Ah, gotcha. That makes sense.

if the A in the antecedent is false, the A in the consequent should be false too — frank

In an interpretation, a sentence is either true or false and not both, and has the same truth value no matter where it occurs in the formulas.

With an interpretation in which A is true:

A is true
~A is false
A -> ~A is false

With an interpretation in which A is false:

A is false
~A is true
A -> ~A is true

/

This is so basic that reading even just an easy Internet article on truth tables would allow you to understand sentential formulas in general.

Thanks.

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

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

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

Per our definition, this argument is not valid becasue all the premises are true and that conclusion is false because you also indicated:

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

I fully understand that the conclusion is also true, so there's that, but that's the nonsense of contradictions. That is, these arguments both meet and do not meet the definition of "valid."

this argument is not valid becasue all the premises are true and that conclusion is false.. — Hanover

They're not all true. One of them is false. Either it is raining or it is not raining.

But if it were the case that both "it is raining" and "it is not raining" were true then it would be the case that "George Washington is made of rakes" is true (and that "George Washington is not made of rakes" is true).

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

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

It is what I'm saying.

The above is not the definition of 'valid argument' but it is a consequence of the definition.

(1) Two equivalent definitions:

(1a) Df. An argument is valid if and only if every interpretation in which all of the premises are true is an interpretation in which the conclusion is true.

(1b) Df. An argument is valid if and only if there is no interpretation in which all of the premises are true and the conclusion is false.

Therefore:

(2) Th. If there is no interpretation in which all of the premises are true, then the argument is valid.

Right, so you're talking about the principle of explosion?

Given that frank and I were talking about the definition of "valid", I (mis)understood him as claiming that you were saying "an argument is valid if and only if there is no interpretation in which all the premises are true".

He claimed that you and I were giving different reasons for why the argument in the OP is valid.

You say that because you're not linking your first argument to your second. That is, I consider Argument 1 to be "an interpretation" of Argument 2, not as two seperate arguments. This is one argument with 2 conclusions, both Q and ~Q. The premises must be true because they are taken as givens. Given P1 and P2, both Q and not Q are implied. The conclusion can be shown to be false by analysis of the same premises.

Given that frank and I were talking about the definition of "valid", I (mis)understood him as claiming that you were saying "an argument is valid if and only if there is no interpretation in which all the premises are true". — Michael

I didn't say "if and only if." I just said that since there are no cases where both premises are true, the argument is valid.

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. — Michael

They are different, but (1) follows from (2).

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

Th. If there is no interpretation in which all the premises are true, then the argument is valid. (Proof: see Df.)

You say that because you're not linking your first argument to your second. — Hanover

Why would I? Every argument is its own thing. If the conclusion deductively follows from the premises then the argument is valid.

The fact that two contradictory premises entail two contradictory conclusions does not mean that neither argument is valid. It says it right there in the Wikipedia article:

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.

I agree, but this was the specific exchange:

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. — frank

What he says certainly follows from what you said, but it isn't what you (literally) said (at least not in the quote he posted), and isn't the definition of validity.

You and I don't have different definitions of validity.

That's what I was trying to clarify.

That's what I was trying to clarify. — Michael

Ok. What I was trying clarify is that he's not talking about explosion. It's simply that if there is no interpretation in which all the premises are true, the argument is valid.

What I was trying clarify is that he's not talking about explosion. It's simply that if there is no interpretation in which all the premises are true, the argument is valid. — frank

That is explosion.

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

We interpret by assigning a truth value to each sentence letter. In sentential logic, that's all there is to it.

Each row of a truth table represents an interpretation and a determination of the sentence based on that interpretation. For example:

Suppose there are two sentence letters, P and Q. Then there are four interpreataion. (In general, if there are n number of sentence letters, then there are 2^n interpretations.)

interpretation 1: P is true and Q is true
interpretation 2: P is true and Q is false
interpretation 3: P is false and Q is true
interpretation 4: P is false and Q is false

Each row of the last four rows here represents one of the four interpretations with two sentence letters:

P Q
T T
T F
F T
F F

Now determine the truth value of a sentence (such as P -> Q) per each interpretation:

P Q ... P->Q
T T .......T
T F .......F
F T .......T
F F .......T

That is explosion. — Michael

Explosion is that any proposition can be proven from a contradiction. What Tones is explaining is that if you have an argument in which there is never a case where both premises are true, the argument is valid.

Explosion is that any proposition can be proven from a contradiction. What Tones is explaining is that if you have an argument in which there is never a case where both premises are true, the argument is valid. — frank

That's the same thing.

That's the same thing. — Michael

If you have an argument in which there is an interpretation where both premises are false, but there are no cases where both premises are true, then the argument is valid. That wouldn't be a case of explosion.

If you have an argument in which there is an interpretation where both premises are false, but there are no cases where both premises are true, then the argument is valid. That wouldn't be a case of explosion. — frank

The reason that there is no interpretation where both premises are true is because the premises are inconsistent, i.e. that their conjunction is a contradiction. As such the argument is valid whatever the conclusion (i.e. anything follows).

The reason that there is no interpretation where both premises are true is because the premises are inconsistent, i.e. that their conjunction is a contradiction. As such, any conclusion follows and the argument is valid. — Michael

You may be right. Nevertheless, what Tones is pointing out is that anytime there are no cases where both premises are true, the argument will be valid. The premises don't have to be inconsistent for that. They're just never both true.

The premises don't have to be inconsistent for that. They're just never both true. — frank

If they are consistent then they can both be true. If they can never both be true then they are inconsistent.

Why would I? Every argument is its own thing. If the conclusion deductively follows from the premises then the argument is valid — Michael

Checking the validity of one argument using another is done all the time.

They can never both be true only if they are inconsistent. If they are consistent then they can both be true. — Michael

@TonesInDeepFreeze is this true?

Couldn't it be:

1. The present King of France is bald.
2. The present King of France is wise.

Therefore: Cows bark.

It's valid, right?

Checking the validity of one argument using another is done all the time. — Hanover

Checking the soundness of one argument using another is done all the time.

Here are two arguments:

P1. If my name is Michael then I am 36 years old
P2. My name is Michael
C1. Therefore I am 36 years old

P1. If my name is Michael then I am not 36 years old
P2. My name is Michael
C1. Therefore I am not 36 years old

Both arguments are valid, but only one is sound.

Notice that 1 and 2 are saying the same thing — frank

No, they are not. But (1) is a consequence of (2).

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. — tim wood

They are different but related.

Anyway, yes, my point was about question begging in everyday polemical discourse. But I also contrasted it with the situation in formal logic in which axioms are also theorems.

- In short, it removes it. See:

Any argument with inconsistent premises is valid, according to Tones. Weird indeed. It requires a strained reading of the fine print of portions of definitions of validity, taken out of context. Earlier posters usefully leveraged the word "sophistry."

(Note that this is different from the modus ponens reading of the OP and it is different from the explosion reading of the OP. The effect of explosion requires explicit argumentation. The OP, for example, is susceptible to explosion, but it is not wielding explosion. Tones is just doing a weird, tendentious, definitional thing.) — Leontiskos

Lots of people are not paying attention to the differentiation of arguments for why the OP might be valid. Three options have been given: modus ponens, explosion, and the definition of validity. @TonesInDeepFreeze's is the latter, and it is tendentious but also probably just sophistic. It is very close to this argument:

• That which has a privation of life is dead
• Rocks have a privation of life
• Therefore, rocks are dead

Tones' argument:

• An argument is valid when it is not possible for the conclusion to be false while the premises are true
• An argument with contradictory/inconsistent premises cannot have (all) true premises
• Therefore, an argument with contradictory/inconsistent premises cannot have a false conclusion while the premises are true
• Therefore, an argument with contradictory/inconsistent premises is valid.

This is what Srap usefully called "reliance in argumentation on degenerate cases":

I think, though, we can allow a somewhat negative connotation because reliance in argumentation on degenerate cases is often inadvertent or deceptive. "There are a number of people voting for me for President on Tuesday [and that number happens to be 0]." — Srap Tasmaner

(And I would be willing to explain why this sort of thing deserves a negative connotation even apart from inadvertence or deception.)

What's interesting here is that Tones is literally applying the material conditional as an interpretation of English language conditionals, and he is relying on the degenerate case of the material conditional to try to make a substantive point. He has trapped himself within a truth-functional paradigm, and has convinced himself that his "reliance in argumentation on degenerate cases" is a normative reliance, such that he is, "merely applying the definitions of ordinary formal logic." This is an especially clear case of the deep confusion that results from the excessive formalism of folks like Tones or Banno. They cannot interpret real English; they cannot distinguish absence from privation; they cannot discern rocks from corpses; they cannot recognize that validity involves a relationship between premises and conclusion.

(Cf. , , )

-

Edit:

"Therefore, an argument with contradictory/inconsistent premises cannot have a false conclusion while the premises are true" [Paraphrase of Tones] — Leontiskos

This is a matter of different modal levels, so to speak, or different domains or levels of impossibility. Tones is committing a metabasis eis allo genos. He is committing a category error where the genus of discourse is not being respected. Contingent falsity, necessary falsity, and contradictoriness are three different forms of denial or impossibility. The definition of validity that Tones favors is dealing in the first category, not the second or third. The domain of discourse for such a definition assumes that the premises are consistent. It does not envision itself as including the degenerate case where an argument is made valid by an absurd combination of premises. An "argument" is not made valid by being nonsense.

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": — Hanover

That doesn't make sense and it is not how interpretations and validity work.

An interpretation assigns one and only one truth value to each sentence letter. (If there are n number of sentence letters, then there are 2^n number of interpretations.)

An interpretation then determines the truth value of any formula that uses only those sentence letters.

Then for an argument, per a given interpretation, all of the premises and the conclusion have a determined truth value.

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

A -> ~A
A
therefore ~A

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

An argument is sound per an interpretation if and only if the argument is valid and every premise is true per the interpretation.

A -> ~A
A
therefore A

Ther is no interpretation in which every premise is true. So the argument is unsound per every interpretation.

I mentioned it several posts back, but it seems possible to have an invalid argument with necessarily false premises.

You could construct a syllogism with an illicit negative, exclusive premises, undistributed middle, etc. and an inconsistent premise.

All triangles are not three-sided shapes.
F is not a three sided shape.
Therefore F is a triangle.

Aside from the first premise being necessarily false, this is not a valid syllogism. Even if we assume true premises and a true conclusion we get something like:

All dogs are not reptiles.
Chloe is not a reptile.
Therefore, Chloe is a dog.

these mean two different things:

1. A → ¬A
2. A → (A ∧ ¬A) — Michael

You might want to double-check that.

Tones' is literally applying the material conditional as an interpretation of English language conditionals — Leontiskos

Actually, he isn't. The OP's question was not about ordinary English at all:

1. A -> not-A
2. A
Therefore,
3. not-A.

Is this argument valid? Why or why not? — NotAristotle

I mainly use formal logic for analysing ordinary language arguments, so that's what I've been thinking about, but the original question was not about that.

This shouldn't be about choosing sides.