Friday, August 19, 2011

A puzzle about stipulation

I don't know whether Mars ever had life.

But it seems I can very easily get to know.  Let P be the property of having once had life.  Now, stipulate the predicate "is xyzzy" as follows: if Mars once had life, the predicate expresses P, and otherwise, it expresses ~P.  Use "<s>" to abbreviate "the proposition that s".  Then, plausibly:
  1. (Premise) I know the proposition <Mars is xyzzy>.  (For I know that "is xyzzy" expresses P if and only if Mars has P, and I can do all the logic needed to yield the claim that Mars is xyzzy.)
  2. (Premise) The proposition <Mars is xyzzy> is the proposition <Mars once had life> if "is xyzzy" expresses P.
  3. (Premise) The proposition <Mars is xyzzy> is the proposition <Mars never had life> if "is xyzzy" expresses ~P.
  4. (Premise) "Is xyzzy" either expresses P or it expresses ~P.
  5. So, either I know <Mars once had life> or I know <Mars never had life>.  (By 1-4)
  6. So, either I know the proposition that Mars once had life or I know the proposition that Mars never had life. (Expanding abbreviations)
  7. (Premise schema) If I know the proposition that s, then I know that s.
  8. So, either I know that Mars once had life or I know that Mars never had life. (By 6 and 7)
And so whichever disjunct is true, I know whether Mars once had life!  So, I didn't know it, but once I stipulated "xyzzy" and came to know that Mars is xyzzy, I got to know it.

The argument is valid, but the conclusion is surely false.  So what should we deny?

My inclination is either to deny that it is possible to stipulate predicates in the way I stipulated "is xyzzy" (and hence (1) is nonsense, since it uses a bit of nonsense, namely "is xyzzy", as if it were a predicate) or to allow such stipulation but not allow that sentences using the stipulated predicate express precisely the propositions that (2) and (3) claim they do.  I also feel some pull to denying (7).

An interesting move would be to deny (1) on the grounds that I once know that it is true that Mars is xyzzy, but I do not know that Mars is xyzzy. That sounds odd.

I think Jon Kvanvig may have once used something in the vicinity of this puzzle, but I could be wrong.

7 comments:

Drew said...

I think Alvin Plantinga discussed this in The Nature of Necessity. He mentioned some thought experiment of two climbers. I do not have the book in front of me, but I can look up the section if you'd like.

Heath White said...

Suppose I have two exams where it is clear that either Al cheated off Betty or Betty cheated off Al, but I don’t know which. Stipulate ‘Chi’ as follows: if Al cheated on the test, then ‘Chi’ refers to Al, and if Betty cheated, ‘Chi’ refers to Betty.

1. (Premise) I know the proposition
2. (Premise) The proposition is the proposition if ‘Chi’ refers to Al.
3. (Premise) The proposition is the proposition if ‘Chi’ refers to Betty.
4. (Premise) ‘Chi’refers to either Al or Betty.
5. So, either I know or I know .
6. So, either I know that Al cheated on the test or I know that Betty cheated on the test.

In either case I know who cheated on the test. Obviously something went wrong. Where? I don’t think there is anything wrong with the stipulation; names like “Jack the Ripper” have this quasi-descriptive character. My vote would be for (against) premises 2 and 3: they demonstrate one thing that can go wrong when we think of propositions in abstraction from language. Mutatis mutandis for predicates.

Alexander R Pruss said...

Heath:

You might want to repost your comment using the codes "&gt;" and "&lt;" instead of just typing in the angle brackets. Have a look at premise 2.

I take arguments like the one you give to show that Millianism is false about names. (Millianism about names I take to be the thesis that the contribution of a name to a proposition is just the referrent, so that <F(a)> = <F(b)> if a and b co-refer.) And I am happy to bite the bullets that need to be bitten to say that.

So I want to make an analogous move about the argument I give. I am not completely clear on how to do that, though.

Heath White said...

Ah. Here we go:

Suppose I have two exams where it is clear that either Al cheated off Betty or Betty cheated off Al, but I don’t know which. Stipulate ‘Chi’ as follows: if Al cheated on the test, then ‘Chi’ refers to Al, and if Betty cheated, ‘Chi’ refers to Betty.
1. (Premise) I know the proposition <Chi cheated on the test.>
2. (Premise) The proposition <Chi cheated on the test> is the proposition <Al cheated on the test> if ‘Chi’ refers to Al.
3. (Premise) The proposition <Chi cheated on the test> is the proposition <Betty cheated on the test> if ‘Chi’ refers to Betty.
4. (Premise) ‘Chi’refers to either Al or Betty.
5. So, either I know <Al cheated on the test> or I know <Betty cheated on the test>.
6. So, either I know that Al cheated on the test or I know that Betty cheated on the test.
In either case I know who cheated on the test. Obviously something went wrong. Where? I don’t think there is anything wrong with the stipulation; names like “Jack the Ripper” have this quasi-descriptive character. My vote would be for (against) premises 2 and 3: they demonstrate one thing that can go wrong when we think of propositions in abstraction from language. Mutatis mutandis for predicates.

Matthew said...

A thought:

To say that the predicate “is xyzzy” expresses P could mean:

(Exp1) “is xyzzy” expresses P in some absolute sense, or
(Exp2) “is xyzzy” expresses P to me.

Obviously, if I don’t know that P, then “is xyzzy” cannot express P to me. And if I don’t know that ~P, then it cannot express ~P to me.

Perhaps bearing this distinction in mind could help resolve the paradox.

Josh said...

I started tripping over your stipulation for "is xyzzy". Can you give a non-circular Chisholm-style definition of 'x is xyzzy'? If not, then I don't grasp the puzzle.

Alexander R Pruss said...

Nope, can't do. :-) But we do make stipulations like: "Let x be the person who did the deed" or "Let F be Sam's favorite property." Stipulations like these can be arbitrarily complex. Thus, we can let x be the person who did the deed if the person who did the deed is male or the tallest woman in the world otherwise.

Well, let L be the property of having had life and let N be the property of not having had life. Then stipulate that X is L if Mars had life and X is in N if Mars didn't have life (that's like stipulating X is somebody's favorite property). That may be enough for the puzzle. For I know that Mars has X. But I don't know that Mars has L and I don't know that Mars has N.