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) is truth-functionally valid, but the implication in the first half of the biconditional is not the same implication that is used in a reductio ad absurdum.

The easiest way to see this is to note that a reductio ad absurdum is not formally valid, and we can see this by noting that the reductio in 's post (now highly edited) does not prove his conclusion:

As I noted earlier in response to Tones' reductio, a reductio is an indirect proof which is not valid in the same way that direct proofs are. You can see this by examining your conclusion. In your conclusion you rejected assumption (2) instead of assumption (1). Why did you do that? In fact it was mere whim on your part, and that is the weakness of a reductio.*

* A reductio requires special background conditions. In this case it would require the background condition that (1) is more plausible than (2). — Leontiskos

That the conclusion of a reductio is not formally provable is the first hint that the implication in Banno's formula is not the implication of reductio ad absurdum. If it were then a reductio would be formally provable.

This is related to Lionino's point about the associativity of the → operator in the case of a contradiction:

I think I finally solved my own problem. When translating it to natural language, I was misplacing the associativity of the → operator in this case.
So ¬(A → (B∧ ¬B)) is the same as (¬A) → (B∧ ¬B), which may be read as "Not-A implies a contradiction", it can't read as "A does not imply a contradiction". We would have to say something like A ¬→ (B∧ ¬B), which most checkers will reject as improper formatting, so we just say A → ¬(B∧ ¬B), which can be read as "A implies not-a-contradiction", more naturally as "A does not imply a contradiction". — Lionino

And it is also related to Tone's point

"If A implies B & ~B, then A implies a contradiction" is true, but it is a statement about the sentences, not a translation of them. — TonesInDeepFreeze

If "A implies a contradiction" were a translation of the sentences, then it would mean ¬A. That is, it would be formally equivalent to ¬A. This is what Banno thinks his truth table has shown, and such is in line with 's claim that "(p ^ ~p) is false in classical propositional logic," as if we could formally translate a contradiction as "false" (whatever that is supposed to mean). Banno's response to Tones should therefore be, " <If A implies B & ~B, then A implies a contradiction> is true, and it is a translation of the sentences, not a statement about them."

Or if you think it is only truth-functional if it fits in a truth-table: — Banno

'non-particular' is your word. It's up to you to say what you mean by it. — TonesInDeepFreeze

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, surely you can identify a contradiction that is not particular?

Perhaps now you are beginning to see the point?

(3) "B & ~B" is a particular contradiction, not just "a contradiction". Even though all contradictions are equivalent, a translation should not throw away the particular sentences that happened to be mentioned. — TonesInDeepFreeze

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 yourself. 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).

The analogical equivocity of the particular metabasis under consideration incorporates both particularity and non-particularity simultaneously. This equivocity is precisely what the reductio runs on, for when the reductio is begun the implicit contradiction is considered in particular terms, but by the end of the reductio it has been isolated and re-conceived in its non-particular sense. Or, at the beginning of a reductio when we only suspect a contradiction, that contradiction lives within the system in a particular way and not in a non-particular way. Once it is "ferreted out" it becomes non-particular, a contradiction qua contradiction, and this non-particular sense is what is required for the special reductio inference.

Sorry - falling behind in this thread.

↪Leontiskos I solved my main problem just right above. — Lionino

I don't know if you saw my edit, which may now be redundant:

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

But for the record I do accept this as a valid rhetorical move. However when it comes to propositional logic, from
P1: A
P2: A→contradict
The conclusion can be whatever we want, from explosion — Lionino

Interesting, thanks for digging into this. Actually thanks for digging into all of the stuff you have dug into in this thread. It has helped me piggyback onto a lot of other ideas.

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

Very good. This is another instance of the wrinkle that is created when the contradiction is allowed.

...so I think now it is a bit more clear why ¬(a→(b∧¬b)) is True only when A is True, the second member is always False and the or-operator returns True when at least one variable is True. — Lionino

Yes, but as @Count Timothy von Icarus and I have noted, it seems simpler to say that (¬(p→q)→p). 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.

(a→b) ↔ (¬a∨b)
¬(a→b) ↔ ¬(¬a∨b)
However (a∨b) and ¬(¬a∨b) aren't the same
So ¬(a→b) and (a∨b) aren't the same

(a→(b∧¬b)) ↔ (¬a∨(b∧¬b))
¬(a→(b∧¬b)) ↔ ¬(¬a∨(b∧¬b))
(¬a→(b∧¬b)) ↔ ¬(¬a∨(b∧¬b))
Since ¬(¬a∨(b∧¬b)) is the same as (a∨(b∧¬b))
(¬a→(b∧¬b)) ↔ (a∨(b∧¬b)) — Lionino

Good. This back to 's point about ¬¬a.

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 am usually semi-lucid when I dream, and so I decided to try to change the topography to make it more like it is in real life. As soon as I did this I noticed that this is similar to a reductio. In both cases an additional, uncharacteristic level of will emerges.

The reductio is an uncharacteristically teleological move for truth-functional logic:

(3) A ... toward a contradiction — TonesInDeepFreeze

The one who performs the reductio sees an opportunity to produce a contradiction and then decides to pursue it in order to achieve the inference desired (which inference is, again, a metabasis).

The Medievals would have called the truth-functionality of classical logic something which pertains to the intellect (as opposed to the will). It is supposed to be purely formal, purely intellectual, and in no way willful. As you and I know, this is not entirely true since any human act involves the will, and therefore in any logic the teleological end of the acts at hand must involve the will. Nevertheless, a reductio involves the will over and above the way that direct inferences involve the will. The reductio attempts to leverage a contingency about the problem at hand in order to wield the contradiction and draw a conclusion. Hence the difference between a supposition and a mere assumption is that the supposition acknowledges the teleological motive of the will in a way that an assumption does not (Tones called his move a supposition whereas Banno called the same move an assumption).

So we could say that the metabasis eis allo genos and the essence of a reductio ad absurdum is found not only in the unique inference that concludes a reductio, but also in its starting point: the supposition. This is what separates the supposition from the other assumptions, even though this difference is mental and not formal. And if we pay very close attention we will see that the formal conclusion is different from the teleological conclusion. The formal conclusion is that the system which includes the supposition is inconsistent. The teleological conclusion is that we should reject the supposition rather than a different premise. There is a miniature inference from the formal conclusion to the teleological conclusion, and this tends to be ignored by most students of classical logic. Put differently, the reductio strictly speaking only tells us that something cannot be supposed. It is a second step to say that that which cannot be supposed is in fact false.

Note too that if someone is a strict univocalist with respect to inference then the reductio and all metabasis is disallowed. A reductio is an inference in a sense that is analogous to the way that, say, a modus tollens is an inference. If -inference- cannot function analogically, the reductio cannot succeed. These are fun little wrinkles in the purported truth-functionality of classical logic. Of course some might in fact disallow reductio and prefer a stricter logically system, but this system will be less powerful vis-a-vis achieving natural inferences.

The main problem for me is, why can we read a→(b∧¬b) as "a implies a contradiction" but not ¬(a→(b∧¬b)) as "a does not imply a contradiction? — Lionino

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, then I again think it is because a reductio is not reducible to a truth-functional move. A reductio requires more than negation and falsity.

Edit:

I think I finally solved my own problem. When translating it to natural language, I was misplacing the associativity of the → operator.
So ¬(A → (B∧ ¬B)) is the same as (¬A) → (B∧ ¬B) — 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.

Suppose that the logic concerned is weaker than Peano arithmetic, such that it can prove its own consistency. Then in this case, a proof of ¬¬a metalogically implies that ¬a isn't provable, i.e that a does not imply a contradiction.

But if the axiomatic system contains Peano arithmetic such that the second incompleteness theorem holds, then a proof of ¬¬a does not necessarily imply the absence of a proof of ¬a, since Peano arithmetic cannot prove its own consistency. — sime

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

The negation of a contradiction is always true, and being true it is implied by anything, true or false. — Count Timothy von Icarus

Yes, good. :up: Kreeft's point comes back.

In a normal conversation, we might ask "but what if A really only implies B and not B and not-B?" Or conversely: "what if A only implies not-B but does not actually imply B?" But the way implication works here it is not an additional premise we can reject, we don't assign a truth value to it except in virtue of the truth values of A and B themselves. — Count Timothy von Icarus

Right. As I have been saying it, "falsity incarnate" and "truth incarnate" are reifications. <FALSE> is a new idea, more or less foreign to classical logic.

However, there is a quite good reason not to do this in symbolic logic. Once you start getting into "what 'really' entails what," you get into judgement calls and a simple mechanical process won't be able to handle these. — Count Timothy von Icarus

Yep.

But of course, you still need judgement to make sure your statements aren't nonsense, so you just kick that problem back a level. A proof from contradiction is only going to be convincing if we believe that A really does imply both B and not-B. I know plenty of skepticism has been raised against proofs from contradiction in general, outside of this issue, but for many uses they seem pretty unobjectionable to me. — Count Timothy von Icarus

I think metabasis is useful, but I don't close my eyes to the fact that it is metabasis:

Every time we make an inference on the basis of a contradiction a metabasis eis allo genos occurs (i.e. the sphere of discourse shifts in such a way that the demonstrative validity of the inference is precluded). Usually inferences made on the basis of a contradiction are not made on the basis of a contradiction “contained within the interior logical flow” of an argument. 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

Reductio ad absurdum is useful and important, but it is not formally valid in the same way that direct proofs are (and because of this it is a (useful) metabasis). We need to recognize that we are doing something special with a reductio, and that a reductio-inference to ¬A is quite different from a direct inference to ¬A. So if someone wants to say that ¬A is implied they must put an asterisk next to implied*.

Where — Lionino

Here:

1. A→(B∧¬B) assumption
2. A assumption
3. B∧¬B 1,2, conditional proof
4. ~A 2, 3 reductio — Banno

A reductio is not truth-functional. If we want to stick to strict truth functionality then we cannot accept reductios. In that case we can only think of them as indicating the inconsistency of a system, not as grounds for denying one thing or another. Technically a reductio is a form of special pleading (i.e. “Please let me reject this premise rather than that one, for no particular reason”). I think I can only be wrong about this if there is some principled difference between a supposition and a premise (or an assumption). A reductio is a kind of bridge between formal logic and the real world, and this is because of the background conditions it presupposes. In a purely formal sense no proposition is inherently more or less plausible than another, and therefore there is no reason to reject one premise rather than another in the event of a contradiction.

No, you asked for the rule of inference from classical logic - it's right there, common knowledge in wikipedia. — flannel jesus

I'm sorry, but as someone who thinks that material implication is an example of the principle of explosion you are out of your depth here. Material implication is entirely different from the principle of explosion, and an argument from the authority of Wikipedia is not a sufficient answer to the question I am asking.

Well, if something results in a contradiction, we are able to rule it out, aren't we? — Lionino

Sure, so long as we are recognizing that "to rule it out" is a special move, unique and irreducible to any other move in classical logic. Specifically, the implication that we deny is in this case is no longer material implication.

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

This all goes hand-in-hand with the fact that Banno's reductio has no power, strictly speaking, to draw the conclusion ¬A. A reductio is not a proof in the strict sense, and this is precisely what a metabasis is: a form of non-strict inference. Anyone can deny that reductio without being the worse for wear, logically speaking.

At one point we may find ourselves in contradiction and if all we do is hold to two contradictory principles we'll do nothing but compute them (if that is our true desire), and die. — Moliere

This strikes me as a strawman, but perhaps we can let it stand as a warning. Perhaps you wish to warn, "You may not be doing this, but be sure that you do not do this." This is fine as far as it goes, and I have said similar things:

Truth be told, PDE is an unwieldy principle. There are cases (such as the hysterectomy) where it seems to obviously apply, but it has often been noted that in other cases the principle can be easily abused. Our topsy-turvy discussion in the other thread got at some of the nuance involved. — Leontiskos

This is the sort of ambiguity that seems to always follow the PDE, namely cases which are hard to decide. So this is in line with the tradition of the PDE, and I think it is good to recognize such limitations. — Leontiskos

Now a parable is able to do what a rational argument could never do, and parables certainly have their place in ethics. Yet as I see it, this parable of yours stands, but only on one foot. In the world of parables, it feels a bit flat and one-dimensional, perhaps because its roots go no further than satire; its target has no more depth than the determinist or monomaniac.

The better parable as I see it is not Buridan's Ass, but Balaam's Ass. At times wisdom will speak through the beast, from the source it is least expected, and it will cut through the rationalizing foolishness of the rider. Granted, there is no good reason why Balaam's Ass cannot speak through Buridan's Beast (and yet we have now left syllogistic).

Lastly, I will point out that lessons and parables and warnings have their place, but of all things they are least helped by repetition. To beat the drum of a parable or a warning again and again does no good, especially if it stands only on one foot. It will tire and collapse, and lose what efficacy it might have had. Confusing the parable for a philosophical example causes it to fall prey to this form of repetition.

(You often give voice to a tongue that should not be foreign to philosophy but is nevertheless opaque to the analytic philosophy that dominates English-speaking forums like this one. Your style of pacifism is a potent example. I am not averse to speaking in this tongue, but only rarely would I expect it to bear fruit in a place like this. It's hard to speak about parables in a place like this.)

- We are very far beyond Wikipedia at this point. At this point one can no longer simply appeal to authorities and logic machines. They have to set out the arguments themselves. One must think about the difference between a reductio ad absurdum and a direct proof, as I believe they should have done when the topic came up in logic class.

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

I noticed that there is in fact a second problem with your reductio. You told me that in classical logic contradictions are to be treated as false, but in your reductio you do not treat the contradiction as false. You treat it as a contradiction, as an outer bound on the logic:

...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

You are appealing to the move of a reductio, “Contradiction; Reject supposition.” But there is no rule, “False; Reject supposition.” Therefore you are clearly not being consistent in treating the contradiction as false.

Now you could of course continue your proof and try to use a modus tollens to arrive at (A∧¬A), But not only will this result in the same exact problem, but it would result in the additional problem of utilizing FALSE in the way I pointed out <here>.

(It would seem that you are wrong in claiming that classical logic treats contradictions as false. In fact it treats them as <ABSURD>. is correct that classical logic treats whatever "implies" a contradiction as false. Note carefully that "implies" here no longer means material implication.)

I think calling them both assumptions has led to your confusion. Premise 1 is more of a GIVEN than an assumption. — flannel jesus

Some call it a "supposition," but they are fooling themselves if they think this answers my objection.

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 there is a mystery why we can say it is false in this case. What rule of inference in classical logic are we appealing to? my point has been that the only legitimate rule of inference that we can appeal to itself turns out to be a metabasis. This is to say that such a rule of inference will never be valid in the same way that a direct proof is valid.

You're reaching. :wink:

I have given my arguments, I have already responded to these charges.

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?

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

I read the whole post. — AmadeusD

Not well.

Before tenancy enforcement infrastructure, you would be an absolute moron to try to 'force' your landlord's hand. — AmadeusD

The point here is not that the landlord must, of absolute necessity, honor his promise. That is a strawman form of obligation. The point is that it is rational for him to do so, and therefore it is rational for you to invoke the promise when he says you underpaid, and therefore it is rational for you to write the check for $975 in the first place. — Leontiskos

- Yikes. :yikes: You're doubling down on that?

I wonder if you will have trouble sleeping on such non-existent arguments?

The reductio shows that A→(B∧¬B)⊢~A. As ↪hypericin pointed out. — Banno

Again:

As I noted earlier in response to Tones' reductio, a reductio is an indirect proof which is not valid in the same way that direct proofs are. You can see this by examining your conclusion. In your conclusion you rejected assumption (2) instead of assumption (1). Why did you do that? In fact it was mere whim on your part, and that is the weakness of a reductio.*

* A reductio requires special background conditions. In this case it would require the background condition that (1) is more plausible than (2). — Leontiskos

Are you saying that if your logic professor asked you to justify an answer to my question you would tell him, "This guy on the internet set out an incomplete argument which he doesn't accept, in which the conclusion was ¬A. Therefore we must reject (2) instead of (1)"?

(i.e. "I matched [his] example" ())

Simply because I matched your example — Banno

You think you get to arbitrarily reject (2) instead of (1) because I gave an example of the unaccountable inference that some in this thread are drawing? My whole point is that ¬A should not follow. What I gave is an example of the argument (claim?) that hypericin originally gave <here>.

So are you agreeing with me that the reductio does not prove ¬A?

- That's fair. Neither one of us has really made any arguments in this exchange. My point though, was that arguments had already been made in the exchange with Michael, and in quoting my post I was invoking some of those. @Michael didn't answer my question because he can't both sustain his position and also give a plausible answer:

Is he being irrational in this? Is he deluded and engaged in bullshit? — Leontiskos

I was in effect posing this same question to you, which is why I said that your argument (or assertion) is the same as his but less strong (as you focused on the tenant rather than the landlord). My argument addresses his argument, and therefore it a fortiori addresses yours.

Indeed, while your second example is a case of modus tollens, the first is not. — 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?

1. A→(B∧¬B) assumption
2. A assumption
3. B∧¬B 1,2, conditional proof
4. ~A 2, 3 reductio — Banno

As I noted earlier in response to Tones' reductio, a reductio is an indirect proof which is not valid in the same way that direct proofs are. You can see this by examining your conclusion. In your conclusion you rejected assumption (2) instead of assumption (1). Why did you do that? In fact it was mere whim on your part, and that is the weakness of a reductio.*

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).

Don't put the blame for your poor notation on to me. — Banno

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.

* A reductio requires special background conditions. In this case it would require the background condition that (1) is more plausible than (2).

- It's your second post and you're already out of arguments?

Because legal support exists. Otherwise, no one in their right mind would go to a landlord and try to hold them to their word. — AmadeusD

Nonsense. People succeed in this sort of thing all the time without legal means. @Michael has literally been arguing that the landlord would only honor his promise if he were irrational, which is an even stronger form of the argument you give. Here is the whole post:

Then suppose you invoke the promise and he says, "Oh sorry, I forgot about that. Never mind."

Is he being irrational in this? Is he deluded and engaged in bullshit?

You say that his word is good enough to write the check for $975, but it is not good enough for you to invoke when he says you underpaid. You are contradicting yourself. You wrote the subsidized check on the basis of a promise - a real promise that involved obligations. Without those obligations it would make no sense to write the subsidized check, and given the promise it makes no sense not to invoke it when he says you underpaid.

The point here is not that the landlord must, of absolute necessity, honor his promise. That is a strawman form of obligation. The point is that it is rational for him to do so, and therefore it is rational for you to invoke the promise when he says you underpaid, and therefore it is rational for you to write the check for $975 in the first place.

This sort of thing happens all the time in real life. Compare this to a different person who writes a check for $975 for no reason. Do they have recourse? Of course not. They are in an entirely different situation. The only difference between the two cases is an obligation. — Leontiskos

Working again in the context of this post, consider its first argument:

• A→(B∧¬B)
• ∴ ¬A

Now consider the way that is interpreting this first argument (and I think this is the same way that many others tend to think about this precritically):

• A→FALSE
• ∴ ¬A

This relates to what I said earlier, namely that a logical sentence that contains a contradiction should perhaps not be considered "well-formed" (I preempt an objection <here>). FALSE is not a term in classical logic, and we have no clear understanding of how it is supposed to be used. Banno seems to be wanting to use FALSE as the second premise in a modus tollens argument in order to draw ¬A. Is that permissible? Does this new term FALSE really work as a substitute for the second premise of a modus tollens? I don't think there is a clear answer.

Now suppose we have:

• A→FALSE
• ¬FALSE
• ∴ ¬A

In this case can we also use ¬FALSE to draw the modus tollens? Is this new term that we have introduced into classical logic negatable? There is no clear answer.

Another way to read the first argument, and the one I prefer*, is as follows:

• A→ABSURD
• ∴ "A cannot be affirmed"

This is exactly what I said originally:

You think the two propositions logically imply ~A? It seems rather that what they imply is that A cannot be asserted. — Leontiskos

I am trying not to repeat what I have said elsewhere, but the equivocation between what is false and what is absurd or contradictory was pointed out earlier:

You could also put this a different way and say that while the propositions ((A→(B∧¬B)) and (B∧¬B) have truth tables, they have no meaning. They are not logically coherent in a way that goes beyond mere symbol manipulation. We have no idea what (B∧¬B) could ever be expected to mean. We just think of it, and reify it as, "false" - a kind of falsity incarnate.*

* A parallel equivocation occurs here on 'false' and 'absurd' or 'contradictory'. Usually when we say 'false' we mean, "It could be true but it's not." In this case it could never be true. It is the opposite of a tautology—an absurdity or a contradiction. — Leontiskos

Speaking in natural language, the opposite of false is true, and yet the opposite of a contradiction is not true. Or rather, 'false' and 'contradictory' are opposites of 'true' in entirely different ways.** Thus when Banno says that a contradiction (b∧¬b) is false, does he mean that it is false or that it is FALSE? It could be treated as false just as we treat ¬q as false, but in that case there is no possible modus tollens on the first argument (because in that case (p→¬q) would require a second premise that negates the consequent, i.e. ¬¬q or just q). In this case the opposite of false is true and everything carries on fine*** at least for the present moment, for as soon as we interact with (b∧¬b) again we are again susceptible to treating it as FALSE instead of false.

But if Banno instead treats (b∧¬b) as FALSE then the opposite of FALSE is ...? We don't know. This is really the opposite of ABSURD or the opposite of a real contradiction, and the opposite of such a thing is not simply 'true'.

Introducing ABSURD in the way I did above destroys the LEM of classical logic. Introducing FALSE significantly complicates the LEM of classical logic, and it is possible that it also destroys it.

* The one I prefer assuming we allow contradictions in our logical formulas, which I rather doubt that we should.

** I would want to say that the opposite of 'contradictory'/'absurd' is 'coherent' or 'consistent', not 'true'.

*** Fine except for the odd wrinkle that a cousin of an exclusive-or is produced where there would otherwise always be an inclusive-or (link).

Your loss. — Banno

I read his responses to Lionino, but many of those posts are just completely blank. He deletes what he wrote. His ready-made approach doesn't answer the questions that are being asked, and you and Tones are two peas in a methodological pod.

Then the thread is in erorr. (p ^ ~p) is false in classical propositional logic. — Banno

This answer proves that you do not understand the questions that are being asked. If one wants to understand what is being discussed here they will be required to set aside their ready-made answers. They will be required to examine the logic machine itself instead of just assuming that it is working.

Whether or not we affirm the negation of the consequent... — Leontiskos

Nowhere in that post do you affirm (B∧¬B). — Banno

I never said I did. Read again what you responded to. "Whether or not we affirm the negation of the consequent..."

You are telling me that (B∧¬B) is false, and that this is presumably the reason why we can draw ¬A in the first argument. But in the second argument we can draw ¬A because of ¬(B∧¬B). So I ask again: How is it that something and its negation can both [function as the second premise of a modus tollens]? By calling (B∧¬B) is false you are presumably thinking of the first argument (and therefore both arguments) as a modus tollens.*

* You are thinking of the first argument as a modus tollens enthymeme, which is how I was conceiving of it earlier in the thread as well.

As has been explained at length, in classical propositional logic contradictions are false. — Banno

I've been ignoring Tones, as he is a pill and he inundates me with an absurd number of replies (15 in just the last 24 hours). Presumably he is the only one you believe has "explained this at length"?

As has been explained, in classical logic a contradiction is false. — Banno

I think the thread shows that this is not true. The problem here is that your answer lacks specificity, and contains the very ambiguity that is creating problems. Are we to consider a contradiction as if it were a variable that just happens to be false (e.g. ¬p), or are we to consider a contradiction as if it is a simple (non-complex) falsity (e.g. "If a conditional is false and its antecedent is true, then its consequent is by definition false")? The problem is that we can only be pretending to consult the truth table of a contradiction, and classical logic is premised upon the consultation of truth tables.

Edit:

I would suggest reading my post here:

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.

Each 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.

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. It does not make much sense to speak of "the notion of contradiction in its entirety". — Banno

See, for example:

If this is the nub then the problem is, as I said earlier, that truth-functional logic does not give us any instruction for how to handle contradictions. — Leontiskos

If you think that I am speaking about, "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," then pray tell how a contradiction is to be dealt with in classical propositional logic...? You seem to be implying that there are correct answers to the quandaries in this thread. For example, you seem to be implying that, according to the logic, one person is right and one person is wrong when they disagree about whether a given instance of (b∧¬b) should be treated as a proposition/variable or as a simple truth value. What, then, is the right answer?

I freely admit that we could draw up additional rules to avoid the problems that are here arising. I have even proposed that we disallow "contradictions" from logical sentences. But I think the thread testifies to the fact that no such rules are generally acknowledged.

Edit:

As has been explained, in classical logic a contradiction is false. — Banno

I think the thread shows that this is not true. The problem here is that your answer lacks specificity, and contains the very ambiguity that is creating problems. Are we to consider a contradiction as if it were a variable that just happens to be false (e.g. ¬p), or are we to consider a contradiction as if it is a simple (non-complex) falsity (e.g. "If a conditional is false and its antecedent is true, then its consequent is by definition false")? The problem is that we can only be pretending to consult the truth table of a contradiction, and classical logic is premised upon the consultation of truth tables.

Because he told me to, and it's rational to pay less if the person asking you for money asks for less. — Michael

Then suppose you invoke the promise and he says, "Oh sorry, I forgot about that. Never mind." — Leontiskos

*Crickets* again?

You say that his word is good enough to write the check for $975, but it is not good enough for you to invoke when he says you underpaid. You are contradicting yourself. You wrote the subsidized check on the basis of a promise - a real promise that involved obligations. Without those obligations it would make no sense to write the subsidized check, and given the promise it makes no sense not to invoke it when he says you underpaid. — Leontiskos

You are contradicting yourself. You know it is rational to invoke your landlord's promise, and you would do so in real life, but in your TPF sophistry-mode you just bury your head in the sand instead of facing up to the irrationality of your position.

Another side of accidents that touches upon consequences well beyond our view is reflected in Aristotle saying there could be no science of them. That is oddly echoed in Chaos theory and the delicate efficacy of the butterfly effect. The big garden is not being tended. — Paine

But if one knows about the butterfly effect, are such effects still accidental? In any case, it would be difficult to know about the butterfly effect without going mad.

I do think the sublimity of Christ’s teaching comes in at this point. It seems clear that Christ wants us to produce second-order goodness (exactly opposite of what is produced when the boss strikes his employee). That is, we are to become the sort of people who produce goodness in excess without knowing it.

Christ upholds the paradox, again and again, wherein we are not to know it, and are certainly not to focus upon it. Consider:

• So you also, when you have done all that is commanded you, say, ‘We are unworthy servants; we have only done what was our duty.’ (Luke 17:10)
• "The Pharisee stood and prayed thus with himself, ‘God, I thank thee that I am not like other men, extortioners, unjust, adulterers, or even like this tax collector. I fast twice a week, I give tithes of all that I get.’ But the tax collector, standing far off, would not even lift up his eyes to heaven, but beat his breast, saying, ‘God, be merciful to me a sinner!’ I tell you, this man went down to his house justified rather than the other; for every one who exalts himself will be humbled, but he who humbles himself will be exalted.” (Luke 18:11-14)
• “Truly I tell you, this poor widow has put in more than all of them; 4 for they all contributed out of their abundance, but she out of her poverty put in all the living that she had.” (Luke 21:3-4)
• (Simon Tugwell's book, The Beatitutes, is very good on this topic)

At times he ups the ante, pushing the paradox very near absurdity:

• “You have heard that it was said, ‘You shall love your neighbor and hate your enemy.’ But I say to you, Love your enemies and pray for those who persecute you, so that you may be sons of your Father who is in heaven; for he makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust. For if you love those who love you, what reward have you? Do not even the tax collectors do the same? And if you salute only your brethren, what more are you doing than others? Do not even the Gentiles do the same? You, therefore, must be perfect, as your heavenly Father is perfect." (Matthew 5:43-48)
• “You have heard that it was said, ‘An eye for an eye and a tooth for a tooth.’ But I say to you, Do not resist one who is evil. But if any one strikes you on the right cheek, turn to him the other also; and if any one would sue you and take your coat, let him have your cloak as well." (Matthew 5:38-40)
• "And when those hired about the eleventh hour came, each of them received a denarius. Now when the first came, they thought they would receive more; but each of them also received a denarius. And on receiving it they grumbled at the householder..." (Matthew 20:9-11)
• “Or what woman, having ten silver coins, if she loses one coin, does not light a lamp and sweep the house and seek diligently until she finds it? And when she has found it, she calls together her friends and neighbors, saying, ‘Rejoice with me, for I have found the coin which I had lost.’ Just so, I tell you, there is joy before the angels of God over one sinner who repents.” (Luke 15:8-10)

The key to all of this is that we are to become like unto God, and not only like unto Plotinus’ God, but unto a god who is very strange, even foreign to the second temple Jews. "I am the vine, you are the branches. He who abides in me, and I in him, he it is that bears much fruit, for apart from me you can do nothing" (John 15:5).

I don’t think the butterfly effect was as foreign to the ancient world as it is to ours. They understood that the spheres interrelate and interpenetrate. They understood that no one thing acts independently of the rest. Christ’s teaching comes into this bed of knowledge and raises it up to a pitch unheard.

…Of course, to the modern mechanistic mind, both the ancient worldview and Christ’s teaching are rubbish, or at best useful fictions. For them it looks to be a science of accidents, for the ancients it is a science of what is known only with great difficulty, and for the Christians it is a science that could not have been known if The Scientist had not shown it to us.

Christ’s teaching does not contradict or invalidate Aristotle‘s, but it does go beyond it. Today we face the odd reality of a West which would reject its Christian inheritance but which unknowingly continues to hold deeply Christian principles while at the same time failing to recognize their sublimity, and especially their paradoxical nature. When a high morality is not seen to be high, and when the paradoxes it contains are not properly recognized, it is wielded with devastating effects.

There doesn't have to be a standard for there to be a spectrum. — Pantagruel

Right, but that's why your original objection doesn't hold. All that is needed is a spectrum. I would actually consider a spectrum a standard.

I personally know lots of people that live their lives recklessly and whose "intentions" routinely cause all kinds of havoc and produce all kinds of "unintended consequences". — Pantagruel

This is emotional reasoning. The problem is that you apparently don't know anyone who struggles with the opposite vice of scrupulosity, and so you run to the opposite extreme. Just as there are people who think too little about the effects of their actions, so too there are people who think too much about the effects of their actions. As I noted in my first post:

I think Aristotle's mean is very important. People think happiness is about chasing pleasures and avoiding pains, but they also fail to observe the mean in explicitly moral thinking. For example, I was recently having a discussion with Joshs over his idea that all blame/culpability should be eradicated from society (link). This is a common contemporary trope, "Blame/culpability is bad, therefore we should go to the extreme of getting rid of it altogether" (Joshs takes the culturally popular route of saying that everyone is always doing their very best, and therefore it is illogical to blame anyone for anything).

For Aristotle it is never that simple. We can't just run to the extreme and call it a day. Things like blame and anger will involve a mean, and because of this there will be appropriate and inappropriate forms of blame and anger. The key is learning to blame and become angry when we ought to blame and become angry, and learning not to blame and become angry when we ought not blame and become angry. — Leontiskos

Modern moral theories always forget about the mean. That is what you are doing. You think "Conscientiousness = Good and Indifference to Effects = Bad." What you don't see is that too much conscientiousness and too little indifference to effects is its own vice: scrupulosity. It's not black and white.

Yeah, this is weird stuff. Much of it goes back to what I said earlier, "When we talk about contradiction there is a cleavage, insofar as it cannot strictly speaking be captured by logic. It is a violation of logic" ().

A contradiction is a contradiction. It is neither true nor false. It is the basis for both truth and falsity.

Thus one can’t pretend to represent a contradiction in the form of a proposition and then apply the LEM to it as if it makes sense to call a contradiction true or false. As soon as a reified contradiction is allowed to enter logical discourse, a wrinkle is introduced which can never ultimately be ironed out. It is hazardous to try to answer the OP's question with a formal proof, because formal proofs cannot represent the notion of contradiction in its entirety. Granted, natural language also cannot do this, but it is at least capable of not-trying to do this. It is capable of apophaticism.

The tokens "(b∧¬b)" and "¬(b∧¬b)" are neither true nor false if by 'true' and 'false' we are talking about something that goes beyond validity and invalidity (i.e. something which the logical system presupposes and is transcended by). The first is invalid but not false. The second is valid but not true. Too often in logic we conflate validity with truth.

But, as my second phrase in this post, I got confused as to whether ¬(a→(b∧¬b)) means a variable that is in contradiction with (a→(b∧¬b)) or simply that (a→(b∧¬b)) is False. — Lionino

Yes, exactly.

¬(a→(b∧¬b)) is only ever True (meaning A does not imply a contradiction) when A is True. 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

Right, one could approach it from two different directions. This is what I was trying to get at earlier, "I think Lionino was somehow seeing this through the overdetermination of the biconditional with respect to ¬¬A" ().

Material implication is supposed to prescind from the reason why the implication is true or false, and therefore it is supposed to prescind from causal and temporal considerations. But in this case the temporal order in which one evaluates a complicated material conditional affects its outcome,* and I can't see how this would ever happen without having that contradiction in the "interior logical flow" of the argument. Similarly, in my last post:

When are we supposed to reduce a contradiction to its functional truth value, and when are we supposed to let it remain in its proposition form? The mind can conceive of [claims] like (b∧¬b) and ¬(b∧¬b) in these two different ways, the manner in which we conceive of them results in different logical outcomes, and symbolic logic provides no way of adjudicating how the [claims] are to be conceived in any one situation. — Leontiskos

* Or perhaps it is not the temporal order, but rather the simple decision of whether to conceive of the different things as propositions or as truth values. A problem of ordering rather than temporal ordering, as it is the combinations that affect the outcome rather than the succession-order in which they are applied. If this is the nub then the problem is, as I said earlier, that truth-functional logic does not give us any instruction for how to handle contradictions. ...and again, there is no such thing as "handling contradictions," so this is to be expected. The problem is that we are trying to handle something that cannot be handled.

Here is something I wrote when my internet was out, and before I was able to read this post of yours. I will read what you have written after posting this (and therefore there may be overlaps or redundancies):

———

If (a→b)∧(a→¬b) is False, ¬A is False, so A can be True or not¹. — Lionino

Note, too, how the LEM operates here.

On the LEM, (a→b)∧(a→¬b) must be either true or false, and therefore a→(b∧¬b) must be either true or false. If a→(b∧¬b) is true then ¬a must be true and (b∧¬b) must be indeterminate. It must be this way in order to avoid affirming the contradiction (and never mind the odd question of whether, in this case, saying “¬a must be true” is the same as saying “a must be false”).

But on the other side of the LEM, if a→(b∧¬b) is false then by material implication (a) must be true and (b∧¬b) must be false. This is different from (b∧¬b) being indeterminate. It is to affirm ¬(b∧¬b), which is the same as (¬b∨b) with an inclusive-or. The inclusive-or provides the second-order possibility of a true contradiction.

Another way this comes out is seen by comparing the truth table of (p→q) with the truth table of a→(b∧¬b). If you run the table “forward,” by using the value of (b) to compute the value of (b∧¬b), then (b∧¬b) is always necessarily false. But if you run the table “backwards,” by using the value of (p→q) to compute the values of (p) and (q), then in that singular case where the entirety of the conditional is false we get (a ^ ¬(b∧¬b)), which again results in the inclusive-or (¬b∨b).

In one way, in saying that the inclusive-or is true we are admitting the possibility that both disjuncts could be true, which would be impossible. Put differently, negating this conjunction results in an exclusive-or rather than the usual inclusive-or, just as affirming the special conditional of (a→(b∧¬b)) results in (a) automatically being false, even though there is obviously no rule that ((p→q)→¬p). Special cases and special exceptions are popping up.

What is happening, I think, is that the two different kinds of falsity are coming into play, for there are two different ways of calling (b∧¬b) false. We can mean that it is false inherently, even without a negation (and this is reflective of the first argument I gave <here>).* Or else by saying it is false we can mean that we logically negate it, viz. ¬(b∧¬b) (and this is reflective of the second argument I gave <here>). When are we supposed to reduce a contradiction to its functional truth value, and when are we supposed to let it remain in its proposition form? The mind can conceive of propositions symbol-strings like (b∧¬b) and ¬(b∧¬b) in these two different ways, the manner in which we conceive of them results in different logical outcomes, and symbolic logic provides no way of adjudicating how the propositions symbol-strings are to be conceived in any one situation. When we introduce contradictions into the logic—even in minor ways such as in the consequents of conditionals—the logic becomes underdetermined. Different people are legitimately interpreting the propositions symbol-strings in different ways.

* This is reflective of what I have been calling “falsity incarnate”

What is the definition 'analogical equivocity'? — TonesInDeepFreeze

It is the kind of equivocity present in analogical predication, where a middle term is not univocal (i.e. it is strictly speaking equivocal) but there is an analogical relation between the different senses. This is the basis for the most straightforward kind of metabasis eis allo genos. The two different senses of falsity alluded to above are an example of two senses with an analogical relation. I was using the term earlier because I believe @Count Timothy von Icarus has an understanding of it.

I know you want a strict definition, but the wonderful irony is that someone like yourself who requires the sort of precision reminiscent of truth-functional logic can't understand analogical equivocity or the subtle problems that attend your argument for ¬A. As we have seen in the thread, those who require such "precision" tend to have a distaste for natural language itself.

Isn't this a fairly big problem given that (¬¬A↔A)? I take it that this is the same thing I have pointed out coming out in a different way? Namely the quasi-equivocation on falsity? — Leontiskos

Compare:

• A→(B∧¬B)
• ∴ ¬A

With:

• A→(B∧¬B)
• ¬(B∧¬B)
• ∴ ¬A

Whether or not we affirm the negation of the consequent, the antecedent still ends up being false. In the first case the consequent is pre-false, in the second case our explicit negation makes it false. In the first case it is treated as "falsity incarnate," whereas in the second it is treated as a proposition. This demonstrates the analogical equivocity with respect to falsity that I pointed out earlier, with help from . I think Lionino was somehow seeing this through the overdetermination of the biconditional with respect to ¬¬A, but it's slippery to grab hold of.

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).

-

Around a third of folk hereabouts who have an interest in logical issues cannot do basic logic. — Banno

And about two thirds of folk hereabouts who are good at manipulating symbols "have no way of knowing when their logic machine is working and when it is not" ().

Aristotle always wins in the long run, as he could both use the machine and understand when it was misbehaving. :wink:

If (a→b)∧(a→¬b) is False, ¬A is False, so A can be True or not¹. — Lionino

But (a→b)∧(a→¬b) being False simply means that A does not imply a contradiction, it should not mean A is True automatically. — Lionino

Isn't this a fairly big problem given that (¬¬A↔A)? I take it that this is the same thing I have pointed out coming out in a different way? Namely the quasi-equivocation on falsity?

...Or is it simpler: ¬(a→(b∧¬b)) → a

Back to material implication:

As noted in my original post, your interpretation will involve Sue in the implausible claims that attend the material logic of ~(A → B), such as the claim that A is true and B is false. Sue is obviously not claiming that (e.g. that lizards are purple). The negation (and contradictory) of Bob's assertion is not ~(A → B), it is, "Supposing A, B would not follow." — Leontiskos

Given material implication there is no way to deny a conditional without affirming the antecedent, just as there is no way to deny the antecedent without affirming the conditional.

- He confuses what is achievable with what is deemed to be achievable in the same way that he earlier confused what is wrong with what is deemed to be wrong in the conversation about penalty/punishment. This is standard Moorean confusion.

- Thanks Lionino. Good reminders and clarifying points.

Edit: I underestimated your post. Like your first post, this is far above and beyond anything else that is occurring in this thread.