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.

The OP's question was not about ordinary English at all. — Srap Tasmaner

Tones is interpreting English-language definitions of validity according to the material conditional, not merely the OP. He himself now recognizes this:

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

Edit:

And now explicitly:

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

He thinks the consequence relation of logic (∴) is the material conditional, such that a contradictory set of premises automatically makes an argument valid, irrespective of any explosive argumentation within the argument.

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.

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 several posts back, but it seems possible to have an invalid argument with necessarily false premises. — Count Timothy von Icarus

I agree, but Tones is talking about assignment or inconsistency, not necessary falseness. A (formal-propositional) contradiction is necessarily false, but not everything that is necessarily false is a (formal-propositional) contradiction.

"All triangles are not three-sided shapes," is necessarily false, it is contradictory, but it is not contradictory in the formal-propositional sense. I think this goes somewhat to my edit about levels of modality. Your earlier post about the relevance of matter and form within abstract fields like mathematics also gets at this point. See:

Edit:

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. — Leontiskos

Well, we also have cases like:

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 — TonesInDeepFreeze

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. Yet you think that merely knowing something about A—that it is inconsistent—proves validity. If an isolated fact about A proved validity then validity would not be a relationship between A (premises) and B (conclusion). This is another source that excludes your view. The other (single-sentence) sources you presented favor my view but do not exclude your tendentious view. — Leontiskos

. . .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. — TonesInDeepFreeze

Validity is a relationship between premises and conclusion. 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

Your interpretation changes the ordering of the conjunction and condition, and probably also the nature of the condition. 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. There is no need to look at the inferential structure.

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.

- Can you spell out your point for me? It looks to me like a good example of why a sentence is different from an argument. I don't think it is possible to translate your point into an argument, is it? If I am right, that's in part because the material conditional and the consequence relation do not operate in the same way, particularly when the antecedent contains a conjunction in that way.

This whole thing is an unwieldy topic in general. For example, can premise (1) of the OP be assigned a true value? And can both premises of the OP be assigned a true value? I suspect that the answers to these questions go beyond the purview of standard propositional logic, and creep into the space of Frege's judgment stroke. So it's not even obvious that Tones is right when he says that the premises of the OP cannot both be assigned a true value, although I have no real dog in that fight.

You might want to double-check that. — Srap Tasmaner

Good catch. Trying to translate English into proportional logic is hard.

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.

Tones is interpreting English-language definitions of validity according to the material conditional — Leontiskos

Is this what you mean:

'Validity' is being defined as a concept that applies to arguments which have the form

$P_0 \land P_1 \land ... \land P_n \to C$

when it should be defined for some other relation than →, because → does not properly capture the root intuition of logical consequence, or "... follows from ...", or whatever.

There are a couple issues here, I think.

One is at least somewhat technical, and I hope @TonesInDeepFreeze can figure out what I'm trying to remember. There is a reason 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. In natural deduction systems, if you assume A and then eventually derive B, you may discharge the assumption by writing 'A → B'; this is just the introduction rule for →, and it is exactly the same as the '→' that might appear in a premise.

Thus the form for an argument above is, I believe, exactly the same as writing this:

$\{P_0, P_1, ... P_n\}\vdash C$

That is, we lose nothing by treating an argument as a single material implication, the premises all and-ed together on the LHS and the conclusion on the RHS. (And I could swear there's an important theorem to this effect.)

the material conditional and the consequence relation do not operate in the same way — Leontiskos

Okay, so yeah, this is what you were saying, but in formal logic identifying the consequence relation with material implication is not an assumption or a mistake but a result. I believe. Hoping @TonesInDeepFreeze knows what I'm talking about.

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

Yep. :up:

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... — Leontiskos

There is a reason 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

I mean your post does use two different operators?

In fact there are a few that come to mind:

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

As a specific example:

1. I am a penguin → My name is Michael
2. I am a penguin ⊢ My name is Michael

(1) is true and (2) is false.

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."

- Good post. This is a very broad and pervasive topic that perhaps deserves its own thread someday.

In natural deduction systems, if you assume A and then eventually derive B, you may discharge the assumption by writing 'A → B'; this is just the introduction rule for →, and it is exactly the same as the '→' that might appear in a premise. — Srap Tasmaner

This is a source of the disagreement. I don't disagree that you can "discharge" the consequence in that way, but it avoids the crucial matter of the degenerative case of the material conditional, and this is precisely what Tones wants to rely upon. It seems to me that the only reason people tend to substitute consequence with → is because arguments de facto exclude the degenerative case that Tones wants to re-introduce. An argument is a teleological act that aims at legitimate validity, not degenerative validity. Validity in logic is desirable, not undesirable.

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.

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

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.

I mean your post does use two different operators? — Michael

Yes that's probably necessary, but something I overlooked.

Here's the sort of thing I was trying to remember. It's Gentzen's stuff.

The standard semantics of a judgment in natural deduction is that it asserts that whenever[11] A 1 , A 2 , etc., are all true, B will also be true. The judgments

A 1 , … , A n ⊢ B

and

⊢ ( A 1 ∧ ⋯ ∧ A n ) → B

are equivalent in the strong sense that a proof of either one may be extended to a proof of the other. — wiki

And similarly

The sequents

A 1 , … , A n ⊢ B 1 , … , B k

and

⊢ ( A 1 ∧ ⋯ ∧ A n ) → ( B 1 ∨ ⋯ ∨ B k )

are equivalent in the strong sense that a proof of either sequent may be extended to a proof of the other sequent. — same

What I forgot is that you move the turnstile ⊢ to the left of the whole formula, with an empty LHS.

So the result I was trying to remember was probably just cut-elimination. I never got very far in my study of Gentzen, so the best I can usually do is gesture over-confidently in his direction.

- What does footnote 11 say? Because the whole dispute rides on that single word, "whenever."

"There are a number[11] of people voting for me for President on Tuesday — Srap Tasmaner

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.

What does footnote 11 say? Because the whole dispute rides on that single word, "whenever." — Leontiskos

Here, "whenever" is used as an informal abbreviation "for every assignment of values to the free variables in the judgment" — same

Actually I expected the footnote just to be a reference to Gentzen, but it was glossed!

I mean, I don't think you can turn it into an argument that doesn't sound very stupid at any rate.

My thoughts were just that an argument isn't considered valid just because there is no way for the premises to be true and the conclusion false.

Consider:

A is not A
B is not A
B is A

This cannot have true premises and a false conclusion because one premise is necessarily false. But surely we don't want to claim that the fallacy of exclusive premises is true just in cases where it is possible for its premises to be true.

To be sure, one might use disjunctive syllogism to prove that B is A from the contradiction, but that doesn't make the form of the above valid.

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.

reductio? — Leontiskos

I'm taking this out of context, for the sake of a comment.

I'm a little rusty on natural deduction but I think reductio is usually like this:

A (assumption)*
...
B (derived)
...
~B (derived)
⊥
━━━━━━━━━━━━
A → ⊥ (→ intro)*
━━━━━━━━━━━━
~A (~ intro)

Not sure how to handle the introduction of ⊥ but it's obviously right, and then our assumption A is discharged in the next line, which happens to be the definition of "~" or the introduction rule for "~" as you like.

Point being A is gone by the time we get to ~A. It might look like the next step could very well be A → ~A by →-introduction, but it can't be because the A is no longer available.

What you do have is a construction of ~A with no undischarged assumptions.

#

We've talked regularly in this thread about how A → ~A can be reduced to ~A; they are materially equivalent. We haven't talked much about going the other way.

That is, if you believe that ~A, then you ought to believe that A → ~A.

In fact, you ought to believe that B → ~A for any B, and that A → C for any C.

And in particular, you ought to believe that

P → ~A (where B = P)
~P → ~A (where B = ~P);

and you ought to believe that

A → Q (where C = Q)
A → ~Q (where C = ~Q).

If you combine the first two, you have

⊤ → ~A

while, if you combine the second two, you have

A → ⊥.

These are all just other ways of saying ~A.

#

Why should it work this way? Why should we allow ourselves to make claims about the implication that holds between a given proposition, which we take to be true or take to be false, and any arbitrary proposition, and even the pair of a proposition and its negation?

An intuitive defense of the material conditional, and then not.

"If ... then ..." is a terrible reading of "→", everyone knows that. "... only if ..." is a little better. But I don't read "→" anything like this. In my head, when I see

P → Q

I think

The (probability) space of P is entirely contained within the (probability) space of Q, and may even be coextensive with it.

The relation here is really ⊂, the subset relation, "... is contained in ...", which is why it is particularly mysterious that another symbol for → is '⊃'.

The space of a false proposition is nil, and ∅ is a subset of every set, so ∅ → ... is true for everything.

The complement of ∅ is the whole universe, unfortunately, and that's what true propositions are coextensive with. When you take up the whole universe, everything is a subset of you, which is why ... → P holds for everything, if P is true.

Most things are somewhere between ∅ and ⋃, though, which is why I have 'probability' in parentheses up there.

The one time he did — Moliere

Which is the interesting point here.

"George never opens when he's supposed to."
"Actually, there was that one time, year before last ― "
"You know what I mean."

Ask yourself this: would "George will not open tomorrow" be a good inference? And we all know the answer: deductively, no, not at all; inductively, maybe, maybe not. But it's still a good bet, and you'll make more money than you lose if you always bet against George showing up, if you can find anyone to take the other side.

"George shows up" may be a non-empty set, but it is a negligible subset of "George is scheduled to open", so the complement of "George shows up" within "George is scheduled", is nearly coextensive with "George is scheduled". That is, the probability that any given instance of "George is scheduled" falls within "George does not show up" is very high.

TL;DR. If you think of the material conditional as a containment relation, its behavior makes sense.

((Where it is counterintuitive, especially in the propositional calculus, it's because it seems the only sets are ∅ and ⋃. Even without considering the whole world of probabilities in fly-over country between 0 and 1 ― which I think is the smart thing to do ― this is less of a temptation with the predicate calculus. In either case, the solution is to think of the universe as being continually trimmed down to one side of a partition, conditional-probability style.))

- Interesting, but it doesn't adjudicate the question. I don't expect the question to be adjudicated on these sorts of grounds (and Tones involves himself in petitio principii when he claims that his sources favor his interpretation). 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.

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

I don't know what you mean. Example?

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'.)

To be sure, one might use disjunctive syllogism to prove that B is A from the contradiction, but that doesn't make the form of the above valid. — Count Timothy von Icarus

Yes, that's my point.

Tones thinks it is valid by definition, because any argument with inconsistent premises is (trivially) valid.

Now the question arises: is it invalid? I don't claim that.

But surely we don't want to claim that the fallacy of exclusive premises is true just in cases it is possible for its premises to be true. — Count Timothy von Icarus

Not sure what you mean by this.

Wrong — TonesInDeepFreeze

Why is it wrong? There is no interpretation where both premises are true.

- Yeah, you're giving me flashbacks to Flannel's thread.

TL;DR. If you think of the material conditional as a containment relation, its behavior makes sense. — Srap Tasmaner

That was a really interesting post, and it presents an interesting attempt to bridge propositional logic and real-world reasoning. I am reading Burnyeat on Aristotle's Enthymeme, which is closely related to your discussion of George. Unfortunately I've already spent too much time on TPF today, so I am not going to say a whole lot more.

My take on material implication:

Material implication is the way it is for much the same reason that humans are the way they are given Epimetheus' mistake. When the logic gods got around to fashioning material implication they basically said, "Well if the antecedent is true and the consequent is true then obviously the implication is true, and if the antecedent is true and the consequent is false then obviously the implication is false, but what happens in the other cases?" "Shit! We only have 'true' and 'false' to work with! I guess we just call it 'true'...?" "Yeah, we certainly can't call it 'false'."

I haven't thought about this problem in some time, but last time I did I decided that calling the vacuous cases of the material conditional 'true' is like dross. In a tertiary logic perhaps they would be neither true nor false, but in a binary logic they must be either true or false, and given the nature of modus ponens and modus tollens 'true' works much better. It's a bit of a convenient fiction. This is not to say that there aren't inherent problems with trying to cast implication as truth-functional, but it seems to me that an additional problem is the bivalence of the paradigm. — Leontiskos

The purpose of material implication is inferences like modus ponens and modus tollens. Degenerative uses are improper. The consequence relation can appropriate the material conditional without any risk of degenerative use (at least until you do the weird stuff Tones is doing, in which case the risks are re-introduced).

See also:

...Soon after this, Frege expresses frustration that 28 years after he introduced the material conditional mathematicians and logicians continue to resist it as something bizarre! — Leontiskos

When a formalist takes up logic, they neglect its teleological character, and when logic has no teleological character there can be no degenerative or non-degenerative uses. That is the problem, methinks.