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…


13 thoughts on “Re-Inventing the Wheel

  1. The truth conditions for []Θa, Θa, and a are all exactly the same. In each case, its V(a,r)=T. There is no model where one is true and one of the others is not. Hence, they all “mean” the same thing. Ditto with B(k,a) and ΘB(k,a).

    Since the whole discussion begins with everyone agreeing that B(k,a) –> a, I dont understand how “showing” that ΘB(k,a)–> []Θa shows anything different than what you begin by assuming. Its completely trivial that this follows from the assumption, since the left and right sides of it and of the assumption say exactly the same thing.

    writing “[]Θa” makes it sound like you’ve found a necessary truth, but the Θ cancels out the [].

  2. To build on Eric’s point, note that (3) is presumably just as problematic as (5). Fortunately, they are not really problematic at all. Just because things necessarily happen in the actualized world-state ‘@’, doesn’t mean that the universe itself couldn’t have been different. It could have actualized a different world-state besides @, after all.

  3. To put a slightly different spin on it, Richard (Brown) tells us that the semantics for [] are “the usual ones”, but its not clear what this means once Θ is introduced. []x is true at w just in case x is true in all possible worlds that w can see. Ok, but what does it mean that Θa is true at world w (~=r)*? I guess it means that, in world w, a is true at world r. But what does that mean? We arent told. If it just means a is true at world r, then 3 and 5 are completely trivial. Because the [] in []Θa, or in any []Θx, is just decoration. But if its meant to mean anything else, (i.e. if 3 and 5 are meant to say something substantial), then we need to hear, precisely, what that something else is, and then we need to hear an argument that 3 and 5 hold under that specification.

    *r is the real world.

  4. Hi Eric,

    Yeah, I take your point about Θa and []Θa…but I don’t see that the same is true of a and Θa. It is true that if one is true at the actual world then so will the other be, but their modal properties will be different. There is a model where a is false and Θa is true; it is any model where a is true in the actual world and there is a possible world where a is false. At any rate the challenge is a good one and I will have to think about it…it might be the case that the kind of necessity that I have in mind in not the necessity of possible worlds but rather temporal necessity…God’s knowing at time t1 that I will do action a at time T2 temporally determines (at T1) that at T2 the action will be done. I am going to have to do some research and get back to this.


    I guess I would try to give the same kind of answer to your point as well. If the past determines the future, then God’s infallible knowledge of what I will do before I do it means that it is true that I will do it before I do it…but again I will have to give this a lot more careful thought.

  5. “Yeah, I take your point about Θa and []Θa…but I don’t see that the same is true of a and Θa. It is true that if one is true at the actual world then so will the other be, but their modal properties will be different. ”

    This is a minor point, but remember that modal logic is not truth functional. So its true that a and Θa have different modal properties, but it doesnt follow from this that they have different truth conditions. a is true (tout court) just in case a is true in the real world. and Θa is true just in case a is true in the real world.


    “There is a model where a is false and Θa is true; it is any model where a is true in the actual world and there is a possible world where a is false.”

    No, there is no model where a is false and Θa is true. There is a model where Θa is true and where a is false in some worlds. but a is true _tout court_ just in case it is true in the real world, i.e. just in case Θa is true.

    But yes, the much more important point is that the [] in []Θa is either undefined, or its just decoration.

  6. If the past determines the future, then God’s infallible knowledge of what I will do before I do it means that it is true that I will do it before I do it

    Yes, that’s the starting point for the problem. The question is whether this implies any significant kind of necessity. I don’t think standard modal logic will help, because there’s always going to be the counterexample of the possible world where you act differently (and God had correspondingly different knowledge).

    Tense considerations seem vital: if it is true now that p (where ‘p’ is some proposition about the future), then there seems a sense in which it is impossible for us from our present situation to bring about not-p. The only futures accessible from our current state are p-worlds. Does this capture your worry?

    (The standard response, I guess, is that “mere truth” is not so modally packed. The not-p worlds are still accessible; you could do otherwise, it’s just that – as a contingent matter of fact – you won’t, and that’s why p is [already] true.)

  7. Hey Eric,

    Isn’t everything that you say about Θa and []Θa also true of (a=a) and [](a=a)? There isn’t a model where one is true and the other is not, so they in some sense ‘mean’ the same thing also, right? But we wouldn’t want to say that the ‘[]’ in [](a=a) is just decoration, would we?


    Yes, I think that captures my worries. And I agree that ‘mere’ truth may not be so modally packed, but that is why I introduced ‘Θ’….once Θa is true the not-p world branchs close and are no longer accessible…normally this would not be a problem because ◊Θa and ◊Θ~a but in the case where God knows beforehand Θa is true beforehand and so the branch closes too early and not due to anything that I do….

  8. But it is never the case that ◊Θa and ◊Θ~a. They’re inconsistent. If a is true-at-@ at any possible world, then it is so at every possible world. That’s why introducing ‘Θ’ doesn’t get us anywhere. (What matters is not what’s possible for the actual world-state ‘@’, but what’s possible simpliciter, i.e. what world-states could have been actualized. Cf. my comment on your earlier post, re: the ambiguity of “the actual world”.)

  9. Richard,

    I would say the difference between []Θa and [](a=a) is that, in the latter, there are two place holders, and its only when you put “a” in twice, that you automatically get the same truth value. put in, say, “a” and “b”, and its a different story. But in the former case, no matter what you plug in, you get the same truth value.

    Perhaps this makes it more clear:

    If I write

    []Θ. and Θ., where “.” is a placeholder (not a variable, a placeholder), they have the same truth value in all models, regarless of how the place holder is filled in.

    But that’s not true of [].=. and .=.

    If I put in “the number of letters in my first name” and “4” then I get true for the second one, but false for the first one. you cant do that with
    []Θa and Θa.

  10. Richard, I think I mispoke. Though Guess I can see that there is a kind of semantics that would allow it…namely one where there were multiple actual worlds simultaneously, maybe some string theorist who think that there are parralell universes that are actual but seperated from us dimensonally…but other than that, I take yor point

    What I was trying to say was that ◊Θa or ◊Θ~a (exclusively). Only one of them can really be true. But from some proir point in past they are both accessible. The world could have gone either way. This seems to me to be true on both meanings of ‘actual’ that you distinguish. But when God knows beforehand this is no longer true. One is already actually true…

  11. The world could have gone either way.

    Here it’s the ambiguity of ‘world‘ (universe vs. world-state) I’m more worried about. The universe could have gone either way, but the “actual world” (if by this we denote the actualized possible world, a.k.a. ‘w42’) could not have!

    Consider: P is true in w42, and so P is true in reality. The universe could have been different, which is just to say that not-P is possible, i.e. not-P is true according to some other state (“possible world”) that the universe could have had — w7, say.

    Now, note that ‘ΘP’ really just says ‘P is true in w42’. (It could have meant something else, just like ‘water’ could have meant something different, rendering that non-English sentence “water is XYZ” true in the appropriate possible world. But that’s quite different from our sentence “water is XYZ” being possibly true. Likewise with ΘP and its modal iterations [◊ΘP, etc.] They’re all logically necessary.)

    Possible worlds, including the actual one (w42), are immutable. They could not have been different. (There is no possible world according to which P is false at w42, for example.) What could have been different is the universe, but the moment you introduce the operator ‘Θ’ you’ve stopped talking about the universe, and starting talking about the mere state (that happens to be the state of the universe) — and we should not be surprised by the immutability of that.

  12. (Aside: you might be interested in 2-Dism, which allows an alternative semantics for ‘actual’, sort of like your last comment hints at. It still wouldn’t have the metaphysical implications you’re after, though.)

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