On The Two-Dimensionalist Reductio and The Response from Ideal Conceivability

According to an argument that was recently formulated by Robert J. Howell, there is a statement whose conceivable truth is enough to show that Chalmers’ two-dimensional approach to semantics is incorrect and directly self-refuting.[1] The statement Howell has in mind is this:

(SN) The space of metaphysically possible worlds is more limited than the space of conceivable worlds.

This statement is of course incompatible with Chalmers’ two-dimensional approach to semantics since, on that approach, the space of metaphysically possible worlds and the space of conceivable worlds coincide; they are in effect one and the same. Now, before I try to explain why Howell thinks that the conceivability of SN is enough to show that Chalmers’ two-dimensional approach to semantics is incorrect, I should introduce Howell’s epistemic notations ‘conceivable1’, ‘conceivable2’, and their modal counterparts ‘possible1’ and ‘possible2’.[2] Whenever a statement is said to be conceivable1 or possible1, what is meant is that it is conceivable or possible with regard to its primary intension. That is to say, the statement in question is conceivable when we treat other possible worlds as if they are actual and then evaluate the statement relative to those worlds. Conversely, when a statement is said to be conceivable2 or possible2, what is meant is that it is conceivable or possible with regard to its secondary intension. In other words, the statement in question is conceivable when we treat other possible worlds as if they are counterfactual and then evaluate the statement relative to those worlds. Clearly then, Howell’s distinction between conceivability1 and conceivability2 is equivalent to David J. Chalmers’ distinction between primary and secondary conceivability and his distinction between possibility1 and possibility2 is in turn equivalent to the distinction between a priori metaphysical possibility and Kripke’s a posteriori metaphysical possibility. With these clarifications firmly in mind, we can now turn to Howell’s two-dimensionalist reductio: