How To Think About Paradox

Philosophers often grapple with paradoxes. But what is the best way—or at any rate, a good way—of thinking about such entities? Here’s one proposed answer to this question, a framework for thinking about paradoxes.

A set of propositions is strongly paradoxical (or a strong paradox) iff its members jointly imply a contradiction, and weakly paradoxical (or a weak paradox) iff it is not strongly paradoxical, but is such that its members jointly imply a falsehood.

We don’t really need the notion of weak paradoxicality. This is because, for any weak paradox {A1, …, An}, there is a corresponding strong paradox {A1, …, An, -B}, where B is the falsehood implied by A1…An. For example, corresponding to the weak paradox {All Russians are smokers, Vladimir Putin is Russian}—whose members jointly imply the falsehood that Vladimir Putin is a smoker—there is the strong paradox {All Russians are smokers, Vladimir Putin is Russian, Vladimir Putin is a nonsmoker} (whose members, of course, jointly imply the contradiction that Vladimir Putin is both a smoker and a nonsmoker). (This example highlights the interesting point that for each deductively valid argument from premises A1…An to conclusion B, there is a corresponding strong paradox {A1, …, An, -B}.) From here on out, I will therefore concentrate on strong paradoxes.

Also, let a properly strong paradox be a strong paradox which is such that none of its proper subsets (i.e. those sets whose members are all also members of the paradox, but which are not identical to the paradox itself) is also a strong paradox. This is, among other things, to weed out “pseudo-paradoxes” that are not interestingly different from other paradoxes. For example, {All Russians are smokers, Vladimir Putin is Russian, Vladimir Putin is a nonsmoker, Paris is the capital of France} is just as strongly paradoxical as is {All Russians are smokers, Vladimir Putin is Russian, Vladimir Putin is a nonsmoker}, but Paris is the capital of France is superfluous. It does not add anything to the paradox.

We have skirted the issue of what makes some paradoxes worth talking and thinking about. Why is it that the Liar Paradox and Zeno’s paradoxes of motion have been debated and thought about since Antiquity, while no conferences or books will be dedicated to my “Putin paradox”? Intuitively, the beginning of a decent answer is that with the “Putin paradox,” it’s perfectly clear which of its elements we should reject (namely, All Russians are smokers). Call a paradox like the Liar Paradox or Zeno’s paradoxes—one that seems worth talking and thinking about, qua paradox—an interesting paradox, and one like the “Putin paradox” uninteresting.

In light of what was just said, it’s tempting to define an interesting paradox as a paradox which is such that it is unclear which of its members should be rejected. But this definition admits of non-paradoxes. For example, it would make {Jack the Ripper was a physician, Jack the Ripper was not a physician} an interesting paradox. And although the question of which of these propositions is true is interesting, it doesn’t seem paradoxical in any way.

So what should we do? A clue might be gleaned from the common definition of a paradox (given, e.g., in Quine’s “The Ways of Paradox”) as an argument from plausible premises to an implausible conclusion, combined with my previous observation that every deductively valid argument has a corresponding strong paradox. Suppose we have a valid argument from some plausible premises A1…An to an implausible conclusion B. On the arguable claim that the negation of an implausible claim must be plausible, the corresponding strong paradox, {A1, …, An, -B}, will then consist of propositions that are individually plausible but jointly unsatisfiable. A plausible proposition can be defined as one whose truth is is more likely than not, all available evidence considered. We can now tentatively define an interesting strong paradox as a strong paradox whose members are all individually plausible. Admittedly, the notions of plausibility and implausibility probably stand in need of further clarification, but this is at least a good starting point.

Note that the elements of an interesting paradox needn’t be equally plausible. Both The Pythagorean theorem is true and Lee Harvey Oswald murdered John F. Kennedy are (in my view, at least) plausible on this definition, though the former is far more plausible than the latter.

We are not, strictly speaking, rationally required to disbelieve an element of each proper strong paradox. Instead, we can suspend judgement about several of its elements. (Three more definitions: To believe p is to believe that p, to disbelieve p is to believe that not-p, and to suspend judgment about p is to believe neither that p nor that not-p. Thus, e.g., theists believe the proposition that there is a God, atheists disbelieve it, and agnostics suspend judgment about it.) In the case of the aforementioned (properly strong) “Jack the Ripper paradox,” for example, we would not disbelieve any element of it, but plead ignorance about the truth-values of both elements. However, if we believe every element of a proper strong paradox but one, it does seem that we are rationally required to disbelieve that last element, for the following reason. Let A be the conjunction of every element of a proper strong paradox, B a conjunction of every element of that paradox but one, and p the element of the paradox that is missing from B. I take it that we are rationally required 1) to reject every proposition that is logically unsatisfiable, 2) to believe everything our beliefs logically imply, and 3) to take the same attitude (i.e. belief, disbelief, or suspension of judgment) towards any two propositions that are logically equivalent. Now, suppose we believe B. Which of the three attitudes, if any, are we then rationally required to take towards p? In the first place, we cannot believe p. We already believe B, so also believing p would mean believing the conjunction of B and p. But the conjunction of B and p is equivalent to A, which is unsatisfiable, and by the first requirement, we are thus required to disbelieve it, which means, by the third requirement, that we are also required to disbelieve the conjunction of B and p. So, belief in p is incompatible with the requirements of rationality. This leaves agnosticism and disbelief—so what about agnosticism? Well, we disbelieve A, which, as mentioned, means we must also disbelieve B & p, which is to say believe -(B & p). And B—which, remember, we believe—together with -(B & p) logically implies -p, which means, by the second requirement, that we are required to believe not-p, which is to say disbelieve p. So disbelief seems to be the only option.

Now, given our definition of an interesting paradox, we want, on the face of it, to believe as many elements of such a paradox as we rationally can, because ex hypothesi, they are plausible. In the case of proper strong paradoxes, this means every element of the paradox but one. So, in practice, our first resort when trying to solve interesting, strong, proper paradoxes should be to find that element of the paradox that can be disbelieved at the lowest cost, or, perhaps equivalently, the element that is least plausible. In some cases, though, it may turn out that suspending judgment about several elements of the paradox is less costly than disbelieving one.

Why is all this logic-chopping useful? Because it clarifies what we do (or what we should do) when we grapple with philosophical paradoxes. Most such paradoxes are (or are usefully expressible as) strong, proper, interesting paradoxes. For example, the (in)famous “liar paradox” may be considered the set of the following propositions:

 

(1)   The statement “This statement expresses a false proposition” (henceforth, L) expresses a proposition

(2)   If L expresses a proposition, p, and p is true, then p is false

(3)   If L expresses a proposition, p, and p is false, then it is false that p is false

(4)   For any proposition p, if it is false that p is false, then p is true

(5)   Every proposition is either true or false

(6)   No proposition is both true and false

 

I won’t begin to grapple with this paradox here, but I do think the above exposition clarifies what we must do when faced with it.

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