S4 to 5D
The modal logical axiom 4, □P → □□P (“If necessarily P, then necessarily necessarily P”), is widely accepted. It is the characteristic axiom of the modal logical system S4, which is subsumed under the most popular modal logical system S5. Axiom 4 is equivalent to ◊◊P → ◊P (“If possibly possibly P, then possibly P”), which requires that the accessibility relation between worlds be transitive.
There is a powerful argument (Hugh Chandler 1976, Nathan Salmon 1981, 1989) against axiom 4. It rests on the thought that an ordinary object could have had a slightly different origin from its actual origin but could not have had an origin very different from its actual origin. By constructing a sorites-like sequence of possible worlds at which the origin of a given object shifts incrementally along the sequence, the argument concludes that accessibility is not transitive, i.e. that what is possibly possible may not be possible.
A recent attempt to defend S4 from this argument (Murray and Wilson 2012) proposes that we abandon the absolute notion of possibility and instead accept a world-indexed notion of possibility; each world comes with its own version of possibility.
I offer a different defense of S4, which preserves both axiom 4 and the absoluteness of possibility. Its key move is to postulate objects as extended not only in physical space-time but in logical space as well, that is, as “five-dimensional worms.” Since S4 and the absolute notion of possibility are very intuitive, quite useful, and widely well regarded, and since my proposal saves both of them, I take the proposal to constitute an argument in favor of “five-dimensionalism.”
The Modal Status of Laws: In Defence of a Hybrid View
Three popular views regarding the modal status of the laws of nature are discussed: Humean Supervenience, nomic necessitation, and scientific/dispositional essentialism. These views are examined especially with regard to their take on the apparent modal force of laws and their ability to explain that modal force. It will be suggested that Humean Supervenience and the nomic necessitation approach struggle to explain the modal force of laws, whereas scientific/dispositional essentialism fares slightly better. However, none of the three views, at least in their strongest form, can be maintained if some laws are metaphysically necessary, but others are metaphysically contingent. Some reasons for thinking that such variation in the modal status of laws exists will be presented, with reference to physics. This drives us towards a fourth, hybrid view, according to which there are both necessary and contingent laws. The prospects for such a view are studied.