Leontiskos

This is perhaps my favorite proof for the modus tollens thus far. The question is whether that second step justifies the modus tollens. Does the "law ...

<I believe the reductio has failed and that the only strict way to draw ¬A is by using the modus tollens.> A?(B?¬B) ? ¬A {modus tollens} Now there is ...

As I see it, the problem is that this is a misunderstanding of a reductio. A zero-premise reductio makes no sense, and a one-premise reductio misunder...

What's at all wrong with this?: (S?¬P)?(B?¬B) ¬P ? ¬S We're going in circles. Time to go <meta>.

I was already convinced that RAA is insufficient. That as you say: RAA will not prove ¬A. The logic of the RAA proves (¬S v P), and the RAA choses one...

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 itse...

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

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:

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 t...

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

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 differe...

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 inva...

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 ...

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 ...

This faces the same problems that the modus tollens faces, as your second premise would function just as well for the second premise of the modus toll...

Yes, this seems right to me. I would add that if we have to choose between A and A?¬B?B, I will choose A every time. That is, if for some reason we mu...

- I understand perfectly well why many people call you a troll, but it isn't exactly the right word. The search for the right word goes on...

- I think it is silly, too. That's why I coined the term in a silly way. But you are the one who requires that sort of thing for the ¬A you wish to dr...

See the original post where I already explained this interpretation:

- Haha - it will take awhile to wrap my head around that, but "modus tollendo ponens" looks fun. :smile:

Because Lionino's second premise was also a quote. It is no coincidence that we are using quotes to express this special kind of modus tollens. Additi...

Thank you. As I put it: Yes, or rather I would want to say that a reductio is not involved at all. The modus tollens is what is really operative. --- ...

First argument A?(B?¬B) "(B?¬B) is false"* ? ¬A Second argument: A?(B?¬B) ¬(B?¬B) ? ¬A These are both modus tollens arguments. One could construct a r...

The proof still exists from your heavily-edited post. Why are you editing posts long after they have been responded to? See: I literally just gave you...

Yeah, you said you preferred the reductio to the modus tollens. Clearly @"Lionino" is following the conversation I am having with you much better than...

Right. It's the thing I've been going on about the whole time. One involves a supposition and one does not. The indirect proof (reductio) strictly spe...

Here's the quote: Am I not allowed to inquire and apply my disjunction as to what you might mean when you say that "in classical logic a contradiction...

Right. As I said, "Perhaps it is right to say that the contradiction introduces exceptions to invalidity but not to validity" (). Still, it seems to m...

I literally said it was an interpretation, not a translation. I still see it as the better option. This does not contradict what I have been saying. M...

Is this just obvious, ?

Such as your example indicates here: . Does classical logic not presuppose that such substitution is truth-preserving? Yes, indeed. The problem is tha...

This is why I think it is more interesting to compare the sense of a reductio ad absurdum to ((a?(b?¬b)) ? ¬a). Common language is equivocal in a way ...

Right. And I think this would always hold with a contradiction. A contradiction could be replaced by B if a second premise stipulates ¬B. By "each of ...

Right, so it's another case of abnormal behavior occasioned by the contradiction. Obviously the same thing arises: ((A?B)?¬A) is not valid. ((A?(B?¬B)...

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 t...

I don't know if I missed this or if it was an edit, but: The conclusion of a reductio is like, "This is an apple." Namely, "This instantiates the gene...

I didn't. Does that mean you are lying here? Instructively, it does not, because lying is not the same as saying something that is false. Even if, arg...

@"TonesInDeepFreeze" - The problem here is not so much that you do not know what you mean by 'particular.' No one in this thread has been able to unde...

Yes, exactly right. :up: And the point is presumably that the statement in the object-language does not translate the statement in the meta-language, ...

Nonsense, and you entirely failed to answer the question: I think your charges of "misrepresentation" are all bosh, but if you want to prove that you ...

You said: In saying this are you saying, among other things, that these two claims are not equivalent? "If A implies B & ~B, then A implies a contradi...

That was my interpretation of Banno, not Banno himself. See the post just above, posted a few seconds ago, for more detail on Banno's view. For instan...

Has everyone agreed by this point that 's truth table does not fully capture what a reductio is? (See bottom of post for truth table) ((a?(b?¬b)) ? ¬a...

If you know what you mean by 'particular', then surely you know what you mean by 'non-particular'? If you can identify a particular contradiction, sur...

This is part of the difficulty. If (b?¬b) is a particular contradiction, then what is a non-particular contradiction? That is what you must ask yourse...

Sorry - falling behind in this thread. I don't know if you saw my edit, which may now be redundant: Interesting, thanks for digging into this. Actuall...

Regarding reductio ad absurdum, last night I was having a dream. I was walking a trail I know well and I noticed that the topography was inaccurate. I...

Are you interpreting "a does not imply a contradiction" as the basis of a reductio (i.e. "Suppose a; a implies a contradiction; reject a")? If so, the...

Thanks. This is what I was trying to remember but could not find online (i.e. the complexities surrounding proofs of ¬¬a).

Yes, good. :up: Kreeft's point comes back. Right. As I have been saying it, "falsity incarnate" and "truth incarnate" are reifications. <FALSE> is a n...