David Corfield 2-D semantics

Idea

It is claimed that to represent certain sentences, two indices are needed corresponding to worlds, times, etc. For instance,

It is possible for everyone who actually survived the Titanic’s maiden voyage to have died on the maiden voyage.

Here it is intended that all of those who survived the 1912 disaster would in another eventuality have died.

If we are to do this via a notion of possible worlds, it seems that we are to assume a considerable degree of stability across them. The same people are present in each world, the same ship sails for the first time in each, and so on.

In this stable case, we have a type of worlds, $W$, and a type of people, $P$. Then we have two properties which say of a person, $p$, in a world, $w$, whether they survive or die on the Titanic’s maiden voyage:

$$w: W, p: P \vdash s(p, w), d(p, w): Prop.$$

($Prop$ is the type of types which are propositions, i.e., subsingleton types.)

Then we have

$$w, w': W, p: P \vdash (s(p, w) \to d(p, w')): Prop.$$

Let $a: W$ be the actual world. Then

$$w': W, p: P \vdash (s(p, a) \to d(p, w')): Prop.$$

Applying dependent product on $P$ yields

$$w': W \vdash \prod_{p: P} (s(p, a) \to d(p, w')): Prop$$

Then dependent sum on $W$ yields the non-dependent type,

$$\vdash \sum_{w': W} \prod_{p: P} (s(p, a) \to d(p, w')): Type.$$

The original statement now declares that this type is inhabited: there is a world where all of those who survived the Titanic in the actual world died.

Other understandings of the situation could be accommodated, e.g., with a world-dependent type of people, with a limited accessibility relation of worlds, etc.

Water is $H_2 O$

In Putnam’s Twin Earth thought experiment, assume $H_2 O$ is fixed type, as is $XYZ$. $Water$ has an index to indicate dependence on the world in which it is formed: $water(w)$. Let $a$ be the actual world and $b$ be Twin Earth.

We turn $H_2 O$ into the constant element in $[W, Type]$.

Then ($water(a) = H_2 O(w)$) is a world-dependent proposition, which is true at $a$ or $b$, so necessarily true.

The 2-D form of this proposition is $(water(w) = H_2 O(w'))$, The diagonal of Stalnaker is $(water(w) = H_2 O(w))$, true at $a$ and false at $b$.

Multi-indexed situations can easily be accommodated. Indices are typed variables.

Once everyone then alive would be dead.

References

• Rabern, B. 2012, ‘Propositions and Multiple Indexing’, Thought: A Journal of Philosophy, vol. 1, no. 2, pp.116-124. doi, pdf