A reductio is as much a proof in classical propositional logic as is modus tollens. — Banno
I am concerned that logicians too often let the tail wag the dog. The ones I have in mind are good at manipulating symbols, but they have no way of knowing when their logic machine is working and when it is not. They take it on faith that it is always working and they outsource their thinking to it without remainder. — Leontiskos
...but they have no way of knowing when their logic machine is working and when it is not. — Leontiskos
and soEach of these systems sets out different ways of dealing with truth values. How the truth value of a contradiction is treated depends on which of these systems is in play. — Banno
Asking, as you do, how to treat the truth value of a contradiction apart from the system that sets out how a truth value is to be dealt with makes little sense. — Banno
Maybe not as much as you think.I have already responded to these charges. — Leontiskos
I'm not seeing a salient point here. Pretty demonstrably, you have made a series of claims that have been shown to be in error.At this point you either have an argument for "∴¬A" or you don't. Do you have one? If not, why are you still saying that ¬A is implied? — Leontiskos
In your conclusion you rejected assumption (2) instead of assumption (1). — Leontiskos
I still can't make sense of it. — Lionino
This is one of those funny places where symbolic logic seems to take a detour from what we mean in natural language. — flannel jesus
Do you think it is correct to translate this as: when it is not true that A implies a contradiction, we know A is true? — Lionino
I don't think there is any mystery around (A→(B∧¬B)) |= ¬A, if something implies a contradiction we may say it is false. — Lionino
I think calling them both assumptions has led to your confusion. Premise 1 is more of a GIVEN than an assumption. — flannel jesus
What rule of inference in classical logic are we appealing to? — Leontiskos
What rule of inference in classical logic are we appealing to? — Leontiskos
in classical propositional logic contradictions are false. — Banno
1. A→(B∧¬B) assumption
2. A assumption
3. B∧¬B 1,2, conditional proof
4. ~A 2, 3 reductio — Banno
...Or in other words, the metabasis is usually acknowledged to be a metabasis. As an example, when we posit some claim and then show that a contradiction would follow, we treat that contradiction as an outer bound on the logical system. We do not incorporate it into the inferential structure and continue arguing. Hence the fact that it is a special kind of move when we say, “Contradiction; Reject the supposition.” In a formal sense this move aims to ferret out an inconsistency, but however it is conceived, it ends up going beyond the internal workings of the inferential system (i.e. it is a form of metabasis). — Leontiskos
So, if you KNOW that A doesn't imply (C and ~C), but you also know that if A was false, A has to imply (C and ~C) by the fact that anything follows from falsehood, then you must know that A must be true.
This makes sense in the universe of classic symbolic logic, where everything has explicit truth values and implication means what it means there. — flannel jesus
This makes sense in the universe of classic symbolic logic, where everything has explicit truth values and implication means what it means there. — flannel jesus
I think there is a mystery why we can say it is false in this case. — Leontiskos
We are very far beyond Wikipedia at this point. At this point one can no longer simply appeal to authorities and logic machines. — Leontiskos
I'm interested in a system of symbolic logic that doens't deviate that drastically from what we normally mean by those expressions - a system of logic where you can say "I don't think A implies (C and ~C)" without simultaneously saying "A is true". — flannel jesus
Another way to think about it is, "The only way you can be CERTAIN that A doesn't apply a contradiction is if you know A is true." — flannel jesus
But I think it might be we are putting the horse before the cart. It is not that ¬(a→(b∧¬b)) being True makes A True, but that, due to the definition of material implication, ¬(a→(b∧¬b)) can only be True if A is true. — Lionino
Well, if something results in a contradiction, we are able to rule it out, aren't we? — Lionino
Perhaps my idea is that if someone engages in these sorts of inferences then there should be added an asterisk to their conclusion on account of the fact that this form of metabasis is highly questionable. I mostly want attention to be paid to what we are doing, and to be aware of when we are doing strange things. — Leontiskos
No, you asked for the rule of inference from classical logic - it's right there, common knowledge in wikipedia. — flannel jesus
material implication is an example of the principle of explosion — Leontiskos
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