Wednesday, January 23, 2008

A Thought About Underlying Logics

My apologies for the weeks between this and the last post. I'm back in Miami, slugging through my reading list, and I should be posting at a much more frequent clip from now on.

Meanwhile, I have a thought (not really a full thought, but at least the beginning of one), not about dialetheism per se but about what Graham Priest calls the "second grade of paraconsistent involvement."

Just for future reference, his "grades" are:

1st: "Gentle-strength paraconsistency" (you reject the principle that anything follows from a contradiction)

2nd: "Full-strength paraconsistency" (you think there some inconsistent but interesting, non-trivial theories)

3rd: "Industrial-strength paraconsistency" (some of those theories may be true)

4th: "Dialetheism" (some of those theories *are* true)


Standard apologetics for the usefulness of paraconsistent logic often include historical examples of inconsistent but non-silly theories. It generally goes something like this. "Impressive Scientist X believed P and he also believed Theory Q, and he knew that Theory Q entails that not-P, but he didn't derive just any claim R, so the underlying logic he was using was clearly not classical."

Now, on the face of it the only sort of explosive logical rule that this situation would be any kind of challenge to would be a claim in the language of epistemic logic that Bxp & Bx~p entailed Bxq for any q. [Of course, (Bxp & Bx~p) is not a contradiction. Only (Bxp & ~Bxp) would be a contradiction.]

Sadly, this has nothing to do with what the historical examples are there for. The historical claim here is not that according to any sort of classical logic, it should follow from Scientist X believing P and ~P that Scientist X will believe that Q for any Q. Rather, the point is that if Scientist X believes that P and he believes that ~P, then classical logic would give him permission to draw the conclusion that Q. The fact that he never exercised this privilege is then seen as evidence that he was (albeit unconsciously, when we're talking about figures who predated the development of paraconsistent logic) operating according to paraconsistent rules of inference in which there are strict limits on what you can derive from a contradiction.

Now, I have severe doubts about the very idea that either scientific practices (or, worse yet, as is sometimes claimed, natural languages) have "underlying logics," but for the moment I'm going to put that to one side.

Instead, let's go for a simple analogy.

Einstein believed in the Special Theory of Relativity.

Einstein never drew the conclusion that "if it is not the case that it is not the case that it is not the case that it is not the case that it is not the case that the Special Theory of Relativity is true, then it is not the case that it is not the case that it is not the case that it is not the case either that Hitler won World War II or that the national flag of Australia is a pirate flag containing a jewel, that, under magnification, is revealed to be a detailed map of the surface of Mars with an X marking the spot where Elvis has high tea every afternoon at four with the Mad Hatter and a ghostly semi-physical representative of the Hegelian world-spirit."

But wait!

According to classical logic, the entire quoted claim is entailed by the truth of the Special Theory of Relativity. (Unless, of course, I slipped up while counting the number of negations.) If the STR is true then it follows that if you put an odd number of negation signs in front of the proposition that it is true, you have a false claim, and any conditional statement with a false antecedent is true, regardless of whether the consequent is true or false. Einstein never drew that conclusion, or any of the infinite number of other similar conclusions classical logic would have given him permission to draw. Does it therefore follow that his "underlying logic" must have been some alternative non-classical logic, where strict rules are in place to reign in the sorts of consequents that can be put on these conditionals?


Daniel Lindquist said...


I'm not sure why that shouldn't be a perfectly cromulent response, here.

Priest begins his "Introduction to Non-classical Logic" by pointing out that the material conditional seems to do a poor job of representing "if..., then..." statements, since it would license all sorts of weird cases that seem ridiculous in plain English. The various non-classical logics he considers are supposed to do a better job at having a "conditional" that works more like our "if..., then..." statements, since they allow fewer weird inferences. One of the ways he does this is by concocting examples similar to the one you gave; I can't seem to find one where he infers "If P, then Q" from "~P" with the material conditional, but he at least pulls the trick in 4.7.1 with the strict conditional: "If there is not an infinitude of prime numbers, then Brisbane is in Germany" seems to be false, but if "if..., then..." is read as the strict conditional, then its truth follows from the impossibility of their being a finite number of primes, ergo one might doubt that the strict conditional does a proper job at showing the job "if..., then..." statements do.

So, I'm not sure how your analogy is supposed to make Priest's Impressive Scientist X apologetics look any worse. If anyone finds the Impressive Scientist X schtick plausible as a case against explosion, then they should find your Elvis-on-Mars schtick plausible (albeit overwrought) as a case against the material conditional.

It is perhaps worth noting that what's relevant (both in your example and in Priest's Impressive Scientist X one) is not merely that so-and-so did not actually draw the conclusion which classical logic would've allowed, but that if they had drawn said conclusion, it would seem that they had done so wrongly -- the inference licensed by classical logic appears to be a bad inference, not merely one that it would be strange to draw, or that historically no one ever has in fact drawn.

FWIW, I am also uneasy with talk of "underlying logics", but I don't see how this counts against the paraconsistentist's argument, here. However we want to talk about the relationship between our symbolic logics and natural language, it would seem that these sorts of arguments show that classical logic does a worse job at capturing our inferential practices than a paraconsistent logic would.

Ben said...


Fair points as far as it goes about conditionals.

First of all, though, something I guess I should have been clearer about:

To the best of my knowledge, Priest has never offered the kinds of Impressive Scientist X apologetics I'm criticizing here. I could be wrong about that, but I've never read anything by him where he employs this kind of argument.

It is, rather, the kind of thing I've heard and read from people stuck at his First Grade of Paraconsistent Involvement, the "I'm not a dialetheist, but..." sort of enthusiast for paraconsistent logic. See, for example,

In fact, I doubt this line of argument would hold much appeal for a dialetheist, since they wouldn't need to resort to this kind of thing. Their claim, after all, is that regardless of the history, there are actually true claims of the form (P & ~P), so if we want to be right about how the world is, we'd best be using paraconsistent logic. No need to bother with historical apologetics about understanding the underlying mechanisms of past theories.

OK, so that's the preliminary note. Priest is not the target here. The non-dialetheist paraconsistentist is.

Second, I think that even if we assume that Einstein's inferential practices are best represented by some sort of alternative logic where conditionals are reigned in, I would still imagine that we could find some sort of long, strange, esoteric inference that logically you would be licensed to make from "the Special Theory of Relativity is true," no?

For example, even if you're working with a logic that deals with conditionals differently, if you still have v-addition, as lots of paraconsistent and otherwise non-classical logics do, then you could still infer from the truth of the STR that "either the Special Theory of Relativity is true or the national flag of Australia is a pirate flag containing a jewel, that, under magnification, is revealed to be a detailed map of the surface of Mars with an X marking the spot where Elvis has high tea every afternoon at four with the Mad Hatter and a ghostly semi-physical representative of the Hegelian world-spirit."

One explanation for why Einstein never inferred this may be that his "underlying logic" was one that eliminated or put various restrictions on v-addition. I don't think, however, that this would be the only or even the most obvious explanation.

Daniel Lindquist said...

The Stanford Encyclopedia article you link to is at least co-authored by Priest; he's named at the bottom, with Koji Tanaka. So, Priest does seem to say this sort of thing. There is also 4.8.2 in "Introduction to Non-Classical Logic", where Priest repeats much the same material, nearly verbatim. (Bohr is an Impressive Scientist.)

I'm not sure why a dialetheist wouldn't want to have nice apologetics for paraconsistent logic (which might or might not be short of full dialetheism). Every little bit helps in the fight against the Old Guard, so to speak. This is certainly Priest's strategy in "What's So Bad About Contradictions?" at least.

I'm inclined to agree with you that "Einstein's underlying logic didn't allow just any v-addition" is not a very plausible explanation for why he never inferred some particular "PvQ" despite holding that P. But again I want to say that if Einstein had drawn this inference (for whatever reason), it would not strike us as an illicit move (though it might be a strange one). Whereas Bohr's (hypothetical) inference that Atomic orbits are rectangular from the other parts of his theory seems not just strange, but irrational -- it doesn't look like he should be able to infer that from his inconsistent atomic theory. Whereas the Elvis-on-Mars-v-STR example you give seems like an allowable inference, albeit one that is unlikely to ever be worthwhile to draw.

(It also occurs to me that one could attack the classical account of disjunction by means of examples like yours. I actually think that Brandon, the Siris guy, brought this up in an earlier comment thread here; I would look this up, but I am late for lunch.)

Ben said...

Ha! Thanks. Catching dumbass mistakes like that one before I repeat it anywhere important is one of the things that makes blogging worthwhile.

Brandon did pursue the "maybe we could get around explosion by banning v-addition" route, but my point here was just that even in standard paraconsistent logics, the inference to that disjunction would be legal, so if the fact that Bohrs didn't conclude any random Q from P & ~P is evidence that he was working with an underlying logic in which this would not be permitted, it should similarly follow that given that Einstein didn't conclude the fanciful disjunction mentioned above from the truth of the STR is evidence that he was working with an underlying logic in which it wouldn't be permitted.

I think a more satisfying explanation in both cases is that being an Impressive Scientist doesn't mean that you know enough about logic to be aware of everything that logically follows from your theories.

Unknown said...

Daniel is exactly right about the material conditional, that is clearly the weakness in your Einstein example. It takes more than a false antecedent to make an implicature true.

Priest brings up his "scientist believed x" schtick at length in Doubt Truth to be a Liar (2006). The thing I find interesting about his favorite case, Bohr's atomic theory, is that it doesn't rely on the sort of thing you attribute to him: to wit, an argument beginning "Bohr believed contradictory thing, therefore..." Rather, the case of Bohr's atomic theory is interested because the inconsistency is endemic to the theory (it's not that Bohr just happened to be invested in separate and unrelated beliefs that are mutually inconsistent). Not only that, but Bohr's atomic theory was wildy successful theory in terms of predictive power over its competitors.

So what? Well, I think the connection to paraconsistency comes from thinking about the normativity of validity. Some thing follow from other things. These are relationships of valid inference. The facts of validity, in themselves, tell us nothing about how to reason. But seeing as valid inference preserves truth and truth is the goal of inquiry, it seems like sound advice to follow valid inference patterns. We ought to reason validly. Bohr had a great theory for its time, he was a brilliant scientist, hence a paradigm of rational inquiry, and yet despite the inconsistency of his theory he never drew arbitrary conclusions from it. Putting these things together: either Bohr was irrational, or he was reasoning correctly, but according to a paraconsistent logic. I'll let you decide which disjunct is more likely true.

Ben said...


If the material conditional is the weakness of the original Einstein example in the post, how about the modified Einstein example in the comments? Do you find classical disjunction just as problematic as the material conditional?

In any case, on Borh as paradigm of rational inquiry, I'd say that if he was knew that his atomic theory and the underlying Maxwell equations were inconsistent, and he literally believed both to be true, he was being, to that extent irrational, and in general that it's entirely possible to be one in certain respects, but not in others. Isaac Newton was certainly a brilliant scientist and one of the all-time iconic paradigms of rational inquiry *in some ways,* but he also wrote extensively about numerology, witchcraft, etc.

I think a useful way to think about the Bohr believing an endemically inconsistent theory is to ask what he would have done if the "anything follows from a contradiction" issue had been brought to his attention. (This eliminates the possibility that I've been pushing with the two Einstein examples, that if something only follows from a theory according to some slightly obscure logical steps, it's not surprising or interesting that even a brilliant scientist who was unaware of the formal details wouldn't make the inference, and says nothing about any weakness of the logic in capturing intuitions since there will be obscure derivations in every logic.) Let's say a wild-eyed logician had stumbled into Bohr's office, told Bohr he was being inconsistent and that logically, it's possible to derive anything at all from a contradiction. Perhaps, with a mischevious gleam in his eye, our trivialist logician would have urged Bohr to start deriving all sorts of crazy, unrelated physical claims from the core contradiction in his theory.

I'd expect Bohr, if he didn't just kick the guy out of his office and get back to work, would have demanded to see the proof that everything follows from a contradiction. At that point, I'd like to believe that Bohr, being a smart and discerning guy, would have noticed that one of the steps in the proof (Disjunctive Syllogism) is obviously not truth-preserving if you remove the assumption that the Law of Non-Contradiction is true. At that point, Bohr would have had at least three options:

(1) He could have done what Priest would have liked him to do, and believes he was somehow unconsciously doing, and abandoned DS and other such rules in favor of adopting some sort of paraconsistent logic so that he could continue to put "T"'s under two mutually inconsistent claims.

(2) He could have done what I would have liked him to do, and chosen to keep the Law of Non-Contradiction, hence keep DS and the rest, and abandon any claim that both halves of his theory were true. This would not necessarily involve changing anything in terms of immediate scientific practice, but just a theoretical clarification, like, "my atomic theory, in some new form I haven't discovered yet that doesn't rely on Maxwell's equations, might be true (in which case, the current version is just a sort of instrumentally useful placeholder that we need to be cautious about taking too literally), or Maxwell's equations might turn out to be exactly right, in which case no form of this atomic theory is right. Since I don't have enough evidence to decide between the two, I'll affirm the truth of the disjunction of those two possibilities and vigorously pursue each disjunct as an avenue of research. The one thing I won't do is affirm the actual truth of the conjunction, because whether or not everything follows, I can see that two inconsistent things can't literally be true."

(3) He could have decided, for whatever reason, that not only did the correct atomic theory a counter-example to the Law of Non-Contradiction, but that the Law of Non-Contradiction was itself a dialethia. This would allow him to keep on using Disjunctive Syllogism, since that's only OK if the Law of Non-Contradiction is true, and keep on affirming inconsistent premises about atomic theory or anything else, since that's only OK if the Law of Non-Contradiction is false. If he decided that the LNC was a true contradiction, and *only* if he made that bizarre call, would he have any sort of logical license to derive just anything from an inconsistent theory.

(Thought about this way, it's sort of amazing that Priest & co. see something amazing--proof of underlying paraconsistency--in the fact that he didn't derive just any random thing from his inconsistency. That's, of course, assuming that Priest & co. have got the history right. I don't know enough about Bohr's attitude towards scientific realism, etc., to know if he thought his inconsistent theory was true or only useful.)

J said...

Conditionals are just fancy conjunctions:

P -> Q =

-P v Q (right? same truth table) =

-(P & -Q)

SO: "If a X is a prime number, then it is a natural number which has exactly two distinct natural number divisors: 1 and itself."

Which means you could not have a prime number X AND X is not a natural number which has exactly two distinct natural number divisors: 1 and itself.

Or "IF cat (the actual animal, feline, etc.), then a mammal."

One could not have a cat X and X is not a mammal.

AS long as one works with mathematics (or well-established physical science), and does not use causal relations, time, modality, then conditionals (ie denied conjunctions) are not really an issue. (as some analytical types realized, back when they advocated a separation between analytical and synthetic "truths")

Q e d

Ben said...

OK, but I think I'm missing the relevance to the point at hand. We'd moved on from conditionals.

J said...

No, the point is that physics, especially modern physics, is not analytical or truth functional: it's all about probability, inference, causality. Conditionals do not apply to probability or causality (really induction), except informally. Even Hume's old dicta on cause still holds (and perhaps more so apres-Bohr): yes, an apple dropped seems to obey regular "laws" (ie Newton, or Einsteinian in right circumstance), and one would not likely bet on the apple rising upwards, but those laws are not laws in the sense that the pythagorean theorem is a law, or the law of non-contradiction is a law. There's no violation of the Law of N-C if one day a dropped apple rose upwards--- or when Einstein altered Newton's theory.

Unknown said...

Ben, I agree that I overstated the case for Bohr as paradigm of rationality. And you are certainly right that any of your options are viable ways of responding to an inconsistent theory that we had good reason to believe. Maybe a better way of bringing out Priest's point is to say that it doesn't really hang on anything Bohr believed. It hangs on the fact that at some historical moment, the best going theory in terms of predictive power was an inconsistent theory. So, the suggestion continues, if we found ourselves in such a position it would seem like the rational thing to do is hold the theory and reason paraconsistently. There are obvious reasons we want to be able to believe our most predictive theories, whereas the alternative of rejecting such theories (in some cases) just because they are inconsistent seems under-motivated.

J said...

Ever hear of Kuhn? Kuhn and other philosophers of science have pointed out that
scientific theories are not "true" in analytical sense (i.e the pythagorean theorem) but a process (Hume had already suggested as much). Paradigms (like Newtonian mechanics) are formed, and then are modified, in some cases radically so (via Einstein or quantum physics). That needn't imply some subjectivism or relativism (as some people, mostly crypto-theists or idealists of some sort, accuse Kuhn of). It means science, even hard-headed scientific materialist science as process. The same situation is happening with the theory of evolution.

Attempting to wedge an entire theory into truth functional terms thus is, like, so wrong. Besides, it's the various fact-statements of the theory that are subject to revision (er, and may involve the...V-word, as in verification), not the "Truth" of the entire theory, which is in flux. And even the various comments on truth from hicks like Pierce and James remain somewhat relevant here: one could go on and on about the "truth" of the scientific theory underlying say a proposed cure for hepatitis C, but it don't mean scheisse unless the cure actually works with real humans, and stops the disease. "Truth" is just a word.

Ben said...


I guess this is the heart of the matter here, whether its irrational to disbelieve inconsistencies despite the fact that they are a good fit with the evidence. If we can't make sense of the idea that two inconsistent things could both be true, then that's pretty good motivation for disbelieving inconsistent theories. If we can make sense of it, if in other words, inconsitent truths are not impossible in principle, then the situation is entirely different.

I would argue that this is a globalized problem for how we think of evidence in general that, taken seriously enough, leaves our normal evidential reasoning in an unrecognizable state--in other words, once you start believing that inconsistent truths are possible in principle, and that we should believe them whenever the evidence favors them, I have a hard time seeing what line you draw in principle against seeing *every* case of mixed evidence as evidence for a true contradiction. OJ's DNA matched the DNA at the crime scene? That's evidence that he killed his wife. OJ's hand didn't fit into the bloody glove? That's evidence that he didn't kill his wife. Why not take the fact that there is evidence for both claims as evidence for their conjunction?


Sure, (consistent, non-tautological) scientific theories aren't true (or false) analytically, but surely they're true (or false) synthetically, right?

Unknown said...

Ben said, "I have a hard time seeing what line you draw in principle against seeing *every* case of mixed evidence as evidence for a true contradiction."

Well that sounds to me like a problem about evidence, not about the rationality of inconsistent belief. I guess there are two issues to consider here. First, if we accept the rationality of some inconsistency, why should we care how much there is? That's a serious question worth pondering. Second, how is the problem of settling where the evidence falls with respect to inconsistent pairs of sentences any different from general problems about settling where the evidence falls? It is hard to figure out how to develop our theories in light of the data because the connections are often tenuous. But so what? That just means we have to look at each case, do our best to apply our best methods, and to some extent make choices about what we treat as most salient to the target phenomena, the most natural interpretation, and so on. Why is 'mixed evidence' a special problem?

Ben said...


Mixed evidence is a special problem because evidence is more often than not mixed. If inconsistent truths are possible, then it seems like we should level the epistemic playing field and set the bar for how much evidence it takes to accept true contradictions down as low as it is for true consistencies of the same type. (E.g. we don't need any more evidence for each conjunct to believe the conjunction "Noel was in the grocery store at 3 PM and not in the grocery store at 3 PM" than we would need for each conjunct to believe the conjunction "Noel was in the grocery store at 3 PM and he was at the laundromat at 3:15.") If we start doing that, then the number of true contradictions we start believing in would, I think, get very large, very quickly, to a degree that should make a good dialetheist question whether he can really be as confident as Priest says he is that the statistical frequency of true contradictions is very low.

(This last point is crucial, since only if we have good reason to think that the statistical frequency of true contradictions is very low is there any justification for assigning very low epistemic probabilities to any given contradiction, which is the point on which the "classical re-capture" hinges. If the re-capture fails, and we can't use rules like DS even in a probabilistic sense, well, this doesn't show that dialetheism is wrong or anything, but it does perhaps show that its effect on our ordinary reasoning processes is more dramatic than its advocates would have us believe.)