In discussing the ideas of the previous post with a colleague (Chris Steinsvold) I found out that I had basically re-discovered (a simple kind) of hybrid logic. Here is a nice introductory paper by Patrick Blackburn aptly subtitled ‘A Hybrid Logic Manifesto’….there is even a hybrid logic website! (where, incidently, I got the paper from…there are all kinds of papers there!)

The basic idea behind hybrid logic is that you first add new constants, called nominals and usually represented by i, j, k,… which are supposed to ‘denote’ or name particular worlds, instants, or whatever is required by the modality one is working in. One then introduces the ‘@’ operator, with a nominal subscript. So, @_{i}p says that p is true at the world/instant/whatever that the nominal i denotes.

So, the ‘@’ operator that I introduced ( v(@a,w)=T iff (a, actual world)=T and v(~@a,w)=T iff (~a, actual world)=T) can be understood as the standard ‘@’ of hybrid logic where the nominal is the actual world and so assumed. To keep things simple I think I will change this @ with the assumed actual world nominal to ‘Θ’ this will give me a three operator modal logic [] (true at all possible worlds), ◊ (true at some possible world), and Θ (true at the actual world)…One reason to do this is that I am still not sure whether @_{i}p means that p is true ONLY at i…that might be acceptable for tense logics but I wouldn’t want to take that to be true for Θp…this just says that p is true at the actual world, whether it is true at other possible worlds doesn’t seem to matter too much for the purposes that I want to use it for.

Then the ‘simpler’ proof that God’s omniscience is incompatible with my free will that I gave in the comments of the previous post can be put as follows

1. ΘB(k,a)

(assume that God actually knows beforehand what I will do)

2. Θa (from 1)

(if He actually knows beforehand that a, then a is true at the actual world)

3. Θa –> []Θa

(if a actually happens, then it is true in all possible worlds that a happens in the actual world)

4. []Θa (2,3 MP)

5. ΘB(k,a) –> []Θa (1-4 CP)

(if God actually knows what I will do before I do it then it is necessary that I do it in the actual world)

So, standard modal logic is not expressive enough to capture the intuition that God’s foreknowledge is incompatible with Human free will. We need to move to hybrid logic to do so…but this isn’t worrisome, since Blackburn shows (in the above linked paper) how we can translate any hybrid logical formula into standard first-order logic…