"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:

P → ¬P
∴ ¬P"

Exactly. And if someone wouldn't make such an inference, I am suggesting that that is a logical mistake of some sort, which is a way of saying the argument is not valid.

There's no logical mistake? It's just the case that "if ... then ..." in ordinary English doesn't mean what "→" means in propositional logic.

Can you explain how those meanings diverge?

"I disagree with regards to ordinary language" I'm not quite getting it, what is the disagreement you have concerning ordinary language? You think someone would make an inference from A->not-A to therefore not-A in ordinary language?

A → B means B or not A

If I punch you then you will cry does not mean you will cry or I won't punch you.

I think you mean to say that the one implies the other through logical equivalence. That is different than saying that the expressions mean different things.

In fact, I would say A->B does not "mean" B or not-A.

you will cry or I won't punch you — Michael

This uses the inclusive OR which is also not so standard in English.

Is it worth pointing out, again, that "P→~P" is not a contradiction? If P→~P is true, then P is false.

If that's been said once, it's been said a thousand times... which is not once.

What is going on here is not a pedantic mismatch between English and some esoteric academic exercise. Rather, there are ambiguities in the English use of "If... then...", "...or..." and various other terms that we must settle in order to examine the structure of our utterances in detail.

What is going on here is not a pedantic mismatch between English and some esoteric academic exercise. Rather, there are ambiguities in the English use of "If... then...", "...or..." and various other terms that we must settle in order to examine the structure of our utterances in detail. — Banno

I consider the logical conditional a performative, as exists in an algorithmic way.

Consider, "If X = 4, then Y = 7." That is , if we set X at 4 then Y is set at 7. We could not program if we could not make such statements. If P then Q results in the occurence of Q when P is the case necessarily. I consider this an analytic operation and consistent with computer logic in programming (as far as I know about programming).

I consider the linguistic conditional not an indication of what is or what will be, but a hypothetical counterfactual that does not indicate, but hypothosizes. Because it does not indicate, we don't speak in the indicative mood, but in the subjunctive, as in what we wish, hope, or hypothesize about.

As in: "If I were President, I would lower taxes." This is not represented as P -> T. That would overstate the meaning of my speculative statement. Note the "were," not "was." This is a counterfactual (it hypothethesizes an antecedent that did not occur), not a logical conditional.

"If I was President, I lowered taxes" makes more sense as a formal conditional.

If I was President, I lowered taxes
I was President
ergo I lowered taxes

But not:

If I were President, I would lower taxes
I were President
ergo I would lower taxes

What does it mean that I were President versus I was President? I think the meaning is critical in changing from the formal indicative conditional to the non-formal linguistic subjunctive conditional.

My thoughts at least.

↪NotAristotle
Is it worth pointing out, again, that "P→~P" is not a contradiction? If P→~P is true, then P is false.

If that's been said once, it's been said a thousand times... which is not once. — Banno

I know Banno; I am not disagreeing with the formal validity of that argument.

there are ambiguities in the English use of "If... then...", "...or..." and various other terms that we must settle in order to examine the structure of our utterances in detail. — Banno

I don't disagree with that either. But the argument A → ~A ∴ ~A clearly does not translate into natural language very well (I don't think there is any way to translate it in a way that renders the translation sensible and "logical"). And yet, the argument is valid formally speaking.

Michael suggested that the argument is not sound in ordinary language. I think he may be right. However, even arguments that are not sound can still be valid such that we can understand how the speaker reached their conclusion (though we may point out to them that such-and-such premise is not true). For example, if someone argued:

1. P
2. P→Q
Therefore, Q.

We might correct them, "well, actually ~Q." "Your reasoning is spot on and logical, it just happens to be that ~P, so while your reasoning is valid, the argument you presented is unsound."

On the other hand, "If it is raining, then it is not raining, therefore it is not raining" sounds like an unwarranted leap that is not logical when we consider it in an informal way. The problem isn't just that the initial premise is unsound (within an informal context); the problem is that the argument just doesn't make sense and is not logical, so soundness aside, that is why I call it "not valid" informally.

We might correct them, "well, actually ~Q." "Your reasoning is spot on and logical, it just happens to be that ~P, so while your reasoning is valid, the argument you presented is unsound." — NotAristotle

Yes, there is a difference between an unsound argument that arises from an incorrect fact as opposed to one that arises from a contradiction.

- If I go to the store, I will buy milk, I went to the store, so I bought milk. That's true, unless I forgot to buy milk.

- If I go to the store, I will not have gone to the store, I went to the store, so I didn't go to the store. That statement is never true regardless of what I do. The reason it's never true is because "If I go to the store, I will not have gone to the store" is logically equivalent to "I did not go to the store."

I don't see any reason to introduce modality. It just adds to the confusion.

SO you want to introduce a new form of validity, that depends not on the explicit structure of the argument but on your intuition. Ok.

"I disagree with regards to ordinary language" I'm not quite getting it, what is the disagreement you have concerning ordinary language? You think someone would make an inference from A->not-A to therefore not-A in ordinary language? — NotAristotle

The formal meaning of negation in intuitionistic logic refers to the syntactical inconsistency of the negated sign, rather than to a purported semantic counterexample denoted by the negated sign. Classical logic inherits the same meaning of negation from intuitionistic logic, except for infinitary propositions that appeal to the Law of Excluded Middle, which have no scientific or commonsensical application. So we should stick to discussing negation in intuitionistic logic, before proceeding to other formal logics such as affine linear logic, whose concept of negation is closer to ordinary use. In such cases (A --> Not A) --> Not A is not derivable, corresponding to the fact that Not A obtains the same semantic status of A.

But can we elucidate the meaning of (A --> Not A) --> Not A in the systems for which it is valid, by appealing to the mutually exclusive states of the weather? Suppose that a weather forecaster said "It is raining in Hampshire therefore it is not raining in Hampshire". Jokes about the english weather aside, wouldn't you assume that they were talking about anything apart from the weather in Hampshire? in which case your abstaining from assigning a meaning to their words would resonate with the formal meaning of negation in intuitionistic and classical logic.

As for formalities,

(A --> ~A) --> ~A is little more than the obvious identity relation ~A --> ~A, due to the fact that ~A is definitionally equal to A --> f , where f denotes absurdity. So we at least have

(A --> f) --> ~A

But the only means of obtaining f from A is via the principle of explosion (A And ~A) --> f. And so it is sufficient that A implies ~A.

(A --> ~A) --> ~A

And since the converse direction is immediately true, we could in fact define the negation of A to be the fixed point of the expression X => (A --> X) that Haskell programmers call a Reader Monad.

~A = A --> ~A
~A = (A --> (A --> (A --> ..... ) ))

which serves to highlight the meaning of Negation As Failure (NAF); A proof of ~A amounts to a finite proof that the right hand side doesn't converge, which represents an infinite failure to prove A by random search. But if we haven't managed to prove either A or ~A using our available time and resources, then we are at liberty to declare ~A by decree and reason accordingly, in which case ~A serves to nullify any hypothesized A by turning it into ~A, so as to ensure consistency with our failure to decide the issue, at least for the time being...

Well it seems to me that all we can rely on when it comes to logic is intuition. If logic is just a formal set of rules as to how symbols may relate then anything can be logical and in that case nothing really is "logical," though I take that to be the discussion in the logical nihilism thread.

Our logical intuitions are basic, or foundational for doing logic, much in the way that having a functional ear is foundational for making a musical symphony.

One could argue that P->Q and P together implies not-Q, but translating that into natural language with the conditional spoken as an "if...then..." (or A and not-A therefore A and not-A) will be very difficult and I would say impossible, and that's because logic relies on meaning maybe just as much as meaning relies on logic.

All that to say that, at least informally A->notA therefore not-A may not be valid after all if our starting point is a set of meaningful natural language propositions.

That doesn't imply that formal logic is merely "academic" because it clearly has application to fields like computer science and mathematics.

But it may imply that some definitions we use in formal logic may be reviseable or at least more fungible then we previously thought.

Well it seems to me that all we can rely on when it comes to logic is intuition — NotAristotle

That's really sad.

Then I challenge you to prove that the following argument is logical:

P
P->Q
Therefore Q.

Or that this argument is logical:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

Or that this argument is logical:

If it rained yesterday then the lake is swollen today.
It rained yesterday.
Therefore, the lake is swollen today.

If you don't think those arguments valid, then you haven't understood "validity". But we already knew that from your OP. And since you do not understand validity, there is no point in presenting you with valid arguments, in order to prove anything.

You have placed yourself outside of rational discourse.

Cheers.

While you are proving what exactly is logical, you might as well prove that 2+2=4 and that there is an external world, but I don't want to hear any of that mathematical intuition or logical intuition or perceptual intuition nonsense.

Okay, correct me if I'm wrong, but you are saying that ordinary natural language is "mappable" onto formal classical logic because in formal logic a syntactic inconsistency viz., a negated sign that is present alongside the original sign, results in an argument that is "not derivable" whether the sign and negated sign are explicitly present or present by implication (A->not-A). So just as the ordinary natural language argument is meaningless, so the classical logic argument is underivable.

Can an underivable argument be valid? (I suppose you would say "yes" because the "underived" (unconditioned) constituents of the argument are mere possibilities).

I would think many people would apply a truth table to the argument (A->notA therefore not-A), as I did, and see based on that, that the premise is only true when "not-A." Maybe "infer" is too strong a word for the conclusion of not-A.

The conclusion does not seem to "follow" or be a "logical" conclusion when we apply the argument to ordinary natural language.

So I guess what I'm wondering is whether an underivable (or meaningless) argument may be regarded as logical? Or are soundness and validity insufficient for a logical argument? Or is meaning related to soundness?

It seems, to me, as though what is meant is critical to determining whether an argument is logical.

Your post is hopelessly confused because you don't know the basic concepts.

If you would just read a little bit in an introductory textbook, in print or online, you would know.

All my remarks pertain to ordinary usage such as found in basic textbooks:

There is no such thing as a "derivable argument" or "underivable argument". The expressions "derivable argument" and "underivable argument' make no sense.

An argument is a pair, with the first component called 'the set of premises', and the second component called 'the conclusion'. An argument is valid or invalid, and sound or not sound. There is no such thing as an argument being "derivable" or "underivable".

Derivability pertains to proof. An argument is not itself a proof. An argument is a pair, with the first component called 'the set of premises' and the second component called 'the conclusion'.

What is underivable or not, is a conclusion from a set of premises. And that pertains to proof.

(1) Example:

set of premises:
{P, Q, R -> S, ~(P & R)}

conclusion:
S

That is an argument (it happens to be invalid). Just a set of premises and a conclusion. It's not a proof.

(2) Example:

set of premises:
{Q -> R, S, P, P -> Q}

conclusion:
R

That is an argument (it happens to be valid). Just a set of premises and a conclusion. It's not a proof.


A proof (in Hilbert form) is a sequence of sentences such that each sentence is a premise or follows by an inference rule from previous sentences in the sequence:

(3) For example (with the applications of the rules annotated):

1. P -> Q (premise)
2. Q -> R (premise)
3. P (premise)
4. Q (from 3, 1 by modus ponens)
5. R (from 2, 4 by modus ponens

That is a proof. It's a proof of R from the set of premises {P, P -> Q, Q -> R}. From that proof, we establish that the argument (2) is valid, since the proof has premises only from those of (2) (we didn't need to use the premise S, by the way) and the last line of the proof is the conclusion of (2).

I would expect any statement to be logically consistent under all values of the antecedent. — Benkei

I don't know what that is supposed to mean.

An antecedent is the part of a conditional that comes before '->'.

The argument under discussion:

A -> ~A
A
therefore, ~A

The only conditional there is:

A -> ~A

Its antecedent is:

A

There are two values we can assign to A: true, false

If A is false, then A -> ~A is true.

If A is true, then A -> ~A is false.

But A -> ~A is consistent in either case. It does not prove a contradiction, and it is satisfiable, since there is an assignment in which it is true, viz. the assignment that assigns false to A.

"The fact that logical inference ignores it because under one of the values of the antecedent it does make sense"

Ignores what?

A -> ~A doesn't make sense to you. But we didn't say A -> ~A makes sense only when A is false. A -> ~A makes sense whether A is true or A is false.

So why do we accept as logically valid a premisse that will result in a logical contradiction under one value of the antecedent? — Benkei

You're mixed up as you don't know the basic concepts. Reading just a little in a textbook in the subject would help you.

No sentence that proves a contradiction is valid. And no set of sentences that proves a contradiction is satisfiable.

And a sentence or set of sentences proves a contradiction or does not prove a contradiction irrespective of any assignment of truth values. That is, if a sentence proves a contradiction, then there is simply no assignment in which that sentence is true. And if a set of sentences proves a contradiction, then there is no assignment in which all the members of the set are true.

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

Again, if a set of premises proves a contradiction, then there is no assignment in which all the premises are true.

But, I guess what you mean is this:

Consider all and only those arguments in which the conclusion is not contradictory.

Okay, say an argument is an N-argument if and only if its conclusion is not contradictory. And say an argument is an S-argument if and only if its set of premises is satisfiable.

So "what happens"?

Any argument is an N-argument if and only if it's an S-argument.

definition of negation in intuitionistic logic. — sime

Yes:

Df. ~P stands for P -> f

where 'f' is primitive.

But, just to note, that can be a definition in classical logic too.

A -> ~A makes sense whether A is true or A is false. — TonesInDeepFreeze

I am not clear on how A -> not-A "makes sense" if A is true.

Also, TonesInDeepFreeze, an argument where all the premises are false and the conclusion is false would necessarily be valid; is that correct?

I was thinking of:

P->not-Q
not-P
Therefore,
not-Q.

Assuming that all the premises are false and the conclusion is false, the argument must be valid. Is that correct?

Or even if just one (but not all) of the premises is false and the conclusion is false (I am having trouble thinking of an example that meets this description).

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 — sime

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. — sime

(1) You say "in a consistent deductive system" but your remarks wouldn't apply to ordinary sentential or predicate systems, but rather, more specifically to modal systems. So, your remarks don't obtain as to deductive systems in general.

(2) With ordinary models for modal propositional logic, sentence letters themselves are members/not-members in worlds. But that ~A is an axiom or theorem doesn't entail that A is meaningless in any given model. Rather, as in classical semantics (but by more complicated considerations) A is false if and only if ~A is true.

(3) I don't know your definition of 'tautology' in modal logic. In propositional logic, a sentence is a tautology if and only if it is true in all models. I am not familiar with a notion in modal logic that being assumed makes the sentence a tautology.

(4) You said [paraphrase:] A stands for "wet in some world", then assuming A yields "wet in all worlds". That would be (where 'p' for possibly and 'n' for necessarily):

pA -> nA

And that is not generally (if at all) considered a validity.

[EDIT:] By the way, (A -> ~A) -> ~A is intuitionistically valid, perforce so is ((A -> ~A) & A) -> ~A.

As the argument forms are intuitionistically valid:

{A -> ~A}, therefore ~A, perforce {A -> ~A, A}, therefore ~A.

Classically by classical models; intuitionistically by intuitionistic models.

I am not clear on how A -> not-A "makes sense" if A is true. — NotAristotle

It makes sense in the sense of having a truth value.

an argument where all the premises are false and the conclusion is false would necessarily be valid; is that correct? — NotAristotle

No, quite incorrect. Egregiously incorrect. That you say that shows that you haven't paid attention to the numerous explanations given in this thread, let alone that you haven't paid attention to the most basic articles available on this subject.

I was thinking of:

P->not-Q
not-P
Therefore,
not-Q.

Assuming that all the premises are false and the conclusion is false, the argument must be valid. Is that correct? — NotAristotle

Not correct at all. It goes exactly against the definition of 'valid'.

Therefore
NOT A is true, and A refers to nothing. — sime

Where can one read an account of ordinary modal logic, ordinary intuitionistic logic or basic Kripke semantics in which that is the case?