limiting yourself to truth-functional logician — Leontiskos
You lie again. — TonesInDeepFreeze
...you don't know what you mean by a "particular contradiction." Unless you think you can use the word "particular" without having any idea what it would mean for something to be non-particular? — Leontiskos
In this are you saying that these two claims are not equivalent?
"If A implies B & ~B, then A implies a contradiction"
(a→(b∧¬b))→¬a — Leontiskos
One is a statement in the meta-language and the other in the object language. They are different levels of statement. — TonesInDeepFreeze
1. "If A implies B & ~B, then A implies a contradiction"
2. (a→(b∧¬b))→¬a
(My claim here is that (1) represents a reductio whereas (2) does not, even though ↪Banno thinks his truth table has shown that (2) translates a reductio.) — Leontiskos
Unless you think you can use the word "particular" without having any idea what it would mean for something to be non-particular? — Leontiskos
I think your charges of "misrepresentation" are all bosh — Leontiskos
You lied — TonesInDeepFreeze
Not exactly the model of a sage and wise poster. You came on here with a chip on your shoulder to everyone. I gave you a chance to have a good conversation, but I didn't see a change in your attitude. — Philosophim
There are particular apples and we can generalize about them. There is no apple that is not a particular apple. But we do say things like "If x is an apple, then x has a core". That is not claiming that there is an apple that is not a particular apple, but rather we can make generalizations about apples. — TonesInDeepFreeze
A contradiction is an assertion of Propositional Logic that is false in all situations; that is, it is false for all possible values of its variables. — Tautologies and Contradictions
Let G be a set of premises and a sentence P is not a member of G. And we want to show that G proves ~P. Then we may use any of the members of G in our argument. But, along with members of G, we also may suppose P to derive a contradiction, thus to show that G proves ~P. — TonesInDeepFreeze
From ~(A -> (B & ~B)) we infer that A implies no contradictions.
From (~A -> (B & ~B)) we infer that A implies no contradictions. — TonesInDeepFreeze
From there things start to make more sense. — Lionino
Does this support my claim that what is at stake is something other than a material conditional? The negation does not distribute to a material conditional in the way you are now distributing it. — Leontiskos
So I guess that, in order to say "A does not imply a contradiction", we would have to say instead (A→¬(B∧¬B)). From there things start to make more sense.
Since ¬(A→(B∧¬B)) does not translate to "A does not imply B and not-B". I have to fix my post above. — Lionino
When I said things like:
Thus when Banno says that a contradiction (b∧¬b) is false, does he mean that it is false or that it is FALSE?
— Leontiskos
...or when Lionino distinguished proposition-qua-variable from proposition-qua-truth-value, we were both pointing to this same valence where a material symbol (b∧¬b) has two legitimately different mental conceptions associated with it. In your language we would say that it can be conceived as a particular contradiction or a non-particular contradiction (non-particular being, in my terms, "falsity incarnate," or FALSE, or ABSURD, and in Lionino's earlier phrasing, contradiction-proposition-qua-truth-value, which truth value is necessarily false as opposed to contingently false). — Leontiskos
I understand that you'd think that B∧¬B should be able to be replaced by any proposition P, but that is not the case.
Example:
(A∧(B∧¬B))↔(B∧¬B) is valid
But (A∧C)↔C is invalid. — Lionino
Difference when the consequent is a contradiction:
(¬A→B)↔¬(A→B) is not valid.
(¬A→(B∧¬B))↔¬(A→(B∧¬B)) is valid.
So when the consequent is a contradiction, the ¬ may be pushed in. But when the consequent is a normal statement, you can't. — Lionino
And in each of the invalid cases if "B" could be made necessarily false they would presumably hold. — Leontiskos
(A→B)↔¬A, ¬B does entail however (A→B)↔¬A, even though (A→B)↔¬A is not True for any B, only when B is False. — Lionino
The antecedent of a negated material conditional is always true, and this goes back to my point in the edit you may have missed above. — Leontiskos
I'd like to explore this idea next:
I think that "A does not imply B" can't even be put in terms of logic, because "A does not imply B" conveys no information.
— Lionino — Lionino
I'd like to explore this idea next — Lionino
-Any consequent which is false proves the antecedent
-(B∧¬B) is a consequent which is false
∴ (B∧¬B) proves the antecedent — Leontiskos
Example:
(A∧(B∧¬B))↔(B∧¬B) is valid
But (A∧C)↔C is invalid. — Lionino
some rules of classical logic to come into conflict — Leontiskos
-Any consequent which is false proves the antecedent
-(B∧¬B) is a consequent which is false
∴ (B∧¬B) proves the antecedent — Leontiskos
I don't think that is logically rigorous. As you say, it is not a term in classical logic, and for good reason.A→FALSE — Leontiskos
Another way to read the first argument, and the one I prefer*, is as follows:
A→ABSURD
∴ "A cannot be affirmed" — Leontiskos
Neither do I. This distinction between false and FALSE is not my doing. It seems to be another case of Leontiskos confabulating arguments on the part of those who disagree with him.Banno may speak for himself, but I don't know what difference in reference you mean by spelling 'false' without caps and with all caps. — TonesInDeepFreeze
Presenting a statement that someone has not made is not presenting a translation.That was my interpretation of Banno, not Banno himself. — Leontiskos
I agree with Tones that you habitually misrepresent positions that are counter to your own, here and elsewhere.I think your charges of "misrepresentation" are all bosh — Leontiskos
I'll agree with that. It is incomplete. As Tones pointed out RAA is an inference rule, not a sequent within classical propositional logic. The inference allows one to infer ~ρ given a proof of (μ ^ ~μ) with ρ as assumption, a form displayed in the truth table.Has everyone agreed by this point that ↪Banno's truth table does not fully capture what a reductio is? — Leontiskos
This is rubbish. Given a proof of B and ~B from A as assumption, we may derive ~A as conclusion. This is the form of reductio inferences and is quite valid.The easiest way to see this is to note that a reductio ad absurdum is not formally valid — Leontiskos
Such as? — Lionino
I think that is a valid way to frame it. The thing about (B∧¬B) is that, differently from other formulas, it is always False. — Lionino
I don't think that is logically rigorous. As you say, it is not a term in classical logic, and for good reason.
If you want to say A always implies False, A→(B∧¬B) is good for that. While A→¬(B∧¬B) is "always implies True". — Lionino
If A implies a contradiction, not-A can be stated from LNC.
Dogs are fish. Fish, among other things, is defined as not-mammals. Dog is defined, among other things, as mammal. So we end up with "A mammal is not a mammal". Thus, "dogs are fish" has to be false, so "dogs are not fish" has to be true from LNC.
"... cannot be affirmed" does not stand to me as useful, as the LNC + LEM don't accept a third value. — Lionino
If this is right then (b∧¬b) introduces instances of formal equivalence that are not provable. — Leontiskos
Does classical logic not presuppose that such substitution is truth-preserving? — Leontiskos
(B∧¬B) is ambiguous, and can be interpreted as p or as FALSE (i.e. always-false). — Leontiskos
Why did you reject (1) and not (2) or (3)? The reductio is not formally valid in that tight sense. — Leontiskos
Presenting a statement that someone has not made is not presenting a translation. — Banno
Either inference, ρ→~μ or μ→~ρ, is valid. — Banno
and see that the choice is not in the reductio but in choosing between the conjuncts. — Banno
Ok, Presenting a statement that someone has not made is not presenting an interpretation. :roll:I literally said it was an interpretation, not a translation. — Leontiskos
Quite so. So what? It remains that RAA is a valid inference in classical propositional logic.This does not contradict what I have been saying. — Leontiskos
Yes, it is truth preserving. That
(A∧(B∧¬B))↔(B∧¬B) is valid
But (A∧C)↔C is invalid
does not make me think rules of logic are conflicting, because the equivalence or not with the second term of an «and-operator» is not a rule of logic. — Lionino
As I replied to sime, interpreting (B∧¬B) as P is not a good move, for P can be True or False, (B∧¬B) cannot be True ever. — Lionino
You would object why I rejected 1 instead of 2? I guess I see your point that it is not valid in a tight sense. After all, from A, A→¬B&B, everything follows, not just ¬A. — Lionino
My question then is whether we ever utilize (B∧¬B) without conceiving of it as a kind of P. — Leontiskos
So do we have a proof for ((a→(b∧¬b)) → ¬a)? — Leontiskos
Leo seems to think that choosing between ρ→~μ and μ→~ρ somehow involves an act of will that is outside formal logic. He concludes that somehow reductio is invalid. His is a mistaken view. Either inference, ρ→~μ or μ→~ρ, is valid.
Indeed, the "problem" is not with reduction, but with and-elimination. And-elimination has this form
ρ^μ ⊢ρ, or ρ^μ ⊢μ. We can choose which inference to use, but both are quite valid.
We can write RAA as inferring an and-sentence, a conjunct:
ρ,μ ⊢φ^~φ⊢ (ρ→~μ) ^ (μ→~ρ)
and see that the choice is not in the reductio but in choosing between the conjuncts.
Leo is quite wrong to assert that Reductio Ad Absurdum is invalid. — Banno
I think the only way we can utilize logical inference is by using the modus tollens — Leontiskos
Ok, Presenting a statement that someone has not made is not presenting an interpretation. — Banno
Thus when Banno says that a contradiction (b∧¬b) is false, does he mean that it is false or that it is FALSE? — Leontiskos
Quite so. So what? It remains that RAA is a valid inference in classical propositional logic. — Banno
A reductio without choosing between them is not yet a reductio. — 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.