As regular readers know, I understand paradoxes to be a particular type of argument. In particular, a paradox is an argument:

That begins with apparently true premises That proceeds via apparently truth-preserving reasoning That arrives at an apparently false (or otherwise unacceptable) conclusion.

[An argument is truth-preserving if and only if there is no way for the premises to be true and the conclusion to be false.]

Solving a paradox, then, proceeds either by arguing that one of these three appearances is illusory (i.e. a premise is not true, the reasoning is not truth-preserving, or the conclusion is not false), or by arguing that some concept involved in the argument is faulty.

There is another way that logicians sometime define paradoxes. On this alternative understanding, a paradox an argument:

That begins with true premises That proceeds via truth-preserving reasoning That arrives at a false conclusion.

This kind of definition, which I shall call the alternative definition, will likely be especially attractive to dialetheists – those philosophers who believe that at least some sentences, including those central to many familiar paradoxes, are both true and false (and hence, as a corollary, they believe that some contradictions are true). The reason is simple: on this definition we can prove very easily that the conclusion to any paradoxical argument is both true and false.

Given the first two clauses in the second definition of paradoxes given above, it follows that the conclusion of every paradoxical argument is true: After all, any paradoxical argument begins with true premises, and proceeds via truth-preserving reasoning. Hence, the conclusion must be true. But the definition also entails, via the third clause, that the conclusion of any paradoxical argument is false. Hence, either no paradoxes exist, or the conclusion to any paradoxical argument is both true and false.

For present purposes, we can assume that paradoxes do, in fact, exist (after all, I have been writing this column on paradoxes since late 2014, and it would seem like a silly waste of time if I was writing about something that doesn’t exist). Hence, on the alternative understanding of what it is to be a paradox, dialetheism must be correct.

Now, the non-dialetheists in the room (or staring at the screen) might just take this to be an argument that the first definition – the one involving three occurrences of the word “apparently” – is a better definition of the phenomenon in question. And I wholeheartedly agree with this assessment. But that doesn’t mean that it isn’t worth exploring the alternative definition a bit more to see what additional puzzles we can concoct.

It’s worth noting that what follows is carried out informally – as a result, and as is to be expected, many of the inferential moves made will be rejected on various non-classical solutions to the semantic paradoxes (especially dialethic solutions!)

One question we might ask is whether we can construct paradoxical arguments in terms of this alternative understanding of the term “paradox”. Similar questions have been raised by logicians since at least the Middle Ages. One well-known example, called the pseudo-Scotus paradox, proceeds via reasoning about whether the very argument in question is truth-preserving:

Premise: This argument is truth-preserving.

Conclusion: Santa Claus exists.

Assume, for reductio, that this argument is not truth-preserving. Then there must be some way for the premise to be true and the conclusion to be false. But if the premise is true, then given what it says (i.e. that the argument is truth-preserving), the conclusion must also be true as well. So there is no way for the premise to be true and the conclusion to be false. Contradiction. Hence, the argument is, in fact, truth-preserving. But that is just what the premise says. So the argument has a true premise and is truth-preserving. Hence, the conclusion must be true as well.

Of course, this is an absurd way to prove that Santa Clause exists, and the argument will likely feel familiar to many readers, since it is a version of the Curry paradox carried out at the level of arguments (this general pattern is called the V-Curry in the technical literature).

Of course, once we have seen the general idea, we can construct all sorts of variants. Of particular interest are variants that involve our alternative notion of paradoxicality. For example, consider:

Premise: This argument is truth-preserving but not paradoxical.

Conclusion: This argument is paradoxical.

We can easily show that this argument must be paradoxical: Assume, again for reductio, that the argument is not truth-preserving. Then there must be some way for the premise to be true and the conclusion to be false. Thus, it is possible that the argument is truth-preserving (since this is required by the truth of the premise) and not paradoxical (since this is required by both the truth of the premise and by the falsity of the conclusion). But in such a scenario, the premise would also be true (since it just says it is truth-preserving and not paradoxical). Hence, since in this scenario we have a true premise in a truth-preserving argument, the conclusion must also be true. Contradiction. Thus, the argument is truth-preserving. Now, assume (again, for reduction) that the argument is not paradoxical. But then the premise would be true. Hence, since we have already shown the argument to be truth-preserving, the conclusion must be true as well. Contradiction. So the argument is paradoxical.

This argument is already troubling enough: We have shown that an argument that (i) begins with a premise claiming that the very argument in question is in the best logical standing (truth-preserving but not paradoxical) and (ii) ends with a conclusion claiming that the argument in question is in very bad standing (paradoxical), is not invalid or unsound as we might expect, but is instead paradoxical.

But things seem even more puzzling. According to the alternative definition of paradox, a paradox is supposed to have true premises, be truth-preserving, and have a false conclusion. But the reasoning above shows that the argument in question has a false premise (since the argument is, in fact, paradoxical) and a true conclusion! Clearly, something has gone dreadfully wrong!

[Josh Parsons pointed out the alternative definition of paradoxes, which led me to thinking about the issues raised in this post.]

Featured image: Kandinsky, Jaune Rouge Bleu. Public domain via Wikimedia Commons.