TonesInDeepFreeze

Speaking of pointless execises, you first post to me was one: I gave two proofs. Your point in directing me to do that turned out to be ill-premised, ...

It's not a misinterpretation. To say that P is contradictory is to say that P is a contradiction. A statement is contradictory if it is a contradictio...

(1) I take 'the contradictory statement is P' to mean that P is a contradiction, as a contradictory statement is a contradiction. (2) But maybe you me...

I suspect you don't understand what you wrote.

Your answer is incorrect. You should give up, since Lionino's post is perfectly in accord with what I have said: the pair of statements are not a cont...

Right, it's not a coincidence. That doesn't entail anything about the material conditional in Boolean logic.

Explosion doesn't make the material conditional in Boolean logic used for computing nonsensical.

(1) You changed the sentence. Here is what you wrote: (2) "If lizards were purple then they would be smarter" is not a contradiction, a fortiori not a...

I've not claimed that anything I've said dissolves any difficulties with material implication. I'm thinking of it in context of symbolic logic, inform...

In the same vein as above, 'true', 'sound' and 'valid have definitions in logic. Of course, it's your prerogative to use any sense you like. But it be...

An informal sense of 'contradict' is 'to imply the opposite or a denial of'; and an informal sense of 'denial' is 'a proposition so related to another...

Again, a contradiction is a statement and its negation. If there is a contradiction then you could show that both a statement and its negation are imp...

Because even informally, the statements don't entail a both statement and its negation. I wouldn't use the word 'logical' since that has a certain mea...

As someone pointed out, the use of variables suggest formality. But, of course, we may address the question in both formal and informal contexts. And ...

There's a mistake in the last row. The value of ~A v ~B is T. So there are two rows, not just one, where (A -> B & (A -> ~B) is true.

The crank claims that we may look in a textbook in mathematics to see that mathematics doesn't agree. What textbooks are those? And notice that the cr...

Mathematical logic formalizes the logic used in other mathematics. The explication of '=' in mathematical logic conforms to the use in mathematics. Th...

In ordinary formal logic and classical mathematics, the material conditional obtains. But, of course, there are other natural language senses. In ever...

The answer is 'yes' and they do not contradict each other. You can read the several differently arranged proofs of that in this thread.

There's no note or link needed. You can find out about material implication all over the place. I'm not your linking service. You skipped what I said ...

I am not. I'm treating '->' as standing for material implication as is ordinary. They are very different. Actually, you insulted me. I hadn't written ...

That is what I replied to.

'imply ~A' is not a proposition, and I didn't say that it is, so I don't require it as an assumption.

I don't claim that one may not discuss all kinds of non-formal, formal, alternative formal, or philsophically formal or informal, or mystical New Age ...

I don't require such an assumption. "imply ~A" is not even a proposition.

That would be my leaning too.

Yes, we say 'necessary' and 'sufficient' conditions. But that is not "necessarily implies' or 'necessarily leads to'. If P -> Q, then P is a sufficien...

Actually, your imperative "Then give your proof" was curt, especially as spoken to someone giving you correct information. The proof is so simple that...

It is not trolling to point out an incorrect statement, and it not trolling nor handwaving to suggest that one can look in textbooks to see that the s...

Look in any textbook on symbolic logic.

So what? '->' is ordinarily regarded as standing for material implication that is not "P necessarily implies Q" nor "P necessarily leads to Q".

Both propositions together imply ~A. The conjunction of the propositions implies ~A. The premise set {A->B, A->~B} implies ~A. Right.

I replied exactly to what you wrote. What you wrote is wrong. A -> B or material implication is not "A necessarily implies B".

Are you serious? You don't know how to prove it yourself? Proof: (1) (A -> B) ... premise (2) (A -> ~B) ... premise (3) A ... toward a contradiction (...

Wrong. Material implication does not require necessity.

They imply ~A. I don't know what you mean by 'cleavage' and 'captured' in this context. But in logic systems we can write contradictions. Indeed, we o...

A contradiction is a formula of the form P & ~P, or in other contexts the pair {P ~P}. We don't have to check four different things to see that formul...

If this is about material implication then the answer is utterly simple: A -> B A -> ~B are not together inconsistent, since they are both true when A...

Identity. I don't know how to make it more clear than I already have. x is x. With models: '=' is interpreted as {<x x> | x e U} where U is the univer...

The symbols are standard. The words are ordinary for logic and mathematics, or if personal, they're defined. So maybe it's something else. Most glarin...

Explicity stated in any textbook in mathematical logic.

'=' is interpreted: For any terms 'T' and 'S' T = S is true if and only if the denotation of 'T' is the denotation of 'S'. Consult any textbook in mat...

It is crystal clear that '=' is interpreted as 'is' in mathematics, since it is explicitly stated that '=' is interpreted as 'is' in mathematics. That...

If A is false, then (A->B) & (A->~B) is true. So (A->B) & (A->~B) does not entail a contradiction. It's that simple.

Further Adventures Of The Crank: Sales Clerk: That will be five dollars. / The crank puts four one dollar bills on the counter / Crank: There you go, ...

Mathematics adheres to the law of identity, since in mathematics, for any x, x=x, which is to say, for any x, x is x.

I have two different dollar bills in my pocket. They are not the same dollar bill. But they are equal in value. (1) bill 1 is not the same as bill 2 b...

I was rushing. My mistake.

No, it doesn't. It's called 'the deduction theorem'. For example: P, Q, R |- S is equivalent with P, Q |- R -> S

You haven't paid attention to my answer to that. Now, you're arguing by mere assertion and repeated mere assertion. Moreover, if P5 is deemed in and o...