If you've ever taken an intro logic class and you asked your professor how they know there are no true contradictions, you probably get some variation on the proof that anything follows from a contradiction. This is a neat little proof, and a lot of very smart people have taken it to settle the question. Sadly, it doesn't.

Sure, taken as an argument that "if there are true contradictions, using classical logic is probably not a good way to model how they work," there's nothing wrong with it--it is, after all, a perfectly valid proof--buf if it is taken as an argument against the possibility of true contradictions, it is (a) somewhat beside the point, and (b) radically question-begging.

In case anyone reads this who missed the first two posts, let me emphasize that I don't believe in the possibility of true contradictions. If it were really this easy to refute dialetheism, I'd be a happy man...although I suppose I'd also have wander off and find another topic to worry about. (I can imagine various theistic philosophers thinking the same thoughts when it occurred to them that the ontological argument might really work, that it might be *just that easy* to come up with a knock-down argument for the existence of God.) For better or for worse, it isn't this easy.

John Depoe, who I went to grad school with for a few semesters when we were getting our MAs at Western Michigan, is one of the very smart people I referred to above who apparently takes it as settling the question. I'm going to pick on him, not because he says anything unusual about it, but because he blogged about it not so long ago, so he's easily quotable and because his post is one of the clearest statements of explosion-as-an-argument-against-true-contradictions. Here's what he had to say:

"The possibility of affirming logical contradictions is strongly unintuitive, but this is not the only reason to reject the possibility of affirming logical contradictions. For those who lack this intuition (or perhaps the intuition isn’t very strong), you can consider the following proof, which shows that if you accept any logical contradiction, you can prove anything.." After giving the proof, he concludes:

"...so you can literally prove anything given the assumption of a logical contradiction. This is sometimes called “explosion” because once you grant a logical contradiction, you can repeat this proof over and over indefinitely, potentially proving every possible proposition is true. So, even [if] you lack the inuition that logical contradictions are impossible, this little exercise is a reminder why contradictions cannot ever be affirmed."

This sounds like a pretty devestating argument. I mean, if someone told you that if some other claim P were true, everything else would be true too, that would be a problem. We we all know that there are at least some falsehoods, right? If P entails everything, P must be false itself.

First of all, hold on. As alluded to above--point (a)--I'm worried this may be somewhat beside the point. OK, so if you can derive any Q (or ~Q) from (P&~P), then everything is both true and false. That might be embarassing for a regular, moderate dialetheist who wants at least some things to come out as just-true or just-false, but I'm not sure how much comfort it can really provide those of us who believe reality to be consistent. (This is somewhat analogous to the way that some people treat the fact that some versions of global skepticism are hard to coherently formulate as a great victory for global realism. Does that really give the rest of us any more confidence that we are justified in *our* beliefs about the external world?) Maybe all it proves is that reality really is globally inconsistent.

Perhaps (a) doesn't bother you. You say, "if I can be sure of anything on the basis of my immediate experience of the world, it's that some statements are false (and not also true.) If dialetheism generates the possibility that I'm wrong about that, then dialetheism is wrong."

Fair enough. There is, however, a much more serious objection, the one alluded to as point (b) above. This argument doesn't actually show that if there really were true contradictions, everything would follow from them. Why not? Well, let's take a look at the proof itself.

1. P & ~P

2. P (from 1 and &-elimination)

3. P v Q (from 2 and v-addition)

4. ~P (from 1 and &-elimination)

5. Q (from 3, 4 and Disjunctive Syllogism)

The problem is that step 5 begs the question. If there are any true contradictions, any at all--even if out of the infinite list of contradictory propositions, only one of them is true--then it follows that Disjunctive Syllogism is not a valid rule of inference. Remember, whether your logic is classical or paraconsistent, a disjunction is true as long as at least one of the disjuncts is true. If P is true and Q is false, or if P is false and Q is true, or if both P and Q are true, then (P v Q) is true. So far, so good. But what about the possibility that P is both true and false, while Q is just false? If that's a possibility, then the disjunction of P and Q could be true, ~P could be true and Q could still be false. This is *not* a matter of arbitrarily tweaking the rules or piling on a second absurdity to save the first. Quite the opposite. This result simply follows from the assumption that it is possible that some statements of the form (P&~P) could be true. In that scenario, plugging in Disjunctive Syllogism would at least sometimes result in generating a false (but not true) conclusion from true premises.

Now, of course, you might object that Disjunctive Syllogism *is* a valid rule of inference. After all, since it is impossible for a statement of the form (P & ~P) to be true, there is no possible scenario where (P v Q) will be true, ~P will be true and Q will be false. I agree. This is, however, to assume precisely what is supposed to be proved...that there are no true contradictions. It's not that impressive that, starting from the premise that there are no true contradictions, you get the result that absurd things would be true if there were any true contradictions.

The result could be put like this. It is false that anything would follow from a contradiction. There is, however, one (and only one) contradiction from which anything follows...the contradiction involved in assuming that the Law of Non-Contradiction is true by treating Disjunctive Syllogism as a valid rule of inference, and assuming that it is false by starting the proof with (P & ~P). This is, sadly, a terribly uninteresting result.

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## 33 comments:

Ben,

Fun post. I don't think you've succeeded in proving that contradictions don't explode, though I do think you've succeeded in articulating the reason that one cannot convince a moderate dialethist by means of this argument.

But let's remember Aristotle's first principle of rhetoric and ask a fundamental question: to whom is the argument of Duns Scotus directed? Not to those vegetative souls who have abandoned disjunctive syllogism, but rather to those puzzled neophytes who can see clearly enough that DS is valid but want to know whether contradictions, once admitted, will be benign or metastatic. As the nifty little proof shows, they do metastasize.

That won't help someone who cannot see that DS is a truth-preserving form of inference. So the search is on -- and I take it this is what you're pushing us into on this blog -- to see whether there are enough common principles of reason left between the dialethists and the rest of us to enable us to show them what's wrong with their position. I'm not optimistic. But that fact doesn't make me lose any sleep at night.

Right. Well, I'd want to push the question of what it means to clearly see that DS is a truth-preserving inference. Surely, clearly seeing it is not a matter of blindly assuming that any argument form that gets its own name in a logic book must be truth-preserving, or not being able to think of a counter-example. If clearly seeing it means really thinking through the possible combinations of truth values, then anyone who'd clearly seen that DS was truth-preserving would be, by definition, someone who has already accepted that it is impossible for the same proposition to be both true and false. If that assumption were wrong, after all, then DS would by definition *not* be truth-preserving in all cases.

Now, the claim that false propositions can never also be true could be self-evidently true, but if Duns Scotus' argument is (a) only directed towards those who start by accepting that claim, and (b) intended to show that one should never affirm contradictions, it seems a tad redundant to me.

Interestingly enough, even the moderate dialetheist would accept that DS is truth-preserving in most instances. Graham Priest argues in "In Contradiction" that, as the statistical frequency of true contradictions is very low, it's safe to assign a very low epistemic probability to any given contradiction prior to good evidence coming out that it is one of the few true contradictions. Hence DS and even reductio ad absurdum, while strictly speaking deductively invalid, are still rationally acceptable forms of inference in most domains, albeit reinterpreted as having merely probabilistic force.

(Just as a preview of coming attractions, I would say that I find Priest's arguments for the low frequency of true contradictions to be surprisingly weak. The rest of us have a good reason for not wanting to grant that any given contradiction is true--that we think that it is impossible in principle for the same proposition to be true and false--but that reason is not available to Priest, and I think the separate reasons he appeals to are pretty flimsy. If I'm right about this, then the dialetheist really should be in the odd position of having to say that someone who uses, e.g. DS, in day-to-day reasoning is doing something irrational.)

In any case, I can sympathize with the not-losing-sleep-about-failing-to-refute-dialetheism position, but I would at least argue that it might be worth losing a little sleep over how to plausibly block arguments *for* dialetheism launched from within classical contexts by means of the semantic and set-theoretic paradoxes. That said, if it's possible to defend the fortress of classical logic *and* lob a few non-question-begging arguments to undermine the claims of the barbarian hordes outside the gates, then the latter is just gravy on top.

Hi Ben,

I find Dialetheism fascinating, and I think it's great that you started a blog devoted to the subject. For now, I just have a quick question: Isn't there a proof of explosion in classical logic which doesn't explicitly rely on disjunctive syllogism? I think the following is valid:

1) p&~p

2) ~p ((1), & elimination)

3) p->q ((2), -> introduction)

4) p ((1), & elimination)

5) q ((3) and (4), modus ponens)

I suppose paraconsistent logics would invalidate that as well; it's just that I've never seen that argument for explosion addressed before.

Strictly speaking, to control contradiction explosion you don't need to regard disjunction syllogism as question-begging. That's become popular, but more and more I think it would make mmore sense to reject disjunction introduction instead. There are independent reasons to worry about disjunction introduction, and the dialetheist's argument as you present it assumes that (P v Q) is true if Q is false and P is true & false, despite the fact that this is certainly not something can be assumed. (Indeed,one would expect that P v Q is true and false, since truth-functionally P v Q is false iff both of its disjuncts are false; but disjunctive syllogism does not tell us what we can do with a premise that's both true and false.) For any good reason one might have to reject disjunctive syllogism, there is at least one good reason to reject addition. Actually, you can even allow addition if you do so on the condition that it not be followed by any elimination (such as disjunctive syllogism).

Brandon,

The moves you suggest really do sound like aribtrarily tweaking otherwise unproblematic rules to avoid this particular consequence. If the dialeheist really had to resort to something that extreme in order to get around explosion, I'd rest a lot easier.

(Put more charitably, there may for the sake of argument be excellent reasons to dislike v-introduction, but, whatever the views of individual dialetheists, it's concpetually unrelated to dialetheism per se, right? As such, there'd be no problem using v-introduction in an argument against the possibility of true contradictions.)

Let's put it this way... I'm not concerned with whether the dialetheist has a "way around" explosion. I'm concerned, rather, with whether I (as a defender of the law of non-contradiction) am begging the question if I use the explosion proof as an argument against the possibility of true contradictions. I think I am.

Whether you want ot call (P v Q) just true or both true and false when P is both true and false and Q is just false is orthogonal to the question of the legitimacy of DJ is this is possible. One way or the other, this would be a scenario where (P v Q) is true (whatever else it is), ~P is true and Q is just fale and not true. If this is a possible scenario, DS is not truth-preserving in all cases.

Quintirius,

That is interesting, and I haven't seen that addressed before either, so I'm less confident about the response to this, but...

As before, I think the issue is not so much whether paraconsistent logicians have some clever way of blocking it within their system, but whether it begs the question when taken as an argument against them by relying on the Law of Non-Contradiction. I think it still does, since on the face of it the same problematic combination of truth values as before look like they'd still invalidate the whole thing.

An -> introduction rule would be truth preserving *given* the LNC, since as long as ~P is true, P can't be true, and as long as P isn't true, the conditional P->Q is true whether Q is true or false, since conditionals can only be false by having true antecedents and false consequents. So far, so good. Unfortunately, in the scenario where P was both true and false and Q was just false and not true, ~P would be true and P->Q would come out false, right?

I'm not sure I follow the objection to the argument. The argument is effectively a reductio ad absurdum (RAA) formulated in classical logic CL. The proof is formulated entirely in the object-language of CL and so begs no questions at all. Your objection is metalogical--concerning which are the right object-language theorems--and so raises questions that the proof just doesn't engage. In other words, there is nothing wrong with the RAA in CL. Such a proof does not purport to show that CL includes all and only the theorems we ought to accept. But the latter, it seems, is what you're worried about.

(Actually, let's put that last point even more simply. "~P, therefore P->Q" is universally truth-preserving because if ~P is true, P can't be true as well, and as long as P isn't true, P->Q is true regardless of whether Q is true or fale. Eliminate the assumption that P & ~P can't both be true, and the inference automatically becomes invalid.)

Mike,

It's certainly a valid proof. It begs the question *if* it's taken as a philosophical argument against the possibility of true contradictions, since one of those CL rules in particular is only considered valid because we all agree that there can't be true contradictions. My objection isn't to CL--I like CL--it's to the use of this proof as a philosophical defense of CL's prohibition against contradictions, an alleged demonstration that "anything follows from a contradiction."

*If* there were true contradictions, some things would follow from particular contradictions and lots of things wouldn't. Obviously, CL, which systemically assumes there can't be true contradictions, wouldn't be the right way to model a reality that actually included true contradictions.

Put differently, I quite agree that the proof doesn't engage the metalogical question. That's my point. Like I said above, the only contradiction from which this shows that anything follows is the contradiction involved in simultaneously accepting and rejecting the Law of Non-Contradiction.*

*Accepting and rejecting it in propositional logic, that is. No similar proof of "anything following from a contradiction" is possible in, e.g. Aristotelian categorical logic, where some syllogisms with contradictory premises deliver their conclusions and some don't.

Ben,

Well, it certainly isn't arbitrary tweaking; there is a whole string of paraconsistent logics that keep disjunctive syllogism but reject disjunction introduction. It's the most common approach in computer systems that have to deal with paraconsistency, for instance, because (among many other reasons) it solves the problem with much fewer modifications of classical logic. Because it's a cleaner and simpler solution, it can also more easily be interpreted in non-dialetheist ways, but, of course, there doesn't seem to be any reason why you can't have a dialetheist logic that rejects or restricts disjunction introduction rather than disjunctive syllogism, despite the fact that the latter is more common. There may not be anything in rejection of disjunction introduction as such that is

particularlyappropriate to dialetheism, but the same may be said of rejecting disjunctive syllogism. My reading of the literature is only partial, but I have seen no good reason ever given for choosing to eliminate disjunctive syllogism rather than disjunction introduction -- the reason why rejecting dis. syll. is popular seems to be historical, accidental, due to the direction early pioneers on the subject just happened to take. Of course, if you do run across a good argument for dialetheists to prefer the one to the other, that's good material for a blog post, since it would be useful for understanding dialetheism! :)I don't think you can dismiss the issue of trying to run DS with a premise that is true

andfalse. If it's trueandfalse, it's not as if you can sometimes treat it as true and sometimes not; it's always both. And even in classical logic DS doesn't work with a disjunctive premise known to be false. E.g.,(a) John is wearing shoes or David is eating spaghetti.

(b) It is not the case that John is wearing shoes.

If the disjunction (a) is false, you can't use DS to conclude that David is eating spaghetti. So in the explosion argument you give, one can argue that the real problem is in taking (P v Q) to be true, simply speaking, which is required for DS to work; but, in fact, one can just as easily argue that whether DS can even be applied in this case is undefined, given that (P v Q) arguably has to be both true and false. Of course, the 'arguably' here indicates that whether you think it is really depends on what we are assuming about vI from premises that are posited to be both true and false rather than, as usual, simply true.

My objection isn't to CL--I like CL--it's to the use of this proof as a philosophical defense of CL's prohibition against contradictions, an alleged demonstration that "anything follows from a contradiction."Ben,

If you like CL, then you should find the proof against true contradictions based on CL compelling. I can't tell why you wouldn't. If you like CL, then you presumably believe that only necessarily true propositions and rules form its basis. Anything inconsistent with those necessarily true propositions and rules--including (P & ~P)-- is thereby necessarily false. That is a good argument against (P & ~P) which any adherent of CL ought to find compelling. It's not clear to me why you don't.

Mike,

"If you like CL, then you presumably believe that only necessarily true propositions and rules form its basis. Anything inconsistent with those necessarily true propositions and rules--including (P & ~P)-- is thereby necessarily false."

Of course. And I do think that (P&~P) is false, and that one of those propositions that forms the basis for classical logic--(x)~(X&~X) is true. The problem arises in contexts where I'm arguing with someone who doesn't agree with me that this proposition is true. Even though *I* believe it, it's simply not legitimate for me to appeal to the very proposition my opponent is denying as a premise in my argument, for the familiar reasons that I can't, in arguing, with anyone who doesn't believe that any proposition P is true in any context whatsoever, say that "P must be true because failing to believe it would be inconsistent with P."

Again, I like the proof just fine. I just don't think that it's a proof that if there really were true contradictions, everything would follow from them. I think it's a proof that "if (a) classical logic was the right logic, and (b) there were true contradictions, then (c) a lot of weird things would follow from the conjunction of those two things."

The problem is that, for obvious reasons, no one who believes (b) believes (a).

(Now, I should say that I don't believe that the philosophical point made above--that if there really were true contradictions, plenty of things wouldn't follow from them--is a criticism of classical logic, since I think true contradictions are impossible. Nonetheless, you shouldn't implicitly to the Law of Non-Contradiction in arguing against someone who denies the Law of Non-Contradiction.)

Brandon,

"... the explosion argument you give, one can argue that the real problem is in taking (P v Q) to be true, simply speaking, which is required for DS to work."

Well, remember, it is true, and I'm not sure how the simplicity is supposed to enter into it. Remember, saying that P is both true and false doesn't mean its some intermediary truth value that's sort of like truth and sort of like falsity, it's that it really and fully has both truth values. As such, (P v Q) meets the requirements for being true. One of the disjuncts is true. Even if (P v Q) was both true and false, this would still be enough for DS to fail to be truth-preserving. If a rule allows you in some situations to reason from all true premises (whether or not they are also false) to a false (but not also true) conclusion, it is plainly not truth-preserving in all cases. I really don't see how you can call something truth preserving if there are cases when both of the premises are true and the conclusion fails to be true.

(Graham Priest at one point in "In Contradiction" considers the suggestion that DS could be seen as valid if it were re-worked to be "~P and (P v Q) entail Q if neither of the disjuncts is a dialetheia (i.e. is both true and false)." The problem, which he points out, is that, if there can be dialetheias, and the semantic paradoxes are taken to be the primary evidence for their existence, then it is plainly possible for a proposition to both be and fail to be a dialetheia. For example, "This sentence is not true and it is not a dialetheia" is both a dialetheia and not a dialetheia. Hence, even that reworked version of DS would not be universally truth preserving.)

There may or may not be good reasons to reject v-addition--I haven't seen them yet, and I tend to be very suspicious of innocent-looking changes to classical logic, but I'll grant for the sake of argument that there could be such good reasons--but in any case, while some individual dialetheists may be able to get around an argument that employs that rule by denying the legitimacy of the inference, certainly using it would not beg the question against dialetheism per se. Using DS, however, does.

But there is no reason to think that it is true, truth-value T. The real problem is that there is one and only one reason we regard disjunction introduction as truth-preserving: in vI we already know that (P v Q) is true, truth-value T, because P is true, truth-value T: it therefore satisfies the formula. But in the explosion argument, understood in three truth-value terms, the truth value of P is not true, T, it's true-false, B. And thus it can only make disjunction introduction truth-preserving if we assume that B in a disjunct makes the disjunction true.

But there is no requirement that the dialetheist do so.So it is in fact unclear whether we should regard (P v Q) as true at all, because it is unclear whether disjunction introduction is a good inference rule if we allow propositions to be both true and false.The Priest argument is not, I think, relevant to my point about DS, because the problem is that DS only works if the disjunctive premise has, or can be assumed for the sake of argument to have, truth-value T, simply speaking, rather than F. So the dialetheist is forced to a choice: he can make the argument you are suggesting, but then he is in effect denying that there are more than two truth-values, T and F, since he would have to say that (P v Q) would be both T and F (and not some third truth-value); or he can posit at least one other truth-value, in which case the argument you suggest cannot be made, because dialetheias are then not of truth-value T. I don't see how it would be possible to have it both ways, as you seem to want.

What the dialetheist is essentially doing is suggesting that 'true' is equivocal: there is an unadulterated truth, so to speak, which is classical T; and there is another kind of truth, which is truth as found united to falsehood, which union some dialetheists represent by B. (What makes him a dialetheist is that he holds that the latter really is a kind of truth, and no one else does.) But none of the classical inference rules were created with the possibility of truth a la B in mind, and so they are all written, so to speak, for the truth of propositions that are T and not for the truth of propositions that are B. So the dialetheist actually has an open field of inference rules he can reject by saying, this one does not work properly with B. The only question is, which ones are best rejected from the point of view of building a consistent and defensible dialetheism.

And that's disjunction introduction! ;)

Actually, though, although I think that it really is smarter to drop disjunction introduction, my primary point in all this is that dialetheists aren't committed to one move in the face of the explosion arguments. There are a few, and of the two most viable, rejecting DS is merely the one on which the most work has been done

so far. It is not a move necessary to the dialetheist.Reading my comment, I'm not quite satisfied with my second and third paragraphs, because I think they are phrased poorly. The basic point is this: The explosion argument shows that the dialetheist's logic can't be classical, but must be paraconsistent. That's all, in fact, that it shows. Whether the dialetheist prefers to reject disjunctive syllogism or disjunction introduction is

purelya matter of what allows for the best (by whatever criterion) dialetheism. The problem with my two paragraphs in the previous comment is that, like your claim about disjunctive syllogism, they state a genuinely general point about dialetheism entirely in terms of one particular possible form of dialetheism (in your case, the kind that rejects disjunctive syllogism, and in mine, the kind that prefers the trivalent model); these areparticularchoiceswithindialetheism, and the reason I brought this subject up in the first place is that none of these particular choices areessentialto dialetheism -- the dialetheist has quite a surprising number of options available, even if they haven't all been full explored yet.A better way to put the point I was making in those two paragraphs is this: while we usually find it convenient to put it in terms of truth, in a truth-functional system everything is determined by truth

andfalsehood -- and in a model where there are two and only two truth values the truth values are always constrastive. T is truerather thanfalse, F is falserather thantrue; that is how they are understood, even if we avoid the cumbersome language. And the inference rules themselves are understood in that way. But to introduce trueandfalse is to introduce something that's not at all truerather thanfalse, and we can't assume that everything that can be said of the latter can be said of the former. So if we begin with a premise that's trueandfalse, we can't assume that vI yields something true until we have determined exactly what depends, in classical logic, on true in the sense of true-rather-than-false rather than true in the sense that admits of true-and-false. My primary point is that the explosion argument can be broken earlier than the DS step, namely at the vI step; and once you reject vI the explosion argument fails, and you don't need to reject DS. This is an entirely intelligible approach; it's really the question every dialetheist has to ask: Whatkindof paraconsistency does he want? I'm inclined to think a vI-rejecting kind has many advantages that a DS-rejecting one does not. I may be wrong (outside of computer science there has only been a little bit of work done on vI-rejecting approaches); but DS-rejecting systems really do seem to require more violence to classical logic than vI-rejecting ones, and without any results more useful to the dialetheist.OK, Brandon, I think I'm starting to see your point better now. It's an interesting one, and I like the formulation that the explosion proof just proves that the dialetheist's logic shouldn't be classical.

My sense of this is that your position undersates the extent to which the dialetheist's undersatnding of truth value is a break from classical logic--putting it in terms of your last comment, the violence to classical logic has already been done in the re-conceptualization of truth value, and the result useful is that it allows them to make sense of the claim that propositions can be really true and really false, not just true, false or some third entirely different value--but I'm not entirely sure about this, and it's an important enough problem that I'm moving it into its own post, the next one up, "One Truth or Two?"

I do have a question, though.

"I may be wrong (outside of computer science there has only been a little bit of work done on vI-rejecting approaches)..."

Where could I read up on the relevant computer science work on this?

Hi, Ben,

It is indeed possible that I'm underestimating the dialetheist break. If we're just considering the paraconsistency that the dialetheist needs, then, of course, there are more options than rejecting or restricting DS. But, of course, there are plenty of ways to be paraconsistent without being dialetheist; and it is probable that not all such ways are equally convenient for the dialetheist to use.

I think this can only really be determined by rigorous study of the alternatives. Part of my reason for thinking that vI is very likely a more interesting culprit for the dialetheist than DS is this: the key feature of explosion that worries people is the implication of any proposition. But how does 'any proposition' get into the argument at all? Only by vI, which in classical logic allows us to pull in arbitrary q's to make (p v q), given that we can assume that p is true. It's the non-problematic complement of explosion. DS is (so to speak) just doing its job, if we take (p v q) to be true, and it's a perfectly ordinary sort of job. If we take vI out, so that it can no longer give DS bad information, DS causes us no more trouble at all. We can solve it be taking out DS, too, of course, but taking vI seems to be a cleaner and more vigorous move, like rooting out the weed rather than cutting off its shoot. Of course, this might turn out to be a false impression.

I have a few things (only a few, unfortunately) that I can recommend on the computer science stuff; but I'll have to dig up some notes, so it will be a bit (I probably can get it done by Monday evening, though). Remind me if I haven't gotten them to you by Tuesday.

Thanks.

I agree that people new to logic tend to find the inference from P to (P v Q) puzzling, but I'm not inclined to take that terribly seriously. It *looks* strange to an intro student since there's P, then suddenly there's P, some symbol and a Q from nowhere, but I think a lot of that surface strangeness is parasitical on the fact that to an intro student who's having trouble not seing it all as uninterpreted symbolism, it may take a while to disentangle the real meaning v's and &'s they're seing from more familiar symbols like addition and multiplication they remember from their k-12 education.

After all, once you explain that (P v Q) is to be read as "at least one of these things is true," it does become clear that this is a universally truth-preserving inference, since as long as P is true, whether the unkown proposition Q is true, false, both true and false, neither, some third, fourth, or fifth newly minted truth vaue, or whatever, it will still be clearly true that at least one of those two things is true. Moreoever, this is totally unaffected by the jump to a 'recognition of the transconsistent,' since whether P is just true and not false or both true and false, if P is true it will *still* be true that of P and Q, at least one of those two things is true.

So I'd have the opposite take. In the explosion proof, v-introduction is just doing its job. DS is just doing its job too, it's just that that job description would be fundamentally flawed if it turned out to be possible for true statements to simultaneously be false ones.

Ben and Brandon,

I agree with Brandon, that v-introduction is a weak point. It brings explosion into the classical deduction system itself:

A, therefore A

A, therefore A v B (v-intro)

A & -A, therefore B

FALSE, therefore ANYTHING

There is no need to start from P & -P.

Sorry, if I messed something up.

Well, your version includes (A & ~A), which is not obviously different from (P & ~P). Whichever letter you choose to represent the contradiction, you need (a) a contradictory premise, (b) v-introduction, and (c) disjuctive syllogism, to generate the explosion. There's no obvious conceptual connection between (a) and (b) such that (b) would no longer make sense if (a) were possible, but there *is* a clear conceptual connection between (a) and (c), such that if (a) is possible, (c) would no longer be truth-preserving in all cases. You *could* get around that result of (a) by eliminating (b), but you'd need an independent reason to reject (b), which would have to come from something other than the possibility of (a). By contrast, the reason to reject (c) falls out in a fairly clear way from the possibility of (a).

since whether P is just true and not false or both true and false, if P is true it will *still* be true that of P and Q, at least one of those two things is true.Well, that would depend. You are forgetting my point that falsity is also part of what makes these operations what they are. (P v Q) is always false when both of its disjuncts are false, so if P is true-false it is

unknownwhat the truth value of (P v Q) would be (since Q could be true, in which case no problem, or false, in which case it's a dialetheia, or true/false, in which case it is also a dialetheia) -- and that's not insignificant since DScannotbe used with a disjunction recognized to be false. If we follow your reasoning, then it's not really the case that the explosion argument gives us a reason to reject DS: it, perhaps fittingly, both does and does not, because (P v Q) will in some cases be true-false, which means that DS in the argument is used both legitimately (because the disjunctive premise is true) and illegitimately (because the disjunctive premise is false). (Which, I must concede, would be an interesting result if we could make sense of it.)Well, even if you're right that in the situation where P is both true and false and Q is just false, (P v Q) will be false as well as true, the move from P to (P v Q) would still be universally truth-preserving. Under all possible combinations, it's still true that whenever P is true, (P v Q) will be true (whatever else it is.)

Ben, sorry for changing letters. But this does not change my point: 1) I didn't start with contradictory premise; 2) I didn't use disjunctive syllogism; 3) I got explosion.

Sorry, right. You didn't use disjunctive syllogism.

I'm just not clear how you think you did get explosion.

What you said was:

"A & -A, therefore B"

...which is explosion. You just didn't explain where you derived this from, if not from the usual intermediary steps of deriving A from A&~A, A v B from A, ~A from A & ~A, and B from ~A and A v B.

Certainly, at any rate, there is a contradictory premise snuck in there (A & ~A), although since I'm not clear how you justified the inference "therefore B," I guess I don't know what oher inferential rules were playing a role there if not DS.

Although you included it, I'm not even sure what role v-introduction was even playing in your argument. Can you take me through it one more time?

Well, even if you're right that in the situation where P is both true and false and Q is just false, (P v Q) will be false as well as true, the move from P to (P v Q) would still be universally truth-preserving. Under all possible combinations, it's still true that whenever P is true, (P v Q) will be true (whatever else it is.)A dialetheist has several choices in the situation where propositions, and P in particular, can be both true and false (and this has been my primary point):

(1) He can take (P v Q) and treat it as simply true. This requires a (slightly) refined form of vI, which in its classical form does not give us a clear reason for saying this when propositions can be both true and false. Since (P v Q) is true, DS leads to explosion, and the non-trivialist dialetheist must reject it.

(2) He can take (P v Q) and treat it as simply false. This also requires a slightly refined form of vI, but the argument for it is as good as the argument for (1). Since (P v Q) is false, DS cannot be used, and cannot lead to explosion. It is pointless to reject it.

(3) He can take (P v Q) and treat it as both true and false. Then DS is and is not a legitimate move, and explosion is and is not the result. It's anybody's guess where this gets us.

(4) He can take (P v Q) and treat it as indeterminate in truth value. Then DS cannot be applied, and does not lead to explosion. Therefore it's pointless to reject it.

[Suppose that Q is true. Then the values for (P, Q) are (TF, T). This meets the usual bivalent rule for truth. So, assuming that we don't take the fact that P is both T and F to gum up the works -- we could, there is nothing stopping us from doing so, but assuming that we don't -- (P v Q) is true. Suppose that Q is false. Then the values for (P, Q) are (TF, F). This fits the bivalent rules for both truth and falsity. So we can do one of three things -- and, again, they are all open, nothing stops us from building a system in which any of them is true: (1) Falsity trumps truth, in which case (P v Q) is false; (2) truth trumps falsity, in which case (P v Q) is true; (3) they both carry over, in which case (P v Q) is both true and false. Exactly the same scenario develops if Q is both true and false. And in any of these cases there is another option, namely, to regard the double truth-value as a wrench in the works and the result of vI to be simply indeterminate in truth value.]

(5) He can reject vI. From P we can't move to (P v Q) at all, and therefore DS can't come into the picture at all.

Therefore there is one and only one scenario in which DS is even the problem (and one where it's not clear whether it is or not), and it is only if we are making the assumptions required for such a scenario that the issue arises. My point: in making the claim above you are making

a lotof assumptions that need not be made. They're not unreasonable, and I think many people make them, but when we are dealing with something as radical as dialetheism, we really shouldn'tbe too casual about what we're assuming.Brandon,

Well, what I said was meant to be neutral between (1) and (3). There's a plausible argument that (3) can be derived from (4) in any case, at least if (4) is taken to mean "(P v Q) is not true and it is not false."

So unless (2) can be plausible motivated, my point stands. Whether (P v Q) is taken as true and not false or as both true and false if P is both true and false and Q is just false, we've still got a situation where P is true and (P v Q) is true, but Q isn't true. If an inference rule starts with true premises and doesn't deliver a true conclusion, it is not universally truth-preserving. Hence DS is shown as invalid in this hypothetical *unless* the the truth of a disjunct no longer guarantees the truth of disjunctions in all cases.

As far as (5) goes, I'd point out that (a) there's no obvious conceptual connection between the two issues, since even if it's possible for the same proposition to be true and false, it's still the case that every time P is true (whether it is true and not false or both true and false), (P v Q) will be true (whether it is true and not false or both true and false.) Also, even if you abolish v-introduction, I'm not sure that saves the universally truth-preserving nature of DS in that scenario, depending on what the new rules are for deriving disjunctions.

You just didn't explain where you derived this from... Can you take me through it one more time?Sorry again for not explaining it at once. I used sequent calculus:

P, therefore P (Axiom)

P, therefore P v Q (vR intro; ie. v intro to the right-hand part)

P & -P, therefore Q (-L intro, ie. negation intro to the left)

This is very close to:

A --> A

A --> (A v B)

-A v (A v B)

(-A v A) v B

-(A & -A) v B

(A & -A) --> B

However, note that as B may be any expression, we may deduce that

(A & -A) --> -(A & -A)

(2) can be plausibly motivated in any case where Q is false, since if Q is false (2) meets the truth-functional requirement for the disjunction being false. The only question is whether one

prefersto regard the disjunction as true-false or just false. (Standard conventions for vI don't tell us, of course.) Since Q is arbitrary and therefore can be false, vI doesn't get you a true conclusion, and DS is blocked. (One could, of course, restrict Q to only true propositions in vI. That would block explosion as well.)On (3), again, I point out that on (3) the premise is false and if a premise is false it is illegitimate to apply DS to it in the first place. If it is illegitimate to apply DS to a premise, the inference is blocked without rejecting DS. And, again, legitimacy in this case is not established by standard conventions, which don't envision the scenario of a true-false premise; therefore it cannot be assumed that DS is applied legitimately here. One could just as easily deny that it should not be (that we should assume instead DS can only be applied to premises that are true and not false,

just as it is in classical logic). Remember, the issue is whether DS is legitimately applied at all; its truth-preserving character is not in question until you have decided that it can.You're right that we still have to ask questions about how DS is legitimately applied to premises when we allow propositions to be true and false. But this is true of every rule, as I pointed out before.

Ben,

I'm missing some of my notes, but I can still give you some suggestions for reading. It's not a very long list, because reading computer science papers on logic is very slow going for me; they're sometimes tricky to find, the notation is sometimes unfamiliar, and they sometimes assume things that people outside of computer science wouldn't necessarily recognize right off the bat. But these are ones that I've found are reasonably good, and they might provide starting points for further inquiry, should your work take you in that direction (not all of the following take the same stance with regard to vI, but they all at least put some restrictions on it). Some you might be able to find online.

Besnard and Hunter. Quasi-classical logic: Non-trivializable classical reasoning from inconsistent information. In Froidevaux and Kohlas, eds, Symbolic and Quantitative Approaches to Uncertainty, volume 946 of Lecture Notes in Computer Science. Springer (New York: 1995).

Hunter and Nuseibeh. Managing inconsistent specifications: reasoning, analysis, and action. ACM Transactions on Software Engineering and Methodology 7 (4, 335-367.

Carl Hewitt. Large-scale organization requires unstratified paraconsistency and reflection. in COIN 2007 (PDF).

Miarka, Derrick, and Boiten. Handling inconsistencies in Z using quasi-classical logic. In Bert, Bowen, Henson and Robinson, eds., ZB 2002: Formal Specification and Development in Z and B. Springer (New York: 2002).

Tom,

Looks to me like DS or its equivalent is still doing the work in the case you give us. Even if it's all happening in the consequent of the conditional, the moves are *exactly* the same. You still went from (P v Q) to Q, and you were allowed to make that jump beause and only because of the presence (in the antecedent in this case) of ~P. The basic inference is the same, that if a disjunction is true, and the negation of one of the disjuncts is true, then other (not-negated) disjunct must be true. It is this inference that is clearly not universally truth preserving if P and ~P can both be true, since P could still be the disjunct that made the disjunction true, even if Q is just false.

Brandon,

I think the point to central point here is not what the classical logic handbook tells us to run if (P & ~P) were taken into account as a possibility--ultimately, of course, you’re right that the rules would need to be re-written from floor to ceiling if that were possible--but whether, if that were possible, Q would still be true every time (P v Q) was true and ~P was true. It seems pretty clear that it wouldn’t be, and that any inference that, when applied to even one case where the necessary premises are true, doesn’t generate a true conclusion, isn’t a valid inference.

I’m a bit unclear about how you could non-arbitrarily eliminate v-introduction if it remains the case that disjunctions are true as long as at least one disjunct is true. If that’s the case, then it can’t help but follow from P's being true that (P v Q) will be true, since (P v Q) will be false regardless of the truth, falsity, or [creative alternate truth vaue of your choice] of Q. If the requirements for statements of the form (P v Q) being true are changed--if disjunctions, are say, true only as long as both disjuncts are true, which would make (P v Q) a redundant extra way of saying (P & Q) , or if they’re only true if one disjunct is true and the other is false, in which case (P v Q) would now be a redundant extra way of saying [(P v Q) & ~(P & Q)]--then it strikes me not that we’d be talking about some logic in which v-introduction was no longer universally truth preserving, but rather about one in which we’ve decided to eliminate our ability to talk about disjunction.

Brandon,

Thanks for the reading suggestions! I'll definitely be looking some of that up.

Ben, I agree. But the problem, in my opinion, is not DS, but basic laws of (classical) logic themselves. First, Law of Non-Contradiction is negated. This is possible only together with negation of the Law of Excluded Middle, on which DS is based. And it seems to me, the Law of Identity is negated also, for we cannot deduce any more P --> -(-P), neither P --> P. Or, if P --> P still applies, standard inferences do not. Dialethic logic, if such can be, cannot be treated in logic based on classic laws.

One result is especially interesting in logic, where basic law is: Possible P & -P, where P is non-terminal expression:

Possible [(P & -P) & -(P & -P)],

ie. possibility of negating its basic law is derived from its basic law.

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