Keith’s Conundrums: The Liar Paradox

Keith Douglas

A reader discussed last month’s combination of two classics. He pointed out that we could measure “to the millimeter” where the parties were and that would show us how to handle the case. This is an epistemic view: The crime took place in one rather than another state, but we cannot know which it is. This seems to ignore the temporal nature of the crime.

Also, by using a unit of distance for the thickness of the line involved, it suggests that the state line is not a line in the sense of geometry. This is correct, and in my view it is the way out of the more traditional Peirce’s paradox as well, which involves wondering about a piece of paper, painted two colours in stripes: What colour is the line that forms the boundary between the stripes?

I do like seeing how people focus on one rather than another aspect of the problem; the temporal aspect I find more interesting, for example. Perhaps by only putting that aspect in the question rather than the statement I missed the opportunity to draw attention to it. (Which is not necessarily a bad thing, of course!)

The Liar Paradox

Today we continue with one more classic, and I intend to use it as an extended riff on epistemic charity to others, amongst other things. So we are dealing with the complexities of a classic conundrum and the sort of “metaconundrums” that result from these complexities.

The liar paradox has been alluded to before in this column. Let’s take a look at it in detail this time. The classic statement of the paradox does not use the notion of liar, so I’ll start with that. Consider:

This statement is false.

Is this statement true or false? Assume that it is true. Then what is the case? It says that it is false, so it is true that it is false, so it is false, so it is true, etc. One can run the analysis the other way and an equal “instability” results.

There are several ways that this problem has been tackled. The one that came to me as a 13-year-old, was simply to deny that all statements are either true or false. As it turns out, that approach does not work as such. Call statements without truth value “gappy.” Consider:

This statement is gappy or false.

One can run the same analysis, unfortunately. Assuming that statement is gappy leads one first to conclude that it is true, then gappy or false. If it is assumed to be gappy, then it is concluded to be true and then hence not gappy or false, etc. If assumed to be false, then it is true, etc.

This general problem is called the problem of the revenge liar. It leads us to realize that simple solutions are often too simple. Let’s consider another approach suggested to me recently. This moves from a semantic approach (dealing with truth) to a more pragmatic one. It starts by putting the concern in a larger context. It asks us to argue truth preservingly and if we discover a contradiction to stop and reject the argument, and not worry about truth value. Call that result “rejecting” the argument.

The problem from this lies in chaining arguments, or alternatively applying the “self reference” ideas to them rather than to statements. Consider the following argument:

This argument should be rejected

Therefore, this argument should be rejected.

Evaluating this is hard. One has to figure out what principles govern meta-argumentation. It can be done any number of ways: When I tried I got lost in many possibilities.

What is the lesson here? Logic is hard — very, very hard — to improve. What would one have to do here? A professional logician may typically start with systematizing some ideas through a rough formalism to quickly check their properties, iterate, etc.

As it happens, as I started work on this column, the June 2021 edition of the Review of Symbolic Logic arrived. It contained, coincidentally, “An Expressivist Analysis of the Indicative Conditional With a Restrictor Semantics.” This paper on one family of meanings of “if” was interesting to read as it contains, by chance, a formalization of acceptance and rejection! This is (as expected in a top tier academic journal) a very advanced paper and much is irrelevant to our topic here. However, we can borrow a tiny, tiny piece of what is presented for our purposes.

First thing I will do is assume we can identify arguments with the conjunction of their premisses. My interlocutor in the original discussion did not discuss this matter. It seems reasonable enough, and we need to do this in order to use the accept/reject machinery I mentioned in some way or other: The article makes acceptance an attitude towards a proposition or statement, which was not our topic at first. Reading through the axioms, one is struck by how they are now straying out of the realm of logic proper into a field called “belief dynamics.” I draw attention to this to raise awareness of some items to consider:

1. To be charitable to my interlocutor, how much do I have to work to interpret his suggestion?
2. What familiarity with this sort of field should be required? You have it only on my say so that it looks relevant. I wouldn’t want that, at least long term.
3. In order to formally study acceptance and rejection we are now borrowing heavily from many areas. Should I worry that I have over-complicated matters?

The one most troubling to me is the first: I want to take the suggestion as charitably as possible but when one knows simple solutions are extremely unlikely to work, what does one do? To someone with the background what was proposed was oversimplified to the point of not being understandable as much of a suggestion. My “first thing” was an attempt to understand as much as possible by adopting one possible version of what was claimed. But, continuing from #3:

1. My interlocutor might complain that I have misrepresented his view: that all the machinery of formal logic, semantics, exact epistemology is besides the point, which is (I think) in part a concern about how logic is taught.

I think of what I am doing as akin to making sure we don’t divide by zero or assert that a projectile landed before it was projected, etc. Those of you who have studied physics (at even the high school level) may remember solving a quadratic equation in problems of the latter case — but one has to discard the “unphysical root.” Oddly, this sort of worry only arises because one has an exact mechanics to begin with. Vaguer notions wouldn’t have that problem.

I turn now to one final area that seems to be relevant to our puzzle. This part does not involve state-of-the-art papers. However, it involves a notion that is hard to grasp at first, for some at least. Consider:

This statement is true.

and

Gibbs is brown.

Ignore the truth value of each for the moment. It looks like both are of the form Pa, where P is some predicate, and a is an individual. P in the first statement is “is true” and in the second is “is brown.” The individual in question is “this statement” and “Gibbs,” respectively.

But is that too quick? Return to our liar sentence. It too looks to be of the form Pa — except that P in this case is “is false.” It seems we should therefore conclude: Logical form is relative to an analysis. In order to study our “liar,” we need to introduce more “machinery.” So formalize the “truth predicate”? And then we’re off … again.

I leave with a final thought: Consider the original statement of the liar’s paradox (Titus 1:12-13, from the Bible, of all things):

“As one of their own prophets has said, ‘Cretans are always liars, evil beasts, lazy gluttons.’ This testimony is true.”

1. Is this a paradox at all?
2. Can one interpret those who drew inspiration from this passage charitably so that it is?
3. Does it require any additional “machinery” to do so? Why or why not?
4. The paradox seems to come out of “self reference.”

Harty Field’s Saving Truth from Paradox analyzes three main ways in which this occurs. Curtailing it is tricky; several ways out require revisions to mainstream mathematics. It is difficult to know how much revision is needed. It depends on how unified “in practice” one regards mathematics and indeed human thought in general. Note taking my interlocutor up may well have this effect. What if logic becomes easier but (say) calculus becomes harder? Does it matter that only in a sort of “abstract” sense does the unity exist? I do agree with the idea that it is a worthwhile goal to unify, so the proposal at least would make for a difficulty.