God's infallible knowledge and free will part II

Originally posted by bbarr
Sorry, I just don't see how this follows from the rejection of axiom (5).

Sorry, I just don't see how this follows from the rejection of axiom (5).

It is just a world-semantic representation of what it means to reject 5. But we can reason it out from there. If we reject the axiom "If possibly P, then necessarily possibly P' then we can have a set of worlds in which 'Possibly P and not necessarily possibly P' (which is the counterexample of 5). To say 'not necessarily possibly P' is the same as 'possibly not possible P'. Just looking at that you can see that one world will have P possible and another world in which P will be impossible. That seems fairly clear enough.

To say possibly P will require P is true in some world (say w1) and to assert that not necessarily possibly P will require a world (w2) in which P is false and for which all related worlds P is false (so then we may add another world w3 in which P is false.) For w2 P is necessarily false even though in w1 P is true.

Originally posted by bbarr
I think your intuition may be based on the assumption that we have to characterize the libertarian commitments in terms of possibility (i.e., If L, then it is possible for S to have done otherwise than he did). I agree that this won't work. But what if we characterize the libertarian commitments differently? We could use counter-factuals and the notion of c ...[text shortened]... t T2, make God's belief at T1 false. That's backward causation...

Meh, I don't know.

On the other hand, there are possible worlds in which S refrained from A-ing. Now, I'm assuming we want to set aside considerations of an unholy alliance between adherents of David Deutsch's multiverse theories and modal realists, but even so the fact that god knows that in this world, call it w, you did in fact A does not entail that S lacked the power to not-A and this is demonstrated by the set of possible worlds in which S refrained from A-ing.

Originally posted by Conrau K
I am arguing that simply because something is necessary in one world does not mean that it is necessary across all worlds and, similarly, just because something is possible in one world it does not have be true of that world. I take issue with your claims that P entails that you will A in all worlds and that Q means that you will not-A in the same world.

I'm not sure why you'd want to do this. S5 seems to me to capture our concept of necessity, and a theist can object to bbarr's premise 4 within S5 without problems as far as I can see.

Originally posted by Conrau K
[b]Sorry, I just don't see how this follows from the rejection of axiom (5).

It is just a world-semantic representation of what it means to reject 5. But we can reason it out from there. If we reject the axiom "If possibly P, then necessarily possibly P' then we can have a set of worlds in which 'Possibly P and not necessarily possibly P' (which is ...[text shortened]... rld w3 in which P is false.) For w2 P is necessarily false even though in w1 P is true.[/b]

I'm really not sure why this point needs disputing. Obviously there are meaningful differences between T, K, S4 and S5, which will affect how necessity operates across worlds. I don't see what the point is in listing the individual differences. The point is simply that, using different axioms, we end up with different notions of necessity. The onus on bbar then is to justify his notion of necessity i.e. how it corresponds to what the libertarian wants it to mean.

Originally posted by Lord Shark
I'm not sure why you'd want to do this. S5 seems to me to capture our concept of necessity, and a theist can object to bbarr's premise 4 within S5 without problems as far as I can see.

It captures one notion of necessity but not all. There are many times people may wish to say something is necessary in one world but not true in all worlds. I think this comment in the entry for modal logic on the Stanford Encyclopedia of Philosophy sums up my feelings:

One could engage in endless argument over the correctness or incorrectness of these and other iteration principles for □ and ◊. The controversy can be partly resolved by recognizing that the words ‘necessarily’ and ‘possibly’, have many different uses. So the acceptability of axioms for modal logic depends on which of these uses we have in mind. For this reason, there is no one modal logic, but rather a whole family of systems built around M.

http://plato.stanford.edu/entries/logic-modal/

Originally posted by Conrau K
[b]Sorry, I just don't see how this follows from the rejection of axiom (5).

It is just a world-semantic representation of what it means to reject 5. But we can reason it out from there. If we reject the axiom "If possibly P, then necessarily possibly P' then we can have a set of worlds in which 'Possibly P and not necessarily possibly P' (which is rld w3 in which P is false.) For w2 P is necessarily false even though in w1 P is true.[/b]

O.K., so we reject 5. Since 5 is a conditional, it will be false just in case its antecedent is true and consequent false:

(MP) & (~LMP)

So, we run the negation through the right conjunct and get:

M~MP

and continue running it through to get:

ML~P

Which simply says "It is possible that necessarily ~P".

So we get the final conjunction:

"It is both possible that P and possible that necessarily ~P".

Now if you read that using possible world semantics, you get:

There is a world where both P and necessarily ~P.

But, since ~P simply follows from necessarily ~P, you get the absurd result that there is a world where both P and ~P. So if you hew to possible world semantics, your rejection of 5 yields a contradiction.

Originally posted by bbarr
O.K., so we reject 5. Since 5 is a conditional, it will be false just in case its antecedent is true and consequent false:

(MP) & (~LMP)

So, we run the negation through the right conjunct and get:

M~MP

and continue running it through to get:

ML~P

Which simply says "It is possible that necessarily ~P".

So we get the final conjunction:

" ~P. So if you hew to possible world semantics, your rejection of 5 yields a contradiction.

"It is both possible that P and possible that necessarily ~P".

Now if you read that using possible world semantics, you get:

There is a world where both P and necessarily ~P.

No. The possible world semantic interpretation would minimally be 'there is a world where P is true and a world where necessarily not-P'. They don't have to be the same world. Granted that, we avoid the logical absurdity (you should know you have gone somewhere awry since K and T are perfectly fine without 5.)

This seems to be the same error as with premise (7). The possibility of Q does not mean that you A in w1. I don't know why you lump everything into the same world.

Originally posted by Conrau K
[b]"It is both possible that P and possible that necessarily ~P".

Now if you read that using possible world semantics, you get:

There is a world where both P and necessarily ~P.

No. The possible world semantic interpretation would minimally be 'there is a world where P is true and a world where necessarily not-P'. They don't have to be t ...[text shortened]... does not mean that you A in w1. I don't know why you lump everything into the same world.[/b]

No. The idea is not that that the different conjuncts have to apply to the same world. That would entail that 'possibly P & possibly ~P' entails a contradiction. Rather, the idea is that 'possibly necessarily ~P' entails there is a world where necessarily ~P. This much you grant. What I don't see, and what I think may be incoherent, is the denial of 'if necessarily ~P at any world, then ~P at every world'. This claim, conjoined with 'possibly ~P', entails the contradiction I pointed out above. But you'll claim this is just question begging.

I guess I just don't understand what you mean when you apply modal operators to particular worlds, especially when you seem to want to continue using possible world semantics. The whole point of possible world semantics is to provide a model for modal validity; to provide the truth conditions for modal claims, and thereby provide a clear meaning for the operators. Strictly speaking, conjoining possible world semantics with world-restricted modal operators is just incoherent. If you want to characterize necessity and possibility in some other way, that's your prerogative. But an unmotivated denial of this way of doing things doesn't persuade me, especially when you've nothing else to use in its place.

Originally posted by bbarr
No. The idea is not that that the different conjuncts have to apply to the same world. That would entail that 'possibly P & possibly ~P' entails a contradiction. Rather, the idea is that 'possibly necessarily ~P' entails there is a world where necessarily ~P. This much you grant. What I don't see, and what I think may be incoherent, is the denial of 'if n ings doesn't persuade me, especially when you've nothing else to use in its place.

This much you grant. What I don't see, and what I think may be incoherent, is the denial of 'if necessarily ~P at any world, then ~P at every world'. This claim, conjoined with 'possibly ~P', entails the contradiction I pointed out above.

Well, it's not a very controversial claim at all. LP in some world simply means that for that world, and all related worlds, P is true. Now, for example, in S4, this necessity becomes much stronger (because LP --> LLP) which would entail that P is true in some world and necessarily true in all related worlds (and consequently true in all worlds related to them.) See how adding this axiom strengthens the notion of necessity?

Now you are simply imposing axioms that don't have to be there. You have this notion of necessity and you're being insensitive to how the theist may wish to use it differently. Necessity, in the language of possible world-semantics, minimally requires that P be true for w1 and all related worlds (because from the perspective of that world, there is no foreseeable counterexample of not-P) --- whether it requires it to be true in more worlds needs more axioms. Necessity would only cut across all worlds in S5, which is a much stronger notion of necessity that not everyone wants to accept. If the theist rejects that definition, you can't just impose it on him anyway.

I guess I just don't understand what you mean when you apply modal operators to particular worlds, especially when you seem to want to continue using possible world semantics.

I think you have it the wrong way around. The possible world semantics simply offers an interpretation or representation for modal operators. That's all. It doesn't restrict their meaning (the axioms do that.) We can use the language of worlds and relations in order to capture the essential differences between K, T, S4 and S5 and whatever other modal systems there are. I am not inventing a new possible world semantic; I am simply using it to illustrate how notions of necessity change with their axioms.

Originally posted by Conrau K
It captures one notion of necessity but not all. There are many times people may wish to say something is necessary in one world but not true in all worlds. I think this comment in the entry for modal logic on the Stanford Encyclopedia of Philosophy sums up my feelings:

One could engage in endless argument over the correctness or incorrect ...[text shortened]... ole family of systems built around M.

http://plato.stanford.edu/entries/logic-modal/

I agree, you could specify a set of possible worlds and an accessibility relation in such a way as to capture some other notion of necessity. I just don't think it will capture the relevant one.

My question remains, why not use S5 since it has intuitive appeal with respect to necessity, given that the theist is well equipped therein to oppose the threat to libertarian freewill?

Originally posted by bbarr
No. The idea is not that that the different conjuncts have to apply to the same world. That would entail that 'possibly P & possibly ~P' entails a contradiction. Rather, the idea is that 'possibly necessarily ~P' entails there is a world where necessarily ~P. This much you grant. What I don't see, and what I think may be incoherent, is the denial of 'if n ...[text shortened]... ings doesn't persuade me, especially when you've nothing else to use in its place.

What I don't see, and what I think may be incoherent, is the denial of 'if necessarily ~P at any world, then ~P at every world'. This claim, conjoined with 'possibly ~P', entails the contradiction I pointed out above.

Just to return to this point, the reason we cannot say 'not-P at every world' is because we have just rejected 5 in this hypothetical. The rejection of 5 means, as has been shown, that necessity and impossibility do not cut across all worlds. It would be quite a strange objection to say that the rejection of 5 is absurd because it contradicts 5. That is just a trivial observation.

Originally posted by Lord Shark
I agree, you could specify a set of possible worlds and an accessibility relation in such a way as to capture some other notion of necessity. I just don't think it will capture the relevant one.

My question remains, why not use S5 since it has intuitive appeal with respect to necessity, given that the theist is well equipped therein to oppose the threat to libertarian freewill?

I agree, you could specify a set of possible worlds and an accessibility relation in such a way as to capture some other notion of necessity. I just don't think it will capture the relevant one.

I disagree. I think that a proper restriction of necessity is exactly what the libertarian theist wants. He wants to say that in the worlds in which he does A, God knows necessarily he does A, and, obviously, it is necessary he does A. But obviously he wants to restrict that necessity only to the worlds in which he does A. S5 is too strong a notion of necessity here.

My question remains, why not use S5 since it has intuitive appeal with respect to necessity, given that the theist is well equipped therein to oppose the threat to libertarian freewill?

Well, I am hard put to think of the appeal of S5. It seems to me to be grossly powerful and would hardly ever correspond to ordinary conversation. It also has very dubious consequences (for example, if something is possibly necessary, it is necessary.)

Originally posted by Conrau K
[b]This much you grant. What I don't see, and what I think may be incoherent, is the denial of 'if necessarily ~P at any world, then ~P at every world'. This claim, conjoined with 'possibly ~P', entails the contradiction I pointed out above.

Well, it's not a very controversial claim at all. LP in some world simply means that for that world, and al ...[text shortened]... am simply using it to illustrate how notions of necessity change with their axioms.[/b]

But I think it is controversial. To say of some proposition that it is necessary means that it is true in that world and all related worlds. In S5, where the relation is symmetric, reflexive and transitive, it means that that proposition is true of all worlds, since all worlds are accessible. This was what Carnap was after when he cribbed from Leibniz that necessity is truth in all possible worlds. Now, I get that for some logics, like tensed and deontic logics, this is too strong. But that is because our intuitions regarding necessity differ when dealing with past and future, or with obligation and permissibility. Sure, if we're dealing with physical necessity, for instance, we don't want such a strong accessibility relation. Here, however, we're dealing with metaphysical necessity. You argue as if we start with the model and a set of axioms, and then read off the interpretation of the modal operators. But, methodologically, this isn't how we proceed. We begin with our intuitive commitments regarding logical necessity, metaphysical necessity, or whatever, and use them to justify our axioms. We then construct models in order both to capture and to clarify our commitments, and in order to make sense of notions like truth and validity.

Suppose the theist wants to reject the accessibility relation encapsulated by S5. He wants to allow for a case where a proposition is necessary in W1 but false in W2. He'll need an accessibility relation that makes W2 inaccessible from W1. But tell me what this means, when we're dealing with logical or metaphysical necessity. It is all well and good to say that a theist may wish to use some notion of necessity that allows for a proposition to be necessary here but false there. But in the absence of even a cursory account or a characterization of the intuitions that inform the rejection of S5, this just seems ad hoc. Take a theorem like the law of non-contradiction. Does the theist want an account of the accessibility relation that allows for this law to be false in some world? Or take a putative instance of conceptual necessity, like 'If S knows P, then P'. Or metaphysical necessity, like that there is no backwards causation. Does the theist want an account that allows for the necessity of these propositions and for their falsity in some other world? Fine, but why? Give me a reason to think that our intuitions regarding these types of necessity (which, broadly, are the sort we've been dealing with in this debate) are not adequately captured by a strong accessibility relation. Give me a reason to think that these propositions could be false in some other world. And it is not sufficient as an answer to these questions that when parts of some formal system are jettisoned the falsity of these propositions is a option. That gets everything the wrong way around.

Originally posted by bbarr
But I think it is controversial. To say of some proposition that it is necessary means that it is true in that world and all related worlds. In S5, where the relation is symmetric, reflexive and transitive, it means that that proposition is true of all worlds, since all worlds are accessible. This was what Carnap was after when he cribbed from Leibniz that y of these propositions is a option. That gets everything the wrong way around.

We begin with our intuitive commitments regarding logical necessity, metaphysical necessity, or whatever, and use them to justify our axioms. We then construct models in order both to capture and to clarify our commitments, and in order to make sense of notions like truth and validity.

Absolutely. I am not suggesting that anyone commits themselves to a set of axioms firstly. I learned modal logic in the context of natural language semantics and so I see the axioms (and the possible worlds semantics representing them) rather as a useful way to describe the possible meanings intended by the speaker. So my question is whether your notion of necessity in this case accurately reflects the usage of the libertarian theist.

But in the absence of even a cursory account or a characterization of the intuitions that inform the rejection of S5, this just seems ad hoc.

Well, I believe that is just what I have tried to do. I do not think the theist has an S5 notion of necessity, such that the necessity of God's knowledge cuts across all worlds. What they want to say is that the necessity of God's knowledge (that you will do A) is true only in worlds in which you will do A, allowing the possibility of not-A in other worlds. But obviously this is just a trivial necessity.

Does the theist want an account of the accessibility relation that allows for this law to be false in some world? Or take a putative instance of conceptual necessity, like 'If S knows P, then P'. Or metaphysical necessity, like that there is no backwards causation. Does the theist want an account that allows for the necessity of these propositions and for their falsity in some other world?

Well, just because they want to reject S5 in one context, does not commit them to a total rejection of S5 (I am not sure if that's what your aiming at here though.) These different modal systems are not completely exclusive. S5 may represent a person's meaning in one utterance but not in another. We have to accept that language is ambiguous. The axioms therefore are just useful ways to capture the different meanings of utterances which have some modal import. So a libertarian theist doesn't have to be committed to a whole-scale rejection of metaphysical necessity (just ignore this paragraph if that is not what you mean.)

Originally posted by Lord Shark
I think we are talking about logical necessity. We can say necessarily p if to deny p involves a logical contradiction. One of the reasons that I think S5 is the right system for necessity is precisely because it cuts across all possible worlds. I'm really not sure what necessity involving a subset of possible worlds would mean really. If you could give an example of 'necessity lite', that might help.

It seems to me that all of our discussions so far have invoked 'necessity-lite' rather than absolute necessity.

Let's look at the example you gave in your OP - coffee and toast for breakfast. In this world, let's say you did actually have coffee and toast this morning. Therefore, it must be possible that you had coffee and toast for breakfast.

Now consider a possible world where you were knocked down by a bus and died last night (sorry!). According to the current interpretation (S5?), it would still be possible you had coffee and toast for breakfast this morning - which is counter-intuitive. (Imagine the conversation with your house mate: "It's not impossible that Lord Shark had coffee and toast for breakfast this morning." "But he died last night!" "Nevertheless, there is a possible world where he does so, so it's still possible in our world." "That's crazy!"😉

One way out would be to say that such a world is not a possible world, that it entails a logical contradiction. But there seems to be no inherent (of the analytical variety) contradiction in such a world; any contradiction appears to be one with facts in our world after the hit was supposed to take place. The only possible worlds admissible in this view would be future worlds - which seems unnecessarily restrictive. (This is not a formally rigorous refutation of such a definition of 'possible worlds', just a sort of argument based on counter-intuitiveness.)

(And, now that I look the Conrau-bbarr conversation more closely, the 'necessary-lite' view - which is not really that lite since I think we've been implicitly using it all along - does seem to have something to do with that discussion.)