If P can only be False, yes; otherwise, no. — Lionino
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
I don't see a meaningful difference. — Lionino
a→(b∧¬b)) is True, (b∧¬b)) is False, therefore a is False (from «1»). — Lionino
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
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
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... — Banno
what is "FALSE"? — Banno
Rubbish.you said you preferred the reductio to the modus tollens. — Leontiskos
Rubbish. — Banno
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
But A, A→¬B∧B ⊢ A is also valid — Lionino
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
And what I am after is a straight forward explanation of what "FALSE" is. — Banno
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
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
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
If ¬(B∧¬B) is true, as it must be, then this is not a valid use of modus tollens.A→(B∧¬B)
¬(B∧¬B)
∴ ¬A — Leontiskos
Why is the second line a quote? — Banno
If ¬(B∧¬B) is true, as it must be, then this is not a valid use of modus tollens. — Banno
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
Either inference, ρ→~μ or μ→~ρ, is valid. — Banno
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
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
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
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.