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This lecture by Graham Priest contains an interesting claim, namely that term logic is paraconsistent.

I have two questions about this:

  • Is paraconsistency in this context ever considered a meta-level manifestation of the existential import of universal quantification?
  • How is the paraconsistency of term logic usually analyzed?

I will consistently use modern-all in English to refer to the modern all that lacks existential import and existential-all to refer to the historical all that has it. all without a prefix can mean either depending on the context.

The example given in the lecture is as follows

      Some As are Bs.          AiB
      No Bs are As.            BeA
BAD! --------------      BAD! -----
      All As are As.           AaA

It is well known that term logic has existential import. The following statement is false because dragons don't exist.

 All dragons are beings.  DaB

Is equivalent to ∀d:D.B(d) ∧ ∃d:D.B(d).

To me, at least, rejection of ex falso quodlibet, seems like a consistent meta-level application of universal quantification has existential import.

I'm not sure exactly how to phrase this argument, I'm using the entailment relation in a slightly different way than it's normally used; I'm still using it to mean semantic consequence, but I want to tweak how semantic consequences work by replacing the notion of all that's used to define it. I want to be the underlying truth that successfully captures in decidable rules.

In modern notation, Let refer to the term logic deductive system. is supposed to be sound and complete for the semantics of term logic, so let's assume that it is. Γ ⊢ φ if and only if Γ ⊨ φ. In modern terms, Γ ⊨ φ is true if and only if φ is true in all interpretations when Γ is true.

If we think of our meta-level all as meaning existential-all, then I think a paraconsistent system obtains. If the premises are inconsistent, then there are no interpretations of Γ, and thus φ is not true in at least one interpretation conforming to Γ and thus the whole existential-all sentence is not true.

Rejecting and thus when there are no interpretations that satisfy the conditions on the left-hand side seems consistent with the behavior of the a connective.

How is paraconsistency in term logic usually analyzed?

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    The modern ∀ does imply existence. Only free logics lack this assumption. en.wikipedia.org/wiki/Free_logic Ah, but I see this isn't what you meant by existential import. – causative Jan 3 at 7:19
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    That's a good point. I thought that didn't apply in my case because I'm using the 2-place variant of ∀, ∀x:S.P(x) , which is equivalent to ∀x:D.S(x)→P(x) where D is the domain of discourse. One place ∀ is usually used in contexts where the domain of discourse is constrained to be nonempty, but I thought this restriction came from the context in which we're using FOL, not FOL itself. When S is empty in this example, the statement is true whether we use 2-place ∀ or 1-place ∀, but SaP is not true. – Gregory Nisbet Jan 3 at 7:25
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    In modern logic ∀x (S(x) → P(x)) implies ∃x (S(x) → P(x) but not ∃x S(x) ∧ P(x). See John Corcoran & Hassan Masoud, Existential Import Today (2015) – Mauro ALLEGRANZA Jan 3 at 8:54
  • @MauroALLEGRANZA I use ∀x:S.P(x) to mean ∀x:D.S(x)→P(x) and ∃x:S.P(x) to mean ∃x:D.S(x)∧P(x), I'm treating predicates additionally as sorts that can be quantified over. I'm using existential import to explicitly mean the claim that a sentence can be true only when the thing quantified over is inhabited. I don't make the claim that ∀→∃ is true, but I do claim that AaB→AiB is true in term logic, because a has existential import in its first argument. I was only trying to compare the truth conditions of and a. I might be misusing the the term existential import. – Gregory Nisbet Jan 3 at 16:17
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    The modern idea that term logic has existential import is very confused, see detailed explanation on Logic Museum. It is true that traditional logic took "all dragons are fire-breathing" to imply "some dragons are fire-breathing", but it is false that it read "some dragons are fire-breathing" as "fire-breathing dragons exist". Hence the distinctions in the OP do not reflect the modern/historical difference at all. To do so, one needs to introduce an explicit existence predicate, and separate its role from that of existential quantifier. – Conifold Jan 4 at 8:54

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