Is it not absurd, though?

Davidson against Wittgenstein. — Banno

How do you state that contest?

That's part of the problem - it was one of the topics I briefly considered for a Doctorate I (thankfully) decided not to pursue. A long time ago.

Roughly, Davidson inverts Tarski, keeping truth constant in order to derive meaning. He hoped to translate natural languages into extension formal language - basically first-order calculus. So truth-theoretical.

Wittgenstein dropped meaning in favour of examining what we are doing. This was loosely linked to introduction rules by Gentzen, roughly that they give logical connectives meaning. Proof-theoretical semantics followed.

But if I had the answer, I'd have that PhD.

I do understand the difference between taking 'true' as defined and taking 'true' as primitive.

My apologies. So Davidson took truth as primitive, using it to define meaning, Wittgenstein took use as primitive, using it to talk about truth. Davidson leads to the truth-theoretical approach, Wittgenstein, indirectly, to the proof-theoretical.

I do not see a way around making some kind of distinction here. Either only mathematics (and logic) gets knowledge and deduction ― and everything else gets rational belief and probability ― or there are two kinds of knowledge, and two kinds of deduction. Pick your poison.

Mathematical knowledge and empirical knowledge differ so greatly they barely deserve the same name. Obviously the history of philosophy includes almost every conceivable way of either affirming or denying that claim. — Srap Tasmaner

Good post. This is a clear representation of the variety of univocity that I would oppose. I don't think we have to pick a poison. They are different but not altogether different. I only would have been happier if you had said, "...[they] differ so greatly they don't deserve the same name."

But this should be bookmarked as a jumping-off point for a substantive thread. [Do they deserve the same name?]

So did you change your opinion?

I already did previously. There's no contradiction in that case because it is vacuously true. But that's a bit of a bait and switch.

I guess my gripe is that I would expect any statement to be logically consistent under all values of the antecedent. The fact that logical inference ignores it because under one of the values of the antecedent it does make sense is all very counter-intuitive to the point I feel the need to reject it.

Yes, because I have a better understanding of how to define validity in a formal context. No, because in a non-formal ordinary sense, and in a natural language context, the argument still seems invalid.

If I uttered: "If it is raining then it is not raining." ... If formal logic is "mappable" onto ordinary language, then you should be able to infer "oh okay, it's not raining." But no one speaks like that and no one would make such an inference. At least, no one would consider such an "argument" "valid." That being so, while I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal.

You're not talking about validity there, you're talking about the truth of an "if ... then ..." premise.

In propositional logic "if Michael is American then he is the President" is true, but in "ordinary" language it isn't.

Your real concern is with material implication.

I am not sure what you mean by saying "If I am American then I am the President" is true in propositional logic. But I do appreciate that that conditional is not true in ordinary language.

I am not sure what you mean by saying "If I am American then I am the President" is true in propositional logic. — NotAristotle

If "P" is false then "If P then Q" is true.

I am not American, therefore, "I am American" is false, therefore "If I am American then I am the President" is true.

Okay, but I can actually see how the edited conditional could be true. For instance, if Michael is a really great citizen, then maybe he would end up being President were he American, if so, then in the ordinary sense, the sentence can be "true" based on what it means.

But no one speaks like that and no one would make such an inference. — NotAristotle

And yet here we are. Turns out folk do speak like that.

If this thread is not long enough, then it is long enough.

I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal. — NotAristotle

The word 'valid' is equivocal.

There are different definitions and understandings of the word 'valid'.

One of the formal definitions of 'valid' is a common one. That definition is not equivocal.

Meanwhile, among the everyday senses of 'valid', which has a definition that is not equivocal?

Okay, but I can actually see how the edited conditional could be true. For instance, if Michael is a really great citizen, then maybe he would end up being President were he American, if so, then in the ordinary sense, the sentence can be "true" based on what it means. — NotAristotle

What is the edited conditional?

The conditional is "If Michael is American then Michael is president".

That conditional is true since "Michael is American" is false. There's no need to mention any aspect of greatness of citizenship, nor "maybe", nor "end up".

So why do we accept as logically valid a premisse that will result in a logical contradiction under one value of the antecedent? It seems we do this because of formal rules and in theory we could change those. What would be the implications if we would say for any given argument under all values of the antecedent the conclusion may not result in a logical contradiction or the argument will be deemed invalid?

@Banno maybe you have thoughts about that too?

If I uttered: "If it is raining then it is not raining." ... If formal logic is "mappable" onto ordinary language, then you should be able to infer "oh okay, it's not raining." But no one speaks like that and no one would make such an inference. At least, no one would consider such an "argument" "valid." That being so, while I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal. — NotAristotle

I disagree with regards to ordinary language, because we ordinarily reject contradictory premises for sake of avoiding contradiction; we naturally reject A whenever A implies (B And Not B) for any proposition B.

One isn't inferring Not A in such cases, rather one is establishing a consistent set of premises for subsequent inferencing. This is reflected by the fact that the case you find to be problematic, is actually an alternative axiom used in the definition of negation in intuitionistic logic.

One isn't inferring Not A in such cases — sime

So you would say that a reductio ad absurdum is not an inference in the proper sense?

What is the edited conditional? — TonesInDeepFreeze

Originally I wrote "if I am American then I am the President", but I changed "I" to "Michael" to avoid any debate about indexicals.

So you would say that a reductio ad absurdum is not an inference in the proper sense? — Leontiskos

It is an inference in the syntactical sense of implication, but not in the semantical sense of implication as ordinarily used by scientists and legal practitioners who are in the business of inferring facts as opposed to uninterpretable sentences.

In a consistent deductive system , If the sign "Not A" is either taken to be an axiom, or is inferred as a theorem, then it means that the sign "A" is non-referring and hence meaningless in that it fails to denote any element of any possible world among any set of possible worlds that constitutes a model of the axioms. By symmetry, the same could be said of the sign "Not A" being meaningless if A is taken as an axiom, but by model-theoretic traditional the sign A is said to not denote anything in a model if ~A is provable.

For instance, let the sign "A" denote the proposition that the weather is wet in some possible world. If "A" is deductively assumed or proved, then A is a tautology, meaning that the logical interpretation of "A" is stronger than being a mere possibility and denotes the weather being wet in all possible worlds. On the other hand, if "~A" is provable, then no possible world is wet, in which case the sign "A" fails to refer.

In conclusion, A and ~A can only both be meaningful if they both stand for possible but unnecessary states of affairs, in which case neither are provable. So the OP's problem isn't a problem, because the signs of the implication A --> ~A aren't simultaneously meaningful.

So in common-sense Kripkean semantics,

A --> NOT A says: all worlds that satisfy A also satisfy NOT A.

But in Kripke semantics, a world satisfies NOT A if and only if it doesn't satisfy A. So the set of worlds S that satisfy this condition is empty. A forteriori, there aren't any worlds in S satisfying A. Therefore
NOT A is true, and A refers to nothing.

What would be the implications if we would say for any given argument under all values of the antecedent the conclusion may not result in a logical contradiction or the argument will be deemed invalid? — Benkei

This goes back to my pedantry comments. I can't see how it could matter if we designated a name for that special class of modus ponens described in the OP, where it is structurally consistent with modus ponens but is logically inconsistent. This thread strikes me as more of a primer in formal logic nomenclature than in logic qua logic.

I think you are right that material implication is a problem in the example I stated; the premise appears to not be true (and to never be true).

Still, it also appears that the conclusion is an unwarranted logical leap from the premises, so that is why I think there might be room to argue that the argument is not valid according to some informal definition of logical validity. That is to say, the conclusion doesn't follow or doesn't lead to the conclusion. I understand that this is not the definition of validity formally speaking.

Still, it also appears that the conclusion is an unwarranted logical leap from the premises, so that is why I think there might be room to argue that the argument is not valid according to some informal definition of logical validity. That is to say, the conclusion doesn't follow or doesn't lead to the conclusion. I understand that this is not the definition of validity formally speaking. — NotAristotle

The conclusion logically follows, as has been explained many times.

P → ¬P
∴ ¬P ∨ ¬P
∴ ¬P
P
∴ ¬P

Or more simply:

¬P
P
∴ ¬P

The only issue is that people misunderstand what "P → ¬P" means. It doesn't mean what "if ... then ..." means in ordinary English.

I may be using "equivocal" incorrectly; what I meant is that there may be two senses of the term "valid" in a logical context; one formal, the other informal and that evaluating an argument with either definition may cause different conclusions as to whether a given argument is valid.

I am referring to the "it is raining" example; the conclusion in that argument appears to be a logical leap. I get that the argument is formally valid, that's the entire point - while formally valid, the conclusion does not appear to "follow."

I am referring to the "it is raining" example; the conclusion in that argument appears to be a logical leap. I get that the argument is formally valid, that's the entire point - while formally valid, the conclusion does not appear to "follow." — NotAristotle

If it is raining then it is not raining
Therefore, either it is not raining or it is not raining
Therefore, it is not raining
It is raining
Therefore, it is not raining

Or more simply:

It is not raining
It is raining
Therefore, it is not raining

Michael, the argument is simply this:

If it is raining then it is not raining.
Therefore, it is not raining.

Who in there right mind would conclude the conclusion from the premises in a conversational setting?

(It is a different argument from the original argument in the first post).

Michael, the argument is simply this:

If it is raining then it is not raining.
Therefore, it is not raining.

Who in there right mind would conclude the conclusion from the premises in a conversational setting? — NotAristotle

They probably wouldn't, because the grammar of ordinary language does not follow the rules of propositional logic.

In propositional logic, the following is a valid argument (specifically, it's a tautology):

P → ¬P
∴ ¬P