[Stephen Boydstun]

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.

[Stephen Boydstun]

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

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

From Parsing Existence

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.

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

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.

This post has been edited by Stephen Boydstun: 06 July 2009 - 11:53 AM

[Thom T G]

Stephen Boydstun, on Jul 6 2009, 10:48 AM, said:

[...]
~~~~~~~~~~~~~~~~

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.

This post has been edited by Thom T G: 10 July 2009 - 08:40 AM

[Stephen Boydstun]

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).

This post has been edited by Stephen Boydstun: 07 July 2009 - 07:08 AM

[Thom T G]

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., W_n) 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.

[Stephen Boydstun]

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.

This post has been edited by Stephen Boydstun: 08 July 2009 - 02:58 PM

[Stephen Boydstun]

From Ninth Section of “Induction on Identity”
(1991 – Objectivity V1N3)

IX. Existence is Identity

Then existents are identities. There are two broad types of identity. There is what I shall term particular identity and what I shall term specific identity. The identities of existents would seem always to be of both types. Always look for both. Existence is identity, particular and specific.

Particular identity is the it-ness, or that-ness, of a thing. Sometimes this is called material identity; more often it is called simply identity. This is the identity we trace by indexicals or nouns, proper or common. . . . It is just the existent(s) itself as against all others and through all time of its existence. . . . The relation of a thing with itself at another time is a relation of particular identity.

. . . . A thing under one description stands in a relation of particular identity to itself under another description. Here we have the identity of token and type; the identity of Spot and a dog, as when we say “Spot is a dog.” The is here is first and foremost the is of particular identity; Spot is one of the members in the extension of the concept dog. Particular identity is the identity of correspondence. (Macnamara 1986, 49–62, 69–76, 83–94, 123–36, 14–47; Nozick 1981, 29–37, 99–104).

Mention of Spot being a dog brings us to specific identity. The intension of the concept dog reflects Spot’s specific identity, his what-ness. Classifications, attributions, and descriptions . . . capture specific identity. Here we have conditions and their satisfactions. Specific identity is the identity of contrasts and character (Macnamara 1986, 124–28, 144–45).

Consciousness is specific identification, and consciousness is particular identification. The form of consciousness of special interest to us here is the propositional form. In propositional consciousness, one notices both types of identifications immediately.

The compositions between subject and predicate in a true propositional consciousness capture objective identity relations. There are at least three such relations affirmed simultaneously in every true proposition. Consider the proposition “Spot is brown and white.” The copula is executes a triple coordination. It wires the subject, the proper name Spot, and the predicate, brown and white coloration, to the same existent, namely, the dog who is Spot. This is a relation of particular identity. Secondly, it specifies Spot in respect of his coloration. This is a relation of specific identity. Thirdly, it affirms that this particular identity and this specific identity obtain in reality. This last is a relation of particular identity between the proposition and reality. For the third identity to obtain, for the proposition to be fully true, the proposition’s first particular identity and its specific identity must obtain in reality (cf. Hoenen 1952, 62–84; see also Shaffer 1969).

I suggest unsurely how these identities may change when we are speaking clearly make-believe, as in the proposition “A unicorn is a quadruped.” The first particular identity coordinates the predicate with a merely posited subject. The subject is not posited as real, but as unreal. There is no direct, concretely real subject. (Perhaps horses are the subject in a roundabout way.) The subject has become only a topic. The first particular identity no longer wires a predicate and subject to a real thing; it jumpers a predicate directly to a subject. Propositions in make-believe are more analytical. Attention shifts to terms, to stipulative definitions and descriptions. The predicate still specifies. The third identity, the particular identity between the proposition and reality, does not obtain.

Our view of earnest predication is much like the Aquinas-Aristotle view. The underlying metaphysics is different. Aristotle’s schemes of substance-accident, matter-form, and potency-act are largely discarded in favor of the modern scientific scheme of material substance and a rich metaphysical principle of identity. Yet our view of predication is much like the view of Aquinas-Aristotle, for we too hold that we cognize true propositions about the physical world and that the structures of these propositions coincide with structures in reality. In our view, the structures are identity structures.

Our view, like the Aquinas-Aristotle view, is squarely opposed to the view of Kant. We maintain that, through-and-through, reality is predicative. This is an aspect of Rand’s dictum “Existence is identity.” In Kant’s view, predicative structure is imposed on the world by the mind (Kant 1965, A50–136, A148–62; B74–175, B188–202; Patton 1967; Bennett 1990, 86–125). In our view, the mind conforms itself to predicative structure in the world [see also].

. . .

All the integers, all the rationals, and all the reals exist [mathematically]. . . . What is mathematical existence? How is it grounded in the real world? [See]

It is sometimes said that mathematical entities, qua mathematical, have a purely intensional existence (Hoenen 1952, 176–77; Shaffer 1969, 140–41). In this way they are like unicorns. They are make-believe. There may or may not be any real [real-world] extension falling under the concept of a certain mathematical entity, but when thinking mathematically, we are thinking only in the “as-if” mode. That much seems true . . . .

The intensional existence of mathematical entities differs profoundly, however, from unicorns in a fable. Mathematics is the study of an objective realm waiting to be discovered. We must discover whether “there is a real number such that its cube minus seven is zero” or whether “there are functions that are everywhere continuous and nowhere differentiable.” Mathematicians may posit a new mathematical entity, but they may then pull out of it more than they put into it. . . .

In mathematical discovery, deduction rules. The deductive demonstration is required, and it disposes of the case. Because of this, it is sometimes said that mathematical necessity is of the “if-then” sort (Armstrong 1989, 119–24). Yes, but that—the principle of noncontradiction—is not the only source of necessity in mathematics. The realms of mathematics have their own distinctive, objective characters. In mathematics, too, existence is identity. Nothing but identity, it seems to me.

It is sometimes said that the physical sciences are of actual existence, whereas mathematics is of possible existence (ibid., 124–26). That is true, but what is the sense of possible here? It is not merely the noncontradictory. It might be reasonably said that it may as well be merely the noncontradictory, provided we confine attention to the right sort of intensional objects. What are the right sort? I suggest that mathematics is the study of predicative structure, with focus on quantification and mapping.


References

Armstrong, D.M. 1989. A Combinatorial Theory of Possibility. Cambridge.

Bennett, J. 1990 [1966]. Kant’s Analytic. Cambridge.

Hoenen, P. 1952. Reality and Judgment According to St. Thomas. H. F. Tiblier, translator. Henry Regnery.

Kant, I. 1965 [A–1781 B–1787]. Critique of Pure Reason. N. Kemp Smith, translator. St. Martin’s.

Macnamara, J. 1986. A Border Dispute: The Role of Logic in Psychology. MIT.

Nozick, R. 1981. Philosophical Explanations. Harvard.

Paton, H.J. 1967 [1931]. The Key to Kant’s Deduction of the Categories. Reprinted in Kant: Disputed Questions. M. S. Gram, editor. Quadrangle Books.

Shaffer, J. 1969 [1962]. Existence, Predication, and the Ontological Argument. Reprinted in The First Critique. T. Penelhum and J. J. MacIntosh, editors. Wadsworth.


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

With Measurement

To find a measure-theoretic account of predication right for my Randian measure-theoretic account of concepts (analysis, not genesis) is part of my ongoing research program.

My triple-identity view of predication, which is now being cast in appropriate measurement terms, can be found in §IX Existence Is Identity (scroll down to p. 43) of “Induction on Identity” (1991). I no longer draw the distinction between particular and specific identity in entirely the same way as I did in that early essay. (The distinction was my own, but it has been also landed upon by philosopher John Campbell.) The basic simple distinction remains and is as I stated it in Note 34 of “Universals and Measurement.”

For the journey from logical quantification in predicate logic to numerically definite quantification (which is still purely logical quantification) to sets to arithmetic (which adds to logic and is the arena of absolute measurement [see Note 13 of U&M], not to be confused with absolute geometry), I recommend Quine’s Methods of Logic. For the journey from logic and set theory to measurement theory, one-dimensional and multidimensional, I recommend Foundations of Measurement by Krantz, Luce, Suppes, and Tversky.


On translation of sentences into logical formulas, one good text with exercises is R. L. Simpson’s Essentials of Symbolic Logic. Some advanced contemporary controversies are treated in Logical Form and Language, G. Preyer and G. Peter, editors.

I should mention for a general audience that there are no mathematical formulas that cannot be translated into a natural language such as English. That is something that needs frequent mention. Similarly, the epigrammatic strings I improvised above concerning geometry are nothing but abbreviations of English statements. One true rendering of the string {[(Ordered)Affine]Euclidean} would be “Axioms that imply ordered geometry can be joined with certain further axioms to imply affine geometry, and those combined axioms can be joined with certain further axioms to imply Euclidean geometry.” Another true rendering would be “A Euclidean plane (or space) is composed of affine structure and specific additional structure, and the affine structure is composed of order structure and specific additional structure.”

This post has been edited by Stephen Boydstun: 08 July 2009 - 02:59 PM

[Thom T G]

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.

This post has been edited by Thom T G: 10 July 2009 - 08:40 AM

[Stephen Boydstun]

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.

This post has been edited by Stephen Boydstun: 09 July 2009 - 07:30 AM

[Stephen Boydstun]

Contradiction with the Individual Concrete

“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

[Stephen Boydstun]

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).

Quote

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.

This post has been edited by Stephen Boydstun: 07 August 2009 - 05:44 AM

[Stephen Boydstun]

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.

This post has been edited by Stephen Boydstun: 15 August 2009 - 07:52 PM