Please help me translate this to modal logic notations

[GiantOreo]

Hey,

I'm a beginner when it comes to modal logic, and I was wondering if anyone can translate this (Omiscience paradox) using NOTATIONS.

1.) A being with free will, given 2 options A and , can freely choose between A and B.
2.) God is omniscient.
3.) God knows the being will choose A.
4.) God cannot be wrong since an omniscient God does not have false knowledge.
5.) From 3 and 4, I will choose A and cannot choose B.
6.) From (1.) and (5.), God's omniscience and free will cannot coexist.

Thanks for ANY help! :) :)

[pdjamesc]

You will need epistemic terms as well. I've had a quick bash at it but didn't get too far (hey, it's 3am here). I'll try again in the morning. I found myself needing:

If God knows A then A is necessary.

which is an obvious weakness which isn't in the original argument.

I suspect that whatever I come out with it will be messy. Are you sure parsing it in modal logic will really help?

[GiantOreo]

I was wondering how it would work out. Thanks for the help...its'a a good start.

Can it be expanded further??

[JohannGoodflag]

I am no expert in modal logic, but there seem to be two crucial issues.

* What do you mean by someone "having an option"? Do you mean that it is possible for them to do it? Also, what do you mean by "freely" choosing? Do you mean it is possible that they will do it? If 'yes' to both, then #1 seems to translate to "If x has free will, and it is possible that x will do either A or B, then it is possible that x will do either A or B". It seems to me that the premise of free will is superfluous here. Perhaps, though, free will is necessary for there to be two possible choices; but this gives you an inference "If it is possible that x will do A and it is possible that x will do B, then x has free will", which isn't the direction you're looking for.

* It seems that the possible worlds you're looking for is, given a time t, all of the "possible" world histories including events at times t' > t. That is, that propositions for times t' <= t are all either true or false, but that for times later than t, only propositions which necessarily hold have definite states of affairs. (Note that if anyone can identify which of the possible future-histories will occur, then the world is in fact deterministic, and there is only one possible state of future affairs --- that is, everything which is possible is also necessary. More generally speaking, knowledge which is not in error can only ever be about things which necessarily hold, which for past events includes things which actually did occur, and for future events include things for which no choice is possible.)

Building on this last point: the most reasonable way to translate #2, together with the consistency-of-knowledge constraint of #4, is that a statement is true if and only if god knows it:
(X) if and only if (God knows X)
Given the remarks above about knowledge and necessity, we could sharpen this (for emphasis) to
(It is necessary that X) if and only if (God knows X).
Then, if God knows that x will do A, it is necessary that x will do A: then, if it is not possible for x to do both A and B, it cannot be possible that x will do B. That is to say: either B is not a choice (in the trivial sense I proposed in the first point above), or God does not know that x will do A (because whether or not x will do A is an undetermined, and so unknowable for any infalliable repository of knowledge).

[wiploc]

Hey,

I'm a beginner when it comes to modal logic, and I was wondering if anyone can translate this (Omiscience paradox) using NOTATIONS.


The only kind of modal logic I know is the many-worlds stuff. Is that what you're after?

crc

[Preno]

How is formalization supposed to help? I suppose you could, for instance, write something like: has_free_will(x) <=> (A a,b: are_options_for(x,a,b) -> can_freely_choose_between(x,a,b)), but what exactly would be the point? Formalization just for the sake of formalization is useless.

[kennethamy]

How is formalization supposed to help? I suppose you could, for instance, write something like: has_free_will(x) <=> (A a,b: are_options_for(x,a,b) -> can_freely_choose_between(x,a,b)), but what exactly would be the point? Formalization just for the sake of formalization is useless.

Right. It is a homework question.

[JohannGoodflag]

If it's a homework question, then it's one I wouldn't expect from a logic course: it's seriously ill-defined. Of course, trying to formalize it does illustrate that it is ill-defined, which raises the more interesting question of what to do with it to make it better.

[Dr. Retard]

The way it's set up seems screwed-up to me. Indeed, the argument being gestured at is pretty well-known for committing a modal fallacy. (There are other arguments for the inconsistency of God's omniscience and free will which do not commit this modal fallacy)

Let's call the being P for 'person'. I'll use <> for possibility and [N] for necessity. Step 1 is easy.

1. <>(P will choose A) & <>(P will choose B)

Step 2 is weird because it's unclear whether God is supposed to be omniscient or necessarily omniscient. (Also, I guess there's some controversy in asserting a bound variable ranging over propositions, but whatever)

2. For all p, God believes that p iff p
- or -
2'. [N](For all p, God believes that p iff p)

Step 3 is easy.

3. God believes that P will choose A.

I'll take step 4 to be a conclusion fallaciously inferred from 2' -- notice how it misplaces the necessity operator.

4. For all p, ((God believes that p) ---> [N](p))

The first part of 5 is easy, but the second part doesn't follow because we haven't said that A and B are mutually exclusive options.

5. ([N](P will choose A)) & (~<>(P will choose B))

And 1 and 5 run into a contradiction: <>(P will choose B) and yet ~<>(P will choose B).

Here's a better way to run this bad argument:

1. It is necessary that if God believes that p, then p. [premise, God's omniscience]
2. Therefore, if God believes that p, then it is necessary that p. [from 1 (fallacy!)]
3. God believes that P will choose A. [premise, God's foreknowledge]
4. Therefore, it is necessary that P will choose A. [from 2,3]
5. If P is free with respect to A, then it is possible that P will not choose A. [premise, free will requires alternatives]
6. Therefore, P is not free with respect to A. [from 4,5]

[Dante Alighieri]

What would be a good example of theological fatalism that doesn't commit the modal fallacy?

[Dr. Retard]

Dante, ones that use the necessity of the past. The past is beyond our control. God's past beliefs concerning our future activities are therefore beyond our control. The impossibility of God's beliefs being false is beyond our control. And (this is the 'transfer principle') if something is necessitated in a way beyond our control by something else beyond our control, then that first something is also beyond our control. So our future activities are beyond our control.

Here's how the Stanford Encyclopedia of Philosophy puts it:

(1) Yesterday God infallibly believed T. [Supposition of infallible foreknowledge]
(2) If E occurred in the past, it is now-necessary that E occurred then. [Principle of the Necessity of the Past]
(3) It is now-necessary that yesterday God believed T. [1, 2]
(4) Necessarily, if yesterday God believed T, then T. [Definition of "infallibility"]
(5) If p is now-necessary, and necessarily (p → q), then q is now-necessary. [Transfer of Necessity Principle]
(6) So it is now-necessary that T. [3,4,5]
(7) If it is now-necessary that T, then you cannot do otherwise than answer the telephone tomorrow at 9 am. [Definition of "necessary"]
(8) Therefore, you cannot do otherwise than answer the telephone tomorrow at 9 am. [6, 7]
(9) If you cannot do otherwise when you do an act, you do not act freely. [Principle of Alternate Possibilities]
(10) Therefore, when you answer the telephone tomorrow at 9 am, you will not do it freely. [8, 9]

Also perhaps Nelson Pike-style arguments based on our inability to bring it about that God's beliefs were different than they are, perhaps those don't commit the modal fallacy.

Of course these arguments might have other things wrong with them.

[Dante Alighieri]

Hm, so, there aren't any modal arguments for theological fatalism that aren't fallacious?

[Dr. Retard]

Dante, those arguments make key use of modal concepts, which is the only thing I can think of that would make an argument a modal argument. So I'd say these arguments are also modal arguments.

[Dante Alighieri]

Suffer my ignorance a bit longer:

Doesn't □(Kg → Ap) → (□Kg → □Ap)?

Kg = God knows that Ap
Ap = a person does a particular action

If the theist admits that □(Kg → Ap), then can't we infer (□Kg → □Ap) and avoid a modal fallacy?

[Dr. Retard]

Yeah, it looks like that entailment holds. No worlds where Kg holds and Ap doesn't. So if Kg holds in all worlds, then Ap also holds in all worlds.

But notice that (□Kg → □Ap) doesn't get you □Ap. The theist can just deny □Kg, saying that there are other worlds where God knows something else, those worlds being different from the actual world. And you need □Ap in order to get a threat to free will going.