The principle-based conception of modality: Sullivan's question addressed

Peter Sullivan asks a question about the Modal Extension Principle which is at the centre of the approach to metaphysical modality which I proposed in "Metaphysical Necessity: Understanding, Truth and Epistemology".1 The relevant part of the Modal Extension Principle states that in the case of a concept C which is not de jure rigid, an assignment s is admissible only if the semantic value of C with respect to s is the result of applying the same rule as is applied in the determination of the actual semantic value of C. Sullivan's question is this: how are we to understand the Modal Extension Principle without encountering insuperable obstacles? Sullivan mentions two obstacles, which correspond to the two ways of understanding the Principle which Sullivan considers. On the first construal, the Principle states that s is admissible only if it assigns to C its actual extension. Given the role of admissibility in the principle-based conception, this first construal has the "unintended result" (as he delicately puts it) that no concept could have an extension which differs from its actual extension. On the second construal, admissible assignments are simply restricted to assigning to any given concept a semantic value which it could have. This, says Sullivan, is circular in any explication of metaphysical modality; and I agree. There is, however, a third way, and it is what I intended. As always, we distinguish between a rule and what it applies to-what it bites on, as Sullivan puts it. Take first, for illustration, the special case of a defined concept such as bachelor. Its actual semantic value is determined by applying a certain rule, R let us call it, to the actual extensions of the concepts man and unmarried. In this example, R is simply the operation of taking the intersection. Under the third, and intended, way of understanding the Modal Extension Principle, an assignment s is admissible only if the semantic value it assigns to bachelor is that which is obtained by applying