Actual truth, possible knowledge

Abstract

The well-known argument of Frederick Fitch, purporting to show that verificationism (= “Truth implies knowability”) entails the absurd conclusion that all the truths are known, has been disarmed by Dorothy Edgington's suggestion that the proper formulation of verificationism presupposes that we make use of anactuality operator along with the standardly invoked epistemic and modal operators. According to her interpretation of verificationism, the actual truth of a proposition implies that it could be known in some possible situation that the proposition holds in theactual situation. Thus, suppose that our object language contains the operatorA — “it is actually the case that ...” — with the following truth condition: ⊢ _vA ⌽ iff ⊢_w0⌽, wherew ₀ stands for the designated world of the model — the actual world. Then we can formalize the verificationist claim as follows:

$$A\phi \to \diamondsuit KA\phi .$$

However, while Edgington's introduction of the actuality operator dissolves Fitch's paradox, our troubles are not yet over. When we combine the truth-condition for the actuality operator with the standard truth-clauses for necessity and knowledge, formulated in terms of appropriate accessibility relations between worlds, it turns out that we once again have to accept the absurd claim: all actual truths must be known! Thus, the standard truth-conditions for the actuality operator and for the epistemic operator do not mix: when we try to combine them, they yield absurdities. To get a proper mix, we need a new semantics for actuality and knowledge.

What is distinctive for our semantic proposal is that we give up the idea of afixed actual world (the designated point) and replace it with avariable perspective. The latter is contrasted with areference-world, which is being referred to, or described. We get what is sometimes called a two-dimensional semantics, in which a formula is being evaluated not just at one point (⊢ _v ⌽) but at a pair of points (w⊢_v⌽, wherew is a point of perspective, whilev is a point of reference). Intuitively, a formula says somethingabout the reference-world, butwhat it says is partially determined by the world of perspective. In particular, a formula such asA⌽ is true from a perspectivew at any reference-worldv iff it is true fromw's perspective atw itself. It turns out that, in a two-dimensional semantics, it is possible to formulate verificationism in a non-paradoxical way, provided we treat knowledge as a “variable-perspective” operator. The truth-condition of such an operator does not keep the perspective-world fixed. It does not involve an accessibility relation between worlds, but rather a relation between pairs of worlds:^w⊢_vK ⌽ iff^w′⊢_ν′⌽ everyw′ andν′ such that 〈w,ν〉E 〈w′, ν′〉. The relationE is meant to model epistemic uncertainty that originates from two different sources: The knower's information about the reference world may be more or less limited, and the same applies to his knowledge about the world that constitutes the point of perspective.