Fitch, Humberstone, and an omniscient being

I just read the paper “Omnificence” by John Bigelow[1]. In the preamble he recounts the following argument for an omniscient being

  1. Any fact (true proposition) is knowable by someone. (Premise)
  2. Therefore, every fact is known.
  3. Therefore, someone knows every fact.

Fitch[2] was responsible for showing that (2) follows from (1). One way to see this is a follows: for reductio assume, contrary to (2), that there is some fact p that is not known by someone. Then, p and no-one knows p is a fact, and by (1) is therefore knowable. Therefore, it is possible that someone knows p, and at the same time knows that no-one knows p, which is absurd. Thus (2) follows from (1).

The move from (2) to (3) comes from Humberstone[3]. Again, for reductio assume, contrary to (3) that for every person x, there is some fact p(x), that x doesn’t know. Now let X be the conjunction formed by taking all facts of the form p(x) and x doesn’t know p(x). Clearly this conjunction is true. By (2) it follows that there is some y who knows X. But one of the conjuncts of X will be p(y) and y doesn’t know p(y), and since y knows X, y also knows this conjunct, which is absurd. Thus (3) follows from (2).

I’m not going to argue for (1) here, although I must admit I struggle to imagine that there could be a fact that is in principle unknowable. What I found particularly interesting in Bigelow’s paper was the comment that logical positivism entails (1): take the verification principle of that movement, which said that a statement is meaningful if and only if it is verifiable. Now, only meaningful statements can take on a truth value, so it follows that every fact is verifiable, and therefore knowable.

Notes

  1. J. Bigelow, “Omnificence,” Analysis 65/3 (2005), pp. 187-196.
  2. F. Fitch, “A logical analysis of some value concepts”, Journal of Symbolic Logic 28: 135-42.
  3. I. Humberstone, “The formalities of collective omniscience”, Philosophical Studies 48: 401-23.

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