Yep. But it looks at first glance as if substitution into an ironic statement preserves truth value. A side issue.

In natural language, predication is often not totally univocal, but is also not totally equivocal. — Count Timothy von Icarus

I don't claim to have academic definitions of 'univocal' and 'equivocal', but at a naive level, as I'm merely winging it here, it seems to me that:

'totally univocal' is redundant. An expression is univocal if and only if it has one meaning. That's total.

'totally equivocal' is hard to conceive. An expression is equivocal if and only if it has more than one meaning. What would it mean to say it is totally equivocal?

For example, we might say that "lentils are healthy," or "running is healthy." These are true statements. And we might also say "Tones is healthy." Yet you would not be "healthy" in the same way that lentils are. — Count Timothy von Icarus

Right.

However, neither is the usage totally equivocal. We call lentils "healthy" precisely because (normally) they promote the health of human beings, i.e. the same "health" we refer to in "Tones is healthy." — Count Timothy von Icarus

Hmm, I'm not sold on that. That "Tones is healthy" and "This apple is healthy" are true two in different senses doesn't suggest to me that there's any matter of totality to consider.

/

By 'analogous predication' you mean as with the Tones/Apple example?

Do you mean that

This apple is healthy
is analogous predication with
Tones is healthy

I do understand that.

So
The animal runs
is analogous predication with
The refrigerator runs

It's sort of like how, as far as I am aware, there is no popular formalization of the distinction between quia vs. propter quid demonstrations (i.e. demonstrating "that something is the case," vs. demonstration "why it is the case.") I don't think most people would deny that they're different (although some would), but rather it seems that the difference should be entirely reliant on the arguments' content, not their form (i.e. an issue of material logic). — Count Timothy von Icarus

I would think
'that it is the case' is a matter of giving an argument
but
'why it is the case' is a matter of exposition, not argument

It's interesting that in mathematics, some people demand to know "Why is that theorem true?" And I can't think of an answer other than "Because there is a valid argument for it from true premises, and here it is ..." That is, I can show you the proof, which, at least for me, does answer "why?". I may be able to give real world examples, and abstract analogies, and point to coherency. But those don't fully answer "why" in the same definitive way that proof does.

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

Very good. Thank you.

I can give you a story that comes to mind in which I'd assert something like that -- say I'm commiserating with a coworkers frustration about George not being as reliable as we'd like, even though he's a good enough fellow. — Moliere

Of course, I understand the basis of the sarcasm.

The substitution is there only because the OP starts with A -> ~A and asks for validity, so substitution seems to work as a model for the sarcastic talking. I agree that the person speaking sarcastically does not in any way mean these logical implications, though -- it's only an interpretation of everyday speech to try and give some sense to the original question that's not purely formal. — Moliere

The original post challenges the validity of
A -> ~A
A
therefore ~A

I don't see your substitution capturing irony.

When I say A sarcastically, I mean ~A, of course. And that is equivalent with A -> ~A. But I don't present it like that at all. I just say A and there is an implicit premise that when I say it, I mean its negation. I don't know how even modal logic could capture that. Or maybe, I am saying that A is true in an alternative world and false in the actual world, but even that seems far-flung.

Getting back to @Srap Tasmaner, he's looking for a use of A -> ~A in everyday discourse. I don't think your proposal works, since people don't acutually say things of the form A -> ~A to convey sarcasm. It seems to me that you followed an interesting idea, but it doesn't do the job here.

Though, related to a different kind of formula, people do say things like:

If 'Fear Factor' is great television then I'm the Queen of Roumania.

That is:

If P then Q (where Q is false)

/

And reductio ad absurdum may occur too (aside from mathematics where it is prevalent):

If Jack robbed the store, then has the loot in his car.
He does not have the loot in his car.
So, the claim that Jack robbed the store leads to a contradiction.
So Jack did not rob the store.

Of course, it can also be cast, more tersely, as modus tollens:

If Jack robbed the store, then he has the loot in his car.
He does not have the loot in his car.
So Jack did not rob the store.

I would think
'that it is the case' is a matter of giving an argument
but
'why it is the case' is a matter of exposition, not argument

I think that is likely often true, but it also seems possible in some cases to construct a syllogism that addresses the "why" (as well as syllogisms that do not seems to address it.) Since I just shared some of the relevant sources in another thread I have them on hand:

Knowledge of the fact (quia demonstration) differs from knowledge of the reasoned fact (propter quid demonstrations). [...] You might prove as follows that the planets are near because they do not twinkle: let C be the planets, B not twinkling, A proximity. Then B is predicable of C; for the planets do not twinkle. But A is also predicable of B, since that which does not twinkle is near--we must take this truth as having been reached by induction or sense-perception. Therefore A is a necessary predicate of C; so that we have demonstrated that the planets are near. This syllogism, then, proves not the reasoned fact (propter quid) but only the fact (quia); since they are not near because they do not twinkle, but, because they are near, do not twinkle.

The major and middle of the proof, however, may be reversed, and then the demonstration will be of the reasoned fact (propter quid). Thus: let C be the planets, B proximity, A not twinkling. Then B is an attribute of C, and A-not twinkling-of B. Consequently A is predicable of C, and the syllogism proves the reasoned fact (propter quid), since its middle term is the proximate cause.

From Aristotle's Posterior Analytics I.13:

Aquinas relates this to causes (although his concept of "cause" is Aristotle's four causes, so they might still be invoked in mathematics)

I answer that it must be said that demonstration is twofold: One which is through the cause, and is called demonstration "propter quid" [lit., 'on account of which'] and this is [to argue] from what is prior simply speaking (simpliciter). The other is through the effect, and is called a demonstration "quia" [lit., 'that']; this is [to argue] from what is prior relatively only to us (quoad nos). When an effect is better known to us than its cause, from the effect we proceed to the knowledge of the cause. And from every effect the existence of its proper cause can be demonstrated, so long as its effects are better known to us (quoad nos); because since every effect depends upon its cause, if the effect exists, the cause must pre-exist.

From St. Thomas' Summa theologiae I.2.2c:

Now, I do think this is probably something that has to stay to one side of form. It was long considered part of "logic," but this is logic interpreted broadly as the study of "good reasoning" (even rhetoric was sometimes lumped in with logic on curricula). However, when it comes to more amorphous debates like pluralism and the "correct logic" vis-a-vis some subject matter, it seems possible that an argument could be advanced that states that a certain sort of logic is "correct" because of the nature of the subject matter, in which case content (matter) would inform form (which I guess it always does, just not in a way everyone can agree upon).

So would it be fair to say that 'why' is more amenable to being answered when causality is involved? That goes to the point that in mathematics it is difficult to answer 'why' without finally resorting to showing the proof, which some people might consider to not be an answer to 'why'.

Yes, I think that's fair to say. We particularly care about this sort of thing in the sciences, the large focus on "correlation versus causation" for instance, or as respects states of affairs.

Interestingly, Aristotle and St. Thomas do make some recourse to causes in discussing mathematics. Even though mathematics is the understanding of form abstracted from matter, they include a form/matter distinction within mathematical entities. The essence (or "what-it-is-to-be") of a triangle is its form and the form determines what is true of all triangles. But particular triangles vary according to their dimensions and this variance is attributable to their "matter," which is termed "intellectual matter" due to these abstractions existing in the mind (ens rationis). We might say the lines are the "material" that compose a triangle. And we might say that triangle is a genus with different species, e.g. "isosceles," with there being things that are true for all isosceles by virtue of their species form.

For example, the Pythagorean theorem can be explained in terms of formal cause, whereas the values for a, b, and c will be explained in terms of material cause.

Whether this distinction is useful is another matter. In the mathematics they had available, it seems like it could be helpfully explanatory, but whether one wants to try to bother reforming the concepts for modern mathematics will probably largely depend on if one thinks the rest of the metaphysics backing it is worth developing.

They do have very interesting philosophies of mathematics though, particularly the potential/actual distinction as Aristotle applies it to the notion of the infinite in physics.


* Because of Aristotle's metaphysics, everything except God has both act and potency, and so some analagous form(actuality)/matter(potentiality) distinction.

In this case, there is no interpretation in which all the premises are true. — TonesInDeepFreeze

A premise is defined as an analytic truth. It cannot be false, regardless of its synthetic falsity. If C means "Cows bark," it is irrelevant if they don't for the purposes of formal logic.

My point is simply that if you have an analytically false premise (meaning it cannot be true in any world), it fails to meet the definition of "premise."

An argument without premises is not a syllogism.

That is to say, accepting what you've argued as true, the OP is not a valid argument because it's not an argument at all.

A premise is an assumed truth.

A premise is defined as an analytic truth. It cannot be false, regardless of its synthetic falsity. If C means "Cows bark," it is irrelevant if they don't for the purposes of formal logic.

An analytic truth is true by definition, e.g. "bachelors are unmarried men." Premises need not be analytic or considered so.

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.

Aren't truth and form mixed together in any tautology or contradiction? We wouldn't want to exclude those though, right?

It seems you could do without it too. I hadn't really given it much thought.

A premise is defined as an analytic truth. — Hanover

No, it's not.

An argument without premises is not a syllogism. — Hanover

Yes. So?

the OP is not a valid argument because it's not an argument at all. — Hanover

A rigorous definition:

An argument is an ordered pair such that the first coordinate is a set of statements* and the second coordinate is a statement*.

The members of the first coordinate are the premises. The second coordinate is the conclusion.

If purely symbolic, the premises and the conclusion are symbolic formulas. If in natural language, the premises and conclusion are natural language declarative sentences.

So, written in that form we have this argument:

<{A -> ~A, A} ~A>

The set of premises is {A -> ~A, A}, and the conclusion is ~A.

Written informally (where everything above 'therefore' is a premise and what follows the word 'therefore' is the conclusion.

A -> ~A
A
therefore ~A

* Or more generally, formulas.

A natural language example:

If the Great Pumpkin is orange, then Great Pumpkin is not orange.
The Great Pumpkin is orange.
therefore, the Great Pumpkin is not orange.

/

With your requirement, even the following would not be an argument (not just one of the premises is not analytic, but neither of the premises are analytic):

If Bob has poor eyesight, then Bob wears glasses.
Bob has poor eyesight.
Therefore Bob wears glasses.

Really, you want to disqualify that from being an argument because the premises are not analytic?

/

You've tried a few incorrect arguments, based on misconceptions, that the argument is not valid. And now another one.

tautology or contradiction — Count Timothy von Icarus

Just to note: tautology is semantic and contradiction is syntactic.

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)

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). — Leontiskos

That is the third time in this thread that you've put words in my mouth.

- 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

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 — Leontiskos

Indeed. Equivalent to the definition I've been stating.

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.[1] It is not required for a valid argument to have premises that are actually true,[2] but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. — Validity | Wikipedia

Indeed. Yet another way of saying the definition.

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 — Leontiskos

Indeed. It is a point I've made many times. It is the completeness and soundness of first order logic.

↪TonesInDeepFreeze - 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 — Leontiskos

You're wrong.

The definition of 'valid argument' there is standard.

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

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.

Quite not idiosyncratic.

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

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.

Wikipedia:

"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 impossible to have both the premises true. Perforce, it is impossible for the premises to both be true and the conclusion nevertheless false.

SEP:

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

That is true, though usually it is addressed as a theorem not a definition. It is the soundness theorem. Still:

The conclusion follows deductively from the premises:

1. A -> ~A ... premise
2. A ... premise
3. ~A {1 2 by modus ponens}

Enderton:

"the concept of validity turns out to be equivalent to another concept (deducibility)"

Again, that's not a definition of 'valid' but rather it mentions an equivalence with deducibility (as it applies to first order logic). Still

The conclusion is deducible from the premises:

1. A -> ~A ... premise
2. A ... premise
3. ~A {1 2 by modus ponens}

@Leontiskos is quoting stuff that is consistent with @TonesInDeepFreeze definition, but claiming that they disagree.

Fucksake.

Not just consistent, but equivalent with.

Yep. Sad stuff.

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.)

Grabbed on the fly:

Three equivalent variations:

"Valid: 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]

https://web.stanford.edu/~bobonich/terms.concepts/valid.sound.html

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.

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. — TonesInDeepFreeze

Well done.

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.

"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)

Frank,

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

Spelled out here:

Can you see it now?