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

(Cross-posted at M-Phi)

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.