To: Craig S.

From: Geoffrey Klempner

Subject: The justification of deductive reasoning

Date: 31st March 2010 11:18

Dear Craig,

Thank you for your email of 22 March, with your essay for the University of London Logic BA module, in response to the question, 'Can there be a convincing justification for deductive reasoning?'

This is an excellent essay with which I have few real disagreements. I am pleased to see that you give consideration to Dummett's British Academy lecture, 'The Justification of Deduction' which is an important contribution to this topic.

I'm a little bit wary of the idea that deductive reasoning has a pragmatic justification. I'm reminded of the story of the American state (I can't remember which one -- the story could be apocryphal) where a law was passed decreeing that the value of Pi would henceforth be taken to be 3. The idea was to make trigonometric calculations used in the building industry simpler, with less likelihood of error. The result was, as you would expect, an unmitigated disaster.

If you break the laws of logic (or in this case, Euclidean geometry) the result is likely to be unfavourable. So it is good policy never to break a law of logic if you can help it. However, this way of looking at the matter suggests that there might be some more obscure laws of logic for which the attitude of the legislature of the unknown US State was indeed appropriate. From a pragmatic perspective, you can only judge by results. This seems wrong. (From dim memory, I think there might be something along these lines in one of Lewis Carroll's 'Alice' books.)

Or think of it as analogous to deliberately making a bad move in chess. Sometimes the consequences can be better than if you had made a good move, because your opponent is tempted to overreach himself in punishing your 'mistake'. I can imagine Reginald Perrin's boss saying, 'I didn't get where I am today without breaking the laws of logic!'

The question asks whether there is a convincing justification for 'deductive reasoning' as such. You are right to focus on this. Formal systems attempt to encapsulate our intuitions about the validity of deductive reasoning. However, in some areas of logic, this can be contentious: e.g. the increasing number of systems of modal logic. How do you justify one person's intuitions about formally valid modal inferences against another person's contrary intuitions? Let's say that both systems have adequate consistency and completeness proofs, using different models of possible world semantics.

This is admittedly not quite the same point: with modal logic we are talking about defending particular instances or forms of deductive reasoning against the challenge of rival forms of deductive reasoning, rather than the justification of deductive reasoning as such. However, I think an examiner would consider this sufficiently relevant to the question, if only to throw the broader question into relief.

Dummett in his British Academy lecture focuses on the clash between classical and intuitionist logic. (There is more on this theme in his article, 'The Philosophical Basis of Intuitionist Logic' in 'Truth and Other Enigmas'.) The key point of dispute concerns the Law of Excluded Middle, or equivalently, the double negation elimination rule. Arguments which use Reductio ad Absurdum to prove a proposition P, by deriving a contradiction from the double negation of P are valid in classical logic but not in intuitionist logic. How does one resolve this dispute?

Dummett would say that you have to engage in the hard work of constructing a theory of meaning for the language and defending it against rival theories of meaning: in this case justifying a theory of meaning in terms of verification or proof conditions against a theory of meaning in terms of truth conditions.

On Dummett's view, the only adequate justification for a verification conditions theory of meaning is one which applies globally, and thus constitutes a global justification for intuitionist logic against classical logic. So here we are doing more than just providing an 'explanatory' argument. People who reason using classical logic are wrong. They are reasoning fallaciously. Whereas people who reason using intuitionist logic are not reasoning fallaciously.

From a broader, historical perspective, metaphysics and logic have always been bound up together, ever since Parmenides. Perhaps the most famous (or notorious) case is Hegel's apparent denial of the law of non-contradiction. In present times, we have systems of dialethic logic, which claim to be in some sense more accurate portrayals of the true nature of reality than classical or intuitionist logic. In a messy, enigmatic, paradoxical world, it pays to be less rigid in one's thinking.

One other thing you could have mentioned is Quine's image of a network of beliefs interacting with experience at the edges, while the laws of logic are situated in the middle. In his 'Two Dogmas' essay, Quine notoriously argues that no belief is immune to revision, not even laws of maths and logic. Insofar as we have not needed to revise our logic in the face of experience, it could be said that the laws of logic have an empirical justification. However, this justification is only ad hoc or temporary. Quantum mechanics is one area where classical logic has (allegedly) been put under pressure as a result of scientific advance.

All the best,

Geoffrey