Friday, December 9, 2011

The Sheffer stroke, and the blandest antirealism ever

In college my metaphysics prof said realism was the doctrine that there is a complete true description of the world, though it might take infinitely many sentences to express this description. Do the following incredibly bland considerations about different truthfunctional connectives commit me to "antirealism"?

(1) If you want thinking about propositions to play a role in psychology, then you need to individuate propositions narrowly enough that truthfunctionally equivalent sentences express different propositions.

(2) If you individuate propositions as narrowly as required by (1) then tautologies involving the sheffer stroke will be different propositions than any corresponding sentences which use more standard truthfunctional connectives.

(3) The same argument goes for all the other infinitely many n place truthfunctional connectives.

Conclusion: no language with finitely many basic connectives can express all true propositions. No finite language can be used to give a complete true description of the world. In particular if the a language L has only n place truthfunctional connectives then there will be tautologies using n+1 place truthfunctional connectives that cannot be expressed in L.


  1. Hmm, this seems suspicious. I don't see how your argument rules out the possibility that the world can be completely described using only philosophical English (which we'd probably both agree is a "finite language" I guess -- it has finitely many basic terms, though infinitely many possible sentences).

    Consider assertions like:

    (1) If "X quakfa Y" means "not both X and Y are true," then for any proposition X, "(X quakfa X) quakfa X" is always true.

    Compare this with:

    (2) (X | X) | X [where "|" is the Sheffer stroke]

    Maybe I'm a little naïve about the theory of meaning, but it seems odd to assert that (2) expresses something that is not captured by (1).. wouldn't this be like saying that "bunnies are cute" expresses something different than "rabbits are cute" just because they use different words? And you probably don't want to go down that path and claim that no assertion in Korean can express the same thing as "The sun is yellow" just because "yellow" is not a word in Korean.

    If you accept that (1) and (2) have the same meaning, then it seems that your argument won't work for any language which is sufficiently rich that it has both the basic logical connectives and it can quantify over both natural numbers and logical connectives (which is a kind of weak second-order quantification, I guess).

  2. Hey John,
    Thanks for your post. I see now that my post was a bit unclear about truth funcitonally equiavlent *sentences* vs. truthfunctionally equivaalent *connectives*.

    I fully agree that quafka and ! mean the same thing, and also that `X quakfa Y' and `X | Y' mean the same thing.

    But do you think that "X|Y" and "~(X&Y)" express the same thing? If so do you think "X|Y" and "~Xv~Y" mean the same thing? In order to make sense of the fact that people can believe some tautologies while not believing others we need to say that mere truth functional equivalence is not sufficient for sameness if meaning.

    Now if you buy that expressions involving | mean something different from their paraphrases in terms of ~ and &, then here is my argument.

    Consider a language that just has ~ v and & as truthfunctional connectives. There are some true propositions, those involving |, which this language cannot express. Thus a complete true description of the world can't be given in L. So we expand L to L', including the sheffer stroke.

    But now we can repeat the trick again. Pick some random way of selecting some lines as true and others as false in a truth table with three atomic propositions. This will corresponds a new, three-place truth functional connective *. Buy the argument above there will be propositions expressable in terms of * which are not equivalent to any truth functional paraphrase using only the logical connectives in L'. Thus we need to expand L' to L''...