# The Free Radical Online - Perigo vs. Nola

#### Round Four: Robert Nola Responds

To fully answer Perigo's questions on logic in (1) to (4) requires
studying the subject from the beginning. So I propose the following:
the editor devote several pages of each issue of *The Free Radical*
over the next year to the study of logic. I would be willing to
provide the logic materials (suggestions about texts, programmes,
etc.) for such a venture. In that way informed knowledge will be
spread by TFR — and the number of truths published in TFR will
be vastly increased. Take up the challenge!! Randians love reason
— don't they? Or do they wish to remain logical illiterates?

(B): REPLY TO (4) To illustrate what my proposal would do, consider
my reply to the challenge issued by Perigo in (4). In *Letters
of Ayn Rand* (ed. Berliner, p. 527) we do not have Hospers' correspondence
but we find Rand saying in reply to him: 'You say that "A is
A" does not provide a validation for any particular arguments,
but "all A is B, all B is C, therefore all A is C" does'.
Rand's view is the converse: 'What, if not "A is A", gives
any validity to "all A is B, all B is C, therefore all A is
C"? What is the latter but one of the concrete applications
or derivatives of "A is A"?'. Rand's reply is hopelessly
wrong; from her so-called "Law of Identity", 'A is A',
no such derivations are possible without extra axioms which then
make the Law redundant. She also gives an analogy involving redundancy
as to why she thinks Hospers is wrong; but it is her Law which is
redundant.

The argument form "all A is B, all B is C, therefore all A is C" can be understood in several ways, depending on the interpretation of the symbols 'A', 'B' and 'C' and the word 'is'. Without an accompanying interpretation, symbolic logic becomes an arcane cabalism. We will see that Rand neglects this point in her talk of 'A is A'. Here are three interpretations.

First, interpreted in Aristotelian terms, the form becomes the valid syllogism called 'Barbara'. Aristotle does not attempt to prove Barbara from (a version of) 'A is A', since he knew it did not follow. Rather he shows that, given additional principles of inference, his syllogisms are mutually interderivable.

Second, if 'A', etc. are interpreted as sets (classes) and 'All A are B' is understood as the inclusion of set A in set B, then the formula expresses a law of Transitivity. This can be proved from the axioms of set theory. On this interpretation, Rand's "Law" 'A is A' says that set A is contained in itself; but Transitivity cannot follow from this. In any case the Identity condition for sets is different: two sets are identical just in case they have exactly the same members. Even using this as an axiom it is not possible to deduce Transitivity as other axioms are needed.

Thirdly, the formula can be interpreted in the Predicate Calculus in which there is a domain of individuals (over which the variable 'x' ranges) and 'A', 'B' and 'C' are predicate expressions. Interpreted thus 'All A are B' becomes 'for any x, if x is A then x is B'. Interpreting the two premises and the conclusion this way, Perigo's request for proof can be met without invoking Rand's 'A is A'. (a) Use the rule Universal Instantiation on the premises and the conclusion. This rule says: from a universal claim (e.g., 'for any x, if x is A then x is B') infer a particular case (e.g., 'if a is A then a is B' (where 'a' names some particular object ). [Note: it is not part of the rule that the universal claim MUST be true.] (b) Now use the Reductio Method of proof by assuming the two premises and the NEGATION of the conclusion and derive a contradiction. [Note the involvement of false and contradictory claims in Reductio arguments.] Thus we have proved 'if a is A then a is C'. (c) Use the rule Universal Generalisation which says: providing a is arbitrarily chosen in the proof, from a particular case (e.g., 'if a is A then a is C' where 'a' is arbitrarily chosen) infer the general case (e.g., 'for any x, if x is A then x is C', or 'all A are C'). QED.[In this system Rand's 'A is A' becomes 'for any x, if x is A then x is A'. But it can be shown that as a single premise this truth is just too weak to generate any other valid argument forms without the help of further axioms'; but then her premise plays no role in the proof and is redundant.]

The above accounts all fall within the theory of proof from axioms
not containing Rand's 'A is A'. About 50 years ago a Dutch logician,
Evert Beth, developed an elegant method for building models of premises
and conclusions now known as the 'tree method'. This works explicitly
with the definition of VALID CONSEQUENCE: there is no model for
both the premises and the negation of the conclusion. For a proof
using the tree method, see H. Kahane, *Logic and Philosophy*
(sixth edition) p 246, worked example (4).

See what real insight logic will provide by taking up the challenge in (A)!

(C) Since Rand rarely tells us how to interpret her Identity Law 'A is A', it can appropriately called 'naive'. Sometimes she means no more than Butler's 18th century identity maxim 'everything is what it is and not another thing'; but this is not 'A is A'. A charitable interpretation of Rand's remarks would be to take the 'A' to mean the same as in the argument form which follows it. If so, and assuming 'A' is a term like 'apple', is Rand saying: 'apple is apple? This hardly makes much sense.

With more charity, we could say that Rand has left out the quantifier 'all'; she should say 'All A are A'. Particular cases of this are 'all apples are apples', 'all fruit is fruit' etc. Now these are undeniably true. But, contrary to Rand, from these quite weak (logical) truths one cannot, for example, deduce the stronger (logical) truth: 'All apples are fruit; all fruit are good to eat, therefore all apples are good to eat' (where 'B' is replaced by 'fruit' and 'C' by 'good to eat'.) For reasons such as these I called foolish Rand's claim that from the single premise (axiom) 'A is A' we can establish the validity of all inferences — or, as she also alleges, the falsity of 'all self-sacrifice is happiness'! (ibid., p. 527).

Modern logical syntax can help us in other ways. Rand treats existence,
identity, and consciousness as "axiomatic concepts" which
appear in her 'formal axioms': 'Existence exists — Consciousness
is conscious — A is A. (This converts axiomatic concepts into
formal axioms.)' (*Objectivist Epistemology,* p. 59). But there
is muddle here. If these claims are of the form 'A is A' (where
'is' stands for identity) then we should say 'Existence is (identical
to) Existence', and 'Consciousness is (identical to) Consciousness'.
The 'A' term should be the same on both sides of the identity.

Rand mistakenly treats these claims as identities when they are, from the point of view of surface grammar, of Subject-Predicate form. In 'Existence exists', 'Existence' is either a name standing for a property Existence (on a par with say Apple or Redness), or a general term standing for a number of existing things (on a par with apples or patches of red); and 'exists' is a predicate standing for a property (assuming there is such a property — a contested issue on which logical syntax has something to say) . Similarly 'Consciousness' is a name standing for some item while 'conscious' is a predicate. No Law of Identity is involved here! These claims are often more convoluted ways of saying the banal truths, 'some things exit' (or 'everything which exists, exists'), or 'some things are conscious'.

At the end of his reply Perigo cites Merrill saying that we would enter a new age of thought if Rand's views were to be adopted. However in logic, the philosophy of logic and in metaphysics of the sort just described, we would, if all her ideas were accepted, turn out backs on important pioneers in logic from Frege, Russell, Tarski to Goedel.

I have not replied to (5)! And the editor of TFR is having paroxysms about word length already. So, take up the challenge in (A) in order to see why Merrill's non sequitur is no argument against Nozick, and why the analytic/synthetic bugaboo might not matter here!

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