Corollaries of Irreducible Primaries that are Axiomatic Concepts

[Space Patroller]

In ItOE Miss Rand states that "Exixtence exists" is, If I recall correctly, both an irreducible primary and the axiomatic concept of Objectivism. She also identifies three corollaries. One of these is "consciousness" This could be interpreted that the raison d'etre of Epistemology, like Metaphysics is an irrecuible primary amd it may be argued that Epistemology is Mteaphysicis applied to Mankind's unique trait, with Ethics, Politics and Aesthetics being driviatives of these two, which is how I treat them. For me, Egoism stands because things, both living and non-living, exist individually at some level and the only choice for Man is to think or not to think. It stands to reason that given these, the persons have the moral right to, as individuals, be the beneficiaries of their choices and the actions to which they lead, if fo no other reason, the Law of Parsimony aka Occam's Razor. and any other ethical form would be or require a contradiction of those two facts since if nature requires existence in individual form and Man's unique characteristic is a volitional rational consciousness then the consequences of the actions of that consciousness which is a characteristic of only individual persons, must redound to that individual who acts.

Now, getting back to the question, are the corollaies of irreducible primaries/axiomatic concepts also irreducible primaries and/or axiomatic concepts, bearing in mind that Rand said "Philosophy is primarily Epistemology" an that one's particular Epistemology is totally under one's choice; i.e. Reason, mixed, Mysticism?

I can envison that the corollaries are derivatives of the irreducible primaries and axiomatic concepts to which the attend. On the other hand I can see that if they have values as corollaries, they must share the status of irreducible primaries and axiomatic concepts.

On a related note would not an axiomatic concept have to be an irreducible primary?

[Grames]

Specifying the genus and differentia of an axiomatic concept should be able to sort this out. The genus is irreducible primary. Concretes and perceptions are other kinds of irreducible primaries. The differentia is the descriptor axiomatic. The genus is not concept, that is one level higher than primary and too wide.

If corollaries are reducible, they are not in the genus of irreducible primary so cannot be axiomatic. But Miss Rand's usage does refer to corollaries as retaining axiomatic status so moving from a corollary to its root is not considered a reduction, or at least not necessary to consider a still further derivative concept as fully reduced.

Edited June 14, 2009 by Grames

[Trebor]

The statement, "Existence exists." is not an Axiomatic Concept; it's an Axiom.

Axioms: "An 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. An axiom is a proposition that defeats its opponents by the fact that they have to accept it and use it in the process of any attempt to deny it."

Axiomatic Concepts: "Axioms are usually considered to be propositions identifying a fundamental, self-evident truth. But explicit propositions as such are not primaries: they are made of concepts. The base of man’s knowledge—of all other concepts, all axioms, propositions and thought—consists of axiomatic concepts."

Concepts are denoted by a single word (It's an aspect of the unit economy of conceptualization.)

Concepts: "With the exception of proper names, every word we use is a concept that stands for an unlimited number of concretes of a certain kind."

"Existence exists." is an Axiom. "Existence," "Identity" and "Consciousness" are Axiomatic Concepts.

Existence: "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."

Corollaries: "A corollary is a self-evident implication of already established knowledge." AND: "Many of the most important truths in philosophy are neither primary axioms nor theorems susceptible of discursive proof; rather, they are corollaries—most often, corollaries of axioms."

Before one can grasp a corollary, one must first grasp that which it is a corollary of, which therefore has primacy.

*All quotes are from the online Ayn Rand Lexicon.

Edited June 14, 2009 by Trebor

[Space Patroller]

The statement, "Existence exists." is not an Axiomatic Concept; it's an Axiom.

Axioms: "An 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. An axiom is a proposition that defeats its opponents by the fact that they have to accept it and use it in the process of any attempt to deny it."

Axiomatic Concepts: "Axioms are usually considered to be propositions identifying a fundamental, self-evident truth. But explicit propositions as such are not primaries: they are made of concepts. The base of man’s knowledge—of all other concepts, all axioms, propositions and thought—consists of axiomatic concepts."

Concepts are denoted by a single word (It's an aspect of the unit economy of conceptualization.)

Concepts: "With the exception of proper names, every word we use is a concept that stands for an unlimited number of concretes of a certain kind."

"Existence exists." is an Axiom. "Existence," "Identity" and "Consciousness" are Axiomatic Concepts.

Existence: "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."

Corollaries: "A corollary is a self-evident implication of already established knowledge." AND: "Many of the most important truths in philosophy are neither primary axioms nor theorems susceptible of discursive proof; rather, they are corollaries—most often, corollaries of axioms."

Before one can grasp a corollary, one must first grasp that which it is a corollary of, which therefore has primacy.

*All quotes are from the online Ayn Rand Lexicon.

If I recall correctly. Rand did refer to "Existence Exists" as the axiomatic concept of Objectivism. First it is as you say an axiom. However she described the following "'what do you mean by existince?' ". Making s sweeping gesture across the plane of vision "'I mean this!'". Now since that gesture meant "everything" and not any specific concrete, it is comeptual rather than perceptual or pre-conceptual, which would make the statement both axiomatic and conceptual.

Beyond that, doesnt the genus subsume all of the characteristics of the corollaries, and thank you for the memory jog with regard to "Existence" and "Identity", I'd been trying to recall them. If the answer is "Yes" then "''Existence Exists" would indeed be an axiomatic concept since it subsumes one or more such. If not then the corollaries would have an existence outside and beyond the egenus.

[Trebor]

If I recall correctly. Rand did refer to "Existence Exists" as the axiomatic concept of Objectivism. First it is as you say an axiom. However she described the following "'what do you mean by existince?' ". Making s sweeping gesture across the plane of vision "'I mean this!'". Now since that gesture meant "everything" and not any specific concrete, it is comeptual rather than perceptual or pre-conceptual, which would make the statement both axiomatic and conceptual.

Beyond that, doesnt the genus subsume all of the characteristics of the corollaries, and thank you for the memory jog with regard to "Existence" and "Identity", I'd been trying to recall them. If the answer is "Yes" then "''Existence Exists" would indeed be an axiomatic concept since it subsumes one or more such. If not then the corollaries would have an existence outside and beyond the egenus.

"Ostensive Definition "Ostensive definitions are usually regarded as applicable only to conceptualized sensations. But they are applicable to axioms as well. Since axiomatic concepts are identifications of irreducible primaries, the only way to define one is by means of an ostensive definition—e.g., to define “existence,” one would have to sweep one’s arm around and say: “I mean this.”"

Defining what she meant by the Axiomatic Concept, "Existence," by sweeping her arms all around to point to everything (an Ostensive Definition) is not the same as explaining the meaning of the Axiom, "Existence exists." Saying "Existence" is not the same as saying "Existence exists." (Saying "Ball" is not the same as saying "[The] Ball rolled."] One is a concept, the other is a statement.

"Existence," "Consciousness" and "Identity" are Axiomatic Concepts. They are each a single word, not a statement. An Axiom is a statement, which means it has at least two words, a subject and a predicate. An Axiomatic Concept is not an Axiom.

"Existence exists." is certainly the fundamental axiom of Objectivism. It's not an axiomatic concept. I posted previously to try and make the distinction between an axiom and an axiomatic concept clear. I think you're blurring the distinction and that is leading to confusing statements and questions.

Axiomatic Concepts: "[The] underscoring of primary facts is one of the crucial epistemological functions of axiomatic concepts. It is also the reason why they can be translated into a statement only in the form of a repetition (as a base and a reminder): Existence exists—Consciousness is conscious—A is A. (This converts axiomatic concepts into formal axioms.)"

Certainly, the statement, "Existence exists," is both axiomatic and conceptual. It is a simple statement, an axiom, composed of two axiomatic concepts, two words, "Existence" and "Exists." (Or one axiomatic concept from two different perspectives.)

An axiom is a statement. An axiomatic concept is a type of concept.

Axiomatic Concepts: "The first and primary axiomatic concepts are “existence,” “identity” (which is a corollary of “existence”) and “consciousness.” One can study what exists and how consciousness functions; but one cannot analyze (or “prove”) existence as such, or consciousness as such. These are irreducible primaries."

You ask, doesn't the genus [of the axiom, "Existence exists,"] subsume all of the characteristics of the corollaries? I assume that you were asking about the genus of the axiom, "Existence exists," and not of the axiomatic concept, "Existence," but perhaps not.

Does an axiom have a genus? I do not think so. Concepts, however, do have a genus. A definition (of a concept) is in the form of a genus (the conceptual common denominator) and species (the distinguishing characteristic).

I believ Dr. Peikoff gave a good example of the fact that the axiomatic concepts, "Existence," "Identity" and "Consciousness" are implied in every awareness:

When one sees a tomato, one is aware that it exists (Existence), one is aware that it (Identity) exists, and one is aware (Consciousness) that it exists. There's not more or different data involved; there's just a difference of perspective or focus.

As to the corollaries of the Axiom, "Existence Exists," again:

Existence: "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."

The corollary Axioms (not Axiomatic Concepts) refer to the same facts from a different perspective, not new or different data. The corollaries are statements:

1. Something exists which one perceives.

2. One exists possessing consciousness.

*All quotes are from the online Ayn Rand Lexicon.

Edited June 15, 2009 by softwareNerd
Added attribution

[Thomas M. Miovas Jr.]

I will add something that I think a lot of people overlook, and that is that causality is a corollary to identity: A thing is what it is and does what it does because it is what it is. In former philosophies, Aristotle and Aquinas, this was phrased as a thing acts according to its nature. But this can lead to confusions, especially rationalistic confusions, that somehow "its nature" is super-added to what a thing is. In Objectivism it is recognized that a thing is what it is,including everything that it is, and it being what it is brings about its capabilities -- its potentials, if you will. A ball can roll because it is a ball and not a block of wood or not a glob of jelly or not a gas cloud, etc. It is what it is, and causality is a recognition of identity when a things acts or changes. That is, to underscore an ancient mistake, a thing does not become something different just because it changes -- a ball does not become a not-ball just because it is set in motion. And when wood changes from wood to ash when it is burned, that burning and that transition is possible because it is wood.

In Objectivism, the axioms and their corollaries are given in perception as understood by a conceptual mind. A dog can see a ball rolling and chase after it, but it cannot grasp causality, which is a conceptual level integration of observations.