Syllogisms and Contradictions

I’m full of ideas from a wonderful event organised by Simon Hewitt at the University of Leeds: Approaches to Contradiction, Old and New. The day before I attended a student workshop and learned about plural logic from Berta Grimau and dialetheism and paraconsistency from Graham Priest.

In the workshop I learned of Catarina Dutilh Novaes’s exciting and, I think, highly plausible performative reading of Aristotle’s argument against the Principle of Non-Contradiction (and why it still doesn’t work); I heard Simon Hewitt’s latest thoughts on the ex contradictione nihil principle (which I’ve named the Principle of Implosion); and I watched Graham Priest demonstrate a way of formalizing the Hegelian dialectic and then applying it to concrete Hegelian and Marxist arguments.

I had great conversations with the other participants and students. I only hope I can remember most of the helpful things I learned.

Here I wanted to write about something that came up in a conversation with Graham Priest, before I forget. If anyone has helpful thoughts, I’d like to hear them.

Graham’s lecture mentioned that the first paraconsistent logic, at least in the Western tradition, was the Aristotelian syllogistic. A paraconsistent logic is one in which it is not the case that anything follows from a contradiction. Paraconsistency denies the rule that ex contradictione sequitur quodlibet, also known as the Principle of Explosion (hence ‘Implosion’ for Simon’s ex contradictione nihil). Explosion rules that:

Graham proves that the Aristotelian syllogistic implicitly rejects Explosion and is thus paraconsistent. The proof is by example.

First, take some invalid syllogistic mood, e.g. SaM, MoP ∴ SeP (all S are M, some M are not P, therefore no S are P). Let’s call this vicious mood Karaoke, in honour of a great Leeds tradition for philosophical workshops. De jure, Karaoke is invalid because it has an undistributed middle, but the real source of the invalidity can be illustrated with the following Lewis Carroll diagram:

The boxes for SM’P and SM’P’ are greyed out, ruled out by the first premise. But the second premise leaves us a choice between SMP and SMP’. This shows us a countermodel for Karaoke, namely SMP — the inside top-left box.

But Graham notes that Aristotle allows that two terms in the syllogism can be the same. Thus in Karaoke we can take S for P and have: SaM, MoS ∴ SeS. Now the premises are contradictories, and if Explosion held the inference would be valid. But still this is the invalid Karaoke mood, and syllogistic rules it invalid.

I want first to note that it’s arguable that by this reasoning syllogistic is not only paraconsistent but dialethic — it permits true contradictions. I’m not at all sure about this. But consider first an open Lewis Carroll diagram with P=S:

This has one quadrant for SS’ and another for S’S. If we take the diagram to represent the space of logical possibility — the range of propositions that might be true — then this diagram allows for the possibility that statements to the effect that all S are not S, some S are not S, etc. are true. These are unusual things to say, but so far we have no true contradictions.

But! If we can have S for P in a syllogistic form, I see no reason we can’t also have S for M. And now we have the following diagram of the space of logical possibility:

This has compartments for SS’S, SSS’, etc. And these look like true contradictions: statements to the effect that S both is and is not S. Thus if the diagram represents the space of logical possibility, it reveals the logical possibility of true contradictions.

My question for Graham was different, however. I was curious about the validity of syllogisms where all the terms are the same. As a theory of inference, the syllogistic becomes quite strange and inconsistent once we permit this possibility.

For instance, SiS, SiS ∴ SiS is an invalid syllogism; three particular propositions never make a valid mood. Graham told me that Aristotle might not have minded this; he seems to hold that valid reasoning must allow something to be inferred from something else; A ⊢ A isn’t a valid form. But then we have the problem that SaS, SaS ∴ SaS is an instance of a valid mood — classic Barbara (SaM, MaP ∴ SaP). So an interpretation of a valid mood can be given that isn’t valid.

Or is it so? Perhaps the argument SaS, SaS ∴ SaS is an interpretation of SaM, PaM ∴ SaP: an invalid mood. It seems that the syllogistic gives inconsistent results where all the terms are the same.

The answer is probably that Aristotle doesn’t believe in proofs from identity statements, or from their negation. But then perhaps he accepts Implosion in some cases: from SaS, SoS, nothing follows at all.