Do you intend for this to be a Socratic interview?

Do you intend for this to be a Socratic interview? — TonesInDeepFreeze

No. It's that if A is false, the first premise is trivially true.

So the argument is one in which the first premise doesn't say anything. The argument would be:

1. Trivial truth
2. A.

Conclusion: not-A.

That's not valid.

Wait a minute. If A is false, then the first premise is:

If not-A, then not (not-A)

You can't change one of the A's to false and not the other one. If A is false, they both have to be false. — frank

That is wrong. You are plainly misusing the terminology.

Considering different interpretations doesn't change formulas.

And I'm not making "one A" false and not "the other A".

There is no sense of "another A" and thus no interpretation in which A is false but "another A" is true.

Again, you need to know what 'interpretation' means.

I think you need to know what "trivially true" means.

Yes, so — TonesInDeepFreeze

Per the definition of "valid":

An Argument is valid if and only if it would be contradictory for the conclusion to be false if all of the premises are true.[ — Hanover

Assuming all premises in the OP true, the conclusion of not A is shown to be false because a valid conclusion of A was shown.

Assuming all premises in the OP true, the conclusion of not A is shown to be false because a valid conclusion of A was shown. — Hanover

Yes. In the only interpretation where both premises are true, there's no way to conclude not-A

So the argument is one in which the first premise doesn't say anything. — frank

A -> ~A
says
If A then it is not the case that A

That is equivalent with
it is not the case that A

In an interpretation in which A is false, "it is not the case that A" is true; in an interpretation in which A is true, "it is not the case that A" is false.

1. Trivial truth
2. A.

Conclusion: not-A.

That's not valid. — frank

Yes, that that is not a valid argument. But when you replace A -> ~A by "trivial truth" the new argument is not equivalent with the original.

(1) A -> ~A is not a truth simpliciter nor a falsehood simpliciter. As has been explained to you about 15 times today, it is true in an interpretation in which A is false, and it is false in an interpretation in which A is true.

(2) What you may say is trivial is not the truth of A -> ~A, since it is true or false depending on the interpretation of A, but rather that A -> ~A is true in any interpretation in which A is false, which, if you like, you may choose to call trivial.

(3) 'trivially true' is a term logicians and mathematicians usually use to describe a statement such that the truth of the statement is very obvious. It's not a technical term in a context such as this. So the judgement that a particular statement is trivially true does not count toward the validity of an argument. Validity is determined by the set of all interpretations, no matter whether one considered certain of the statements to be trivially true or not.

(4) A -> ~A has a deductive role in the argument. It's not the same argument if you take it out and replace it with "trivial truth".

(5) We actually do have a defined* constant called 't' such that t is interpreted as true in all interpretations (though 'trivial' is not part of this). In that case there is this argument:

t
A
therefore ~A

That argument is invalid.

Indeed, adding t as a premise to any argument has no effect on validity:

Let G be any set of premises. Let P be a statement. Let the argument be

(5a) From G, infer P.

Now add t. So we have:

(b) From Gu{t}, infer P.

Then (5a) is valid if and only if (5b) is valid.

* Usually, the definition is of the form, for some formula P, P -> P.

(6) So, even if we supposed that the premise A -> ~A were true in every interpretation, we can't just replace it with "trivial truth". That is, if we suppose that A -> ~A has as some property (such as being true, or trivially true, or having four symbols, or being a conditional, or having only one sentence letter, etc.) you can't just replace the formula with a mention of an adjective that applies to it. That would be blatantly fallacious.

Sure. I would encourage you to write out in English the only case where both premises are true, and see if you think not-A makes sense as the conclusion. If it does, great. Bon Voyage.

think you need to know what "trivially true" means. — frank

You think incorrectly.

I well know how logicians and mathematicians use the verbiage 'trivially true' and 'follows trivially' for statements and arguments respectively. Usually it means that the statement is very obviously true or the inference is very obviously valid. In logic and mathematics it doesn't mean that the statement or the argument is otiose (though, sometimes it does mean that the vacuous case needn't be considered since the generalization being proved is not claimed to include the vacuous case).

But 'true' has formal import while 'trivial' does not. There's nothing wrong with saying 'trivially true' or 'follows trivially' but it doesn't count toward evaluation of an argument.

That is, there is no definition:

P is trivially true if and only if [definens]

It's a personal choice for any author whether to use 'trivial' or not as part of the informal prose in which mathematics is often written.

And if we choose to say that "condition is true if the antecedent as false" is trivial, then we can say that about all the connectives:

It's trivially true that a conjunction is true if both conjuncts are true.

It's trivially true that a disjunction is true if at least one of the disjuncts is true.

It's trivially true that a negation is true if it is the negation of a false statement.

It's trivially true that a biconditional is true if either the left and right are both true or both false.

/

Moreover, see my previous post that explains in detail the fallaciousness and irrelevance of your argument.

'degenerate' in a non-pejorative sense as often in mathematics — TonesInDeepFreeze

Yes, that was my meaning, as with the boat example. 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]."

importance in Boolean logic used along the way in switching theory, computation, etc. — TonesInDeepFreeze

Absolutely. I mentioned automated reasoning projects when you need classical logic in full generality with P → P and all. Of course.

But that is not the only sort of application of logic, and as I noted reducing someone's argument to P → P is pointing out that they are "begging the question," generally considered a fatal problem for an argument. That conditional is legitimate in form, and is generally a theorem, but it is fatal if relied on to make a substantive point or demonstrate a claim. It will only happen inadvertently ― in which case, a good-faith discussant will admit their error ― or with an intent to mislead by sophistry.

Logic is a vast field of study, including all kinds of formal and informal contexts. I would not so sweepingly declare certain formulations otiose merely because one is not personally aware of its uses. — TonesInDeepFreeze

A fair point. Let's say, I'm only pointing out forms, or uses of particular schemata, that might raise our suspicions. "Heads I win, tails you lose" may in some cases demonstrate that I'm necessarily right and you necessarily wrong. Hurray! But in some cases it might amount to me stacking the deck against you.

Fundamentally, all we're talking about in this case is arguing from a set a premises which are inconsistent ― in fact, here, necessarily inconsistent. As I keep saying, that's either inadvertent or deceptive. Very often on this forum people attempt to show that a set of premises is inconsistent precisely by making correct inferences that make the inconsistency obvious. But people arguing from inconsistent premises often make inferences that, while in themselves correct, continue to hide their inconsistency. (Sometimes this is because not all premises are explicitly stated, and the inconsistency is in what is "assumed".) In such cases, it is not uncommon for people to insist on the correctness of their inferences. But it's not validity we usually disagree over, but soundness, and inconsistent premises make valid inferences unsound.

None of this news to you, I'm sure.

Per the definition of "valid":

An Argument is valid if and only if it would be contradictory for the conclusion to be false if all of the premises are true.[
— Hanover

Assuming all premises in the OP true, the conclusion of not A is shown to be false because a valid conclusion of A was shown. — Hanover

The argument:

A -> ~A
A
therefore ~A
valid

Another argument:

A -> ~A
A
therefore A
valid

The fact that the premises are inconsistent doesn't vitiate that the argument is valid. Actually the fact that the premises are inconsistent entails that the argument is valid.

(2) A conclusion itself is valid if and only if it is true in all interpretations. An argument is valid if and only if there are no interpretations in which the premises are all true and the conclusion is false.

In the only interpretation where both premises are true, there's no way to conclude not-A — frank

A -> ~A
A
therefore ~A

There is no interpretation in which both the premises are true.

I don't know why you continue to ignore the definitions and explanations given you.

PS. 'vacuous' is more informative for the conditional than 'trivial', since 'vacuous' is specfic while 'trivial' is not. That is, vacuousness is a certain kind of triviality.

Another twist. A conditional may turn out to be vacuously true, but it might be quite nontrivial to prove that the antecedent is false.

A -> ~A
A
therefore ~A

There is no interpretation in which both the premises are true. — TonesInDeepFreeze

If the antecedent in the conditional is false, then the first premise is true. Now say the second premise is true. Then the conclusion does not follow.

If you insult me one more time, we're done. I'm satisfied with ending this discussion.

The fact that the premises are inconsistent doesn't vitiate that the argument is valid. Actually the fact that the premises are inconsistent entails that the argument is valid.

(2) A conclusion itself is valid if and only if it is true in all interpretations. An argument is valid if and only if there are no interpretations in which the premises are all true and the conclusion is false. — TonesInDeepFreeze

The premises are consistent and the conclusions are not.

The conclusion is not true under all interpretations. Sometimes it's A and sometimes it's not A.

Sometimes it's A and sometimes it's not A. — Hanover

Correct

- 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.

"Trivial" has a clear meaning in analytical philosophy.

@Count Timothy von Icarus

Still hunting for a solid example, but in the meantime there's

(1) You only won because you cheated.

The sequence from here is most likely

(2) So you admit I won.

(3) It's not a proper game if you cheated.

And then the argument shifts from whether I got more points than you (or whatever) to whether the rules were followed.

But "the rules" can be surprising. Casinos universally have a rule against "card counting," which amounts to a rule against being too good at playing cards.

Sometimes among children using greater skill or knowledge is treated as cheating. It's clear they have an intuition about what fair competition is, but they mistakenly treat every advantage as unfair, or every unfair advantage as cheating.

What's happening here, broadly, is that the competition continues "by other means." 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.

So we have two versions here:

(4) If you won because you cheated, then you didn't win.
(5) If you "won" because I cheated, then you lost.

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."

The argument:

1. A -> ~A
A
therefore ~A
valid

Another argument:

2. A -> ~A
A
therefore A
valid — TonesInDeepFreeze

If under #1, I assume A (the negation of the conclusion) and I prove A from that (as is shown under #2), then I've proven invalidity by negation because I've shown my negation is true.

Trivial — frank

Feynman had a party trick he used to do, I think in grad school. He could tell whether any mathematical conjecture was true.

What he would do is imagine the conditions concretely, in his mind. Like start with a tennis ball to represent some object; then a condition would be added, and he'd need some explanation of what it means, to know whether to paint the entire ball purple, or half, or maybe add spots or something. He would follow the explanations making changes to his imaginary object and then when asked, is it X?, he could check and see.

But the trick is this: when he got one wrong and the math students explained why, he would say, "Oh, then it's trivial!" which to the mathematicians was always completely satisfying.

I'm restating in my own manner some of the points you've made.

in argumentation on degenerate cases is often inadvertent or deceptive — Srap Tasmaner

I agree that a pedantically correct application of notions in formal logic could be abusive sophistry in a context in which they are not understood But, at least at the moment, an actual example from public discourse doesn't leap to my mind. On the contrary, for example, if you were interviewed by a news outlet about the sunken boat and you tried to pull the stunt you mentioned, you would be pilloried. Or if you tried to make a vacuous argument among your non-logician friends, the best you'd get would be "Huh?" Meanwhile, of course, contentious, especially polemical, public discourse is rife with hideous, obnoxious, downright sneaky and pernicious use of all kinds of informal fallacies.

"begging the question," generally considered a fatal problem for an argument. That conditional is legitimate in form, and is generally a theorem, but it is fatal if relied on to make a substantive point or demonstrate a claim. It will only happen inadvertently ― in which case, a good-faith discussant will admit their error ― or with an intent to mislead by sophistry. — Srap Tasmaner

Question begging happens a lot. But, again, I can't think of an instance in public discourse in which the speaker appealed to P -> P as a tautology in order to convince anyone about anything.

As to complaints about formal logic, people who don't know about the subject often miss the point about such things as axioms proving themselves. In formal logic it is completely open that that one doesn't assert that a non-logical P holds in and of itself without respect to being either an axiom or derivable from the axioms. That is, when we say "Put P on the line as P is an axiom", we don't hide what we are doing. It's a kind of "benign" question begging. On the other hand, informal question begging is malign when, as is usual, it tries to hide the nature of the inference.

Fundamentally, all we're talking about in this case is arguing from a set a premises which are inconsistent — Srap Tasmaner

It is such a case. It is also a case of modus ponens. I've mentioned why I pointed that out.

necessarily inconsistent. — Srap Tasmaner

I don't know what that means.

But people arguing from inconsistent premises often make inferences that, while in themselves correct, continue to hide their inconsistency. — Srap Tasmaner

That is very very common in public discourse. Maybe it's the norm! It's maddening, frightening and ultimately depressing.

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.

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.

A -> ~A
A
therefore ~A

There is no interpretation in which both the premises are true.
— TonesInDeepFreeze

If the antecedent in the conditional is false, then the first premise is true. Now say the second premise is true. Then the conclusion does not follow.

If you insult me one more time, we're done. I'm satisfied with ending this discussion. — frank

It's hardly an insult to say that I don't understand why you are not minding the definitions and explanations. I've patiently given you information and explanations, repeated as has been needed. I am sincere when I wonder why you, in one post, seem to understand, but then blow right past again. As to whether you choose to reply to me, of course, that is entirely your choice

Here we go again:

If, in an interpretation, the antecedent is false, then, in that interpretation, the first is premise true, and the second premise is false not true. In that interpretation, the second premise is false.

You need to learn what an interpretation is. That's not an insult. It is good advice, given lagniappe in addition to all the information and explanations I'm giving you.

You can't deny that A is a conclusion because it is proven by the second premise, which is also A.

To deny A flows from the premises makes the curious argument that a premise has been eliminated by other premises.

In any event, premise 1 is reducible to ~A, so when you couple that with the second premise of A, you then can claim "A and ~A," allowing you to prove whatever you want.

Premise A & ~ A

Inferences:
:
A
~ A
A v C (cows bark)
~A
Therefore C

and so on and on

- 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:

↪TonesInDeepFreeze
"Trivial" has a clear meaning in analytical philosophy. — frank

I'm interested in what that definition is.

Meanwhile, the context here is examination of a particular formal argument. In that context, you fallaciously used the notion of triviality, as I detailed for you.

I would encourage you to write out in English the only case where both premises are true — frank

There are no interpretations in which both premises are true.