I guess you mean there are interpretations where the sentences are uttered in a context where they could be true. — frank

No, that is not what I mean.

In a post, I spelled out in detail what an interpretation is.

It is not valid since there are interpretations in which the premises are true but the conclusion is false.

I affirm that it is valid by any of these considerations:

(1) Apply the definition of 'valid argument'.
— TonesInDeepFreeze

And that is the option we are talking about, nitpicker.
— Leontiskos

Three options have been given: modus ponens, explosion, and the definition of validity. TonesInDeepFreeze's is the latter, — Leontiskos

That gives the impression that I opt for the latter more than the others. But that is not the case:

I started in the thread by pointing out that the argument is modus ponens. Then I was challenged about that and more about the question of validity came up. Then I adduced the definition of validity and showed that the argument is valid. [EDIT: That's not correct. I started in the thread by both an appeal to the definition of validity and that the argument is an instance of modus ponens, and the fact that the premises are inconsistent does not disqualify the argument from being valid. In any case, whatever approach, yes, finally to show the validity of an argument boils down to showing that the definiens of the definition holds for the argument. But my point stands that it's not like I just chose one of the three options, as in subsequent post I especially stressed modus ponens.]

It is not "nitpicking" that I now mention that, you putting words in my mouth, distorting, confused and clueless about basic formal logic, side stepping, intellectually dishonest, would be conversation controlling, tendentious distraction.

From the post you sidestepped:

Your interpretation is mistaken because validity is an inferential relationship between premises and conclusion. You would establish an inferential relationship without examining the inferential structure and relations. To say, "The premises are contradictory, therefore an inferential relationship between premises and conclusion holds," is to establish an inferential relationship without recourse to inferential relations.
— Leontiskos — Leontiskos

I addressed the matter of a 'relation'. You sidestepped that. And a bunch more that you sidestepped. Including that my definition is virtually the same as Mates and equivalent with the others.

The sources I cited include a notion of "follows from," which obviously excludes Tones' approach of relying on the degenerative case of the material conditional. When A is false (A→B) is true, but B does not follow from A. — Leontiskos

Wrong. As been explained to tedium. And you are seriously confused in distorting me:

When A is false, A -> B is true.

And B follows from {A -> B, A}.

I have not at all taken that to provide that B follows from A.

You're ridiculous.

As Enderton notes, validity is about deducibility. — Leontiskos

He said that validity and deducibility turn out to be equivalent. We easily may define (and many or most authors do) 'valid formula' and 'valid argument' without need to mention deducibility. Indeed, Enderton defines 'valid formula' (if I recall, he doesn't define 'argument' in that book) without mention of deducibility. It is only later that he remarks that validity turns out to be equivalent with deducibility.

I refer only to sentences here, not formulas in general, to keep it simple:

Df. A sentence is valid if and only if it is true per all interpretations and assignments for the variables.

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.

Df. A set of sentences G entails a sentence P if and only if there is no interpretation in which all the members of G are true and P is false.

None of those mention derivation (proof, deducibility).

Df. A sentence P is derivable from a set of sentences G, per a set of axioms and a set of inference rules, if and only if there is a sequence of sentences such that each entry is either an axiom or is inferred from previous entries by an inference rule.

Then the key theorems:

Th. If a sentence is derivable from logical axioms alone, then it is valid. (soundness). Equivalently: If a sentence P is derivable from a set of sentences G, then G entails P. (soundness)

Th. If a sentence is valid, then it is derivable from logical axioms alone. (completeness) Equivalently: If a sentence P is entailed from a set of sentences G, then P is derivable from G. (soundness)

So:

Th. A sentence is derivable from logical axioms alone if and only if it is valid. (soundness and completeness). Equivalently: Th. A sentence P is derivable from a set of sentences G if and only if G entails P. (soundness and completeness)

That is what Enderton is referring to.

It is not merely about truth values. — Leontiskos

You, totally cluelessly misconstrue a central matter in logic.

Validity is semantic. It is usually defined with regard to truth values and interpretations. Such definitions do not mention deducibility. In sentential logic, truth tables represent the determination of validity.

Again, what Enderton refers to is the fact that validity and deducibility turn out to be equivalent. But still the definition itself of validity does not require mention of deducibility.

[EDIT: Leontiskos displays typical rank sophistry. He has never read Enderton's book, let alone studied it and understood it. He just cavalierly, unthinkingly picked a quote from it out of context to support his false claim. If he had actually read Enderton, he would see that Enderton's definition does NOT mention deducibility, indeed it is entirely semantic, and that Enderson's point is that it "turns out" that validity and deducibility are equivalent. Enderton didn't say that validity is "about" deducibility. Just as Leontiskos puts words in my mouth, he puts words in Enderton's mouth. Moreover, what Enderton mentioned is just a well known and central proven fact. Anyone familiar with the basics of this subject knows that validity is semantical, deducibility is syntactical, and they have separate definitions, but we prove an equivalence.

Leontiskos also says:

As Enderton notes, validity is about deducibility. It is not merely about truth values. It is about the inferential relationship between premises and conclusion. In order to show that Q follows from P, we have to show how Q is correctly inferred from P, and we need to have evidence that ~Q cannot also be inferred from P. — Leontiskos

That is not what Enderton wrote and not implied by anything Enderton wrote.

(1) Above I addressed the misrepresentation that Enderton wrote that validity is about deducibility.

(2) In order to show that Q is entailed by a set of sentences G (Enderton's terminology is 'logical consequence' rather than 'entailed') it is NOT required to show an inference, especially not an "inferential relationship" (whatever that would mean other than that there is a correct inference) and especially not a requirement to show that ~Q cannot be inferred from G. Rather, it suffices to show that there is no interpretation in which all the members of G are true and Q is false.

It is true that if we show that there is a deduction from G to Q, then Q is entailed by G (that is the soundness theorem). But it is not required that we use that method. We still may use the semantical consideration alone: showing there is no interpretation in which all the members of G are true and Q is false.

And it is true that if G proves Q and G is consistent, then G does not prove ~Q. But it is not true, contrary to Leontiskos's ignorance and tendentious mangling, that, to show that G entails Q, we are required to show that G does not prove ~Q.

Leontiskos is so often in really bad faith when talking about logic. It's fine that he has a different notion of logic, and fine even to critique what he doesn't like, but it is bad faith and destructive to reasoned dialogue that he misrepresents what he critiques and blatantly misrepresents other posters too and then blatantly misrepresents a cite he pulled blindly without reading and understanding the basics of logic to which the cite pertains.]

A key contention of mine is that I am representing the notion of validity in formal logic better than Tones is. — Leontiskos

Hilarious!

I don't need to read a whole article that I've read before. I'm just wondering whether you'd give a particular example.

1. A → ¬A
2. A ⊢ ¬A
3. A ⊨ ¬A
4. A ∴ ¬A — Michael

Good list.

(1) P -> Q ... is a sentence that is interpreted as true if and only if either P is interpreted as false or Q is interpreted as true.

Indeed, sometimes '->' is not primitive but is defined:

Df. P -> Q stands for ~P v Q
or
Df. (P -> Q) <-> (~P v Q)

(2) If G is a finite set of formulas and P is a formula:

G |- P means there is a proof of P from G

(3) If G is a set of formulas and P is a formula

G |= P means P is entailed by G. (where entailment is semantic)

(4) G ∴ P might be the same as (2) or the same as (3) depending on the author.

we don't need an additional implication operator ― that is, one that might appear in a premise, say, and another for when we make an inference. — Srap Tasmaner

No, I do distinguish between what is object-language and what is meta-language.

'->' is in the object language. Or if the object-language is English, then 'if then' is in the object language. And in ordinary formal logic, that is the material conditional.

In the meta-language, we also use 'if then'. I put it here in italics to distinguish from 'if then' in the object-language. But still, also the meta-language 'if then' is the material conditional.

In ordinary formal logic, writers don't stop regarding 'if then' as the material conditional (if then) just because it occurs in the meta-language.

For example:

(2) If the sentence "P -> Q" is false, then P is true and Q is false.

'->'is in the object-language, and 'if then' is in the meta-language. But both of them are the material conditional.

(2) If the sentence "If John went to the store, then John got bread" is false, then "John went to the store" is true and "John got bread" is false.

The outer 'if then' is in the meta-language' and the inner 'if then' is in the object language. But both of them are the material conditional.

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. — Leontiskos

That is equivalent with my argument. But my argument did not mention consistency.

There is a conceptual reason for that. Though, it is not incorrect to mix semantical and syntactical considerations, I prefer the clarity of keeping it to only one of them in this definition. (And of course, there are other times when we do want to use both semantical and syntactical considerations, especially as they relate to one another.)

Tones is talking about assignment or inconsistency, not necessary falseness. — Leontiskos

Again, I used the notions of interpretation (which involves truth and falsehood), not the notion of inconsistency.

This is what I say is the common interpretation of your sources on validity:

1. Assume all the premises are true
2. See if it is inferentially possible to make the conclusion false, given the true premises
3. If it is not possible, then the argument is valid — Leontiskos

Perhaps I've overlooked, but I don't recall any of my cites saying "assume", "see" and "make" - verbs.

You've reinterpreted the definition for your tendentious purpose.

Your interpretation — Leontiskos

My interpretation is literal in an example such as Mates, and virtually literal in certain others, and equivalent with the rest of them.

Your tendentious interpretation is quite a departure as it imposes a routine to be carried out, described with a series of verbs.

The cited definitions don't mention routines to be carried out.

Put my wording next to the cites. Put your interpretation next to the cites. See that yours is nowhere near as close as mine.

Your interpretation changes the ordering of the conjunction and condition — Leontiskos

(1) No, it doesn't. (2) Even if it did, it would be okay as long as the definition were equivalent.

You want to say that if we cannot assume that all the premises are true (on pain of contradiction), then the argument is valid by default. — Leontiskos

I didn't say anything about anybody assuming anything.

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? — frank

Yes.

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? — frank

Wrong. (Even considering the difficulty with the definite description 'the present King of France'.)

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

I don't know what you mean. Example?

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. — frank

Correct that I didn't mention inconsistency.

But "never both true" implies inconsistency.

It is a theorem: If as set of sentences is not satisfiable then it is inconsistent.

The reason that there is no interpretation where both premises are true is because the premises are inconsistent — Michael

That is one way of looking at it. But we don't need to refer to inconsistency (which is syntactical) as we can also just note that semantically, there is no interpretation in which both premises are true.

Either is okay, but I note that in fact, I kept it all semantical.

They cannot interpret real English — Leontiskos

What a stupid thing to say.

The original argument was symbolic. Of course, that could be taken as symbols meant to stand for natural language sentences. But in any case I made clear that my explanation is per ordinary formal logic and that other natural language contexts may differ.

Validity is a relationship between premises and conclusion. — Leontiskos

It is a relation. It is the relation whose members are all and only those arguments that are such that there is no interpretation in which all the premises are true and the conclusion is false.

Three options have been given: modus ponens, explosion, and the definition of validity. TonesInDeepFreeze's is the latter... — Leontiskos

You didn't even read among some of the first posts I made in this thread about modus ponens, and I went on about it. I was the one who pointed out that it is an instance of modus ponens; and there was even an extended discussion about that, as at least one poster disputes that it is an instance of modus ponens. And I even mentioned that it is modus pones just a few posts before yours. For Pete's sake!

I affirm that it is valid by any of these considerations:

(1) Apply the definition of 'valid argument'.

(2) See that it is an instance of modus ponens and note that modus ponens is a valid argument form.

(3) See that the set of premises is not satisfiable, so, by explosion, the argument is valid.

(4) Prove the conclusion from the premises and note that the soundness theorem: "If a sentence is provable from a set of sentences, then the sentence is entailed by the set of sentences."

so you're talking about the principle of explosion? — Michael

Explosion is related, but I didn't mention it or need to mention it for the purpose at hand.

There are both semantical and syntactical versions of principles. These are definitions I use. Different authors have variations among them, but they are basically equivalent, except certain authors use 'valid' to mean 'true in a given interpretation', which is an outlier usage. I mention only sentences here for purpose of sentential logic; for predicate logic we have to also consider formulas in general and some of the definitions are a bit more involved.


Semantical:

Valid sentence: A sentence is valid if and only if it is true in all interpretations. A sentence is invalid if and only if it is not valid.

Logically false sentence: A sentence is logically false if and only if it is false in all interpretations.

Contingent sentence: A sentence is contingent if and only if it is neither a validity nor a logical falsehood.

Satisfiable: A set of sentences is satisfiable if and only if there is an interpretation in which all the members are true.

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

Entailment: A set of sentences G entails a sentence P if and only if there is no interpretation in which all the members of G are true and P is false.

Sound argument (per an interpretation): An argument is sound (per an interpretation) if and only if it is valid and all the premises are true (per the interpretation). Note: When a certain interpretation is fixed in a certain context, we can drop 'per an interpretation' in that context. For example, if the interpretation is the standard interpretation of arithmetic. For example, informally, when the interpretation is a general agreement about common facts (such as that Kansas is a U.S state).

Explosion: For a set of sentences G, if there is no interpretation in which all the members of G are true, then G entails every sentence.


Syntactical:

Proof: A proof from a set of axioms per a set of inference rules is a finite sequence of sentences such that every entry is either an axiom or comes from previous entries by application of an inference rule. (And there are other equivalent ways to formulate the notion of proof, including natural deduction, but this definition keeps it simple.)

Theorem from a set of axioms: A sentence is a theorem from a set of axioms if and only if there is a proof of the sentence from the axioms.

Contradiction: A sentence is a contradiction if and only if it is the conjunction of a sentence and its negation. (Sometimes we also say that a sentence is a contradiction when it proves a contradiction even if it is not itself a conjunction of a sentence and its negation.)

Inconsistent: A set of sentences is inconsistent if and only if it proves a contradiction. (Sometimes we say the set of sentences is contradictory)

Explosion as a sentence schema: For any sentences P and Q, (P & ~P) -> Q.

Explosion as an inference rule: For any sentences P and Q. From P & ~P infer Q.

/

So explosion and "any argument with an inconsistent set of premises is valid" are similar.

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 — Hanover

What in the world? There is no interpretation in which both a statement and its negation are true.

I mentioned it lately only because the matter was raised. It is taken for granted that in such contexts, the material conditional is used. But since the matter was raised, I responded.

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

English as a meta-language regarding formal logic. In that meta-language, 'if then' is taken in the sense of the material conditional.

Indeed, we could even formalize the meta-language, and the formal conditionals would be the material conditional.

In ordinary contexts, including a natural language meta-language, ordinarily, when logicians (since at least the advent of 20th century logic) use 'if then', they use it as the material conditional.

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.

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.

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

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

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

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.)

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.

Thanks.

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.

It was an interesting idea, though.

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?

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.

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

And also an example of modus ponens.

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.

Thanks for that post. It is helpful and I need to look more into the subject.

The idea that it is a relationship already excludes your reading. If a relationship between A and B must be established, then one must know something about both A and B. — Leontiskos

The validity relation is:

{<X Y> | X is a set of sentences & Y is a sentence & there is no interpretation in which all the members of X are true and Y is false}

So, easily we have that an argument is valid if and only if it is a member of that relation.

The validity relation is a relation in the ordinary formal sense of a set of ordered pairs. That is distinct from any of the ordered pairs themself.

It's called paraphrase — Leontiskos

In this instance, the use of quote marks made it look like a quote, and not just a paraphrase.

And even as a paraphrase, it would be incorrect.

These are not conclusive in favor of your reading, and you would need to quote the context around these sentences given the way you have shown yourself willing to ignore context. — Leontiskos

Oh, please. (1) I do not ignore context. (2) They are simple definitions. (3) How much context would I have to type for you? (4) The cites you gave are equivalent with my wording and the wording of the two recent cites I gave.

A reasonable person would see those quotes and say, "Okay, I do see that your definition is used too and that yours and the others are equivalent." Instead, you can't stand to concede even the simplest point.

"it is not possible for the premises to be true and the conclusion false" is not uncontroversially fulfilled by a set of inconsistent premises. — Leontiskos

It's not at all controversial in ordinary formal logic. It is easy to see that if a set of sentences is inconsistent then there is no interpretation in which all the members are true.

And look at your own cites:

"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. In calling an argument valid, we aren’t saying whether the premises are true. We’re just saying that the conclusion follows from the premises – that if the premises were all true, then the conclusion also would have to be true." [bold added]

"In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion." [bold added]

It's remarkable that you can't stand to be wrong - to the degree that you don't heed even your own cites!

/

"it is impossible that all the premises are true and the conclusion is false"

"it would be contradictory (impossible) to have the premises all true and conclusion false"

"there are no interpretations in which all the sentences in Gamma are true and Phi is false"

"impossible for there to be a situation in which all the sentences in the first set are true
and the other sentence false."

"not possible for the premises to be true and the conclusion false"

"there is no interpretation in which the premises are all true and the conclusion is false"


Those are all ways of saying the same thing.

But you can't recognize that which is blazingly clear.

Another one:

"a major topic in the study of deductive logic is validity. This is a
relationship between a set of sentences and another sentence; this relationship holds whenever it
is logically impossible for there to be a situation in which all the sentences in the first set are true
and the other sentence false." [bold added]

https://logiclx.humnet.ucla.edu/Logic/Documents/CORE/LogicText%20Chap%200%20Aug%202013.pdf

From that same source:

"An "argument", in its technical sense, consists of two parts: a set of sentences, called the premises, and a sentence called the conclusion. The term "argument" may suggest a dispute, but in logic something is called an argument whether or not any people ever have or ever will disagree about it. Likewise, the "premises" of such an argument may or may not have been believed or asserted by somebody, and it is sometimes useful to examine arguments whose "premises" would never be believed by any rational person. Likewise, by calling something a "conclusion" we do not suggest that anyone ever has or even should "conclude" this thing on the basis of the premises given."

I ignored nothing. The bolded part is another way of saying the unbolded part:

"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. In calling an argument valid, we aren’t saying whether the premises are true. We’re just saying that the conclusion follows from the premises – that if the premises were all true, then the conclusion also would have to be true."

So, flip the bolding:

"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false. In calling an argument valid, we aren’t saying whether the premises are true. We’re just saying that the conclusion follows from the premises – that if the premises were all true, then the conclusion also would have to be true."

They are two ways of saying the same thing. And:

Gensler:

"An argument is valid if it would be contradictory (impossible) to have the premises all true and conclusion false."

It is impossible to have both A -> ~A and A true. Perforce, it is impossible to have the premises all true and the conclusion false. — TonesInDeepFreeze

Which you ignored.

And you ignored these:

"A sentence Phi is a consequence of a set of sentences Gamma if and only if threre are no interpretations in which all the sentences in Gamma are true and Phi is false." (Elementary Logic - Mates)

"An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false." (The Logic Book - Bergmann, Moor and Nelson).

With bolding for your bolding pleasure:

"A sentence Phi is a consequence of a set of sentences Gamma if and only if threre are no interpretations in which all the sentences in Gamma are true and Phi is false." (Elementary Logic - Mates)

"An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false." (The Logic Book - Bergmann, Moor and Nelson).

/

You've mentioned a hypothetical trial. I don't claim that the formal notion is suitable in all situations.

My point has never been that the formal notion should have dominion over all contexts. Rather, it has been shown that the original argument is valid per ordinary formal logic.

I did not claim that validity requires that there is no interpretation in which the premises are all true.
— TonesInDeepFreeze

And I never said you did (you are falling into the fallacy of affirming the consequent). — Leontiskos

I affirmed no consequent. To be clear, I said I did not say the thing, whether you said that I said it or not.

Meanwhile, I did not say that the inference is "trivial" though you've twice now claimed I did.

Your claim is, "Whenever the premises are inconsistent, the argument is valid." — Leontiskos

That is the second time you put quotes around words I didn't say. The fifth time in this thread you've put words in my mouth.

"If the premises are inconsistent, then the argument is valid" is equivalent with my wording of the definition, but for the purpose of the definition, I don't mix consistency (syntactical) with satisfiabilty (semantical). So you are incorrect when you put those words in quotes and ascribe them to me.

So then any argument that has no true premises is valid. That's weird. — frank

No, that's not correct.

If there is no assignment in which all the premises are true, then the argument is valid.

That is very different from what you mentioned.

"An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false." (The Logic Book - Bergmann, Moor and Nelson).

"A sentence Phi is a consequence of a set of sentences Gamma if and only if threre are no interpretations in which all the sentences in Gamma are true and Phi is false." (Elementary Logic - Mates)

you are turning the consequence relation into a material conditional, and claiming that inconsistent premises trivially show an argument to be valid in the same way that the false antecedent of a material conditional trivially shows the conditional to be true. — Leontiskos

(1) I did not say it is "trivial". That was another poster. I already pointed out to you that I did not say it is "trivial". So this is the fourth time in this thread that you put words in my mouth.

(2) Of course, we can state two equivalent ways:

there is no interpretation in which all the premises are true and the conclusion is false

every interpretation in which all the premises are true is an interpretation in which the conclusion is true.

And, yes, the equivalence is per the material conditional.

As in this quote:

"an argument is valid if and only if it is necessary that if all of the premises are true, then the conclusion is true; if all the premises are true, then the conclusion must be true; it is impossible that all the premises are true and the conclusion is false." [bold added]

Ordinary formal logic adopts the material conditional, not just in the object theory but in the meta-theory too.

The question is whether we should read Gensler as presupposing that the premises are consistent. — Leontiskos

There is no question. He does not presuppose it.

A consequence relation is not established by your, "The premises are inconsistent..." — Leontiskos

(1) The consequence relation is this:

{<X Y> | X is a set of sentences & Y is a sentence & there is no interpretation in which all the members of X are true and Y is false.}

(2) "The premises are inconsistent" is not what I wrote.

(3) I did not claim that validity requires that there is no interpretation in which the premises are all true. Rather, I applied the definition of validity to the case in which there is no interpretation in which all the premises are all true.

I'll spell it out for you again:

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

(5) Now, consider this simple thing:

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

(6) So, if there is no interpretation in which all the premises are true, then the argument is valid.

(7) There is no interpretation in which both A -> ~A and A are true.

(8) Therefore, the argument is valid.