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.
⊨ and thus
⊢ when there are no interpretations that satisfy the conditions on the left-hand side seems consistent with the behavior of the
How is paraconsistency in term logic usually analyzed?