Wednesday, August 29, 2012

Nullary predicates and truth

This post develops an ultimately unsatisfactory deflationary theory of truth. Feel free to skip.

A binary predicate needs two names, or two quantifiers, to make a sentence. A unary predicate needs one name, or one quantifier, to make a sentence. A nullary predicates needs no names—it by itself, with no arguments, makes a sentence. For instance, the English "It rains" is a nullary predicate. It pretends to be a subject-predicate sentence, with the subject "it", but the "it" has no reference.

English allows the stipulative introduction of new names and predicates. Thus, I can say things like:

  1. Let "Cloak" denote Socrates' nose. It is notorious that Cloak is snub.
  2. Let "tigging" denote that which is in fact Sam's favorite activity. There is then a possible world where Sam would rather eat spinach than tigg.
Extend English to allow the stipulative introduction of nullary predicates. Then a number of the tasks for which we use the word "truth" could be replaced by such stipulative introduction. For instance, we can replace:
  1. Kathleen's theory about the origins of the universe is not true.
  1. Stipulate that "xyzz" is a nullary predicate expressing Kathleen's theory about the origins of the universe. It's not the case that xyzz.
By "we can replace", I mean that the communicative tasks accomplished with (3) could be accomplished with (4).

So far this strategy will only handle some uses of "true", namely those where the predicate "is true" is joined to a name or definite description. What about a more complex case?

  1. At least one of Kathleen's astrophysical theories is true if string theory is true.
This we can handle as well with nullary predicate stipulation:
  1. Stipulate that "xyzz" is a nullary predicate expressing the disjunction of Kathleen's astrophysical theories. Stipulate that "strig" is a nullary predicate expressing string theory. Xyzz if strig.

But what I cannot handle using this method are uses of "is true" embedded in modal operators, such as:

  1. Kathleen could have come up with a true astrophysical theory.
Maybe we can handle this if we allow the "true in w" predicate and quantification over worlds:
  1. There is a world w and a proposition p such that Kathleen comes up with p in w, and p is an astrophysical theory in w, and p is true in w.
Now, this seems to get us no further ahead—after all, we've only replaced the "true" in (7) with "true in w" in (8). Actually, we can make the theory stay alive a bit longer: "true in w" need not be a notion that depends on the notion of truth. Suppose, for instance, that worlds are maximal compossible sets of propositions. Then "p is true in w" can be replaced with "p is a member of w".

However, if that's how we understand worlds, then we need substitutional quantification to explain what it means to say that "s in w" (e.g., "Snow is white in w"), where "s" is a sentence. Our bet may be to say: for all s, s in w if and only if the proposition that s is a member of w. But the "for all" is substitutional quantification. But substitutional quantification and truth are probably interdefinable, so if we have to rely on substitutional quantification, the above account fails.

At this point my toy deflationary account of truth in terms of stipulation of nullary predicates comes to a halt. It is modal embedding that brings it to this halt.

It is interesting that modality does not seem to bring to a halt a similar view of A-predicates.

1 comment:

James said...

Isn't "It rains" equivalent to P(x) = "x rains", a unary predicate?