Monday, 16 January 2012

Paradoxes: something's gotta go, but what? And why?

(Cross-posted at NewAPPS)

The coming months, I’ll be teaching a course on paradoxes, which will focus on historical and methodological rather than technical aspects, so it is quite likely that there will be a constant stream of blog posts on paradoxes. We shall see…

One useful definition of a paradox is the one offered by M. Sainsbury in his highly influential book Paradoxes (p.1):

This is what I understand by a paradox: an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises. Appearances have to deceive, since the acceptable cannot lead by acceptable steps to the unacceptable. So, generally, we have a choice: either the conclusion is not really unacceptable, or else the starting point, or the reasoning, has some non-obvious flaw.

Paradoxes create a situation of cognitive dissonance which must be resolved in one way or another. (Moreover, in those fields of inquiry where theories cannot be straightforwardly tested empirically against reality, if a paradox arises, it is often seen as a sign of the inadequacy of the theory.) Different entrenched beliefs are shown to be in tension with one another, so something’s gotta give. But what should give?

Sainsbury’s passage already suggests the three main alternatives: i) one of the premises must be rejected; ii) one of the steps of the reasoning involved must be rejected; iii) the apparent unacceptability of the conclusion must be revised. Different solutions to the Liar paradox illustrate these three approaches:

1) Tarskian approaches reject some of the premises, namely some of the principles guiding a naïve conception of truth.

2) Revisionist approaches (e.g. Field’s) revise the logic underlying the reasoning giving rise to the paradox.

3) Dialethist approaches revisit the unacceptability of the conclusion of there being one sentence (the Liar sentence) which is both true and false.

Naturally, some solutions to paradoxes blend more than one of these approaches, but any proposed solution to paradoxes must take at least one of these three routes to dispel the situation of cognitive dissonance.

I’ve professed elsewhere my sympathy for type 3 approaches, i.e. those which are not afraid to embrace the conclusion after all, in spite of its apparent couter-intuitiveness. Naturally, the point is not simply to uncritically accept the unacceptable conclusion, but rather to view it as possibly a non-trivial genuine discovery; the history of science is full of surprising discoveries.

But rather than arguing for type 3 approaches, today my goal is to elaborate a bit on why I especially dislike type 2 approaches, i.e. revisionist approaches. My distaste for them does not stem from a particular fondness for classical logic, or whatever other well-entrenched system of reasoning. Rather, my issue with type 2 approaches is that such rejections of certain rules of inference typically have a ‘fix-up’ feel to them: this particular rule has served us well until now, seems well-motivated, but in order to avoid paradox, we should just ditch it, for no other reason. However, if a given rule or principle of logic is to be rejected, it seems to me that such a rejection must be based on independent grounds, not only the fact that the paradox may be blocked.

Another reason for rejecting such revisionist approaches (and one which has been articulated by e.g. Stewart Shapiro in presentations) is that you may end up with a logical system that is no longer useful as a tool for reasoning. Try doing mathematics with the ‘logic’ developed by Field to cope with paradoxes! You may be able to avoid paradoxes, but the price is just too high. Or to pursue the ‘pathology’ metaphor which is often used in connection with paradoxes: you may cure the disease, but the treatment is so violent that you are left with a highly dysfunctional organism. (So there might be pragmatic reasons for adopting type 3 approaches as well, something along the lines of ‘stop worrying and learn to love the paradoxes’.)

This way of setting up the issue occurred to me when skimming through the recent paper by Elia Zardini in RSL 4(4), 2011, ‘Truth without contra(di)ction’ (which I haven’t gotten around to giving all the attention it deserves yet!). The abstract says:

I propose a new solution to [semantic] paradoxes, based on a principled revision of classical logic. Technically, the key idea consists in the rejection of the unrestricted validity of the structural principle of contraction.

Contraction is the structural rule according to which if premises A1, A2 … B, B … entail C, then premises A1, A2 … B … also entail C; that is, one of the copies of B has been ‘deleted’ (contracted), and the consequence still holds. Contraction allows for the ‘deletion’ of premises that occur multiple times (as long as at least one copy is left in place), and is a valid rule in most (but not all) logical systems available in the literature. Linear logic is one of the few prominent logical systems which reject contraction.

In section 2.3 of his paper, Zardini briefly discusses the gist of his independent motivation for rejecting – in fact, restricting – contraction. He writes (p. 504):

But what is the intuitive rationale for restricting contraction? What is it about the state-of-affairs expressed by a sentence that explains its failure to contract? […] I believe that in attempting to find these answers one has to step out of the abstract realm of formal theories of truth and engage in some concrete metaphysics of truth.

He then goes on to argue for the idea of unstable states-of-affairs as those where contraction would fail. Others (e.g. Ole Hjortland) have been thinking about the idea that contraction may be the ‘real villain’ in terms of giving rise to paradoxes. But to my knowledge Elia is the first to ask the question of the plausibility for rejecting/restricting contraction outside a formal system, and independently of the goal of blocking paradoxes in and of itself. I still need to think more carefully about his idea of ‘unstable states-of-affairs’, but at least I think he is asking the right questions.

This way to go about type 2 solutions to paradoxes, i.e. revisionist solutions, makes me much happier than the usual ‘fix-up’ approaches that abound in the literature. I sure hope that revisionists will follow Elia’s lead and start engaging in deeper philosophical, not only technical, analyses of why a given logical principle or rule is to be rejected.


  1. Interesting. I would have thought dialetheists were revisionary as well. Don't they need to reject the rule of inference that anything follows from a contradiction?

  2. Jonathan is right, although in my opinion the rejection of explosion (i.e., that everything follows from a contradiction) isn't that much of a problem. I think a truly pressing problem is what, if contradictions can be true, we are to make of proofs by reductio ad absurdum. If we reach a contradiction from the assumption that p, it seems the most a logic like LP (Priest's Logic of Paradox) could tell us is that *either* p is false *or* that the contradiction is true. I'm not sure how useful LP or similar logics would be in comparison Field's, but the weakening of proofs by reductio is pretty bad all by itself.

  3. @Jonathan The difference is that the paraconsistent rejection of explosion is not originally motivated by the need to avoid paradox as such, so it has the kind of independent justification that I refer to in the post. Rather, the motivation was to describe the 'logic' underlying theories that were inconsistent and yet not trivial, in particular some physical theories.

    In Priest's case, accepting that there may be propositions that are both true and false (gluts) then leads to the revision of the logic, but not as a means to block the emergence of paradox. That's why I see this approach as a type 3 approach, even though it also entails a revision of the logic.

    @Jason, quite a few mathematicians are more than happy to do without reductio proofs, for reasons unrelated to paradoxes. As for paraconsistency and reductio, I actually do think you can still have reductio proofs in a paraconsistent setting, it's just that there will be fewer such proofs than in the classical setting. This is because the class of pairs of propositions that a paraconsistent logician is prepared to hold as jointly untenable is smaller than the corresponding class for the classical logician. I defend this point in the only paper on paraconsistency I ever wrote, in the 'Handbook of Paraconsistency' (2007 or 2008, I'm not sure).

    1. Nothing of the independent motivations you mention - dealing with the logic of inconsistent theories - supports dialethism. This is what Priest borrows from relevance systems, but one can see truth value gaps rather than gluts in any such motivation. The only thing that can actually motivate accepting a contradiction is a paradox in the sense you described: an argument from plausible premises that leads to a contradiction.

      Graham certainly tries to distinguish the good from the bad contradictions, and to make sense of how one could ever know the difference, but I deny that he has done so adequately. So giving up the claim that one has lost in the dialogical game if one reaches a contradiction remains, as I see it, a huge revision.

      Beyond this, note that messing with reductio alone won't do it for you. There are paradoxes that don't rely on it. Beyond this, impredicativity as a phenomenon is much broader than semantic paradoxes. This, to my mind, is something that is generally unappreciated by philosophers and a problem with all of these approaches.

    2. This post was not in any way meant to argue for dialethism. I am making the methodological point that approaches to paradox that revise the implausibility of the conclusion have much to be commended for, not arguing in favor of any specific instantiation of such approaches. Also, it is important to distinguish dialethism and paraconsistency; dialethism is a specific variant of the general paraconsistent framework. Personally, I can't say I am a dialethist anyway, so I don't see why I am being asked to defend this position here :)

      Perhaps a less controversial illustration of type 3 approaches to paradox is the one mentioned by Graham himself in his NYT piece, namely Cantor's 'solution' to Galileo's paradox of infinity. At first sight, the conclusion that two infinite can be of the same size and yet one be a proper subset of the other was implausible, but now everyone accepts it.

    3. I was responding to your claim that Graham's type 3 solution is independently motivated - independent of solving paradoxes - while type 2 aren't. The specifics of what sort of paraconsistency one endorses is irrelevant to that point. The way you have set things up, including the definition of 'paradox' makes it pretty close to true by definition that a type 3 solution can only be defended as a solution to paradox.

    4. Well, there is a difference between something not being independently motivated and you not finding the arguments convincing :) Priest takes the idea of there being true contradictions very seriously; he applies it elsewhere (i.e. beyond the specific case of the Liar) and claims that it sheds new light on all kinds of other issues. You may not find his arguments and his program convincing/appealing, but the methodological point I was making still stands.

      Let me quote a passage from his NYT piece:

      "Scientific advances are often triggered by taking oddities seriously. For example, at the end of the 19th century, most physicists thought that their subject was pretty much sewn up, except for a few oddities that no one could account for, such as the phenomenon of black-body radiation. Consideration of this eventually generated quantum theory. Had it been ignored, we would not have had the revolution in physics produced by the theory. Similarly, if Cantor had not taken Galileo’s paradox seriously, one of the most important revolutions in mathematics would never have happened either."

      What I mean by type 3 approaches is not merely to resign yourself to accepting the implausible conclusion; it is also to take the implausible conclusion seriously and test it as a hypothesis elsewhere, so to speak. We could refer to this as 'robust acceptance', as opposed to 'weak acceptance' which would be the acceptance of the conclusion merely because it is the conclusion of the presumably valid and sound reasoning in question. Weak acceptance would be ad-hoc, but robust acceptance would not.

  4. Okay, but then why not have two first-level categories of response to the paradox (revisionist and non-revisionist) and then two sub-categories among the revisionists -- one for ad hoc revisions and one for not-ad-hoc revisions? I guess what I want to know is whether there is anything more to the distinction between types 2 and 3 than just the motivations of the theorists.

  5. The main difference is what I spelled out in the post: type 2 solutions single out one of the inferential steps actually used in the derivation of the paradox, and then declare that it is no longer a valid inferential step. Type 3 solutions do no such thing, as the law of non-contradiction (which must be revised in virtue of accepting the existence of gluts) is not used in your typical derivation of the Liar paradox (although I suppose one could give a formulation of it where LNC does occur). All you need is the naive truth-schema to go from the assumption that the Liar sentence is false to the conclusion that it is true, and from the assumption that it is true to the conclusion that it is false. Again, type 2 revisionist approaches consist in rejecting one of the elements of the reasoning used specifically to derive the paradox in question, not any arbitrary revision of the logic.

  6. (Cross-commented at NewAPPS.)

    Regarding the Liar and the taxonomy (1) - (3), I have an undeveloped strategy in mind which I'm not sure fits (though perhaps it does in (1)?).

    Briefly and loosely: I have found it useful to approach the Liar paradox (and hopefully in turn its kin, such as the truth-teller sentence, the two-sentence version, Yablo's paradox) with the question: what should we say about the Liar sentence? Is it true, false, meaningless or something else? The answer, to my mind, is pretty clearly 'something else'. The problem then is to work out a way to say all that seems salient about such sentences, without saying paradoxical things in the process.

    For example, one might call the Liar sentence 'unevaluable' to mark the fact that, going by ordinary evaluation algorithms, we get into an infinite loop. But one might want to say something which distinguishes it from the truth-teller (something like oscillatory/paradoxical unevaluability vs. "stable").

    Furthermore, one might be led to consider 'This sentence is unevaluable' as unevaluable as well. Now, if an unevaluable sentence is neither true nor false, the statement that the Liar sentence is false will be false, as will the statement that the Truth-teller is true. By contrast, the statement that 'This sentence is unevaluable' is unevaluable will be true.

    One may then be led to posit "honorary truth-values", to mark this difference. But then statements about honorary truth-values will require a second level of honorary truth-values, and so on, leading to something like a type-hierarchy, where one (it may seem) just can't say certain things about everything in the hierarchy. (It was at this point that I stopped thinking about the matter, but this blog post has made me want to go back to it. Scary stuff.)

    Anyway, I'm inclined to call this a 'classificationist' approach or something (note that one needn't say that "level 1 honorary truth", or whatever, is truth in any naive or ordinary sense). I'd be interested to know where people see it as fitting.

  7. Contraction could also be argued to be the villain behind Curry's Paradox, which is in a sense much worse than Russell's --- it won't go away simply by a paraconsistentist revision.

    At any rate, there are many other examples of non-classical logics failing contraction, and there is no need to go fully substructural to find one. Think about Lukasiewicz logic.

    1. Yes, blaming contraction is the latest fashion in the world of Curry, as discussed for example in a paper by JC Beall and Julien Murzi that has been mentioned before here at M-Phi a few times:

      It would be interesting to see in each of these cases where contraction is rejected or restricted what the rationale/motivation is. To repeat something I said in the post, what I am after are independent reasons why contraction is the 'bad guy', besides the fact that it gives rise to paradoxes.