To: Chris M.

From: Geoffrey Klempner

Subject: Is any justification of deduction circular?

Date: 5th May 2009 11:31

Dear Christian,

Thank you for your email of 27 April, with your essay for the University of London Logic module, in response to the question, ''Any justification of deduction is bound to be circular.' Discuss.'

You have canvassed lots of different ideas here -- Goodman's idea of 'entrenched' deductive practice, Quinian holism, inductive and pragmatic justifications of deduction. You have also considered the question whether, given that a justification of deduction would be in some sense 'circular', such circularity is benign or vicious.

Dummett's British Academy lecture, 'The Justification of Deduction' is one of the key texts here. You criticize Dummett (calling his distinction between explanatory and suasive arguments 'deeply flawed') on the grounds that there are 'many forms of deductive rule systems'. I think this is a bit unfair. The distinction between explanatory and suasive justification is Dummett's first word on the question of justification, not his last. It seems there might be room for a 'justification' which does not persuade those who need to be persuaded -- the deductive sceptics, supposing any to exist -- but rather helps us understand better how deduction works, how it can lead to knowledge.

What Dummett goes on to do is explore the dispute between classical and intuitionist logic, where the question of semantic interpretation becomes central. (Admittedly, this was not the focus of original critics of classical logic like Brouwer, whose interest was in the mode of being of mathematical entities, as 'free creations of human thought'.) In Dummett's view, a 'justification' of deduction turns out to be a theory of meaning in terms of a central concept. If the central concept is truth, then the theory of meaning yields a semantic interpretation of classical logic. If the central concept is that of, e.g. verifiability or criteria, then that yields a semantic interpretation of intuitionist logic. The former theory of meaning is 'realist', the latter 'anti-realist'.

In short, you can determine whether someone is a realist or anti-realist about meaning, by seeing whether or not they accept the classical 'law of excluded middle' (LEM). Intuitionists do not reject that law, but rather refuse to assert it, substituting instead the double negation of the LEM. A consequence of this is that it is no longer possible to prove theorems using reductio ad absurdum.

I don't actually accept any of this -- I think that a global anti-realist about meaning and truth is fully justified in embracing the LEM -- however, I thought it was important to set the record straight.

Now, it looks superficially as if the principle of bivalence -- which according to Dummett requires a realist notion of truth -- is just a restatement of the law of excluded middle. Indeed, any semantic interpretation of an argument which shows its validity, e.g. using a truth table to show that a given formula of propositional calculus is a theorem, merely repeats the same rules, but at a 'meta' level. Dummett accepts this. This is the sense in which a justification of deduction is merely 'explanatory'. However, contrary to initial impressions, a certain amount of persuasion is involved, namely, in the dispute between the realist and anti-realist regarding the proper form of a theory of meaning. And this dispute has consequences (according to Dummett, at least) in our actual deductive practice.

On a more technical level, proofs of consistency and completeness are important in logic, especially when the powers of the logic in question exceed our ordinary intuitions (e.g. various systems of possible world logic). These are 'justifications' in a sense which goes beyond merely telling us what we already knew.

Your point about 'visual translation' is interesting. I wonder what you would say about Cantor's famous (or infamous) 'diagonal argument' proving the existence of a non-denumerable infinity (the sequence of cardinals aleph null, aleph one, aleph two etc.). Imagine (for the sake of reductio) that all the real numbers have been 'counted'. Then we can display them in a table (row 1, row 2, row 3 up to infinity). Now add one to the first digit of the first real number (if 9 then make it 0), add one to the second digit of the second real number, and so on. The result is a real number which *cannot* by definition be in this list.

Cantor's proof raised great controversy when it first appeared. Many students coming at it for the first time have the impression that it is just a 'trick'. How do you decide? Do you 'see' it or not? If the final criterion is what we 'see', then it looks as if we are back in the situation described by Goodman, where there is no independent criterion for the validity of our deductive vision.

All the best,

Geoffrey