The grain of truth in @Leontiskos' position is that reductio arguments need to be used with care. If we have a bunch of assumptions that lead to a falsehood, we can throw out any and all of the assumptions.

This has a place in many philosophical critiques. For example, it underpins the Duhem–Quine thesis, and greatly complicated Popper's fablsificationism.

Not quite so in formal systems. In classical logic if assumptions a,b,c,d lead to a contradiction we can only conclude that their conjunct is false.

But this allows is also to conclude certain implications. We might infer for example that a^b^c→~d. That is, if a, b, and c are true, it must be d that is false.

Worth keeping in mind

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This is the path that @Banno and @TonesInDeepFreeze have chosen:

• (a→(b∧¬b)) → ¬a

They have two possible routes which could be used to reach their destination: Mt. Caradhras or the Mines of Moria. Gimli suggest that they take the Mines, but Banno knows that "the dwarves dug too deep in their greed, awakening horrors in the depths." So they try the pass of Caradhras, but it turns out to be unworkable, smothered by snow and storm. Do they dare tempt the Mines of Moria? Viewers must wait and see... :grin:

• Mt. Caradhras = reductio ad absurdum
• Mines of Moria = A modus tollens with only one premise

Note: For those considering the mines, two posts may be especially useful: first, second.

The grain of truth in Leontiskos' position is that reductio arguments need to be used with care. — Banno

But my fuller position is that any inference utilizing strange senses of would-be familiar logical concepts must be used with care. I am not opposed to the Mines of Moria, but I don't think people are taking enough care in traversing them.

But couldn't we just assume B here and get ~A just the same? "If B then ~A," seems to work fine here because the conjunct is still going to come up false.

And then we can do the same thing assuming ~B. That covers all our options assuming LEM.

I guess I'm not seeing the trouble here. I can see the trouble with proofs by contradiction in mathematics that prove things for which no constructive proof exists. That makes sense because, on some philosophies of mathematics, an entity doesn't exist until the constructive proof does (and perhaps it can't exist).

The problem is that this A, A→¬B∧B ⊢ ¬A was given as reductio ad absurdum. But this entails anything. — Lionino

To my mind the explosion only occurs if you don't reject either of the two premises. If you reject either of the two premises via reductio, explosion is avoided, no? When we reject (A→¬B∧B) and accept A, explosion no longer follows. The same is true if we accept (A→¬B∧B) and reject A.

What if we reject (1) instead? Then A is made true, but it does not imply (B∧¬B). — Leontiskos

But my fuller position is that any inference utilizing strange senses of would-be familiar logical concepts must be used with care. I am not opposed to the Mines of Moria, but I don't think people are taking enough care in traversing them. — Leontiskos

And yet your claim that Reductio is invalid is just wrong.

And yet your claim that Reductio is invalid is just wrong. — Banno

What I have consistently said is that reductio is not valid in the same way that a direct proof is. Perhaps I slipped at some point and called it invalid. In any case, I don't see this as a big mistake. As I said earlier, what is entailed is the disjunction, not some subset of the disjuncts. That is, in this case if we are to avoid the contradiction as a reductio requires that we do, then we must reject either (A→¬B∧B) or else A.

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To me, RAA depends on modus tollens. — Lionino

What does "depends on" do here?

Modus Tollens: ρ→φ, ~φ ⊢ ~ρ
RAA: ρ→(φ^~φ) ⊢ ~ρ

Both are equally useable rules of inference.

What I have consistently said is that reductio is not valid in the same way that a direct proof is. — Leontiskos

So, what is a "direct proof"? I gather you think using MT is direct, but RAA isn't? What's the distinction here?

While you are there, what does "FALSE" mean?

But couldn't we just assume B here and get ~A just the same? "If B then ~A," seems to work fine here because the conjunct is still going to come up false. — Count Timothy von Icarus

You're basically preaching to the choir. <This> is the third time I presented that idea. But a proof that requires an additional assumption is different from one that does not.

Edit: Sorry, misunderstood - I guess I'm wondering how a proof by exhaustive cases fits in. That's what you're introducing, and it's a good introduction.

And then we can do the same thing assuming ~B. That covers all our options assuming LEM. — Count Timothy von Icarus

I am wondering if allowing contradictions in this way fiddles with the LEM, but this is just conjecture.

I guess I'm not seeing the trouble here. I can see the trouble with proofs by contradiction in mathematics that prove things for which no constructive proof exists. That makes sense because, on some philosophies of mathematics, an entity doesn't exist until the constructive proof does (and perhaps it can't exist). — Count Timothy von Icarus

The trouble is simply that (b∧¬b) has been consistently creating unexpected behavior in this thread, but I find your approach interesting. Do continue.

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¬c does not entail ¬a

If a → c it does. Contraposition, flip em and switch em (reverse the order and negate both).

Brad always wears his hat (a) on Mondays (c).

If Brad is not wearing his hat (~c) it cannot be Monday (~a).

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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

Modus Ponens is ρ→φ, ρ, ⊢ φ.

Not seeing it in 4.

So, what is a "direct proof"? I gather you think using MT is direct, but RAA isn't? WHat's the distinction here? — Banno

Modus tollens requires no "and-elimination" step. Is that a good way to put it in your language?

Nope.

Straight RAA does not require the "and elimination". It's an additional step when there are multiple assumptions.

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While you are there, what does "FALSE" mean? — Banno

If you don't want to read the posts where I quite sincerely tried to get at this, we could just say that FALSE is what is necessary to get the modus tollens to run with only one premise. It is the sense of the contradiction required for the valid inference.

It is what is supposed to answer this question:

how do you prove that you may derive ~ρ from ρ→(φ^~φ)? — Lionino

I consider it an open question as to whether this question is answerable.

Straight RAA does not require the "and elimination". It's an additional step when there are multiple assumptions. — Banno

I have never seen a reductio that does not have multiple assumptions.

Edit: this is what I think a one-premise reductio would look like:

A→ABSURD
∴ "A cannot be affirmed"

...

Introducing ABSURD in the way I did above destroys the LEM of classical logic. — Leontiskos

I didn't contrapose them when I copied and pasted it, I inverted it.

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.~(b ∧ ~b) → ~a - contraposition (1)
4. ~a - modus ponens (2,3)



~(b ∧ ~b) is given in 2
~(b ∧ ~b) → ~a

That's not contraposed, I do see now that I didn't contrapose them in the post despite referencing it.

It's ~c → ~a.

This is of course assuming ~c because c would be affirming a contradiction.

And the disjunctive syllogism
1. a → (b ∧ ~b)
2. ~(b ∧ ~b)
3.~a V (b ∧ ~b) - material implication (1)
4. ~a - disjunctive syllogism (2,3)

OK. Much clearer Thanks.

(2) amounts to

2. ~(b^~b)

I have never seen a reductio that does not have multiple assumptions. — Leontiskos

:lol:

I give in.

- I'm quite serious. See my edit to that post, which may help you.

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The truth-functionalist is likely to object to me, “But your claims are not verifiable within classical logic!” Yes, that is much the point. When we talk about metabasis eis allo genos, or contradiction per se, or reductio ad absurdum, we are always engaged in some variety of metalogical discourse.

...

How can we start inching towards the difference between ‘false’ and ‘FALSE’? First I should say that the “proposition” (b∧¬b) can be either. It can be interpreted as false or as FALSE each time we touch it with our mind. What this means is that terms like (b∧¬b) or ‘false’ are metalogically equivocal or ambiguous given the question we are considering... — Leontiskos

The problem as I see it is that those who will not move into an analysis of the language are trying to solve a metalogical problem with the logic itself, and this cannot be done. We must move into metalogical discourse, and because of this I would propose analyzing the nature of (b∧¬b) and the attendant inferences using English rather than (redundant) logical translations. The logical translations involving that term seem by this point to be clearly underdetermined.