Wednesday, March 28, 2012

A being that knows much

Here is an argument inspired by the Gale-Pruss cosmological argument.

  1. (Premise) Every first-order truth is knowable.
  2. (Premise) The conjunction of all basic first-order truths exists and is a first-order truth.
  3. (Premise) If all the basic first-order truths of a world w1 hold at a world w2, then w2=w1.
  4. Let p be the conjunction of all basic first-order truths. Let w be a world where there is a being, x, that knows p. (By 1 and 2)
  5. (Premise) Necessarily, if someone knows p, then p is true.
  6. p is true at w. (4 and 5)
  7. w is the actual world. (3, 4 and 6)
  8. There actually is a being that knows the conjunction of all basic first-order truths. (4 and 7)

Note that the restriction to first-order truths makes this not be just the standard knowability paradox.

I am not giving a definition of basicness. The crucial constraint is that the notion be such that both (2) and (3) hold. If one says that every first-order truth counts as basic, then (3) is very plausible, but (2) is less clear because one worries about a conjunction that has itself as a conjunct. Or maybe we could say that the basic first-order truths are truths expressed by literals, namely truths of the form of the proposition that P(a1,...,an) for some predicate P or its negation. This requires a free logic, an existence predicate, and propositions about all possible non-existent entities. Alternately, we might take "basic" to just mean "fundamental".

Note added later: I forgot the word "basic" in (3) in the original argument.


Heath White said...

I think the argument has a problem. Question: is “there is an x, such that x knows p” in step 4 supposed to be a basic first-order truth, or not?

If it is, then step 4 essentially stipulates the existence of x. That’s question-begging.

If it is not, then the inference to 7 fails (I think). For consider two worlds, in both of which p is true, but in one of which there is the all-knowing x and in the other of which x does not exist. Surely we cannot infer, from the truth of p, that we are in the first world and not the second.

Alexander R Pruss said...

A. I think claims about what someone knows are paradigmatic cases of second-order truths.

B. The hypothesis that you're worried about is ruled out by (3), which affirms a supervenience of truth on basic first-order truth.

This supervenience is, admittedly, controversial.