It's not raining and it's raining therefore it's not raining.. So yeah, it's "incoherent" in that its premises are inconsistent.
— Michael
Accepting that definition of "incoherent," — Hanover
Whatever
@Michael meant, I don't take it as a definition. It only states:
If a set of statements is inconsistent, then it is incoherent.
It doesn't say:
A set of statements is inconsistent if and only if it is incoherent.
More generally, an expression may be incoherent but not inconsistent. Expressions that are not syntactical are incoherent but they're not even statements, so they are not even in the category of things that are consistent or inconsistent.
By using 'incoherent' rather than 'inconsistent', we lose the information that the premises are not merely incoherent, but they are, more to the point, inconsistent.
Also,
@Michael, as I understand him, meant scare quotes. Indeed, I don't see the analysis of this particular matter in ordinary formal logic as being in regard to a wider rubric of 'incoherent' (that includes both not-syntactical gibberish and syntactical inconsistency) but rather in regard to inconsistency.
Also, personally, in this context, I like to mention satisfiability rather than consistency, since they are equivalent only in first order logic, and, even more basically, mentioning satisfiability rather than consistency underscores that we don't need to have a particular, or even any, deductive calculus in view.
/
I suggested the neologism 'revonah' for an argument that has an unsatisfiable set of premises.
but maybe a neologism that is more technical sounding would be better:
Df. An argument is sat-premised if and only if the set of premises is unsatisfiable.
Df. An argument is unsat-premised if and only if the set of premises is unsatisfiable.
we have (1) valid and coherent arguments and (2) valid and incoherent arguments [and] (3) valid and sound arguments and (4) valid and unsound arguments. — Hanover
Soundness is per each interpretation. But let's say we're confining to just one interpretation, so we don't have to say 'per the interpretation':
(1t) sat-premised and valid
Not every sat-premised argument is valid.
Not every valid argument is sat-premised.
(2t) unsat-premised and valid
Every unsat-premised argument is valid.
Not every valid argument is unsat-premised.
(3t) sound
Every sound argument is valid.
Not every valid argument is sound.
(4t) unsound and valid
Would you agree that:
A. All 3s are 1s, but not all 1s are 3s?
B. All 2s are 4s, but not all 4s are 2s.
C. No 1s or 3s are 4s or 2s.
D. No 4s or 2s are 1 or 3s. — Hanover
(C) and (D) are WRONG (see below).
These are all CORRECT except those marked WRONG:
(A1) For any argument, if it is (3t) then it is (1t).
(A2) It is not the case that, for any argument, if it is (1t) then it is (3t).
(B1) For any argument, if it is (2t) then it is (4t).
(B2) It is not the case that, for any argument, if it is (4t) then it is (2t).
(C1) For any argument, if it is (1t) then it is not (4t). WRONG.
There are arguments that have a satisfiable set of premises but there is at least one false premise. This is a
key point in ordinary formal logic. Consider:
{"Macron is German"} is satisfiable but "Macron is German" is false. This is a
key point in ordinary formal logic: A set of premises may satisfiable but still have falsehoods. Consider:
"Macron is German" is false per ordinary facts, but there are interpretations in which "Macron is German" is true.
(C2) For any argument, if it is (1t) then it is not (2t).
(C3) For any argument, if it is (3t) then it is not (4t).
(C3) For any argument, if it is (3t) then it is not (2t).
(D1) For any argument, if it is (4t) then it is not (1t). WRONG.
There are arguments that are unsound but have a satisfiable set of premises. This is a
key point in ordinary formal logic: For example:
"Macron is German" is false per ordinary facts, but there are interpretations in which "Macron is German" is true.
(D2) For any argument, if it is (4t) then it is not (3t).
(D3) For any argument, if it is (2t) then it is not (1t).
(D3) For any argument, if it is (2t) then it is not (3t).