Logicalism, Physicalism and David Lewis
In "Tharp's Third Theorem", Lewis agrees with Jackson that "all of us are committed to the a priori deducibility of the manifest way things are from the fundamental way things are (whatever that may be)" (TTT, p.96). His somewhat cryptic argument isn't quite the same as Jackson's though, and it seems that he avoids the mistake I mentioned yesterday.
Note that Lewis doesn't say we're committed to the a priori deducibility of all truths from the fundamental truths. Instead, he speaks of the "fundamental way things are", or from "contingent truths, supervenient on the fundamental way things are" (TTT 96). (In case that's not clear: Like Lewis, I use "truth" for "true sentence", not e.g. for "true proposition".)
Consider again some a posteriori necessary connection between fundamental truths and other truths:
1) if H2O covers most of the earth then water covers most of the earth.
According to Lewis, Jackson showed that this "must follow a priori from contingent a posteriori premises that are made true by the fundamental way the world is" (TTT 95), viz. from
2) if H20 covery most of the earth then the watery stuff covers most of the earth.
(2) is a priori equivalent to (1), and is "made true by the fundamental way the world is". Sure, but so is (1). It would be more interesting had Lewis claimed that (2) is itself a fundamental truth, or at least a priori entailed by the fundamental truths. But he doesn't say that -- and rightly so, as logicalism illustrates. So what's the point about (2)? The point, says Lewis, is that (2)'s A-intension is the same as its C-intension.
To see why this is relevant, recall the two-dimensional analysis of necessary implication and a priori implication Lewis gives in "Reduction of Mind" (p.297 in Papers): Proposition P necessarily implies sentence S iff P is a subset of S's C-intension; P a priori implies S iff P is a subset of S's A-intension. (All these entities are sets of worlds.) Hence if the A-intension of (2) coincides with its C-intension then whatever necessarily implies (2) also a priori implies (2). So the proposition representing the fundamental way things are, which necessarily implies everything, a priori implies (2); and since (2) a priori implies (1), the fundamental proposition a priori implies (1).
This all seems correct to me -- except for the assumption that for every necessary a posteriori sentence there exists an equivalent sentence whose A-intension coincides with its C-intension: if this is meant to quantify over English sentences it looks rather dubious.
But the important point here is that the basis that a priori implies everything is a proposition, not a sentence. What we learn is that if physicalism is true then every truth is a priori deducible from the (horizontal) proposition (aka C-intension) expressed by some physical truth. But as long as we don't know what horizontal proposition our physical sentences express, this won't enable us to deduce all truths from the physical truths. Thus we can't deduce all truths from (x)(F)(Fx <-> ACT(Fx)) even though the horizontal proposition expressed by (x)(F)(Fx <-> ACT(Fx)) a priori entails all truths.
It's precisely this gap that Jackson wants to close by silently attributing to physicalists the further claim that physical truths can always tell us what horizontal proposition our sentences express. Lewis doesn't say that. He leaves the gap open.