If P is false then if P is true then it is true that P is true is a contradiction pretty plain and simple. — Hanover

You're confused.

Look at the truth table by which you will see that if P is false, then P -> ~P is true.

It's ridiculous to argue about it. Just look at it.

You're confused. I'm not "equating" A -> ~A to A -> B.

Let P and Q be metavariables over formulas. Then modus ponens is any argument of the form:'

P -> Q
P
therefore Q

Instantiate P to A. Instantiate Q to ~A. There is no restriction against such an instatiation.

So

A -> ~A
A
therefore ~A

is an instance of modus ponens.

This is where we disgree. — Hanover

It's where you disagree with the definition of 'modus ponens'.

You are welcome to state an alternative logic, but in ordinary truth-functional logic:

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

It's a valid argument only if you allow that A --> ~A is of the form A-->~B. — Hanover

That is incorrect. Validity is semantic.
A -> ~A
A
therefore ~A

is valid since there are no interpretations in which the premises are true and the conclusion is false.

As to form, we prove that anything in the form of modus ponens is valid.

I don't think it follows proper modus ponens syntax. The antecdent and consequent cannot be the same because if they are then it is reducible to simply ~A. — Hanover

(1) It is valid whether viewed as modus ponens or viewed by consideration of the fact that the premises are not satisfiable.

(2) You have a serious misconception of modus ponens,

Modus ponens is any argument of this form:

P -> Q
P
therefore Q

There is no restriction on what P and Q can be.

That includes taking P to be A, and taking Q to be ~A.

A -> ~A
A
therefore ~A

is most certainly an instance of modus ponens.

"If A is true, then A is false" is a necessarily false statement. — Hanover

That is incorrect.

If A is false then "If A is true then A is false" is true.

"If A is true, then A is false" is logically equivalent to "A is false or A is false." This means that A is false. — Hanover

That is correct.

I'd argue A --> ~ A is not of the form A --> B as required as a first premise of modus ponens. — Hanover

Then you'd argue incorrectly

The generic modus ponens syntax requires that the antecedent and consequent be different, meaning that A --> A is not logically equivalent to A -->B because the latter is not reducible to a contradiction. — Hanover

Modus ponens is any argument of this form:

P -> Q
P
therefore Q

There is no restriction on what P and Q can be.

That includes taking P to be A, and taking Q to be ~A.

A -> ~A
A
therefore ~A

is most certainly an instance of modus ponens.

I mean the conclusion is true regardless of the truth value of A. — Count Timothy von Icarus

It's a valid argument, so the conclusion is true in any interpretation in which all the premises are true. There are no interpretations in which all the premises are true. The conclusion is true in some interpretations and false in other interpretations.

An argument is valid if and only if there is no interpretation in which all the premises are true and the conclusion is false.

An argument is not "inconsistent". What are inconsistent are sets of formulas. What is inconsistent here is the set of formulas that is the set of premises.

one of the premises must be false given that they are "inconsistent?" — NotAristotle

There is no interpretation in which both premises are true.

If the interpretation has A as true, then A -> ~A is false.

If the interpretation has A -> ~A as true, then A is false.

And no interpretation has both A and A -> ~A as false.

The argument is valid but unsound you are saying? — NotAristotle

The argument is valid, and there is no interpretation in which the argument is sound.

what makes " A -> not-A " a premise that is not true? Does it have something to do with truth tables? — NotAristotle

A -> ~A is true when A is false and it is false when A is true.

That's correct. [EDIT: except for use of '=' instead of '<->']

So is:

A -> B
A
therefore B

You missed my point.

Of course, there are different ways to show the validity of the argument. But my point is that one of the ways doesn't require appealing to explosion or even contradiction since the argument is in the form of modus ponens.

not-G -> ( not- (P -> A) )
not - P

does not imply

G. — NotAristotle

In classical logic (but not intuitionistic logic),

~G -> ~(P -> A)
~P
therefore G

is valid.

in fact, the premises do not actually tell us anything. On the other hand,

not- G -> ( not- (P -> A) )
not- A

does seem to imply..

P. — NotAristotle

That's wrong.

Or, you're welcome to state your alternative logic.

My view was characterized by posters recently.

I take the problem to be to explain the puzzle: How did we infer a seemingly false conclusion from seemingly true premises with seemingly correct logic?

My answer is that the argument uses two different senses of "if then".

And it is likely that ~(P -> Q) is interpreted by some people with the truth table for (P & ~Q) instead of the truth table for ~(P -> Q). But that is not the answer I provide to the puzzle, which is more general: Different senses of "if then" are used, whether a reinterpretation of the truth table or even an interpretation that is not truth-functional.

I have not necessarily signed on to the views or explanations of other posters.

There must be a difference between implication and deduction — NotAristotle

There is.

An argument is an ordered pair where the first coordinate is a set of formulas (the set of premises) and the second coordinate is a formula (the conclusion). (Or 'statement' instead of 'formula' if the context is less formal.)

A deduction is a certain kind of sequence of formulas (or a certain kind of sequence of formulas alongside numbered sets of previous entries), or tree, or sequent, or tableau, depending on the context).

An implication is a formula of the form 'P -> Q'. Or, an implication is an argument.

As given, 3 is a conclusion.

3 follows from 1 and 2 by modus ponens.

[EDIT:

A -> ~A is not a contradiction.

A is false or not depending on an interpretation.]

In this case we don't need to appeal to the fact that the premises are inconsistent. If the logic includes modus ponens, then the example is valid, even if the logic does not include explosion.

This is the principle of explosion. — unenlightened

That is correct, but it is not necessary to appeal to explosion, since the argument is valid as it is an instance of modus ponens.

A truth table will tell you this is true is Sue is sitting or if she isn't sitting. — Count Timothy von Icarus

I can't parse that.

no way that the conclusion can follow from the premises. — NotAristotle

Define "follows from".

In ordinary logic, a conclusion follows from a set of premises if and only if there is no interpretation in which all the premises are true and the conclusion is false; and P -> Q is true if and only if at least one of these: (1) P is false, (2) Q is true; and ~P is true if and only if P is true. Thereby:

There is no interpretation in which
A -> ~A
and
A
are both true
and
~A is false

Indeed
A -> ~A
A
therefore ~A
is an instance of modus ponens:

P -> Q
P
therefore Q

The fact that {A -> ~A, A} is inconsistent doesn't contradict that there are no interpretations in which both A -> ~ A and A are true but ~A is false, as indeed there are no such interpretations since there are no interpretations in which A -> ~A and A are both true, even without consulting modus ponens.

If you propose a context with different definitions of "follows from" or different definitions for the truth or falsehood of '->' or '~', then you're welcome to state your definitions.

I've been overlooking the fact that real numbers are typically defined as equivalence classes of Cauchy sequences, not just individual Cauchy sequences. — keystone

Cauchy sequences themselves are infinite sets.

PA |- ~Con(PA) v Inc(PA)

equivalently:

PA |- Con(PA) -> Inc(PA)

equivalently:

PA + Con(PA) |- Inc(PA) — TonesInDeepFreeze

Followup to myself:

Yes, each of the above is correct.

It depends on what the purpose of the translation is.

If the purpose is to directly emulate the sentence as literally said, then:

~G -> ~(P -> A)

If the purpose is to provide a reasonable guess as to what was meant when the sentence was said, then:

~G -> (P -> ~A)

I wouldn't assume that the everyday sense of "if then" in the problem has a truth table interpretation.

And, the premise is "If there is no God, then it is not the case that if I pray then my prayers are answered"; the premise is not stated as "If there is no God then if I pray then my prayers are not answered". But if it were stated that way, then, of course

(~G -> (P -> ~A)) & ~P, therefore G

is WRONG and there's not "puzzle" to it.

I took the problem to at least present a "puzzle".

In set theory, equivalence does not imply equality. Here's the most trivial example:

{{0 1}} is a partition of {0 1}. And that partition induces the equivalence relation {<0 0> <1 1> <0 1> <1 0>}. And per that equivalence relation, 0 and 1 are equivalent. But 0 and 1 are not equal.

But, of course, one may posit a different mathematical approach by which certain equivalences imply equality. That's a matter of stipulation.

Though if a metaphysical or philosophical ruling on the question is sought, then that is yet another matter and would not be settled by mere mathematical formulations or stipulations.

@Michael made much the same point. — Banno

Then he's right. It takes only a moment to see that the salient feature of the argument is that it shifts from one sense of "if then" in one place to another sense of "if then" in another place.

Of course, "~G -> ~(P -> A) and ~P, therefore G" is classically valid. But what is interesting about the problem is that it has seemingly true premises and valid logic that lead to a conclusion that doesn't seem to follow from the truth of the premises.

"If there is no God then it is not the case that if I pray then my prayers are answered" seems true. It seems true based on an everyday sense of "if then" by which a conditional may be false when its antecedent is false.

But the inference "If there is no God then it is not the case that if I pray then my prayers are answered, and I do not pray, therefore there is a God" is valid based on a different sense of "if then" by which a conditional is false if and only if its antecedent is true and its consequent is false.

Noting that shift from one sense to another is a decisive and incisive explanation of how seemingly true premises and valid logic seem to lead to a conclusion that does not seem to follow from the truth of the premises.

Actually, I think you're the sinkhole. You seem to enjoy destructive conversations. — keystone

Keep digging your sinkhole deeper.

I didn't ask for a definition of 'planar graph'. You didn't read what I said about this a few posts ago. You are a sinkhole.

It's ironic that you became distant right after I went back, carefully studied, and addressed your comments on topology. — keystone

Apparently, you don't recall the post in which I said that I'm willing to indulge you only up to the point that you go past the process of definitions.

You don't need to concern yourself with my decisions about how I spend my time and energy. Instead, you need to start by at least getting a grasp of the basic ideas of primitive, definition, axiom and proof.

1D analogue of the established term "planar diagram" — keystone

You need to define "1D analogue of the established term "planar diagram"" in terms that don't presuppose any mathematics that you have not already defined and derived finitistically and such that it justifies such verbiage as about "embedding in a circle".

But don't bother if it is to re-enlist me. I was willing to take it step by careful step with you. But you can't discipline yourself to do that, as instead you just jump to whole swaths of handwaving. I said that at the very first point you invoked anything not previously justified by you then I'm out. I don't need to waste my time and energy on you. You are BS.

You haven't identified any falsity in my current position.

You haven't even defined enough to get the stage of consideration of truth or falsity.

Please, give me a chance. — keystone

I have! Many times! And previously too. But you abuse my time and effort. I'm done.

If your offer to help was sincere — keystone

How dare you question my sincerity that has been demonstrated over and over in careful attention to details, in my labor to explain things for you, in this thread and in one several months ago? Get a load of your narcissistic self. You are full of yourself and full of BS ... though that is redundant.

There's been a bunch of these around recently, so here's one that is actually valid...

If God does not exist, then it is false that if I pray, then my prayers will be answered. So I do not pray. Therefore God exists.

Attributed to Dorothy Eddington.

~G→~(P→A)
~P
G — Banno

That relies on conflating two different senses of "if then": an everyday sense and the material conditional. I'll use '-->' for the everyday sense and '->' for the material conditional:

(1) Everyday sense:

((~G --> ~(P --> A)) & ~P) --> G

If ~G is true, then to have ~G --> ~(P --> A), even in the everyday sense, ~(P --> A) must be true. But why is ~(P --> A) true? Only because, unlike the material conditional, the everyday sense allows that a conditional may be false even when its antecedent is false.

(2) Material conditional:

((~G -> ~(P -> A)) & ~P) -> G

That is a tautology. Because, unlike the everyday sense, a material conditional is false if and only if its antecednt is true and its consequent false.

/

With (1) we nod agreement with ~G --> ~(P --> A)) based on an everyday sense of the conditional by which a conditional (such as P --> A) may be false even when its antecedent is false.

With (2) we don't nod agreement with ~G -> ~(P -> A) since the material conditional (such as P -> A) is true when its antecedent is false.

There's an important distinction between handwaving and BS. Handwaving involves vagueness or imprecision, where the core idea might be sound but lacks detail or rigor in its current form. BS, on the other hand, is fundamentally incorrect—an argument that doesn't hold up under scrutiny and lacks substance from the start. — keystone

That's BS. BS includes nonsense, doubletalk and falsity. And handwaving is not necessarily just lack of rigor to be supplied later. And you presume that your "core ideas" are "sound".

I said I'd be willing to check you out to the extent that we could turn your ruminations into primitives, definitions and axioms. I predicted that right after the first round you would resort to yet more undefined handwaving and I said that I would drop out when that happened. Indeed, with the very first predicate 'is a k-continua' still not fully defined, you've piled on a big mess of more of undefined terminology and borrowing of infinitistic objects while you claim to eschew infinitistic mathematics. You disrespect my intellectual interest that way, just as occurred several months ago with a different half-baked and self-contradictory proposal of yours. You are a sinkhole of a poster. You need to obtain an understanding of the basic concepts of primitive, definition, axiom, and proof. I'm done with providing you assistance of this kind.

You're conflating non-equivalent theorems.

Theorem 1. The set of natural numbers is well ordered by the standard less than relation on the set of natural numbers.

That does not require the axiom of choice.

Theorem 2. Every set has a well ordering.

That is equivalent with the axiom of choice.

/

I don't know what is being asked by "what is the object behind/described by" a theorem.

Do you want to know what object a theorem is? A theorem of a theory is a sentence provable in the theory. A sentence per a language is a certain kind of sequence of symbols. A sequence is a certain kind of function. And symbols themselves may be taken to be certain mathematical objects.

Though, when I say 'object' I profess no particular metaphysical sense. I only mean 'object' in the ordinary sense of 'a thing' or 'something mentioned' or 'the referent of a pronoun such as 'it'', as I don't profess to be able to explicate that notion more than as a basic presupposition for talking about, well, things, as in "the number 1 is something different from the number 2".

You seem not to understand how the mathematical method of handwaving works. It's not ZF or PA or one of those; it's the theory BS. You need to familiarize yourself with its advanced techniques as exemplified by @keystone.

All graphs are 1D drawable (in that each can be embedded in a circle without any of its edges crossing) — keystone

Thank you. You saved me a lot of time and effort. Because my prediction that you would resort to half-baked handwaving is confirmed, so I am done with trying to help you formulate your stuff into mathematics.

A circle is an infinite set. But you say you're going to do this without infinite sets. You don't know what you're doing and you don't know what you're talking about. And if you were to say something like "a circle that's only a potential infinity" or whatever, then that would not cut it, since you haven't given a mathematical definition of such a thing. Just as I mentioned, you start to give mathematical definitions, but they finally end up relying on even more complex notions that are themselves presumptuously undefined and only gestured at with half-baked handwaving. You're a vortex.

I don't think it's a theorem in PA, it's a theorem about PA. PA + some additional axiom could make cons(PA) a theorem, but that wouldn't be a theorem in raw PA. — fdrake

Yes, the incompleteness theorem, when about PA in particular, is a meta-theorem about PA.

If PA is consistent, then:

PA |/- G

PA |/- ~G

PA |/- Con(PA)

But, if I'm not mistaken, there are certain statements that PA can prove about itself.

@Tarskian claims ('Inc' here for 'is incomplete'):

PA |- ~Con(PA) v Inc(PA)

equivalently:

PA |- Con(PA) -> Inc(PA)

equivalently:

PA + Con(PA) |- Inc(PA)

I tend to think that that is correct, though I'm not sure.

If at some point I have time enough, I'd like to refresh my knowledge to organize notes on what are some things PA does prove about itself.

Gödel's incompleteness theorem proves that PA is inconsistent or incomplete. — Tarskian

Of course, that is just sentential logic from:

If PA is consistent then PA is incomplete.

That is a perfectly legitimate theorem in PA. — Tarskian

I'm not sure about that; I'd have to think about it.

It does not prove that PA is incomplete. — Tarskian

Correct.

That is a theorem in PA + Cons(PA). — Tarskian

If Con(PA) then PA is incomplete. That is: If PA is consistent then PA is incomplete.

the Gödelian statements that cannot be expressed by language. There are uncountably many of those. — Tarskian

In a given countable language, there are only countably many sentences.

But there are uncountably many languages and systems, so it's trivial that there are uncountably many true but unprovable sentences.

But there are uncountably many languages and systems, so it's trivial that there are uncountably many true and provable sentences across the class of all languages.

I did not say that your remark would be wrong — Tarskian

Here's what you wrote:

it is not possible to preclude the second disjunct. — Tarskian