Friday, 21 April 2017

Modern Quantificational Logic Doesn't Subsume Traditional Logic

It seems to be a received view about the relationship of traditional Aristotelian logic to modern quantificational logic that the inferences codified in the old-fashioned syllogisms - All men are mortal, Socrates is a man, etc. - are all, in some sense, subsumed by modern quantificational logic. (I know I have tended to assume this.)

But what about:

P1. All men are mortal.
C. Everything is such that (it is a man  it is mortal)?  

This is a logical inference. It is not of the form 'A therefore A'. It embodies a very clever logical discovery! P1 and C are not the same statement. Talk of 'translating' the former by means of the latter papers over all this.

Modern quantificational logic does not really capture the inferences captured by traditional logic, any more than it captures this link between the two. It does capture inferences which, given logical insight, can be seen to parallel those codified by traditional logic, but that is not the same thing.

1 comment:

  1. I've only got some basic logic under my belt, but I suspect that there is a difference between these two statements, but that this difference can be captured by contemporary logic. I would translate your two statements as:

    P1. All men are mortal. -> For all x (where the domain of x is men), x is mortal. (AxMx)

    C. Everything is such that (it is a man ⊃ it is mortal). -> For all x (where the domain of x is anything), if x is a man then x is mortal. (Ax(Ex->Ox))

    Because they are logically equivalent and the latter can be easier to work with, I think we default to the latter, but I think the former captures the semantic difference you're picking up on.