I think I see what the discussion is turning around. The matter with modus tollens is that q could be otherwise, while in reductio it is not the case, by definition. Then again, I don't think it is meaningful or interesting. I am quite happy with my natural language conversions.

, what is "FALSE"?

It's your term.

This conversation is increasingly inane. Again, I seem to have reduced you to reciting gobbledygook.

If P can only be False, yes; otherwise, no. — Lionino

Right.

The matter with modus tollens is that q could be otherwise, while in reductio it is not the case by definition. Then again, I don't think it is meaningful or interesting. — Lionino

It's the thing I've been going on about the whole time.

I don't see a meaningful difference. — Lionino

One involves a supposition and one does not. The indirect proof (reductio) strictly speaking arrives at the conclusion of a disjunction, whereas a bona fide modus tollens does not: <p→q; ¬q; ∴¬p>.

What I claim is that what is at stake is not a bona fide modus tollens:

a→(b∧¬b)) is True, (b∧¬b)) is False, therefore a is False (from «1»). — Lionino

Note how you require natural English to give this argument, namely with the premise, "(b∧¬b)) is False." That was part of my point in <this post>. Compare the modus tollens you gave to this one:

• A→(B∧¬B)
• ¬(B∧¬B)
• ∴ ¬A

Are they both valid? As I said:

How is it that both (B∧¬B) and ¬(B∧¬B) can have the exact same effect on the antecedent, allowing us to draw ¬A? How is it that something and its negation can both be false? This is key to understanding my claim that two different senses of falsity are at play in these cases. — Leontiskos

So I ask again: How is it that something and its negation can both [function as the second premise of a modus tollens]? — Leontiskos

Edit:

Earlier quote in context:

We can apply Aristotelian syllogistic to diagnose the way that the modus tollens is being applied in the enthymeme:

((A→(B∧¬B))
∴ ¬A

Viz.:

Any consequent which is false proves the antecedent
(B∧¬B) is a consequent which is false
∴ (B∧¬B) proves the antecedent

In this case the middle term is not univocal. It is analogical (i.e. it posses analogical equivocity). Therefore a metabasis is occurring. — Leontiskos

i.e. It is unclear that "false" means the same thing in both premises. One is necessarily/always false, one is not. Does modus tollens care?

I think Leontiskos is talking about choosing between the conjuncts, while Banno is correctly stating that reduction ad absurdum is formally valid. — Lionino

I'm glad you followed this.

So I ask again: How is it that something and its negation can both [function as the second premise of a modus tollens]? — Leontiskos

It doesn't. Explained last time you made this claim...

Oh, and what is "FALSE"?

It doesn't. Explained last time you made this claim... — Banno

Yeah, you said you preferred the reductio to the modus tollens. Clearly @Lionino is following the conversation I am having with you much better than you are following the conversation I am having with him. I am discussing the modus tollens with him (alongside the reductio).

what is "FALSE"? — Banno

I tried to look at this here:

you said you preferred the reductio to the modus tollens. — Leontiskos

Rubbish.

Leo, what is "FALSE"?

Rubbish. — Banno

The proof still exists from your heavily-edited post. Why are you editing posts long after they have been responded to? See:

It might as well be a Reductio, although even there it is incomplete. It should be something like:

1. A→(B∧¬B) assumption
2. A assumption
3. B∧¬B 1,2, conditional proof
4. ~A 2, 3 reductio — Banno

Leo, what is "FALSE"? — Banno

I literally just gave you a link to a post where I examine that. Did you even read past the first sentence? You're acting like Michael. :roll:

First argument

• A→(B∧¬B)
• "(B∧¬B) is false"*
• ∴ ¬A

Second argument:

• A→(B∧¬B)
• ¬(B∧¬B)
• ∴ ¬A

These are both modus tollens arguments. One could construct a reductio ad absurdum similar to either one, but these arguments contain no supposition. They are meant to be direct proofs, not indirect proofs.

The first argument is treating (B∧¬B) one way, and the second is treating it a different way. Both arrive at the same conclusion from a diametrically opposed second premise. What I have been saying from the beginning of this discussion is that there are two different senses of falsity at play, which are not being properly differentiated. Surely the diametrically opposed second premises here demonstrate those two senses.

The second premise of the second argument is straightforward, as it treats (B∧¬B) like any other negatable proposition and then draws the standard modus tollens conclusion. In this case the falsity of the consequent is based on negation.

The second premise of the first argument is not straightforward, as it treats (B∧¬B) unlike any other proposition and then draws an exceptional kind of modus tollens conclusion with but a single premise (it is thought that the second premise is redundant, and in any case it is a premise written in English). In this case the consequent is thought to be false even though not negated. This is what I have been calling FALSE (link).

* This is an instance of FALSE

(@Lionino)

ρ,μ ⊢φ^~φ⊢ (ρ→~μ) ^ (μ→~ρ) — Banno

When we do a reductio
A, A→¬B∧B ⊢ ¬A is valid

But A, A→¬B∧B ⊢ A is also valid

So the question is: how do we choose between either? Isn't it by modus tollens?
p→q is True, q is False ⊢ p is False
a→b∧¬b is True, b∧¬b is False, ⊢ a is False

The post does not show that I said I 'preferred the reductio to the modus tollens".

And what I am after is a straight forward explanation of what "FALSE" is. Referring back to old posts that do not give a clear explanation anyway is in no way helpful.

But A, A→¬B∧B ⊢ A is also valid — Lionino

Thank you. As I put it:

What if we reject (1) instead? Then A is made true, but it does not imply (B∧¬B). Your proof for ¬A depends on an arbitrary preference for rejecting (2) rather than (1). — Leontiskos

So the question is: how do we choose between either? Isn't it by modus tollens? — Lionino

Yes, or rather I would want to say that a reductio is not involved at all. The modus tollens is what is really operative.

---

And what I am after is a straight forward explanation of what "FALSE" is. — Banno

The irony here, Banno, is the post where you pivoted from the modus tollens to the reductio (you literally rejected my modus tollens interpretation of your desire to draw ¬A). In the post you were responding to I attributed the modus tollens argument to you, and you then claimed that I had misattributed that argument. Whatever the case, <that post> still stands. Here is what I told you when you rejected my interpretation:

I am attributing the modus tollens to you because you are the one arguing for ¬A. If you are not using modus tollens to draw ¬A then how are you doing it? By reductio? — Leontiskos

That post is meant to be a tu quoque. I have been arguing against drawing ¬A. You have been arguing for drawing ¬A. That post is meant to say, "If you want to draw ¬A, you will have to tell us what you mean by FALSE." And recall:

What is at stake is meaning, not notation. To draw the modus tollens without ¬(B∧¬B) requires us to mean FALSE. You say that you are not using a modus tollens in the first argument. Fair enough: then you don't necessarily mean FALSE. — Leontiskos

What you have said is that, "in classical logic a contradiction is false," () and this is the basis of my questions about your position. "A contradiction is false; therefore we can draw ¬A." How so??

Edit: More precisely:

Thus when Banno says that a contradiction (b∧¬b) is false, does he mean that it is false or that it is FALSE?... [argument continues on] — Leontiskos

A→(B∧¬B)
¬(B∧¬B)
∴ ¬A — Leontiskos

If ¬(B∧¬B) is true, as it must be, then this is not a valid use of modus tollens.

Again, as i pointed out previously, you are comparing two very different arguments. But further your case is not helped by your not setting out the inferential steps in each case. Indeed, your new first example is not well-formed in classical logic. Why is the second line a quote? Was that in error (if so, then I suggest you edit it...). And if you were to modify it so as to be well-formed, what would it look like, if not ¬(B∧¬B)?

And how is "(B∧¬B) is false" and example of FALSE? Because of the quotes? What do they do?

What you write here is just a muddle.

Reductio:

Why is the second line a quote? — Banno

Because Lionino's second premise was also a quote. It is no coincidence that we are using quotes to express this special kind of modus tollens. Additionally, I pointed to this very fact in my own post. You aren't following along very carefully.

:lol:

I think you have here managed to set out Leo's confusion far more clearly than has Leo.

Can you see the answer?

Can you see the answer? — Banno

Can I? Yes. Do I? No.

- Haha - it will take awhile to wrap my head around that, but "modus tollendo ponens" looks fun. :smile:

Well, first, ρ,μ ⊢φ^~φ⊢ (ρ→~μ) ^ (μ→~ρ) is nonsense. I stuffed up. Am I allowed to edit it? :wink:

It should be
(ρ^μ) →φ^~φ⊢ (ρ→~μ) ^ (μ→~ρ)

If ¬(B∧¬B) is true, as it must be, then this is not a valid use of modus tollens. — Banno

See the original post where I already explained this interpretation:

Note that we could also do other things, such as treat the second premise as truth incarnate, but this is harder to see:

A→(B∧¬B)
¬(B∧¬B), but now conceived as "true"
∴ ¬A does not follow

...that is, if we conceive of the consequent as a proposition and the second premise as truth incarnate, then ¬A does not follow from the second premise (or from the consequent, absent a premise that negates the consequent qua proposition). — Leontiskos

truth incarnate... — Leontiskos

:grin:

Presumably that's TRUE? Or is it 'TRUE'?

- I think it is silly, too. That's why I coined the term in a silly way. But you are the one who requires that sort of thing for the ¬A you wish to draw. My point in all this is that we should not allow contradictions into sentences.

I don't see where I require anything like that in that post. After all, it's your post.

It might help if you explained what FALSE is.

As it stands, I can't take much more of this gobble. It's clear that reductio is valid as used in classical logic.

- I understand perfectly well why many people call you a troll, but it isn't exactly the right word. The search for the right word goes on...

Either inference, ρ→~μ or μ→~ρ, is valid. — Banno

And since we said ρ earlier, we choose ρ→~μ?

Even if that solves the problem between the conjunct, A, A→¬B∧B can entail quite literally anything. So I wonder why we choose ¬A as an entailment instead of quite literally anything, if not by modus tollens?
The image I posted doesn't help me much.

Reductio: — Lionino

P→(Q∨R), ¬¬P, ¬Q, ¬R entails ¬P
But it also entails anything one wants, including Q, P, ¬J.

So here's the apparent problem:
A, A→¬B∧B ⊢ ¬A
A, A→¬B∧B ⊢ A
It seems we can infer both A and ~A from the same thing. But that's because the two assumptions, A and A→¬B∧B, are inconsistent.

1. a → (b ∧ ~b)
2. If b is true (b ∧ ~b) is false. If b is false (b ∧ ~b) is false, so (b ∧ ~b) is false.
3.~a → ~(b ∧ ~b) - contraposition (1)
4. ~a - modus ponens (2,3)

No modus tollens required. You can do it as a disjunctive syllogism too. For the second premise you can assume the b is true or false. Doesn't matter because the conjunction will come out false.

But what I am more interested in would be a system that takes:

a→b
a→~b
a→ (b • ~b)

As premises with their own truth value, judging whether or not the implication is actually valid (i.e. that the two are even related). I suppose this is what relevance logic is for. It would make tables very long, but it doesn't seem that complicated.

It seems we can infer both A and ~A from the same thing. But that's because the two assumptions, A and A→¬B∧B, are inconsistent. — Banno

Yes, this seems right to me. I would add that if we have to choose between A and A→¬B∧B, I will choose A every time. That is, if for some reason we must take the path of the reductio, why would we choose to affirm A→¬B∧B?

1. a → (b ∧ ~b)
2. If b is true (b ∧ ~b) is false. If b is false (b ∧ ~b) is false, so (b ∧ ~b) is false.
3.~a → ~(b ∧ ~b) - contraposition (1)
4. ~a - modus ponens (2,3) — Count Timothy von Icarus

This faces the same problems that the modus tollens faces, as your second premise would function just as well for the second premise of the modus tollens.

What is at stake here is premise (2) (which I have called FALSE) along with using FALSE in a standard inference. As @Lionino has shown, it is not safe to assume that an inference which works with P will also work where P=(b∧~b).*

* From the start I have argued that this is because (b ∧ ~b) is a metabasis on falsity (when interpreted as a truth value), and any inference that uses it in this way is involved in this metabasis.

See:

Note that the (analogical) equivocity of 'false' flows into the inferential structure, and we could connote this with scare quotes. (B∧¬B) is "false" and therefore the conclusion is "implied." The argument is "valid." — Leontiskos