Re-Inventing the Wheel

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, @ip 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 @ip 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…

Third Time’s the Charm (or: This Time I Really Got It!!!)

OK, so I am basically obsessed with this stuff about God’s omniscience and Free Will. I have been having some very interesting, and helpful, discussion about whether Plantinga’s defense, which I take it is the standard defense, of their compatibility is any good or not. I have a sneaking suspicion that the two are incompatible and I have been trying to construct a poof to that effect, with mixed results…but I think I got it this time…if it turns out that I don’t then I promise that I will give up!

It seems to me that the problem is that “If God knows what I will do before I do it then it is necessary that I do it” does not really capture what the person who says that God’s foreknowledge is incompatible with our free will is trying to say. This is because, as we have seen, it must be the case that all my actions are necessary, but this doesn’t sound right at all (however, I do think some people are committed to it).

So, to make it clearer what I am actually trying to say, let me introduce a new modal operator ‘@’  with the following truth condition, where ‘v(x,w)’ is the valuation of x at world w,

v(@a, w)=T iff v(a, actual world)=T

v(~@a,w)=T iff v(a, actual world)=F

this says that @a is true if a is in fact true at the actual world and ~@a is true if a is in fact false at the actual world (~@a<–>@~a) so there is no need to introduce a fourth operator). ‘[]’ and ‘<>’ are given their usual interpretations.

Then I can say that God actually knows before I do a certain action that I will in fact actually do it. To avoid getting involved in tense logic let us introduce a predicate ‘B’ for before (though I think we could define ‘B’ in terms of the standard tense operators F, P, H, and G). Let ‘k’ be ‘God knows that’ and ‘a’ be some action of mine, then I can symbolize ‘God actually knows before I do action a that I will in fact actually do action a’ as @B(k,a), then the proof goes as follows

1. @a & @B(k,a)   (this says that God actually knows what I did before I did it)

2. []@B(k,a) –> []@a  (necessary truth)

3. @a –> []@B(k,a)      (necessary truth)

4. @a       (from 1)

5. []@B(k,a)     (4,3 MP)

6. []@a     (5,2 MP)

7. (@a & @B(k,a)) –> []@a   (1-6 conditional proof)

Since 7 says that if it is the case that I actually do a and God knows beforehand that I actually do a then it is necessary that I actually do a, and God’s actually knowing that I do a entails that I actually do a (7) reduces to

7′ @B(k,a) –> []@a

which says that if God actually knows what I will do beforehand then it is necessary that I actually do it.

Now one may wonder what the difference between ‘a’ and ‘@a’ is. Ordinarily there will be no difference, but there will be a huge difference when we examine the modal properties of the two. []a will be true iff a is true in all possible worlds, whereas []@a will be true if @a is true in all possible worlds, or in other words if it is the case that at every possible world it is true that, in the actual world, I do a. This is why (3) above is a necessary truth but (3′) is not,

(3′) a –> []B(k,a)

(3′) says that if I do a then in every possible world God knows beforehand that I will do a. This can be false because there are possible worlds where the antecedant turns out false because in that world I do not do a and so God does not know it. But (3) can’t be false. For if it were then it would be the case both that I actually do a and that God did not actually know beforehand that I did a, which is just to deny that God is omnicient (so enigman will be happy).

Whew! So, if this is right then God’s foreknowledge is indeed incompatible with my having free will. If not then I will finally have to admit that there is at least one metaphysical interpretation on which it can both be true that God knows what I will do before I do it and that I am free…and I will then actually be very depressed!

What God Doesn’t Know

 So, I have been thinking a lot about Free Will and Omniscience, and though it has been rewarding I am less than happy with the results because I have been convinced that the proof I gave is only valid if one accepts metaphysical ssumptions that I do not accept…drat! I suppose there is some solice in knowing that it can be used to attack people who do hold those metaphysical views…but in the course of thinking about this stuff I came to realize that there are some problems with the claim that God is omnicient…for instance consider (1),

1. God knows that this sentence is false.

If (1) is true then God knows that the sentence is false, but if he knows that it is false then it is false and if it is false He doesn’t know it and so is not omniscient. So if this sentence is true then God is not omniscient. However if the sentence is false then God doesn’t know that the sentence is false and so he is not omnicient.

So God can’t know everything. But one may think that this is only due to the fact that (1) is contradictory and no one can know contradictions…but what about (2),

2.God can’t know that there is something that He doesn’t know

If (2) is true then there is something that God doesn’t know, namely that He doesn’t know something. If the sentence is false, then he can know that there is something that He doesn’t know, and since he knows this, that means that there is something that He doesn’t know and so can’t be omniscient.

hmmm…..

Over at Brains…

In case anyone here doesn’t know, I am also a contributor to Brains a group blog in the philosophy of Mind, Psychology and Cognitive Science…if anyone is interested here are some links to my posts there…

1. When Platitudes Collide 

2. The Qualitative Character of Conscious Thoughts

3. Do Horses Sense Their Riders desires?

4. Brain States Vs. States of the Brain

5. Multiple Realization  Vs. Multiple Instantiation

6. Just Like Home

(I Think) I Got It!

If you have been following the discussion in Plantinga on Free Will and Omiscience you will have seen that I have been struggling to construct a proof of (1), which says that if God knows that I will do some action before I actually do it then it is necessary that I actually do it, from (2), which says that it is necessary that if God knows what I will do some action in advance then I will actually do it.

(1) K(G,R,a) –> []D(R,a)

(2) [](K(G,R,a) –> D(R,a)

So far the two attempts that I have made have both been invalid because of some bonehead mistakes. This has been driving me crazy for the past couple of days, but now I think I got it, in fact it almost seems too simple (which probably means I made another bonehead mistake!)…

I thought that it would be easier if I did not include quantifiers, but I think that is what actually confused me. So, what I really want to prove is (1′), which says that for any action x if God knows that I will do it in advance then it is necessary that I actually do it, from (2′), which you can figure out for yourself.

(1′) (x)(K(G,R,x) –> [](D(R,x))

(2′) (x)[](K(G,R,x) –> D(R,x))

this actually turns out to be quite easy (I *think* :) ).

1. ~(x)(K(G,R,x) –> []D(R,x))          assume as a theorem

2. (Ex)~(K(G,R,x) –> []D(R,x))         1, by definition

3. (Ex)~~(K(G,R,x) & ~[]D(R,x))   2, by def

4. (Ex) (K(G,R,x) & ~[]D(R,x))       3, by def

5. K(G,R,a) & ~[]D(R,a)               4, EI

6. K(G,R,a)                          5, CE

7. []K(G,R,a)                       6, necessitation

8. ~[]D(R,a)                       5, CE

9. (x)[] (K(G,R,x) –> D(R,x))           assumption (2′)

10. [](K(G,R,a) –> D(R,a))              9, UI.

11. []K(G,R,a) –> []D(R,a)              10, distribution

12. ~[]D(R,a) –> ~[]K(G,R,a)         11, contraposition

13. ~[]K(G,R,a)                                 8,11 MP

14. []K(G,R,a) & ~[]K(G,R,a)          7,13 CI

15. (x)(K(G,R,x) –> [](D,R,x))      1-14 reductio

What I didn’t notice before was that since we are assuming 1 as a theorem and we can get K(G,R,a) from that then we can use the rule of necessation, which says that if phi follows from a theorem then phi is necessary, to get []K(G,R,a).

So, free will is incompatible with God’s omniscience…

Plantinga on Free Will and Omiscience

I have been preparing for a phil religion class I will be teaching in the Fall and so have been thinking about a lot of these issues. Today I was reading Platinga’s defence of Human free will with God’s foreknowledge of what we will do. He formulates his defense in terms of a scopal difference so, (1) is true but 2 is false (where K(x)=God knows that I will do X in advance and D(x)=I actually do x)

1. [] (K(x) –> D(x))

2. K(x) –> []D(x)

2 is false according to him because there is a possible world (not the actual world) where I do not do X even though God knows that I will do X in the actual world. One thing that worries me is that (1) is equivelent to (1′)

(1′) []K(x) –> []D(x)

and (1′) does seem to entail that I am not free…

He then considers Pike’s modification of this argument which claims that it is an essential trait of God that he is omnicient. Pike’s version of teh arguement is that if God knows at T1 that I will do X at T2 then, if I do not do X at T2 I make the belief that God has a T1 false and this means that God is not omniscient. Plantinga’s response is to point out that to say that I could have done otherwise is to say that there is a world W where I do do otherwise but that this does not mean that God holds a false belief in the actual world.

The problem with this line of argument is that it assumes a view of possible worlds (i.e. Plantinga’s view!) that I find objectionable. To say that I could have done otheriwse is NOT to say that there is a possible world where I do do otherwise and that that world might have been actual! It is to say that I, in this world, might have done otherwise. Adapting Kripke’s humphry objection we can say that it is cold comfort to be told that my being free means that there is someone else who could have done something different than I did. That is nonsense! What it means to be free is that I, myself, could have done otherwise. So if I could have done otherwise in the actual world then God cannot know at T1 what I would do at T2 or if He does then I am not free. So I just don’t see why Plantinga’s response is anything more than question beging…unless I am missing something? 

Logical Skepticism

I was reading this (older) post on the Law of Non-Contradiction over at Philosophy, et cetera. Is it really the case that it is not rational to question LNC? One might think that (P v -P) is an analytic truth. Indeed, I think that it is. But this is true only if one is operating within the confines of a two-valued logic. If one takes a many-valued logic then -(P & -P) is not analytic at all! So then the issue is whether or not classical two-valued logic is The One True Logic or not. How would we know this? I take it that someone like Richard would say that classical logic is what a maximally rational ideal agent would subscribe to. But how could we possibly know what kind of logic such an agent would subscribe to?

In an earlier post (Why Does 1+1=2?) I argued that there is nothing that could possibly decide between whether or not mathematics is an empirical science or an a priori one. The same seems true of logic. The basic laws of logic only seem rational to us because of the kind of experience that we have in this world filled with ‘medium sized dry-goods’ and this means that coherence as a criterion of truth is only valuable because of the world we live in and the kind of experience that we have. So the fact that the LNC seems to us to be denied at our own peril cannot possibly be evidence that it cannot be denied. We have no way of knowing whether it is actually ideally rational or not, since what we can concive of is limited to the kind of experience that we have.