Wednesday, August 29, 2012

A (failed?) Platonist deflationary theory of truth

Propositions are nullary relations, and properties are unary relations. Suppose the Fs are a plurality.

  1. For any object x, there is the property Dx of being distinct from x.
  2. For any n-ary relation P, there is an (n+1)-ary property P+ such that x1,...,xn,xn+1 stand in P+ if and only if x1,...,xn stand in P.
  3. For any pair of properties P and Q, there is the property P∨Q of having at least one of P and Q.
  4. If for every x among the Fs there is a property Px, then there is a property that is the conjunction of the properties Px as x ranges over the Fs, a property that is had by y provided that y has Px for every F x.
Now, for any proposition p, define the property Pp=Dp∨p+ (by 1-3; note that because a proposition p is a nullary relation, p+ is a property). Let T be the conjunction of the Pp over all propositions p. Then a proposition p has T if and only if it is true.

But we cheated in 2. For suppose P is nullary (i.e., a proposition). Then the second part of the biconditional says that an empty list of objects stands in P, which is just a fancy way of saying P is true.

Still, it is interesting that we can define a property of truth as long as we're given the arity-raising operator (·)+, plus some plausible (abundant) property formation rules.

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1 comment:

  1. Alexander R PrussAugust 30, 2012 at 9:04 AM

    I was too quick to criticize my theory.

    Any decent account of abundant property formation is going to have to have a slew of place-rearrangement operators. For any function f from {1,...,n} to {1,...,m} (I will for simplicity assume that functions are defined to come along with a codomain), there will have to be an operator (.)^f such that if P is n-ary then P^f is m-ary and and (x_1,...,x_m) stand in P^f iff (x_(f(1)),...,x_(f(n))) stand in P. E.g., otherwise, we won't be able to form the property of being self-loving from the property of being loving (let f:{1,2}→{1} be defined by f(i)=1 for all i and let L be the property of being loving; then L^f is being self-loving).

    Our arity increase operator (.)^+ is then defined by the identity function f from {1,...,n} to {1,...,n+1}.

    Now, of course, in order to explain what the (.)^f and (.)^+ operators are, one has to make use of semantic language like "is true" or "stands in". But that no more makes for a vicious circularity than using conjunction in one's theory of truth, despite the fact that if one has to explain what conjunction is, one will use a truth table.

    I think the infinite conjunction operator implicit in 4 is more problematic than then the arity-increase operator, but even that may not be a killer for the account.

    Of course there are still the paradoxes of truth, but the same is true for just about every other theory of truth.

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