# The Free Radical Online - Perigo vs. Nola

#### Round Six: Robert Nola Responds

The local Randians have wheeled out their big American gun, David Kelley, leader of one of the USA Objectivist factions and author of the elementary logic text The Art of Reasoning, to continue the debate on logic with me. But he campaign is a fizzer given the area of agreement between Kelley and myself. As I suggested in my last exchange with Perigo, he should read a logic book, even Kelley's. And so should Robert White who in TFR #33 expressed his ignorance of symbolic logic while casting aspersions upon my colleague Rod Girle and his logic text. White can get relief from his stupefaction by logical symbols through Kelley Chapter 13.

(A) Contrary to Perigo, Kelley and I agree that valid arguments can contain true as well as false statements; validity is not impugned by falsity. This much is standard logic.

(B) Despite appearances, Kelley and I do not even disagree (2) about logic's independence from 'facts of world history' (but not these alone since mathematical claims can appear in arguments). Consider Kelley's MODUS PONENS 'If p then q, and p, THEREFORE q'. And let 'p' = 'it is now summer in NZ' and 'q' = 'it is now winter in the USA'. When these substitutions are made, the argument is true of the world, or has to do with reality, because the component statements are all true. But the world does not make this argument valid. (C) Where Kelley and I depart is that valid logical consequence, indicated by '... THEREFORE ...', is not determined by reality, or as Kelley puts it by a 'metaphysical fact about the nature of existence as such'. There is no need to appeal to such arcane facts (of which there would have to be uncountably infinitely many). Moreover logical principles also hold in (possible) worlds with no things in them.

In his book Kelley informally defines 'valid consequence' as 'if the premises are true then the conclusion MUST be true' (the word 'must' being linked to the notions of possibility and necessity). For Propositional Logic Kelley cashes out 'must' in terms of truth tables. Other ways of defining 'logical consequence' say that it is a matter of valid logical form. In turn this is understood as: there is no model for interpretation of the form such that the premises are true and the conclusion is false. (Proofs are needed.) This still allows that valid arguments need not have true premises and invalid ones need not have false conclusions.

Since I am limited to a 900 word reply, see the survey of issues
on VALID CONSEQUENCE in Stephen Read, *Thinking About Logic*,
Chapter 2. To establish logical consequence, there is no need to
appeal to metaphysical facts about existence; instead the notion
of form and certain mathematical devices are enough. This view is
quite objective, but may not be 'Objectivist' (not that I care).

(D) Kelley seems to appeal to the 'Classical Laws of Thought', Principles of Identity, Non-Contradiction and Excluded Middle. These are obscure principles, not clearly described in Rand. Within the system of Classical Propositional Logic they are expressed: 'p is equivalent to p'; 'not both p and not-p'; and 'either p or not-p'. In other systems they would differ, if present at all. Thus within non-classical propositional systems all contain the first, but para-consistent systems reject the second, while multi-valued, intuitionistic and fuzzy logics all reject the third.

Moving to Predicate Logic, as I discussed in TFR #32, p 24, Rand's own attempt to supposedly derive from the ill-expressed Identity 'A is A' (it ought to be: 'for all x, x is A is equivalent to x is A') the expression 'all A is B, all B is C, therefore all A is C', is a failure.

(E) But Kelley suggests, and I am well aware, that these Principles have also been understood as meta-Principles about systems of logic and not formulae within systems. Kelley adds that these Principles are 'grounded in reality' and that 'valid inference is grounded in these laws' (3). But what is the obscure 'grounded' relation?

Suppose as meta-Principles they say: (1) Everything is self-identical (Randians have a misleading tendency to go beyond mere self-identity to having identity conditions or a nature). This is an old Principle which Butler expressed as 'every thing is what it is and not another thing'. I and most philosophers accept it. (2) No proposition can be both true and false. (3) Every proposition is either true or false.

Now these three meta-Principles underdetermine logical systems. They hold of systems as diverse as: classical logic: Kelley's own term logic (a departure from classical Predicate Logic developed by Sommers which allegedly has a grammar closer to natural language than Predicate Logic); the many modal and tense logics.

But (3) does not hold for logics that rival classical logic such as intuitionistic, multi-valued, fuzzy and several relevant logics. Meta-Principle (2) fails for logics with contradictions in them, something Randians find hard to understand but which have an important applications in AI in dealing with distributed data systems. Finally, there are Free Logics, so-called because they abandon the existence assumptions associated with the classical stance. Though classically important, the three Principles determine (ground?) very little in logic, widely understood. But this is not to say there is no Philosophy of Logic.

Robert Nola, PhD

Department of Philosophy

University of Auckland

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