What you're after is a more robust relationship between premises and conclusions, something more like grasping why it being the case that P, in the real world, brings about Q being the case, in the real world, and then just representing that as 'P ⇒ Q' or whatever. Not just a matter of truth-values, but of an intimate connection between the conditions that 'P' and 'Q' are used to represent. Yes? — Srap Tasmaner

...And I want to say that an argument is supposed to answer the "why" of a conclusion. Inferential argumentation is an explanation for a proposition/conclusion. Validity is one aspect of the goodness of such an explanation.

Similar to what I said earlier about the genus of discourse, some arguments are apparently neither valid nor invalid:

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

Probably they are not "arguments" at all.

To give another example using Srap's color idea:

• Everything which is not white contains pigment
• Numbers are not white
• Therefore, numbers contain pigment

That is the sort of thing that is occurring when one tries to claim that any argument with inconsistent premises is trivially valid. The domain of discourse when speaking about validity is arguments, and arguments do not contain premises that are known to be inconsistent. Some arguments have premises that are inconsistent but are not known to be inconsistent, and that is where reductio comes in. Are these latter kind truly arguments? Not in any perfect or ideal sense, but they are in the sense that the arguer believes the premises to be consistent.

Well, what do we say here ― leaving aside whether color exclusion is a tenable example? What you're after is a more robust relationship between premises and conclusions, something more like grasping why it being the case that P, in the real world, brings about Q being the case, in the real world, and then just representing that as 'P ⇒ Q' or whatever. Not just a matter of truth-values, but of an intimate connection between the conditions that 'P' and 'Q' are used to represent. Yes? — Srap Tasmaner

These are interesting topics that Aristotle also takes up, but I don't think I'm being overly greedy in what I desire. I am not requiring a special kind of aitia/account/explanation. Here is what I said above:

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

...

...validity is an inferential relationship between premises and conclusion. — Leontiskos

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.

A key contention of mine is that I am representing the notion of validity in formal logic better than Tones is. I don't even need to advert to real-world cases, like that of color. Even within propositional logic itself, validity has to do with "follows from" and deducibility.

- 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.

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.

- 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.

- 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

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

- 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.

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

- 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.

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.

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

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.

- 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.

"a major topic in the study of deductive logic is validity. This is a relationship..." — 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.

That is the second time you put quotes around words I didn't say. — TonesInDeepFreeze

It's called paraphrase, and we both know you hold to the paraphrased proposition. You should be a lawyer given the way you constantly complain, nitpick, and manage bizarre readings interpreted via a form of legalese.

"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). — TonesInDeepFreeze

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.

"It is not possible for the premises to be true and the conclusion false" is not uncontroversially fulfilled by a set of inconsistent premises. You are literally interpreting English conditionality via the idiosyncrasies of the material conditional, which is ironic given the way you protest labels which reduce your thinking to truth-functional categories. It is curious to me that you do not recognize the way your argument rests on a mere technicality.

I'm really not convinced this is going anywhere given how many times you have now repeated yourself, but the issue here has to do with consequence or inference vs. the material conditional. I gave examples of sources which agree that a valid argument requires that the conclusion follows from the premises, and everyone knows that the idiosyncratic/trivial case of the material conditional, where a false antecedent automatically makes the conditional true, is not a case of "follows from." This is why logicians refused to admit the material conditional for many decades after Frege had attempted to introduce it.

"Hanover's defense was logically inconsistent, therefore his conclusion follows from his defense," is not correct. B does not automatically follow from A whenever A is incoherent.

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

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.)

There is no question. He does not presuppose it. — TonesInDeepFreeze

There is no question that he would reject your tendentious interpretation, which fully ignores the bolded sentence of Gensler's.

Suppose you are on the jury. @Hanover presents his defense. It is a garbled mess of incoherent and contradictory gibberish. He concludes, "...Therefore, the defendant is innocent." The jury goes into deliberation. You say, "Well, we must first recognize that Hanover's defense was a piece of valid reasoning." The rest of the jury looks at you with blank stares. You continue, "His premises were inconsistent, and any argument with inconsistent premises is necessarily valid." The blank stares only become more protracted.

Now it would not help you in any way if Gensler and Enderton were fellow jurors. Even more than the other jurors, they would think you were confused. Gensler might say, "Did you read past the first sentence of my explanation of validity? Very few people would construe it in the bizarre way you have, but even so, I went on to clarify the concept in the following sentences."

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). You think that any argument with inconsistent premises is automatically valid, not that every valid argument has inconsistent premises. Here is Gensler:

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. — Gensler, Introduction to Logic, Second Edition, p. 3

Validity has to do with the conclusion following from the premises. Your claim is, "Whenever the premises are inconsistent, the argument is valid." But inconsistent premises do not show that the conclusion follows from them. Hanover's defense is not valid reasoning just because it is confused.

(Now you can hold to your tendentious position if you like, but it is not the position of Gensler, or Enderton, or SEP, or Wikipedia.)

And the argument is valid by Gensler, Enderton, SEP and Wikipedia. — TonesInDeepFreeze

Let's take the first:

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

The question is whether we should read Gensler as presupposing that the premises are consistent. You want to say, "The premises are inconsistent, therefore the argument is valid," and you want Gensler to agree with you. But the quotes I gave from Gensler (and everyone else) do not support your interpretation:

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. — Gensler, Introduction to Logic, Second Edition, p. 3

Your interpretation flies in the face of the bolded sentence. Gensler is talking about a consequence relation between premises and conclusion. A consequence relation is not established by your, "The premises are inconsistent..."

(As I've pointed out, 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.)

And with the argument mentioned in the original post, it is the case that there is no interpretation in which all the premises are true. — TonesInDeepFreeze

And that does not make the argument valid for Gensler, Enderton, SEP, or Wikipedia.
But it does for you.
Because you are leveraging an idiosyncratic notion of validity.

- Wrong again:

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

In this case there are no interpretations in which all the premises are true. Perforce, there are no interpretations in which all the premises are true and the conclusion is false. So the argument is valid. — TonesInDeepFreeze

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

In this case there are no interpretations in which all the premises are true. Perforce, there are no interpretations in which all the premises are true and the conclusion is false. So the argument is valid. — TonesInDeepFreeze

What you've done is imported the artificial truth-functionality of the material conditional into the consequence relation itself. You have contradicted ↪Hanover's "flows from." You are effectively saying, <Any "argument" with nonsense premises is "valid."> — Leontiskos

As I said, in this particular regard, I'm merely applying the definitions of ordinary formal logic. — TonesInDeepFreeze

Ordinary formal logic does not define the consequence relation as identical to the material conditional. — Leontiskos

Here is Gensler speaking about validity in his introductory chapter:

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. — Gensler, Introduction to Logic, Second Edition, p. 3

Here is Enderton:

What is surprising is that the concept of validity turns out to be equivalent to another concept (deducibility)... — Enderton, A Mathematical Introduction to Logic, p. 89

Here is SEP:

A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises.

...

...the argument is valid [when] the conclusion follows deductively from the premises... — Logical Consequence | SEP

Here is Wikipedia:

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. — Validity | Wikipedia

@TonesInDeepFreeze wants to say that an argument is definitionally/trivially valid if it its premises cannot all be true (i.e. if it is inconsistent). He says that he is "merely applying the definitions of ordinary formal logic." Except the reputable sources and logicians simply do not define validity in such a way.

(@Hanover)

analogous predication — Count Timothy von Icarus

I don't know what you mean. — TonesInDeepFreeze

Aristotle calls such a thing a "pros hen" homonym.

• Analogy and Analogical Reasoning | SEP
• Medieval Theories of Analogy | SEP

Well, while I think Srap has a good point about our being able to live without A→~A in most situations, I think it is important that statements like "nothing is true," are able to entail their own negation—that logic captures how these claims refute themselves. — Count Timothy von Icarus

But does logic really capture how these claims refute themselves? I don't think so. It merely defines a formal notion of contradiction and shows that a contradiction has occurred. The how/why question is beyond the logic (as is the reductio-remedy), and I believe you yourself pointed earlier to the logical simplification of 'contradiction' (i.e. an all-false truth table).

It may seem bizarre that a valid argument could have at least one premise that is necessarily false at first glance, but I think it is fairly intuitive if one thinks in terms of truth-preservation. If the premises were true, it would preserve truth. But the "truth" of a false premise cannot be preserved.

And it's a good thing that it is valid because we often can reason from necessarily false conclusions in valid arguments to identifying false premises. — Count Timothy von Icarus

Why would it be a good thing? It is good that we can reason from non-necessarily false conclusions in valid arguments to identifying false premises. An argument from a necessarily false conclusion is a reductio, and the question of whether an absurdity is valid is part of the very question at hand.

It may seem bizarre that a valid argument could have at least one premise that is necessarily false at first glance, but I think it is fairly intuitive if one thinks in terms of truth-preservation. — Count Timothy von Icarus

Validity in propositional logic involves a relativization of truth-values with respect to inference-relations. Inference-relations are held steady, and if the truth-values cash out given the stable inference-relations, then we call it "valid." The inference relations are conceived as meaning-stable, and the variables are conceived as meaning-variable (i.e. truth-variable). But in this case what is at stake is the meaning and stability of the inference-relations themselves. The contentious move is to claim that the consequence-relation involved in the OP is the stable, familiar consequence relation of modus ponens. It isn't. That's that place to start.

To claim that it is involves:

...prioritizing truth-functional process over logical telos. — Leontiskos

Put differently, the notion of validity assumes a truth-functional context where truth and form are entirely separable. Yet when we think deeply about inferences themselves, such as modus ponens, truth and form turn out to be less separable than we initially thought. When we stop merely stipulating our inferences and ask whether they actually hold in truth, things become more complicated.

Yes. I edited that post. It's just weird that any argument that can't have all true premises is going to be valid. — frank

We could say with that if the conclusion flows from the premises then the argument is valid.

1. P→Q
2. P
3. ∴ Q

4. A→~A
5. A
6. ∴ B

Now one could say that (3) flows from (1) and (2); and that (6) flows from (4) and (5). But this latter use of "flows from" is very different from the former. 's contention that they are the same use is not "merely applying the definitions of ordinary formal logic."

As I said, in this particular regard, I'm merely applying the definitions of ordinary formal logic. — TonesInDeepFreeze

Ordinary formal logic does not define the consequence relation as identical to the material conditional.

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

In this case there are no interpretations in which all the premises are true. Perforce, there are no interpretations in which all the premises are true and the conclusion is false. So the argument is valid. — TonesInDeepFreeze

What you've done is imported the artificial truth-functionality of the material conditional into the consequence relation itself. You have contradicted 's "flows from." You are effectively saying, <Any "argument" with nonsense premises is "valid.">

Just trying to think of real world examples of a formula like "A → ~A", likely dressed up enough to be hard to spot. Excluding reductio, where the intent is to derive this form. What I want is an example where this conditional is actually false, but is relied upon as a sneaky way of just asserting ~A.

I suppose accusations of hypocrisy are nearby. "Your anti-racism is itself a form of racism." "Your anti-capitalism materially benefits you." "Your piety is actually vanity." — Srap Tasmaner

Isn't that reductio?

I would say that, like argument, contradiction also requires a kind of middle term, and is therefore never direct. For example:

A→B
B→~A
A
∴ B — Leontiskos

People can only make this inference because they do not see that they are being inconsistent. When there is neglect we hold them responsible for the mistake.

So A→~A is never a self-conscious premise.

I'm sure there are more convoluted ways to go about it, but does that satisfy your objection? — Hanover

Your "disjunctive syllogism" is different than my A→A, so in that sense, sure. You are effectively saying that A flows or follows from the contradiction, not from itself.

So a second objection would be that nothing flows or follows from a contradiction (which is the flip side of saying that everything flows or follows from a contradiction).

- I added an edit to that post, which might help. My point about conditionals and arguments would also apply to "proves," "flows," etc.

Only arguments are valid, and "A, therefore A," is not an argument. Argument, at the very least, involves rational movement. — Leontiskos

-

Ergo:

• A→A
• A, ∴A
• A flows from A
• A proves A

These are all based on the same error, that of a non-inference inference.

There is too little knowledge of Aristotle on these forums, and that is why we don't seem to understand what arguments are. :grin:

Another argument:

A -> ~A
A
therefore A
valid — TonesInDeepFreeze

I realize a lot of people like this claim, but I don't think it is right. You are confusing consequence or inference with identity.

Even on a very formal reading, this is invalid. "A→A" and "A, ∴A" are not the same statement. Even so, there is a dispute here about what '→' and '∴' mean. In that way it is the same problem of trying to hold to truth-functionality (turtles) all "the way down."

Only arguments are valid, and "A, therefore A," is not an argument. Argument, at the very least, involves rational movement.

-

The core of truth in @Hanover's variegated posts is that "A→A" is not a conditional and "A, ∴A" is not an argument. If you admit such things to the bar of conditionals and arguments, you are fudging the meaning of "conditional" and "argument." You are prioritizing truth-functional process over logical telos.

The losing party, in one sense, grants that they lost, but continues in the competitive spirit, which means they have to shift ground from whether they "officially" or "technically" lost to whether that was a "real" loss, or whether there had a been a "real" competition in the first place. — Srap Tasmaner

I'm not sure what post you are responding to, but there is of course a substantive issue here. It is the difference between rules-as-arbitrary and rules-as-substantive, and logic-as-arbitrary and logic-as-substantive. There are true charges of cheating and false charges of cheating, and it's not always easy to disentangle the two.

The move is always to a meta-level. What is the game? What is the competition? What is logic? Our world has a remarkable tendency to try to avoid those questions altogether, usually for despair of finding an answer.

As Banno notes, validity is determined by asking if the conclusion flows from the premises, and so he argues under mp, it does, so it is valid.

The wiki cite adds criteria, namely (1) that the negation of the conclusion cannot also flow from the premises for validity and (2) the premises under any formulation must also reach the same conclusion. — Hanover

Right: the conclusion must flow from the premises. The premises must provide the aitia for the conclusion. A contradiction is not an aitia.

As I argued at length in Flannel's thread, contradictions and inconsistencies are not meaningful. To pretend they are meaningful is to become lost in the logical abyss. If you feed the "argument" of the OP into the propositional logic machine, the answer is neither "invalid" or "unsound." It is, "Does not compute."

- Good posts. :up:

But it's not validity we usually disagree over, but soundness, and inconsistent premises make valid inferences unsound. — Srap Tasmaner

In cases of inconsistent premises what happens is that the person arguing arbitrarily makes use of some premises while conveniently ignoring others. For example:

• A→B
• B→~A
• A
• ∴ B

Or a reductio, which has been shown elsewhere to falsify one side of a contradiction rather than the other side for no necessary reason. Is the argument above or a reductio valid? Are they sound? Neither answer is obvious. We can't just say, "Ah, it's cut and dry. The argument is valid but unsound."

Similarly, the arguments over the OP turn on the nature of modus ponens, which is not a simple question. If modus ponens is just a matter of symbol manipulation then the OP is valid. If modus ponens is more than that then the OP is probably not even valid.

- Why do you think dialetheism relates to the consequence relation? Presumably you think the LEM is tied to the consequence relation, and that dialetheism therefore interferes with it, but I'm not sure you have given an argument in that vein.

But I don't really intend to continue this conversation about dialetheism, especially given my earlier demonstrations of the incoherence of the "Liar's paradox." From what I have seen, people are dialetheists for the same reason they dye their hair purple. :grin:

I find this particularly unconvincing as respects "afterlife" beliefs because many ancient visions (and the dominant modern vision) of the afterlife seem significantly more unpleasant than just ceasing to exist. — Count Timothy von Icarus

Yep. :up:

I don't think it's that hard to define at all. — Count Timothy von Icarus

I haven't seen anyone define any of the positions in a clear and non-vacuous way, much less go on to argue in favor of one or another.

Their argument is roughly that the intuitive/informal notion of logical consequence is multiply-realizable (granted it is more technical in its details). — Count Timothy von Icarus

"There are multiple formal ways of realizing the informal notion of logical consequence." I suppose this gives us something, but I don't think it is very substantial. If, for example, everyone agrees that Aristotelian syllogistic and propositional logic are two ways of formalizing the informal notion of logical consequence, then where does the actual disagreement lie?

Again, what is needed is someone who believes they disagree and is willing to set out a substantial argument. The polemicists disagree without substance, and the rest of us are not sure what we are supposed to be disagreeing about.

Were debating whether to call certain formulations "modus ponens." — Hanover

I figured this would be an interesting thread. This is the standard set piece where Banno and Tones think logic is arbitrary symbol manipulation and others think it has to do with correct reasoning, but this thread brings it out quickly.

For my money the question here is whether modus ponens is arbitrary or non-arbitrary. (Whether what is at stake is a mere matter of definition.)

The basic idea is "formally correct but misleading". Akin to sophistry. Or to non-cooperative implicature, like saying "Everyone on the boat is okay" when it's only true because no one is left on the boat and all the dead and injured are in the water. — Srap Tasmaner

Yep. :up:

- This is elsewhere referred to as deautomatization.