Yep - well beyond my level, but I wish it wasn't.

So rounding back to your chat with , I'm reticent to place any firm boundary between formal and natural languages. Of course we could specify such a boundary, arbitrarily. That's cheating.

In one view, we have a formal object-language, and an informal or formal meta-language that includes the formal object-language. — TonesInDeepFreeze

Yes, and I rather like that. But as I understand it, Kripke's theory of truth involves one language, avoiding separating a meta language from an object language. It does this by only assigning a truth value to certain formulae, not to all.

The point is not that Kripke's theory of truth can be used as a basis for arithmetic (That seems to be a topic of some discussion amongst the academicians). So while fDrake is quite right,

Consistent systems capable of first order arithmetic can't contain their own truth predicate, so we don't put the predicate in. — fdrake

Kripke's system plays with consistency, creating a formal language that contains it's own truth predicate.

Hence, it might be premature to use "not containing it's own truth predicate" as a way to demarcate between formal and natural languages.

And a further point, whatever such demarcation might be offered, some clever logician might find a way to undermine it.

All speculative.

Consistent systems capable of first order arithmetic can't contain their own truth predicate — fdrake

I'm not so sure of this, since Kripke's theory of truth contains it's own truth predicate, and there is considerable work around its relation to arithmetic, I don't think we can yet rule out a Kripke-style first order arithmetic. I might be mistaken.

That is, the creativity of logicians is such that it might be better not to specify such a demarcation between formal ind informal languages, lest they invent a counter instance.

Yep. Curious how rendering them into a more formal language detracts from their use.

, Isn't formal language a part of natural language?

And Mathematics, also?

We understand each by what we do with it. Or rather, to understand a language is to be able to make use of it. Sets are not only defined by rules, but by our actually putting things into groups.

PI §201, yet again. There is a way of understanding a rule that is not found in setting it out but in following it.

...there is life... — Janus

Yep, nice. But is "There is life" then the negation of "There is death"? If we pars "There is life" as "there is something that is alive" then it's negation is "it is not the case that there is something that is alive", which is not the same as "there is something that is dead". There are things that are neither alive nor dead, so being dead is not the negation of being alive.

Or do we pars the whole first assumption as "everything that is alive will die" in which case we have an implicit temporality in "will" and need to include time. But then nothing is both alive and dead at the same time.

That is, in setting the passage out as a series of proposition, the negation dissipates.

Only if you agree to write the preface. — Srap Tasmaner

Leon and Hanover are more of an inspiration for Tones. They bring forth his best work.

Let’s publish all of Tone’s posts as an introduction to logic, and use the proceeds to fund the forum.

Let's just say it was pleasing to learn that at least one of the mods knows a bit about logic.

I think it shows that 'not-A' has at least two different senses. — Janus

Propositional logic deals in propositions. Your piece has the form of a modus ponens, but doesn't deal in propositions. That makes it interesting in several ways. But "not-a" is pretty well defined in propositional logic, in various equivalent ways. And by that I mean that the things we can do with negation in propositional logic are set. There are not different senses of "not-A" in propositional calculus.

The argument in the OP is for all intents a propositional argument. It is an instance of the application of modus ponens. And it is valid. The thread should have finished at 's post. The subsequent discussion displays ignorance of basic logic rather than any failing of that logic.

1.Life therefore death
2.Life
Therefore
3.Death. — Janus

is not an example of 'not-A', nor of propositional logic, although it is a striking example of the creativity of language.

Formal logic can set out some of the structures we might wish to find in natural languages.

The confused ignorance on display hereabouts might turn folk off looking at logic on detail, or encourage them to think it useless. The lesson from this thread, if there is one, might be that if folk begin by misunderstanding logic, they cannot conclude that logic tells us nothing about language.

Those who pretend to be defending a supposed common sense logic that is incompatible with formal logic are doing a disservice to rationality.

@TonesInDeepFreeze and to a lesser extent @Michael have presented a patient, consistent and correct account of the validity of the argument in the OP in the face of some extraordinary rubbish from folk we might have expected to know better.

A sad thread, this one. A low point in the history of the forums.

Let’s be careful and precise. We are philosophers here. — Fire Ologist

:rofl:

If you look at the argument in the OP, there can never be a case where both premises are true. — frank

Yep.

Any argument with inconsistent premises is valid, according to Tones — Leontiskos

Compare (A & ~A) → B. Of course Tones is right, because anything follows from such a contradiction.

Leon has no idea.

If A then not-A necessarily implies A and not-A, — Benkei

But that is not so. It doesn't. And I don't have Tone's patience. Have a think about what happens when A is false.

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

The level of dishonesty here is extraordinary.

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?

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.

Yep. Sad stuff.

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

Fucksake.

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

Interesting. Can we substitute salva veritate into an ironic statement? Seems to work.

If George is George Atkins then "George is going to open the store tomorrow" is true iff "Atkins is going to open the store tomorrow".

:wink:

It violates the LNC, which is foundational and introduced by Aristotle before modus ponens so he certainly didn't intend that the inference can work. — Benkei

Perhaps I misunderstood you. I had taken "it" and "the inference" to be the argument in the OP. Hence it appeared you were saying the argument in the OP was invalid.

Good to see that the number of folk who think that argument in the OP valid has dropped from a third to a quarter.

Still sad that it remains that high.

Neat.

Not quite propositional, though, using modus ponens on a mass noun.

You've been busy...

In this case, for instance, it is suggested that we conclude ~A by modus ponens. — Srap Tasmaner

Folk who understand that the argument is valid yet not sound will make no such conclusion.

The argument presented in the OP is valid, and has the form MPP. Pointing out the error of thinking otherwise has more of disabuse than abuse.

1. Meaning what exactly? — Srap Tasmaner

Exactly that. If you modify the substitution rule to remove substitution of the same variable on both sides of a function, can you demonstrate that the resulting calculus will be complete? Can you prove A→A, for example?

Yep.

Is the answer to (1) something I should care about? — Srap Tasmaner

That's entirely up to you. But you are on this thread, so forgive my presumption. Failing to see that the argument in the OP is valid is an indication of a lack of understanding of basic logic. Refusing to give an opinion says something else.

But what do you mean by 'abusive'? — TonesInDeepFreeze

Some folk think that pointing out an error os abusive. Odd, sad, but true

Again, 1 is false, and your argument (1-3) valid but unsound.

But (A->~A) & A is a contradiction. — Srap Tasmaner

Yep.

Would there be any harm in requiring that the conditional in a modus ponens have fresh variables on the right hand side? — Srap Tasmaner

Well for a start you would no longer be dealing with a complete version of propositional calculus...

But around here we're more interested in the practical use of logic, — Srap Tasmaner

Too often this is an excuse for poor logic.

Back to the question: Do you, Srap, agree that the argument in the OP is valid?

The contradiction is 1. ~A, 2. A. — Hanover

Only line 1 is not, ~A. It's A→~A.

It's
1. A → ~A (assumption)
2. A (assumption)
3. ~A (1,2,MPP)

Not

1. ~A (assumption)
2. A (assumption)
3. ~A (1)

If the OP uses propositional logic, it doesn't use propositional logic. — Hanover

This is false. It corresponds to line two of the truth table given above.

The OP is a factual question, not an issues of opinion. The one-third of folk who think that the argument is invalid are wrong. As wrong as if they had asserted that 2+2=5.

One-third of folk who have at least enough interest in logic to respond to the OP do not have a basic understanding of validity.

That's pretty sad. On a philosophy forum, it's pathetic. That is, it arouses pity.

It seems folk think A → ~A is a contradiction. It isn't.

Here is the truth table for the contradiction A and not-A:

Notice that the column under A and not-A is false for every assignment to A. That's why it is a contradiction: it is always false.

Here is the truth table for A→ ~A

Notice that the column for A→ ~A is not false if A is false. A → ~A is not a contradiction. Rather, it says that A is false.

If A is true, then A is false. Therefore A cannot be true.

, 's error, perhaps.

The OP uses propositional logic. In propositional logic, the argument is valid.

...but you may insist that it is as it is. — Hanover

Allowing substitution of any well-formed formula is not a personal foible. It is how propositional logic works. (φ, φ →ψ ⊨ ψ ) for any well-formed formula φ and ψ. Nothing says they must be different.

https://thephilosophyforum.com/discussion/11395/a-counterexample-to-modus-ponens/p1

Not the sort of thing I had in mind. Nor, frankly, am I inclined to go into details here, where simple substitution is apparently contentious. More agreement is needed before we might proceed to such other disagreements.

A thread of mine attempted amongst other things to discuss plausible cases in which modus ponens might not apply. It was lost in misunderstanding, which is a shame but perhaps not a surprise.

I've used ChatGPT occasionally to check things, usually nomenclature, sources, who first proposed an idea, or such. This case is a reminder to be aware of confirmation bias. ChatGPT gave @Hanover too great a confidence in his error.

It makes as much sense to define MP as excluding instances where A and not A coexist. — Hanover

As Tones explained, it's not MP you have misunderstood, but substitution. MP is a rule of inference, saying that if you have φ and φ →ψ, then you also have ψ, where φ and ψ are whatever formulae or propositions or sentences you are discussing. That includes substituting the same formula for both, and the negation of φ for ψ.

You are mistaken. Sorry.

No, but it might make for shorter threads. As it stands the acrimony will only build. Good for thread length, of course. Hanover has not understood substitution, as you have succinctly explained, and hasn't understood validity. We might allow some space for them to learn.

Logic is generally handled very badly here - as if it were a question of opinion as to what is valid and what is not, rather than of structure. That a third of folk think the argument in the OP is invalid... that's cause for concern.