Keith’s Conundrums: The Grim Paradox

Keith Douglas

Last month, I posed three puzzles surrounding time. No responses were received. Admittedly, the last one was very challenging. In fact, I have yet to see it discussed in anything but the philosophy literature proper — i.e., in no popularizations. This is not surprising given the calculus prerequisite. I still have no clear answer other than my worries about continuity in general. I have never managed to articulate these worries precisely so I will ignore them for now and leave the puzzle as is.

As for the lamp, one way out in the literature is to simply hold that there is a failure of what most philosophers call “determinism” here. This should not actually be so surprising; there are lots of “pathological” such failures even in classical mechanics. In this case, then, the idea is that there is no fact of the matter whether the lamp will be on or off after the so-called supertask is complete.

As for the Futurama-derived puzzle, my hints give away most of what I take to be the answer. However, one thing to note: Fry and his father do not actually have to be genetically identical, I don’t think. His genome just has to be “undoable” that way. In other words, if Fry’s father underwent mutation X transforming gene G to G’, Fry has to undergo X’ to do G’ back to G.

The Grim Paradox

Last time I promised something new. Sorry, this is something borrowed. We have a very interesting head-scratcher related to the liar paradox, which we addressed a while ago. I am going to call this the “Grim paradox” as it is a bit, shall we say, depressing. (Warning: It might not be a paradox!) Also, it is due to Patrick Grim, a noted contemporary philosopher and is discussed at great length in his 1991 book The Incomplete Universe.

Is there a set of all and only truths? Here you can read “set” either theoretically or in ordinary language — and in either case with varying ideas in mind. Let’s tackle the first with an initial exploration. Then I leave to you to find a way out by varying the meaning of “set” or trying other alternatives that might be similar and still allow one to talk and reason about “all truths.”

This set would putatively contain a bunch of statements (finesse these however you want to make each of them “completely true”). Call the set T and put:

T0: Iron rusts in the atmosphere over time.

T1: Joan plays the cello.

T2: Most birds fly.

T3: …

Now consider the statement (*):

(*) is not a member of T.

If (*) is true, then we’re done: It truthfully states a truth not in the set of all truths. So there’s no such set. Alternatively, suppose (*) is false. Then it is a member of T. But falsehoods do not belong in a set of truths, so that’s a nonstarter. It also leads straight to contradiction, because we have both (*) is not a member and yet is.

Grim uses this oddity to explore many other philosophical puzzles. Here’s one that I draw  attention to because many CFIC members are interested in philosophy of religion.  Consider the statement (**):

God believes that (**) is false.

If we assume an omniscient god, we are driven straight into paradox here — try it! But I also wonder, as the non-theist also has trouble with matters like these. In other words, both these and the other subjects of the book, if he is correct, lead to a form of incompleteness similar to those proved by Gödel.

However, Grim’s results, unlike those of Gödel, are “outside of math,” which is a bit disconcerting. If one wants to repair our semantics, what do we do here? What does it say about the world we purport to describe? Grim seems to think that this reflects something about the way the world is more generally. He may well be right; but if the Gödel theorems tell us that math is “indefinite” in some way, that’s fine for us mathematical fictionalists. But I don’t think that works at all for the world itself! And this has nothing to do with realism, it seems, either, since the antirealist is in the same bind, as far as I can tell.

Note one final problem: If one manages to repair as mentioned, does that repair the notion of omniscience?