But in this case, they're the same variable. They're both A. — frank

No, P is A. Q is ¬A.

No, P is A. Q is ¬A. — Michael

Ok. I see. But then, what about the second premise? If A is false, wouldn't the second premise actually be not-A?

Ok. I see. But then, what about the second premise? If A is false, wouldn't the second premise actually be not-A? — frank

I'll rephrase it into English for you.

1. If Socrates is mortal then Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal

Given that P → Q ↔ ¬P ∨ Q, this can be rephrased as:

1. Socrates is not mortal or Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal

This can be simplified to:

1. Socrates is not mortal
2. Socrates is mortal
3. Therefore, Socrates is not mortal

The argument is valid (as per the principle of explosion) but is unsound because (1) and (2) cannot both be true.

If we exclude necessarily false premises can we still demonstrate explosion? Or does keeping contradictory premises out of valid arguments remove explosion?

I see. I don't think that's what Tones was saying though. He was saying that since there are no cases where both premises are true, the argument is valid.

Yes, the argument is valid as I said. But it isn't sound because one of the premises is false.

Yes, the argument is valid as I said. — Michael

You're giving a different reason for why it's valid versus Tones.

I can't see that we are.

We both agree that the argument is valid because the conclusion deductively follows from the premises, i.e. that if the premises are both true then the conclusion is true.

I can't see that we are. — Michael

You are. He's just using the definition of validity:

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

There is no interpretation in which all the premises are true. Therefore, the argument is valid.

if the premises are both true then the conclusion is true

And as previously mentioned, P → Q ↔ ¬P ∨ Q. So the above can be rephrased as:

a. One of the premises is false or the conclusion is true.

And (a) is true because one of the premises is false.

There is no interpretation in which all the premises are true. Therefore, the argument is valid. — frank

That's not what he's saying.

That's not what he's saying. — Michael

All he had to do is say that there aren't any cases where both premises are true, therefore it's valid.
— frank

I said it over and over and over for you.

All you had to do is read the replies given you. And that's hardly the only point I explained for you. — TonesInDeepFreeze

It hinges on the definition of validity. It's weird, but according to Tones, that's how it works.

He's not saying what you think he's saying. These are two different claims:

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

You are claiming that he is asserting (1), when in fact he is asserting (2), as am I.

You are claiming that he is asserting (1), when in fact he is asserting (2), as am I. — Michael

Notice that 1 and 2 are saying the same thing: The argument is valid if there is no interpretation in which

All the premises are true AND the conclusion is false.

There aren't any interpretations where all the premises are true. So it's valid.

There aren't any interpretations where all the premises are true. So it's valid. — frank

That's not what he's saying. I don't know how to explain this to you in an even simpler way.

That's not what he's saying. I don't know how to explain this to you in an even simpler way. — Michael

You may be right. Let's double check with him. @TonesInDeepFreeze

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

It depends on the length to which we "interpret" an argument and how you interpret "interpret."

P1. P->~P
P2. P
Conclusion: ~P

can be interpreted as:

1. ~P (P1, which is equivalent to ~P v ~P)
2. P (P2)
3. ~P v P (1, 2, this is correct as either v or &)
4. P - > P (3)
Conclusion: P (2,4 )

These two arguments are interpretations of each other because they maintain truth throughout based upon the premises provided.

Interpreting the same argument, we arrive at contradictory conclusions, which violates the definition of "valid."

This is the explosion issue. Everything follows from a contradiction. The question of validity versus soundness doesn't typically contemplate the contradiction, but it instead contemplates synthetic falsity of contingent premises yet valid logical structure (e.g. All cats can fly, I have a cat, my cat flies, valid but unsound because cats don't fly versus If all cats can fly then all cats can't fly, I have a cat, my cat can't fly.).

I'll put this to rest if someone can find an article outside our blabbing that actually considers the issue of the "validity" of the OP.

I don't quite understand what you're trying to say here. I'm just explaining very basic terminology.

If the conclusion follows from the premises then the argument is valid. If the argument is valid and the premises are true then the argument is sound.

The argument in the OP is valid because the conclusion follows from the premises, but it's unsound because one of its premises is false.

Question begging happens a lot. But, again, I can't think of an instance in public discourse.... As to complaints about formal logic, — TonesInDeepFreeze

Small point. Public discourse often (usually/always?) uses rhetorical logic - Rhetoric. Not ever to be confused with "formal" logic. Not because one better than the other, but because they're different tools for different purposes. And confusion can happen because in some ways they overlap.

..

I don't quite understand what you're trying to say here. — Michael

I'm saying that if you can interpret the same argument and obtain contradictory conclusions, then the argument is not "valid" under this definition of "valid":

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

If we interpret it under my first iteration, we receive the conclusion ~P.
If we interpret under my second iteration, we receive the conclusion P.

We therefore have an "interpretation" in which all the premises of #1 are true and the conclusion is shown to be false via interpretation #2.

the reductio shows that the first premise is unsound but why is it unsound? It's unsound because it's logically contradictory. If A then not-A necessarily implies A and not-A, which tells me the argument must be invalid. — Benkei

What reductio?

A premise is not sound or unsound. An argument is sound or unsound.

Df. An argument U is sound if and only if U is valid and all the premises of U are true.

But, to be more rigorous, note that 'true' is relative to an interpretation. So, to be more rigorous:

Df. An argument U is sound per an interpretation M if and only if U is valid and all the premises of U are true per M.

Note that if the set of premises is inconsistent, then there is no interpretation in which all the premises are true, so the argument is unsound per every interpretation.

/

It is not correct that A -> ~A implies both A and ~A.

Rather, (A -> ~A) along with A implies both A and ~A.

So the set of premises is inconsistent. So there is no interpretation in which all the premises are true. EDIT: That is going along with you mentioning inconsistency, though my earlier arguments have not mentioned consistency.

/

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

Note that if there is no interpretation in which all the premises are true, then there is no interpretation in which all the premises are true and the conclusion is false.

A -> ~A
A
therefore ~A

There is no interpretation in which all the premises are true. So the argument is valid and unsound.

I really don't understand what you're trying to say. Have a look at this.

The following argument is valid:

It is raining
It is not raining
George Washington is made of rakes

And the following argument is valid:

It is raining
It is not raining
George Washington is not made of rakes

And the following argument is valid:

It is raining
It is not raining
It is raining

And the following argument is valid:

It is raining
It is not raining
It is not raining

As the article says, "this arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical because although the above is a logically valid argument, it is not sound (not all of its premises are true)."

This is just what the word "valid" means. I think you think it means something else.

The argument is an example of the principle of explosion — Michael

And also an example of modus ponens.

An example where we might want to argue that both premises are true might be instructive.

Suppose A = "This sentence is false."

We might suppose that "A → ~A." And, because, if the sentence is false, it will thereby be true, we might also want to assert "A" as a second premise.

The argument preserves the truth of our two (assumedly) true premises. If A is true it implies that it is also false, but if A is false it is true. Our problem here is a "truth glut," we have too much truth for A on account of both A and its negation being (arguably) true.

The inferences in OP's argument are right in line with the idea that if A is true then it is also false. And so the argument is valid in that, if we want to maintain the truth of both premises, the conclusion will follow.

However, it seems possible to construct an argument where you have a necessarily false premise that is nonetheless invalid.

For example:
All cats are mammals.
Samuel Clemens is not Mark Twain (necessarily false because they are the same person)
Therefore, Mark Twain is a mammal.

We have no middle term (and none implied, i.e, this is not an enthymeme) in something that looks like a syllogism. My first thought is that we could also probably construct an undistributed middle fallacy with a necessarily false premise.

The inferences in OP's argument are right in line with the idea that if A is true then it is also false. — Count Timothy von Icarus

This is the misunderstanding.

A → ¬A does not mean "if A is true then A is also false".

As I said above, these mean two different things:

1. A → ¬A
2. A → (A ∧ ¬A)

"if ... then ..." in propositional logic does not mean what it means in English.

tautologies might be semantic, e.g. "bachelor's are unmarried men," or "triangles are three-sided." — Count Timothy von Icarus

I don't mean that sense of 'tautology'. I mean the sense: a tautology is a sentence that is true on every row of the truth table. (If we are confined to just propositional logic, then that is equivalent with: a tautology is a sentence that is true in every interpretation, i.e. a tautology is a valid sentence).

(p→q) ⇔ (~q→~p) — Count Timothy von Icarus

Of course, (P -> Q) <-> (~Q -> ~P) is a tautology.

Of course, (A -> ~A) <-> (~A v A) is a tautology.

At least in propositional logic, my understanding is that tautologies are defined in terms of form — Count Timothy von Icarus

That is a common notion and quite fine. But there is a different, though compatible, notion: a tautology is a sentence that is true on every row of the truth table.

You are correct. I was speaking to our intuition about: "This sentence is false." If it is true it is false, yet we say also because it is apparent that if it is false it is also true.

A → ¬A does not mean A ∧ ¬A. It means ¬A ∨ ¬A
— Michael

How do you figure that? — frank

A choice of three ways to figure it:

(1) Prove it in the sentential calculus.

(2) Show it as an instance of an already proven theorem schema (as @Michael did).

(3) Show it on a truth table.

Would you please at least learn the most basic thing, which is to write truth tables?

It was an interesting idea, though.