The topic of metaphysics, in my meaning, is concrete existence, whether those concretes are entities, attributes, actions, or relations, whether actual or potential. This conception of metaphysics is consistent with Rand’s views that all existents have particular and specific identities (AS 1016, 1035–37; ITOE 6, 240), that “everything that man perceives is particular, concrete” (ITOE 1, 199), and that “‘things as they are’ are things as perceived by your mind” (AS 1036).

To fully comprehend the concrete, we require the abstract. To fully comprehend the actual and the potential, we require the possible.

Rand conceived of logic as “the art of non-contradictory identification,” and she held that “logic rests on the axiom that existence exists” (AS 1016). Recall that Rand’s first axiom for metaphysics is the affirmation that existence exists. ($) Two further axioms are manifest to one in the act of grasping the statement “existence exists.” These are that something exists which one perceives and that one exists and possesses consciousness of existing things (AS 1015). We have the following:

I concur with the preceding, and to Rand’s E-axioms I would add a related supposition, which I call my epsilon-premise:(E) Existence exists.

(E1) Something exists which one perceives.

(E2) One exists and possesses consciousness of existing things.

(

*ε*) There is nothing in existence whose existence cannot be asserted.

If something exists, its existence can be asserted. Then because (E1) and (E2) are implicit in the act of grasping the assertion that any particular thing exists, there is nothing in existence that is not potentially the subject of acts of consciousness.**

Rand also takes as axiomatic that “to exist is to be something, as distinguished from the nothing of nonexistence, it is to be an entity of a specific nature made of specific attributes” (AS 1016). A thing is itself, it is what it is. An object is of one sort or another, it possesses certain attributes and not others, and its actions are certain ones and not others. In three words,

Every existent is with identity. If something exists, then it is not without identity.(I) Existence is identity.

The identities of existents exist with them. Adding (

*ε*), we may conclude (

*ι*): all identities of existents can be asserted. Any identity of an existent that is so can be asserted to be so. Implicit in the act of grasping the statement that a certain identity is so, of a certain existent, is the fact that some identity holds which one perceives and that one exists as a consciousness of identities.

Concerning consciousness, taken as the act of perceiving that which exists, Rand poses the further axiom:

The art of non-contradictory identification is logic, in Rand’s conception of it, and “logic rests on the axiom that existence exists” (AS 1016). Then it is not a logical possibility that nothing exists (further).(C ) Consciousness is identification.

Rand presents (E) expressly as an axiom. She does not present (I) and (C ) expressly as axioms, though she strongly suggests that they are axioms. They are introduced as completing the traditional law of identity, the principle that a thing is itself, or A is A (AS 1016; further – A, B). Rand takes (I) and (C ) to state primary facts and to be immediate and most important elucidations of the three concepts she takes to be axiomatic expressly:

*existence, identity,*and

*consciousness*(ITOE 55–56).

Rand takes the proposition “Existence is identity” to express a primary fact. This assertion is a fundamental proposition composed with the concepts

*existence*and

*identity*(further), which concepts, along with the concept consciousness, Rand takes as axiomatic. Other propositions expressing primary facts are “Existence exists” and “Consciousness is of existence not only itself.” Rand constructs arguments to show that these two propositions are indeed axiomatic (AS 1015–16, 1039–40).

I have come round to taking the proposition “Existence is identity” as an epigram encapsulating identity postulates on existence. A few years ago, led by Rand’s text, I began drawing forth specific identity postulates on existence contracted into “Existence is identity.” I have constructed arguments showing that some of these postulates are indeed axiomatic. I will state these propositions and my proofs of their axiomatic status in subsequent posts to this thread.

A philosophic axiom “is a statement that identifies the base of knowledge and of any further statement pertaining to that knowledge, a statement necessarily contained in all others, whether any particular speaker chooses to identify it or not” (AS 1040). Rand’s axioms, then, cannot be proven without circularity. Her axioms affirm such things as the fact of existence, the fact of one’s consciousness, and the fact of one’s existence. These cannot be proven without recurrence to themselves if Rand has found axioms that truly meet her requirement of absolute fundamentality in reality and in any knowledge of reality.

The E-axioms are implied in any conscious action, including in any claims denying those axioms (AS 1016). More generally, for any of her axioms, Rand maintains that one has to “accept it and use it in the process of any attempt to deny it” (AS 1040). One cannot deny the truth of these axioms without some form of self-contradiction. Pose any such axiom as false, there will be contradictions with what one retains as real or there will be contradictions with presuppositions of the activity of rational deduction.

(See further.)

Notes

*That does not mean the sciences traditionally not counted among the physical sciences should be thought of as subdivisions of the physical sciences.

**There are things no longer in existence which were not subjects of consciousness during their existence and which are now no longer potential subjects of consciousness.

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

Beyond the Axiomatic – Identity with Measurement

I have held with Rand:

Furthermore,(Im) All concretes have measurable relations to other concretes.

The thesis (Cm) is, in my view, a claim to be set within the popular approach of analyzing cognitive systems as computational systems.(Cm) Cognitive systems are measurement systems.

Rand’s measurement-omission analysis of concepts implies a distinctive magnitude structure for metaphysics. The argument supporting that claim was given in the sixth paragraph of my essay Universals and Measurement (The Journal of Ayn Rand Studies 5(2): 271–305). What is that argument? It is simple: All concretes can be placed under concepts. All concretes can be placed within some concept-class or another. For all concretes, some of those concept-classes will be of the Randian measurement-omission form. Then all concretes must stand in some magnitude relations such that the Randian form of conceptual rendition is applicable to them.

The preceding argument does not rely on a supposition that all one’s concepts are formed by a process of measurement omission. The argument says only that any concept you give me can be reformed into a measurement-omission form, or if not, at least this much is true: all the concretes falling under your concept fall under some concept(s) or other for which we can discover its measurement-omission form. That last, modest premise is all I need to conclude that Rand’s measurement-omission form of concepts implies that all concretes must stand in certain minimal magnitude relations.

Rand’s concept theory—not her formation theory, but her analysis theory—implies a specific, meager (but nontrivial) magnitude structure for all concretes, which is to say, a distinctive magnitude structure for metaphysics. In my “Universals and Measurement,” I uncovered that minimal structure and characterized it in three ways: by its automorphisms (#13), by its mathematical category, and by the types of measurement it affords.

What is meant by a magnitude structure? That means an ordered relational structure. These structures are not only abstractions. They can be concretely realized relations. The most accessible example is geometry. Not analytic geometry (with coordinate systems, calculus, and all that), but the plain old synthetic geometry such as one learns to weave in a high school geometry course. The various geometries are all ordered relational structures, but some have all the structure of others plus more. Here are some geometries in their cumulative hierarchy of structure:

{[(Ordered) Affine] Euclidean}

{[(Ordered) Absolute] Euclidean}

{[(Ordered) Absolute] Hyperbolic}

{[(Projective) Affine] Euclidean}

[(Projective) Elliptic]

These layers of ordered relational structure are an objective matter. They have been discovered, not arbitrarily constructed (a, b).

I should mention for a general educated 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.”

The minimal magnitude structure implied (by measurement-omission concept analysis) for all concretes is metaphysical structure. It is structure beyond logical structure; it is constraint on possibility beyond logical constraint. Yet it is structure ranging as widely as logical structure through all the sciences and common experience.

So when I said above “to fully comprehend the actual and the potential, we require the possible,” the structure of possibility is not only logical, but mathematical. The contrast of concrete existence to abstractions over it is to be cashed, in terms of mathematical structures, as the contrast of synthetic geometry to analytic geometry.

In “Universals and Measurement,” I sought and found the specific minimal magnitude structure the world must have such that Randian conceptual rendition of the world is possible. To ask for such structure conditions for the possibility of conceptual rendition sounds suspiciously similar to Kant’s quest for the conditions of possible experience or his quest for the conditions of possible cognition (

*KrV*B138). There is a great difference between what I was seeking in “Universals and Measurement” and what Kant was seeking in his famous questions. The magnitude structure I captured is not something that our cognitive system (specifically, our conceptual faculty) prescribes for the knowable world. Rather, however pervasively that structure is in the world, it is there independently of our cognitions.

Ours is not a Kantian program. We do not say that because our conceptual faculty works necessarily in such-and-such way we must find the world everywhere conforming to that way. We do not say that our concepts must all necessarily be susceptible to being cast in measurement-omission form, and that therefore we must find the world affording that form. Rather, we leave open to trial whether we shall find the world everywhere congenial to the measurement-omission form of concepts.

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.

Rand took the thesis (Im) to be axiomatic in that she took it to be entailed by her axiom (I). A thing not measurable in any way “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, . . . in short, it would not exist” (ITOE 39). Rand is supposing that anything bearing some relationship to the rest of the universe bears some measurable relationship to the rest of the universe. I think that this supposition, which is tantamount to (Im) (all concretes have measurable relations to other concretes), is a postulate additional to the axiomatic postulate (I) (existence is identity). I do not regard the postulate (Im) to be axiomatic; unlike the axiom (I) (i.e., specific postulates that fall under the epigram and are axiomatic), the postulate (Im) can be denied without self-contradiction and is therefore open to possible restriction by counterexamples. Like Rand, however, I take (Im) to be an unrestrictedly true postulate.

Return now to the proposition (I) itself: Existence is identity. Rand demonstrated that some other propositions in which axiomatic concepts appear are axiomatic, but she did not demonstrate the axiomatic status of this proposition. She challenged anyone to make any claim of knowledge that does not presuppose (I). One cannot claim to know how to lift water to the roof of a skyscraper without knowing the identities, the natures, of gravity, water, and pipes (AS 1036). Let one “who does not choose to accept the axiom of identity, try to present his theory without using the concept of identity or any concept derived from it” (AS 1040).

If “Existence is Identity” (I) were axiomatic, then any knowledge presupposes it and no counterexample to it should be forthcoming. But inability to come up with a counterexample to a proposition is not enough for demonstrating the proposition to be an axiomatic one; absence of counterexample obtains for any postulate applicable to and true of all concretes, not only for the axiomatic postulates. To show the postulate to be axiomatic, we must show its denial to be contradictory to itself or its presuppositions.

*Continued below*—

—Exclusions of Non-Contradiction: Entities

—Exclusions of Non-Contradiction: Actions

—Exclusions of Non-Contradiction: Attributes

**Edited by Stephen Boydstun, 04 February 2010 - 09:02 AM.**