will omit them below for ease of reading.)

“Walking is slowest gait of legged locomotion.”

—Definition

“Socrates is a man.”

—Man is the species necessary for and a presupposition of this individual instance Socrates.

“Red is a color.”

—Color is the species for, . . . red.

“White is not black.”

—Black and white are species of color. Not black is necessary for and a presupposition of white.

“Fire is hot.”

—Hot is necessary for, essential to, and a presupposition of fire.

“Locomotion is of animal.”

—Locomotion presupposes animal movement specifically. Animal is necessary for locomotion.

“Problems are of life.”

—Problems presuppose life, a genus specific to problems. The concept life is necessary for the concept problems.

EXISTENTIAL SYNTHETIC, NON-NECESSARY

“Some soup is hot.”

—Hot is part of the specific identity of some soup. Hot is not necessary to soup ranging free of particular identity.

“Man is bipedal.”

—Bipedal is part of the specific identity of man. Bipedal is not necessary for man. Bipedal presupposes animal, but not man, though man presupposes locomotion.

“Socrates is white.”

—White is part of the specific identity of Socrates. White surface presupposes surface, but not surface of man specifically. Additionally, changing this specific identity of Socrates would leave specific and particular identity sufficient to yet be Socrates. And Socrates does not presuppose white surface, though Socrates presupposes body and its color potentials.

“Socrates is barefoot.”

—Barefoot is part of the particular identity of Socrates (when he has his druthers, which is perhaps always). Barefoot is not necessary for or a presupposition of Socrates. Similarly it goes for “Kant was the son of a harness maker.”

EXISTENTIAL SYNTHETIC, NECESSARY

“Kant was the son of Johann Georg and Anna Regina.”

—The predicate says a particular identity of Kant, one necessary to his specific identity as a biological offspring (alongside his sister), which is a specific identity necessary for man.

“Kant succeeded Hume.”

—Where Kant and Hume are considered not in biological generation, but as makers of philosophy: The predicate says a particular identity of Kant necessary to his particular identity as the maker of transcendental idealism, which required the earlier philosophy of Hume (Treatise), where transcendental idealism is the species of philosophy that is Kant’s.

Where any of these existential statements are true and objectively validated, their denial is a contradiction. To deny an existential analytic statement is to contradict specific identities and their conceptual dependencies, their relations of presupposition. To deny an existential synthetic statement is to contradict non-neccessary specific identities or particular identities.

Existential specific identities are not to be held as conceptually necessary of subjects until they are proven to be so. With growth in mathematical formulation of concepts and growth in knowledge of causal relations, statements of existentially specific identity formerly synthetic may become analytic. Hypothetical example would be “Snow is white” (snow purely H2O in sunlight). Historical example would be our concept of the sun as supporter of life on earth. The circumstance that the yellow light from the sun is higher in frequency than the infrared radiation returned to dark space from earth has been proven to be necessary for life. That the incoming solar photons are higher in frequency than earth’s photons sent to dark space is a consequence of the circumstance that the sun is much hotter than the darkness of space (Penrose 2011, 76–79). That characteristic of the sun (which happened to have been necessary in the concept sun already) has thereby been shown to be necessary in the concept of the sun as supporter of life on earth.

One challenge to my analytic-synthetic distinction for existential statements can be seen by reflecting on the following argument.

....Only animals are naturally locomotive.

....Man is naturally bipedal, which is a form of natural locomotion.

....Therefore: Man is an animal.

Is not “Man is an animal” known analytically, by genus-species relation? How is it that an existentially analytic statement is deducible from existentially synthetic statements? The conclusion as conclusion of this argument is existentially synthetic, with the relation of subject and predicate being one of specific identity. It leaves open whether that relation is also, beyond this argument, a relation of genus and species.

“All men are mortal.” Is that proposition analytic or synthetic under my division? Consciousness and intelligence cannot obtain without physiology. Man cannot be man without consciousness and intelligence. A living body with its physiological processes is a machine subject to the principles of physics, chemistry, and more. Machines are subject to the second law of thermodynamics. They will wear and eventually disintegrate. Machines of repair and replacement will also wear and eventually disintegrate. Therefore, man cannot be man without mortality. “All men are mortal” is an existential analytic statement.

Part of our concept of what man is necessarily by one genus is that he is a machine. Part of our concept of what man is necessarily by another genus is that he is a form of life. From Rand we learn that a process in which there is no active perpetuation of an individual against the possibility of destruction is not the process that is life. Whether a single-cell or a multi-cellular organism, that circumstance of self-preservation against disintegration obtains necessarily to live and to be life. A being without potential mortality would not be a human being for it would not be a living being (Rand 1957, 1012–14; 1961, 15–17; Peikoff 1991, 189–93, 208–11; Gotthelf 2000, 70–71, 80–81, 85n7; cf. Touchstone 1993; further, Nozick 1971, §§1–2; Den Uyl and Rasmussen 1978, §§1–2; King 1984; Nozick 1981, 413–28, 441–44, 515–17, 555–66). The concept human presupposes the concept mortal. “All men are potentially mortal” is an existential analytic statement. That result is also implied patently by the analyticity of “All men are mortal” established in the preceding paragraph.

I gave an example of an existential non-necessary synthetic statement having its status turned into existential analytic by the growth of knowledge. The existential analytic statement “All men are mortal” might undergo metamorphoses with the growth of knowledge, while retaining its analytic status. Say it is found that there is some third thing equally elementary with energy and entropy and this third thing must figure into all our thermodynamic analyses of systems. Say the result is that our present second law of thermodynamics obtains not in all physical situations, but only in the vast range of situations we know of today. Then the analytic statement “All men are mortal” might need to be refashioned into the analytic statement “All men are mortal unless they can get out of such-and-such class of situations.” I note in passing that it would remain true and analytic that “All men are potentially mortal.”

Some existential analytic statements may continue analytic with the growth of knowledge only by contracting their previous scope. An historical example would be Kant’s analytic statement “All bodies are extended” (KvR A7 B11). What had been a synthetic statement for Kant “All bodies are heavy,” meaning all bodies have weight, is today an analytic statement, suiting our wider context of elementary particle physics and general relativity. (We should put it more exactly “All matter has nonzero rest mass.”) Because our particle physics allows that leptons, such as electrons, are not extended, the qualified form of Kant’s analytic “All bodies are extended” would be something along the lines of “All baryonic matter is extended.”

The most basic element in the conception of analytic propositions, in the many varieties of the conception, is that in such propositions the proposition can be known to be necessarily true if only one understands the concepts that are its subject and predicate. The necessary logical tie between subject and predicate of the analytic proposition may be plain from the proposition alone, but it need not be. For the latter, what is required for analyticity is construction of deductive proof of the proposition from the specific identities captured in its subject and predicate concepts. Under the wing of deductive proof, we should include mathematical induction. It is analytic that all odd counting numbers squared, minus one, are evenly divisible by four. The necessary truth of that proposition is not obvious. Neither can it be shown by reductio ad absurdum, that is, by self-contradiction upon supposing it false. But this necessary truth can be established using mathematical induction. It is a truth whose truth and its necessity can be shown by specific-identity analysis of the subject and predicate in the proposition.

Similarly, though without needing to employ mathematical induction in its proof, it is an analytic truth, in my system, that the chance each die of a pair rolls a four when thrown together is one-in-six times one-in-six, yielding a chance of one-in-thirty-six. Is there a computer program guaranteed mathematically to generate chances of outcomes of throws of a perfectly fair pair of dice? (e.g.). If so, then statements about the chances of outcomes in the output of the machine are analytic, and they are analytic in my scheme notwithstanding the elements of natural law (classical, chaotic, and quantum) and boundary conditions entering into their capability for perfectly fair outcomes. In contrast, it will never be analytic to say “This particular pair of dice in my hand is such-and-such degree of closeness to being a perfectly fair pair.” If I roll the dice a thousand times, I will only be able to say that the pair is fair to some particular degree of probability. Add another thousand times, I will have a better idea of how likely it is the pair is fair. At no point is my statement of that likelihood analytic, for that statement cannot be known necessarily true by knowing the meaning of the terms in the statement. That is how it goes with existential statements of particular identity. They are permanently synthetic, never analytic. Unlike synthetic existential statements of specific identity, such statements cannot become analytic with the growth of knowledge.

Existential statements in which the concepts and the relations between them have been characterized mathematically can be analytic only if they are statements about classes. Once it has been established that length and frequency of any pendulum on the surface of the earth stand in a certain mathematical relation with each other, the law and its physical necessity is a part of the relations between the concepts pendulum, length of pendulum, and frequency of pendulum oscillations. It is also analytic that a pendulum of such-and-such particular length will oscillate with such-and-such frequency at the surface of the earth. Say I have a bob suspended from a string I hold in my hand. Measuring the length of the string to be within such-and-such range, we know, if I get the assembly oscillating in a plane, the frequency will be within such-and-such range. That length measurement gives us existential information beyond the information we have in the concepts and their lawful relation. Particular existential identities are synthetic and cannot be known without observation of particular values of traits had by the particular existent. Observation of others in its class will not do. Its specific identity will not suffice to know which length the particular pendulum has, hence which frequency its oscillations will be.

In future installments, I shall consider what various thinkers in the history of philosophy of predication should say about my analytic-synthetic distinction. At the same time, we shall see how the distinction criticized by Quine, Goodman, White and others up to the time of Peikoff’s 1967 essay took the forms it did take. It appears so far that my analytic-synthetic division of types of predication is not in substantive disagreement with the position of Rand and Peikoff in 1966–67, though I contradict them nominally. We shall see if this appearance can be shaken off. Then too we should compare my formula with other revised formulas of analyticity being defended in recent years. Fresh perspective of the situation of Rand’s metaphysics and epistemology in the history of philosophy is to be expected.

My distinction between existential analytic statements and existential synthetic statements has taken some cues from Aristotle, especially from Categories. My division does not coincide with the Aristotelian division according to essential or accidental predication, which can be summarized as a division according to whether a thing is or has what is predicated (Code 1985, 103–4). I follow Rand in taking existents in all cases as both being and having identity.[7] My distinction is closer to Aristotle’s distinction between per se and per accidens predication than to essential-accidental predication. Aristotle’s necessary, per se predications include more than his essential predications (Witt 1989, 104–8).

When Aristotle looks for what something is, he is looking for what it is essentially, which for him is most fully being, with inessential being as less being. In step with Rand, I say that what something is includes all of its specific identity—including not only its kinds, but its attributes and actions—and all of its particular identity. When we focus on what something is as to specific identity, a thing is simply all the specific identities it is, whether non-necessary, necessary, or essential. There is no ascending degree of being in that sequence in Rand’s view or mine. Yet right placement of a specific identity among those three is, for us as for Aristotle, comprehension of the wheels of the world.

(To be continued.)

Notes

5. “The admission of secondary substance as substance is, in fact, a survival in Aristotle of the Platonism from which he could never entirely free himself” (Kneale and Kneale 1962, 31). Rand replaced Aristotle’s secondary substance with concepts of entities, which concepts are analyzed in term of shared measurable dimensions along which measure values of items in a similarity class are clustered (Rand 1966–67, 13–15; further, Kelley and Krueger 1984, 49–58, 63–64; Kelley 1984, 332, 339–52; Boydstun 2004, 280–85, 291–93;* cf. Jetton 1991;* 1998, 63–72;* 2011, 215–21, 225–27). The same sort of replacement obtains in Rand’s measurement analysis of concepts corresponding to the said-of standing of Aristotle’s things not only said of, but in primary substance, that is, in Rand’s analysis of concepts of attributes and actions of entities.

6. There are two broad types of identity, specific and particular. The identities of existents are always of both types. Existence is identity, specific and particular. Specific identity answers to what, to character such as kind, form, attribute, capability, or susceptibility. Particular identity answers to that and which, to where and when, and to how much. The distinction between specific and particular identity is my own, first drawn in 1991.

7. I notice Chris Sciabarra quotes an oral remark of Rand’s as follows: “An attribute is something which is not the entity itself. No one attribute constitutes the whole entity, but all of them together are the entity—not ‘possessed by’ but ‘are’ the entity” (Rand 1969–71, 276; Sciabarra 1995, 146). That suggests Rand held existents are identities, but they do not also have identity. Certainly Rand stressed being identity over having identity (Rand 1957, 1016, 1036; 1966–67, 56; Peikoff 1991, 6–7; Machan 1992, 34). I do not take her to have denied existents have identity along with affirming that they are identity. She said, for example, a human possesses consciousness and has volition (1957, 1015, 1041). Consider also the remarks of Nathaniel Branden, “not to possess an identity, not to possess a nature . . . means: not to exist,” and of Barbara Branden, “things possess identity” (Branden c. 1968, 28, 166). Allan Gotthelf shares my understanding that for Rand existence both is and has identity (Gotthelf 2000, 40).

References

Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton.

Barnes, J., editor, 1995a. Metaphysics. In Barnes 1995b.

——. 1995b. The Cambridge Companion to Aristotle. Cambridge.

Boydstun, S. 1991. Induction on Identity. Objectivity 1(3):1–56.

——. 2004. Universals and Measurement. The Journal of Ayn Rand Studies 5(2):271–305.

Branden, N. c. 1968. Lectures The Basic Principles of Objectivism. In The Vision of Ayn Rand. 2009. Cobden.

Code, A. 1985. On the Origins of Some Aristotelian Theses about Predication. In How Things Are. J. Bogen and J. E. McGuire, editors. D. Reidel.

——. 2010. An Aristotelian Puzzle about Definition: Metaphysics Z.12. In Lennox and Bolton 2010.

Den Uyl, D. J., and D. B. Rasmussen 1978. Nozick on the Randian Argument. In Paul 1981.

Gerson, L. P. 2005. Aristotle and Other Platonists. Cornell.

Gill, M. L. 2010. Unity of Definition in Metaphysics H.6 and Z.12. In Lennox and Bolton 2010.

Gotthelf, A. 2000. On Ayn Rand. Wadsworth.

Jetton, M. 1991. Formation of Concepts. Objectivity 1(2):95–97.

——. 1998. Pursuing Similarity. Objectivity 2(6):41–130.

——. 2011. The Sim-Diff Model and Comparison. The Journal of Ayn Rand Studies 11(2):215–32.

Kant, I. 1781, 1787. Critique of Pure Reason. W. S. Pluhar, translator. Hackett.

Karamanolis, G. E. 2006. Plato and Aristotle in Agreement? – Platonists on Aristotle from Antiochus to Porphyry. Oxford.

Kelley, D. 1984. A Theory of Abstraction. Cognition and Brain Theory 7(3–4):329–57.

Kelley, D., and J. Krueger 1984. The Psychology of Abstraction. Journal for the Theory of Social Behaviour 14(1):43–67.

King, J. C. 1984. Life and the Theory of Value: The Randian Argument Reconsidered. In The Philosophic Thought of Ayn Rand. D. J. Den Uyl and D. B. Rasmussen, editors. Illinois.

Kneale, W., and M. Kneale 1962. The Development of Logic. Clarendon.

Lennox, J. G., and R. Bolton, editors, 2010. Being, Nature, and Life in Aristotle. Cambridge.

Lloyd, A. C. 1981. Form and Universal in Aristotle. Francis Cairns.

Machan, T. 1992. Evidence of Necessary Existence. Objectivity 1(4):31–62.

Nozick, R. 1971. On the Randian Argument. In Paul 1981.

——. 1981. Philosophical Explanations. Harvard.

Paul, J., editor, 1981. Reading Nozick. Rowman & Littlefield.

Peikoff, L. 1976. Entity. In The Ayn Rand Lexicon. H. Binswanger, editor. 1986. New American Library.

——. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.

Penrose, R. 2011. Cycles of Time. Knopf.

Rand, A. 1957. Atlas Shrugged. Random House.

——. 1961. The Objectivist Ethics. In The Virtue of Selfishness. 1964. Signet.

——. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. Meridian.

——. 1969–71. Transcript from Rand’s Epistemology Seminar. Appendix to Rand 1966–67.

Sciabarra, C. M. 1995. Ayn Rand: The Russian Radical. Penn State.

Smith, R. 1995. Logic. In Barnes 1995b.

Touchstone, K., 1993. Can Art Exist without Death? Objectivity 1(5):1–28.

Witt, C. 1989. Substance and Essence in Aristotle. Cornell.

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If epistemological suppositions don't inform or better inform scientific methodologies, they are only unto themselves. Those have to do with acquiring knowledge and validation of knowledge or invalidation of what was once considered to be true. This is not necessarily two steps forward one step back, for Einstein didn't invalidate Newton, now, did he?

--Brant

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Let objective analytic statements be those validly stating necessary specific identity, formal or existential. Their predicates are necessary to their subjects being specifically what they are. Let objective synthetic statements be existential ones validly stating non-necessary specific identity or stating particular identity. Existential synthetic statements of particular identity can be further divided according to whether they are necessary for necessary specific identity.

The distinction between necessary and non-necessary properties has existed since Aristotle (if not before) and is sound. I regard the analytic-synthetic distinction as utterly useless.

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If epistemological suppositions don't inform or better inform scientific methodologies, they are only unto themselves. Those have to do with acquiring knowledge and validation of knowledge or invalidation of what was once considered to be true. This is not necessarily two steps forward one step back, for Einstein didn't invalidate Newton, now, did he?

--Brant

Brant,

Logic and theory of knowledge are part of philosophy, and these have informed physical science. I would add, however, that geometry and number theory (and, later, group theory) have also informed some sciences. We think of ancient astronomy, optics, and harmonics; we think of Aristotle, Euclid, Hipparchus, and Ptolemy.

Using Aristotle’s approach to developing physical science, Theodoric of Freiberg* was able to advance understanding of the rainbow. Descartes was able to advance theory of the rainbow even further, using his own approach to the science of optics, which was in some contrast to Aristotle’s general conception of what physical science requires. Neither Theodoric’s method of science nor Descartes’ method of science is the method of modern science. We can examine their work in theory of the rainbow and say how it was they made their advances in terms of our understanding of what is science and its methods. (Feyerabend and Rand disagreed, to put it mildly, on what is modern science. All the physicists I have known, including the theoretical ones, would have sided with Rand’s elementary points against Feyerabend in that essay Kant v. Sullivan.)

I would emphasize that the methods in modern science have been developed largely by the scientists in new reaches for further scientific understanding. Along the way, the scientists have received new mathematical tools, which can bring new light. Modern science still uses the ever-present constraint and license provided by logic, including syllogistic logic. But Aristotle’s way for physical science in Posterior Analytics is not our way.

There is a two-way street between philosophy and science in theory of knowledge. This has been so since Greek philosophy and science.

Just as Descartes had a picture of what scientific epistemology he was operating under in his scientific investigations, so too Newton and Einstein had a picture, our more modern picture, of what they were doing. Newton and Einstein each had some influence on subsequent epistemology of physics, both in the practice itself and in philosophers’ formulations of the epistemology of modern science. Einstein had some influence on logical positivists in their efforts to formulate a new divide between analytic and synthetic truth. And certainly the disciplines of geometry and Newtonian physics informed Kant in his formulation of such a distinction.

Rand’s epistemology and my further developments are in gear with the conceptual continuities and alterations seen in the advance of science. Einstein together with Planck carefully suited special-relativity dynamics to Newtonian dynamics. That fit and others of special relativity to classical mechanics are something over which a Randian epistemologist can smile. As you and I do. But notice two points.

Firstly, scientists such as Einstein and Planck get their training in scientific epistemology primarily through their training in physics, although I would allow that some past philosophic thought has contributed to methods of physics. Were such scientists to consider what a contemporary philosopher such as Philip Kitcher says about conceptual advance in science, they might say “Yeah, looks familiar, looks about right.” The main type of learning from philosophy by science in a contemporary dialogue would seem to be like, though less definite than, the learning going on when a speaking and literate person is first taught the grammar they are already using to a considerable extent. Another analogy would be the knowledge a first course in deductive logic might impart to the student who is thinking all along the course “Yes, this is correct; and no, to do that would be incorrect,” in agreement with logical doctrines, because the student is already able to think.

Secondly, there are some important guidelines that the sciences seem to have developed for themselves which do not seem to have been found by philosophy as such, at least not with the same definiteness. I’m thinking of the idea that invariance is key to objectivity, as in the modern, post-Einstein physicists’ conformance of innovations to certain invariances of mathematical forms under transformations, where those mathematical forms and their transformations reflect physical reality. Nozick and Suppes have appreciated and assimilated into wider perspectives this strain of epistemology in modern science. More generally, it would seem that the incorporation of modern mathematics (geometric, topological, algebraic, and statistical) into the epistemology of modern science has proceeded without much specific assistance from philosophy.

Merlin,

How far the necessities of specific identity—including necessity of natural law—that I have placed into my class analytic* coincide with necessities conceived in the philosophies of Aristotle or others is something I hope to uncover in the sequel. The reasons for which various philosophers have proposed their various analytic-synthetic distinctions should also be examined in this study.

One purpose for which a class of truths called analytic has been upheld has been to demarcate “philosophy as a mode of inquiry that is fundamentally distinct from, and in some interesting sense prior to, that of the natural sciences, with its [philosophy’s] own distinctive methods, sources of knowledge, and objects of knowledge” (Juhl and Loomis 2010, x). Philosophers, such as Quine, who reject first philosophy altogether would naturally aim to dissolve a distinct analytic class of truth. Other philosophers, such as Kripke, have rejected any distinct usefulness of the familiar concepts of analyticity, arguing that concepts of the a priori and of the necessary can do any work for which then-current concepts of analyticity were being maintained.

The concept of analyticity I have crafted—objective analyticity—is intended to fit tightly Rand’s metaphysics, epistemology, and conception of logic. Naturally, I will be dissecting the Rand-Peikoff writings on past analytic-synthetic distinctions in the sequel. The relation my analyticity proclaims between scientific knowledge and those philosophic areas of knowledge, and between scientific knowledge and mathematics, will also be addressed. I shall address the question, too, of how, if anyhow, my new segregation of necessary truth warrants the description analytic rather than only the name necessary.

I hope to address Greg Browne’s work also.* It turned out that the direction I was heading for a new A-S distinction while discussing his work at OL five years ago is not at all the distinction I have ended up embracing.

Thank you both for the stimulating feedback.

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...

A is A is called tautology in logic and among philosophers, without necessarily prejudging the repetition to be needless to their cognitive purposes. Indeed, to the purpose of valid deduction, tautologous repetition is the seal (Carnap 1934, 44). With logic and philosophy, whether and in what ways A is A is meaningless are further questions, unlike ordinary parlance of tautology (e.g. Wittgenstein 1918, §§ 4.46, 6.1; Ayer 1946, 41, 81, 87, 135; Pap 1958, 438).

...

I won't quote the whole thing, of course, but the entire article was very helpful. Our local Objectivism Meetup is reading OPAR. We just got started reading the week before last; and last week we discussed the first half of "Reality." For tonight, before we conclude the chapter, we are asked to discuss Tautology. Is "Existence exists" a meaningless statement. I believed not for some of the reasons given here in Boydstun, but, clearly, this investigation is very far beyond and much deeper than my own. Thanks!

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If you want to know if the animal before you is a man, check the genome. That will tell you for sure.

A one cell sample is sufficient.

Ba'al Chatzaf

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On Logic, Aristotle and Rand

Rand set out:

To exist is to be something, as distinguished from the nothing of non-existence, it is to be an entity of a specific nature made of specific attributes.[1] Centuries ago, the man who was—no matter his errors—the greatest of your philosophers, has stated the formula defining the concept of existence and the rule of all knowledge: A is A. A thing is itself. You have never grasped the meaning of his statement. I am here to complete it: Existence is Identity, Consciousness is Identification. (AS 1016)

By “greatest of your philosophers,” Rand meant Aristotle. Unlike moderns such as Leibniz, Baumgarten, Kant, or Rand, Aristotle did not connect a “law of identity,” in so many words, with his principle of noncontradiction.[2] Aristotle also did not connect the law of identity that speaks to the distinctive natures of things with a formula such as “A is A” or “A thing is itself.” Aristotle would say “A thing is itself” is nearly empty and useless, and he would not connect that proposition to “A thing is something specifically,” which he thought substantive and important.[3]

Aristotle observed that saying “Man is man” or “The musical is the musical” and so forth are all occasions of affirming that a thing is itself. Someone who had said “Each thing is itself” might have meant “Each thing is inseparable from itself; and its being one just meant this” (Metaph. 1041a1). This meaning in A is A states a truth of all existing things—an existing thing is one, one with itself—and that is, I’d say, sufficient for some tare weight of objective meaningfulness in the statement. That much of A is A is a background assumption in all other meaningful statements.

In mathematics if we can show that an equation can be reduced to the equation 1=1 or 2=2 or sinq=sinq, and so forth, we have proven that the initial equation (apart from any physical application) is true. That is a usefulness of “A thing is itself.” In science one aim (and thrill) is discovery of A is A sleepers. This has been accomplished through the join of observation, mathematics, and induction. Examples are the discovery that the evening star is the morning star and that light is electromagnetic radiation (within a certain range of radiation frequency).

For Leibniz “the primary impossibility is this: A is not A; just as the primary necessity in propositions is this: A is A” (1678, 187). The identity of which Leibniz speaks as a basis of logical necessity is the identity of sameness. A demonstration establishes a sameness between the subject and predicate in the conclusion. It shows that the conclusion’s predicate is contained in the conclusion’s subject. Among the premises could be observations or intellectual truths.[4] The demonstration proceeds by recognizing definitions and by substitution. For effective use in proofs, definitions must not contain contradictions, manifest or concealed. It is not enough that we understand what we say in a definition, for it can still be the case that our definition is of something impossible.[5] Natures of things are implicated in deductive proofs by the observations, intellectual truths, and definitions employed in the demonstration.

Loemker writes of Leibniz: “Contradiction, . . . or the principle of impossibility, is implied in identity, and the two are opposite aspects of the same law, which Leibniz sometimes calls the basic law of being” (1969, 24). In Leibniz’ view, the law of identity entails that predicates of affirmative propositions are contained in their subjects. As with Aristotle, with Leibniz the primary form of being is substance. Identity-containments by subjects of their predicates record the relation of substance to its modifications. Such identity-containments look rather like Rand’s identity of character, or nature.

Rand did not continue with Aristotle’s central concept of substance, rather, she made existence most fundamental and made natured entity the primary form of existence. Entity, not substance, takes the role of bearing attributes and actions. Rand’s conception that “logic is the art of non-contradictory identification” and that “logic rests on the axiom existence exists” embed logic in her fundamental metaphysics: Existence exists and is identity; consciousness is of existence and is identification. In amplification of her compact statement “Existence is identity,” Rand goes on to say that the law of identity (and lack of contradiction) applies to objects, to attributes, to actions, and to their compositions into larger wholes.[6] In Rand’s metaphysics, identity as to nature is tied at the most basic level to identity as self-sameness. For Leibniz identity as self-sameness is the deeper reality of the two. Not so for Rand.

Rand shared with Leibniz the view that the principle of noncontradiction rests on the law of identity. In the 1960’s lectures Basic Principles of Objectivism, Nathaniel Branden held forth and explained Rand’s idea that the law of identity is the basic principle of metaphysics and of epistemology.

The three . . . laws of logic are: The Law of Identity, the Law of Contradiction, and the Law of Excluded Middle. The last two are merely corollaries or restatements of the first.

[The law of identity] is the link between the two sciences, the bridge between existence and consciousness, between reality and knowledge.

As a principle of metaphysics, the Law of Identity tells us that everything which is, is what it is. As a principle of epistemology, it tells us that contradictions cannot exist, that a thing cannot be A and not-A. (Branden c. 1968, 66–67)

Those relations on identity and noncontradiction were also presented by Leonard Peikoff in his 1972 lectures on the history of philosophy, where he indicated historical philosophic puzzles resolved by these Randian conceptions. In Objectivism: The Philosophy of Ayn Rand, Peikoff writes:

The law of identity acts as a bridge linking existence and consciousness, or metaphysics and epistemology. The law acts as a bridge in a second respect also. The law defines the basic rule of method required for a conceptual consciousness to achieve its task. In this regard, the law tells man: identifications must be noncontradictory.

. . . Aristotle’s law of contradiction states . . . nothing can be A and non-A at the same time and in the same respect. This is not a different fact from the law of identity. It is a corollary of the latter, a restatement of it for the purpose of guiding human cognition. (1991, 118–19)

Rand’s sense of identity basing noncontradiction goes beyond Leibniz to include natures at the most fundamental level of identity. Identities of nature or character are not modifications of existence. Existence is identity, not only identity of sameness, but identity of character.[7] There are no existents without both of those aspects of identity.

There are occasions in which one could say “Man is man” by way of stressing that some stunning actions of man are among human capabilities, parts of human nature. Seeing the first man walk on the moon, one might say “Man is man.” Seeing Romeo take his life, one might say “Man is man.” The reader of Atlas Shrugged finds Ayn Rand proclaiming “Man is man” with a meaning along these lines. Her proclamation was to stress that, notwithstanding his freedom of mind, man has a definite nature, that he is nothing but man, and that he is one.

In Rand’s fundamental Existence is identity, the identity of an existent includes its that/which and its what. Rand states her finer structure for the law of identity as follows:

Whatever you choose to consider, be it an object, an attribute, or an action, the law of identity remains the same. A leaf cannot be a stone at the same time, it cannot be all red and all green at the same time, it cannot freeze and burn at the same time. A is A.

. . .

A contradiction cannot exist. An atom is itself, and so is the universe; neither can contradict its own identity; nor can a part contradict the whole. (AS 1016)

Aristotle was the founder of logic, and his great contribution thereto was his theory of correct inference, which is largely his theory of the syllogism. Though he did not realize it, the formula “A is A” in the form “Every A is A” can be used to extend the kingdom of the syllogism. By about 1240, Robert Kilwardly was using “Every A is A” to show conversions such as the inference “No A is B” from the premise “No B is A” can be licensed by syllogism.[8] Aristotle had taken these conversions, like the first-figure syllogistic inferences, to be obviously valid and not derivable.[9] Aristotle takes first-figure syllogisms to be obviously valid and the paragons of necessary consequence. The mere statement of these syllogisms makes evident their necessary consequents. Using conversions as additional premises, Aristotle shows that all syllogisms not first-figure can be reduced to first-figure ones. Their validity is thereby established, by the obvious validity of the first-figure ones and by (what he took to be) the irreducible obvious validity of the conversions.[10]

We find Leibniz, four centuries after Kilwardly, illustrating the utility of Some A is A for concluding Some A is B from All A is B via a syllogism, third mood of the first figure. “I offer these examples . . . to show that identities do indeed have a use and that no truth, however slight it may seem, is completely barren; on the contrary, . . . these identities contain the foundations for all the rest” (Leibniz 1679, 226; see also 1705, 362–63).

There are places in which Aristotle connects “A thing is something specifically” or “A thing is what it is” with the principle of noncontradiction: “The same attribute cannot at the same time belong and not belong to the same subject in the same respect” (Metaph. 1005b19–20). Though not given the pride of place given it by Rand, there is some recognition that Existence is identity in Aristotle: “If all contradictories are true of the same subject at the same time, evidently all things will be one . . . . And thus we get the doctrine of Anaxagoras, that all things are mixed together; so that nothing exists” (1007b19–26).[11] Aristotle realized too that any existent not only is, but is a what.[12]

Rand acknowledges the greatness of Aristotle particularly for his laws of logic, as they are called in elementary logic texts of today and the last few centuries: the laws of noncontradiction,[13] excluded middle,[14] and identity. Those are important principles of logic, though, as we have seen, Aristotle was not securely on board with that last one. It is not clear that Rand was cognizant of the even greater importance for logic of the theory of correct inference that Aristotle invented with his theory of syllogism.

The tremendous importance of the laws of noncontradiction, excluded middle, and identity, in Aristotle’s sense of them or in Rand’s, is importance for metaphysics, thence all appropriate thinking on account of that metaphysical structure.[15] Rand praised Aristotle also for his identification of “the means of human knowledge.” That was in a postscript to her Atlas Shrugged. Presumably, this point of praise refers not only to the role of the three laws of what may be called metaphysical logic,[16] but to Aristotle’s general picture of how we obtain knowledge by reasoning on sensory experience, not by mentally contacting a transcendent platonic realm of forms.[17] Rand writes also that Aristotle’s “incomparable achievement lay in the fact that he defined the basic principles of a rational view of existence and of man’s consciousness: that there is only one reality, the one which man perceives—that it exists as an objective absolute . . .” (1961, 22).

This particle has the set of properties A.

Particles having the set of properties A are electrons.

Therefore, this particle is an electron.

That is a case of a first-figure syllogism, or anyway, excluding singular terms from syllogisms, it is a close relative of that first-figure, third-mood syllogism: Some a’s are b, and all b’s are c; therefore, some a’s are c. On account of a syllogistic inference such as my electron one, Rand could sensibly say “the process of subsuming new instances under a known concept is, in essence, a process of deduction” (ITOE 28). This deduction looks every bit like a case of identification, based on Rand’s kind-sense of identity, though without reliance on the principle of noncontradiction. So I’d say Rand got the genus right, though the differentia wrong, when she defined logic as the art of noncontradictory identification.[18] Perhaps “the art of perfectly truth-preserving inferential identification” would be a better definition of deductive logic devolving from Rand’s conception that logic is slave of existence,[19] that existence is identity, and that consciousness is identification.

The inferences of first-figure syllogisms are, I maintain, licensed directly by identity alone, in Rand’s full sense of identity, and without recourse to noncontradiciton. Nathaniel Branden and Leonard Peikoff erred in trying to support Rand’s definition of logic, with its differentia of the noncontradictory, by appeal to noncontradiction rather than directly to identity as basis of the inference in a certain first-figure syllogism.[20] It is the inference-form of my electron example, but in the familiar case: Socrates is a man, all men are mortal, and therefore, Socrates is mortal. They rightly point out that denial of this inference would lead to contradiction,[21] but that is not to the point of first basis: One already knows that these first-figure inferences are valid, that their conclusions necessarily follow, just as Aristotle observed.

Rand took thinking and logical inference to be volitional cognitions. “To think is an act of choice. . . . The connections of logic are not made by instinct” (AS 1012). Logical inference is consciously directed, in Rand’s view, and that seems right to me in consideration of the process of bringing forward and latching onto logically relevant reasons for some target proposition and the process of finding implications of some target proposition.

Rand wrote further: “The pre-conceptual level of consciousness is nonvolitional; volition begins with the first syllogism” (1961, 15). I rather think volitional thinking, with action- and image-schemata, is in the repertoire before attaining first uttered word (at about one year), which word is co-referential and incorporated into schemata (and later into sentences). But the thing of present interest is Rand’s notion that volition begins with the first syllogism. That would be a deductive inference, whereas the abstractive process of getting one’s first worded concepts would really be, in her view, an induction. “The process of observing the facts of reality and integrating them into concepts is, in essence, a process of induction” (ITOE 28).

At times Rand seems to use syllogism in a super-broad, rather emblematic way to mean simply any logical inference, deductive or inductive.[22] Pellegrin points out that at 92a28 of Post An. “the term syllogism is taken in a broad and non-technical sense” (2010, 131n15). In the Barnes translation of Post. An., the term syllogism does not appear in this line. Rather in Barnes, Aristotle’s allusion reads “if you produce a demonstration in this way.” In a translation appearing in Code 2010, the allusion reads “for this manner of syllogism” (78). Taking demonstration in an ordinary, nontechnical way, I suggest that the places in which Rand uses syllogism in a broad and nontechnical sense, she means demonstration (or validation) in a broad and nontechnical sense. That is, in a sense broad enough to include demonstrations by deductive inference (say, mortality of man from mortality of animals) as well as demonstrations by induction (say, from various sets of evidence to the mortality of all animals, to the roundness of the earth, or to the cause of the tides).

Rand’s broad use of syllogism is a bit grating in speaking of the right way of changing adult minds (coordinate with their autonomy and with objective facts, rather than by substituting the will of one person with the will of another by force).[23] Logic or reason would have been better. Her use of syllogism in connection with acquisition of one’s first concepts is grating to the point of a jam. Some sort of abstractive induction is prize principle of that day, and even demonstration (or validation) would seem out of order as characterization of what is happening in that day of the child’s development: (i) personal grasp and naming of some class of items grouped by similarity facts and (ii) boost in personal power of communication by that acquisition.

Notwithstanding such fumbling, I incline to think Rand has proposed worthwhile extension and reform of Aristotle in philosophy of logic; with her conception of logic and all cognition as identification; with existence (ever with identity) supplanting being; with entity supplanting substance; with identity supplanting form; and with essential characteristic(s), as relative to a context of knowledge, supplanting absolute essence. I expect all these shifts imply further differences, yet to be discerned, between logic in Aristotle and logic as it should stand in the metaphysics and epistemology of Rand.

Notes

[1] Cf. Avicenna 1027: “It is evident that each thing has a reality proper to it—namely, its quiddity” (I.5.10). I shall use what or whatness in place of the traditional quiddity (quidditas); see e.g. Gilson 1939, 199.

[2] Leibniz 1678; Baumgarten 1757 [1739], §11; Kant 1755, 1:389; 1764, 2:294. Rand, in the “About the Author” postscript to AS, and N. Branden, in Basic Principles of Objectivism, erroneously thought Aristotle held the tight bond of identity and noncontradiction that had actually come to be recognized only with Leibniz and his wake.

[3] Aristotle, Metaph. 1030a20–24, 1041a10–24.

[4] Intellectual truths such as “Nothing is greater or less than itself.”

[5] Leibniz 1684, 293.

[6] AS 1015–16.

[7] Cf. Peikoff:

Aristotelians seem committed, in spite of themselves, to the view that particulars qua particulars are unknowable by man. Every determinate characteristic of a particular, and thus everything knowable about it, is placed ultimately on the side of Form; Matter in itself is the unorganized, the indeterminate, the nothing-in-particular; it is, as . . . Aristotle put it, “unknowable.” But if all we can ultimately know of a particular is Form; if the individualizing element, the principle of individuation, is in itself unknowable; does this not suggest that the individuality of things is in itself unknowable, i. e., that particulars qua particulars are unknowable? (1964, 214)

[8] First mood of the second figure; Kneale and Kneale 1962, 235–36; see also Kant 1800, §44n2.

[9] Lear 1980, 3–5.

[10] Lear 1980, 1–14.

[11] See also Aristotle, Metaph. 1006b26–27, 1007a26–27. Let EI designate Rand’s “Existence is Identity.”Aristotle, Avicenna, Henry of Ghent, John Duns Scotus, Francis Suárez, Spinoza, Leibniz, Baumgarten, Kant, and Bolzano also reached principles close to (EI), though not the Randian rank of (EI) or near-(EI) among other metaphysical principles. A Thomist text Rand read had included: “What exists is that which it is” (Gilson 1937, 253). That is a neighbor of Rand’s “Existence is identity.” Neighbor Baumgarten: “Whatever is entirely undetermined does not exist” (1757, §53).

[12] Metaph. 1030a20–24; Post. An. 83a25–34.

[13] De Int. 17a33–35; Metaph. 1011b26–27; Plato, Rep. 436b.

[14] De Int. 17b27–29; Metaph. 996b26–30.

[15] Cf. Bolzano: “The name of laws of thought is justified to a certain extent because laws of things as such are also laws of our thinking of those things” (1837, §45).

[16] Cf. Rödl 2012, 22, 39–43.

[17] Aristotle, Post. An. 99b35–100b5; Salmieri 2008; 2010.

[18] AS 1016.

[19] “Logic rests on the axiom that existence exists” (AS 1016). That does not imply that logic is not a tool of right inference concerning propositions containing only terms referring only to nonexistent things (specified by mention of existent things).

[20] Branden c.1968, 67; Peikoff 1991, 119. Leibniz errs in this way as well; 1678, 187. But on another occasion, Leibniz writes, after listing some “Propositions true of themselves” (such as A is A), he writes “Consequentia true of itself: A is B and B is C, therefore A is C” (quoted in Kneale and Kneale 1962, 338).

[21] See further Buridan 1335, 119–20.

[22] The first definition of syllogism in my American Heritage Dictionary is as in any elementary logic text. The second definition is: “Reasoning from the general to the specific; deduction.” The definitions are very like these in my Webster’s Unabridged.

[23] AS 1022–23.

References

Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor (1984). Princeton.

Avicenna 1027. The Metaphysics of The Healing. M. E. Marmura, translator (2005). Brigham Young.

Baumgarten, A. 1757 [1739]. Metaphysics. 4th ed. C. D. Fugate and J. Hymers, translators (2013). Bloomsbury.

Bolzano, B. 1837. Wissenschaftslehre. P. Rusnock and R. George, translators (2014).

Branden, N. c. 1968. The Basic Principles of Objectivism Lectures. Transcribed in The Vision of Ayn Rand (2009). Cobden.

Buridan, J. 1335. Treatise on Consequences. S. Read, translator (2015). Fordam.

Code, A. 2010. An Aristotelian Puzzle about Definition: Metaphysics Z.12. In Lennox and Bolton 2010.

Gilson, E. 1937. The Unity of Philosophical Experience. Ignatius.

——. 1939. Thomist Realism and the Critique of Knowledge. M. A. Wauk, translator (1986). Ignatius.

Kant, I. 1755. A New Elucidation of the First Principles of Metaphysical Cognition. In Walford and Meerbote 1992 (WM).

——. 1764. Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality. WM.

——. 1800. The Jäsche Logic. J. M. Young, translator. 1992. In Immanuel Kant – Lectures on Logic. Cambridge.

Kneale, W., and M. Kneale 1962. The Development of Logic. Oxford: Clarendon Press.

Lear, J. 1980. Aristotle and Logical Theory. Cambridge.

Leibniz, G. W. 1678. Letter to Herman Conring – March 19. In Loemker (L) 1969.

——. 1679. On the General Characteristic (L).

——. 1684. Meditations on Knowledge, Truth, and Ideas (L).

——. 1705. New Essays on Human Understanding. P. Remnant and J. Bennett, translators (1996). Cambridge.

Lennox, J. G., and R. Bolton, editors, 2010. Being, Nature, and Life in Aristotle – Essays in Honor of Allan Gotthelf. Cambridge.

Loemker, L. E., translator, 1969 [1952]. Gottfried Wilhelm Leibniz: Philosophical Papers and Letters. 2nd edition. Kluwer.

Peikoff, L. 1964. The Status of the Law of Contradiction in Classical Logical Ontologism. Ph.D. dissertation.

——. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.

Pellegrin, P. 2010. Definition in Aristotle’s Posterior Analytics. In Lennox and Bolton 2010.

Plato c. 428–348 B.C. Plato: Complete Works. J. M. Cooper, editor (1997). Hackett.

Rand, A. 1957. Atlas Shrugged. Random House.

——. 1961. For the New Intellectual. Signet.

——. 1990 [1966–67]. Introduction to Objectivist Epistemology. Expanded second edition. H. Binswanger and L. Peikoff, editors. Meridian.

Rödl, S. 2012. Categories of the Temporal – An Inquiry into the Forms of the Finite Intellect. S. Salewski, translator. Harvard.

Salmieri, G. 2008. Aristotle and the Problem of Concepts. Ph.D. dissertation.

——. 2010. Perception, Experience and the Advent of Universals in Posterior Analytics II.19. In From Inquiry to Demonstrative Knowledge – New Essays on Aristotle’s Posterior Analytics. J. H. Lesher, editor. Academic.

Walford, D., and R. Meerbote, translators, 1992. Immanuel Kant – Theoretical Philosophy, 1755–1770. Cambridge.

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On Logic, Aristotle and Rand

Normativity of Logic – Kant v. Rand

Normativity of Logic – Robert Hanna

Between False, Invalid, and Meaningless

Meaningless Tautology

Analytic-Synthetic Distinction
Part 1 – Quine
Part 2 – White and Rand-Peikoff
Part 3 – Objective Analyticity

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