## Term Logic Paraconsistency -- Is it just a meta-level manifestation of existential import?

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

• 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
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. https://en.wikipedia.org/wiki/Free_logic Ah, but I see this isn't what you meant by existential import.

– causative – 2021-01-03T07:19:58.040

1That'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 – 2021-01-03T07:25:46.350

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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 – 2021-01-03T08:54:18.053

@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 – 2021-01-03T16:17:48.433

I've only omitted the "domain" D "sort" for simplicity. The above are "standard" results with classical logic (semantics with non-empty domains). According to it, the aristotelian rule AaB→AiB is not valid (exactly because "universal" S may be empty, in which case ∀x:S.P(x) is vacuously true while ∃x:S.P(x) is false). The paper linked above gives an interesting condition under which the A's inference holds in classical modern logic: when ∃x S(x) is valid. – Mauro ALLEGRANZA – 2021-01-03T16:43:07.700

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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 – 2021-01-04T08:54:01.997

@Conifold I'm not convinced that the article you reference in Logic Museum is correct. For example, Stephen Read in "Aristotle and Lukasiewicz on existential import" Journal of the American Philosophical Association, 1:535-544 (2015) argues that Aristotle, Avicenna, Ockham and Buridan among others did indeed hold that universal affirmatives presuppose existence of the subject. In the Museum article, it is scarely possible to understand what an I proposition is supposed to mean. – Bumble – 2021-01-04T21:43:14.943

@Bumble The meaning of I seems straightforward to me: among dragons, whether they exist or not, some are fire-breathing. Perhaps it seems puzzling because we are too used to multitask ∃ for both "some" and existence. But Kant's opinion was not widely shared before Brentano, and formal systems that introduce a separate existence predicate E! are now well-known, e.g. Berto's, Parsons's, Zalta's. SEP discusses them. Even on Read's reading, Aristotle admits empty terms, meaning translation with E! is more appropriate.

– Conifold – 2021-01-05T12:34:50.683