Thursday, May 27, 2010

Ockham on the paradoxes of entailment

While looking through an edition of Ockham, I came across this interesting text:

Other rules are given:
(10) From an impossibility anything follows.
(11) What is necessary follows from everything.
Therefore this follows: 'You are a donkey, therefore you are God'. This also follows: 'You are white, therefore God is triune'. But these consequences are not formal ones and they should be used much, nor, indeed are they used much. (Summa totius logicae III, III, C. XXXVI)

What is interesting to me is (a) that the consequences are not formal ones (don't follow in relevance logic?) and especially (b) that these rules shouldn't be used much. Why not?

Presumably, the only time you establish an impossibility in a sound argument is as part of a reductio, and if you do that, you don't use (10) next—you close the subproof and use reductio ad absurdum.

And using (11) leads to arguments that are not as perspicuous as they could be. For if you've established necessarily(p), it is more perspicuous to conclude p from necessarily(p) by axiom M of modal logic than to do something like:

  1. necessarily(p).
  2. 2+2=4.
  3. p. (By (12), (13) and rule (11))

However, while (10) and (11) are useless considered as rules of inference, as true propositions they can be quite useful, and do in fact occur in philosophical discussion, for instance in providing counterexamples (suppose you say that x depends on y if and only if exists(x) entails exists(y); then if numbers are necessary beings, everything depends on the number 49, which may seem to be absurd). So we need to distinguish between (10) and (11) as truths and (10) and (11) as rules of inference.

This distinction is needed anyway for modus ponens; for if modus ponens is simply the universally quantified truth:

  1. For all p and q, if (p is true and it is true that if p, then q), then q is true,
then to apply modus ponens given the truth of P and of if P then Q, you will need to do universal instantiation on (15) to get:
  1. If (P is true and it is true that if P, then Q), then Q is true.
And then using the antecedent of (16) as a premise to get to the conclusion Q, one will have to use modus ponens, which lands one in a vicious regress. (This is an argument of Sextus Empiricus against the very idea of rules of logic. But Sextus confuses truths qua truths and rules. Not that I know exactly how to draw the distinction either.)

1 comment:

Douglas said...


Do you think that modus ponens is 15?

To my ear, it sounds odd to identify inference rules with items that are truth-apt. Why not hold that 15 is true if and only if modus ponens is sound? (Loose analogy: an argument is valid if and only if its corresponding conditional is a logical truth.)

Against Sextus, one might argue that rule circularity is not vicious in the case of logic.