The Compatibility of Omniscience and Freedom

I say even if God knows what you are going to do tomorrow, this does not stop you being free to act otherwise. You won’t act otherwise, but you could.

Let’s set out a few definitions. You are free to do an action if it is possible that you do it, and if it is possible that you don’t do it. If either of these options is removed, you are no longer free. So if ‘p’ is ‘you will do x’, then you are free to do x if and only if (iff):

It is possible that p, and it is possible that not-p

Alternatively, we will write this as follows (where ‘◊’ means ‘possibly’):

◊p &  ◊~p

The problem is that this freedom condition seems to be ruled out by the idea of God’s foreknowledge. The reasoning is that if God already knows that p, then it is necessary that p. We can write this as follows, where K = God knows, and □ = necessarily:

If Kp, then □p

And if it is necessary that you are going to do x, then it is not possible that you will not do x. If necessarily p, then it is not possible that not-p:

If □p, then ~◊~p

So let’s put this into an argument that seems to show that freedom and omniscience are incompatible by deriving a logical contradiction:

 

Premise 1) I’m free to do x     (i.e. ‘it is possible that p and it is possible that not-p’)

Conclusion 1) Therefore, it is possible that not-p.

Premise 2) God knows that p.

Premise 3) If god knows p, then p is necessary.

Conclusion 1) Therefore, p is necessary.

Premise 4) If p is necessary, then it is not possible that not-p.

Conclusion 3) Therefore, it is not possible that not-p

Conclusion 4) Therefore, it is possible that not-p, and it is not possible that not-p.

 

We can write exactly the same argument in symbols as follows (in the right I give whether each line is an assumption or how it follows from something previously assumed):

 

Premise 1) ◊p &  ◊~p                         (assumption)

Conclusion 1) ◊~p                              (from pr. 1, and conjunction elimination)

Premise 2) Kp                                      (assumption)

Premise 3) If Kp, then □p                 (assumption)

Conclusion 2) □p                                (from pr.2 and pr.3, and modus ponens)

Premise 4) If □p, then ~◊~p             (definition of □ and ◊)

Conclusion 3) ~◊~p                            (from con.2 and pr.4, and modus ponens)

Conclusion 4) ◊~p & ~◊~p                (from con.1 and con.3, and conjunction introduction)

 

So we have derived a contradiction; it is possible that I will not do x, and it is not possible that I will not do x. This means we have to either reject the truth of one of the premises, or reject the validity of the argument form. Now the validity is easy to address, as it uses nothing but inference rules from classical propositional logic and the duality of necessity and possibility (i.e. □ = ~◊~ and ◊ = ~□~). There is nothing controversial at all here. So we must reject the truth of at least one of premises 1, 2 or 3, on pain of having to accept a contradiction.

Premise 1.

We said that being free to do x requires that it is possible to do both x and to not do x. Not all definitions of freedom require this. In fact this is a strong condition, and ‘compatibilists’ (like Spinoza, or Frankfurt) will contend that one can be free even if only one option is possible, just so long as that option is chosen. So the prisoner is free to stay in the cell, even though it is not possible to leave, for example. So it is possible to reject this premise. I think we can keep it however, and still avoid the consequence. We do not have to be ‘compatibilists’ to argue that God’s foreknowledge is compatible with freedom.

Premise 2.

This says that God knows what will happen tomorrow. To deny this means either giving up on God’s omniscience, or on the fact that there is a truth about the future (i.e. giving up on the principle of bivalence). We could go the second route, and retain omniscience, given that there is no truth about the future for him to not know. It should be noted that if we go this route, we have to also also hold that God is located in time. In this case, he would find out what happens tomorrow with the rest of us. A timeless God cannot ‘find out what happens’, as this would be a temporal activity. Anyway, we do not have to reject bivalence or require God to be in time, as I say we can avoid the contradiction even if premise 2 is true.

Premise 3.

This, as I see it, is where the confusion sets in. It says that ‘If God knows that p, then it is necessary that p’. Why would we think this premise is true? One reason is as follows. If you know something, anything, then it has to be that it is true. After all, you can’t know something false. It’s part of the definition of knowledge that it is of something true. God, who is infallible, only makes this force stronger; he couldn’t be wrong about anything. So if he knows something is going to happen, it is definitely, necessarily, going to happen. How could he be wrong?

Well, we need to be careful about the logical form of what we are saying. It is necessary that God knows p, and truth is a necessary component of knowledge; but this doesn’t mean that what God knows is necessary. Here is the sentence that is doing all the heavy lifting conceptually:

If God knows that you will do x, then it is necessary that you will do x.

I agree that everything God knows is true, i.e. he is infallible, and that everything true is known by God, i.e. that he is omniscient. But this only amounts to the following:

Kp iff p

This says that ‘God knows that p  if and only if p’. I can even go all the way and say that this is a necessary truth:

□(Kp iff p)

Now, we can derive a conditional which is very similar to premise 3 (which I will call 3.1) from this, namely:

3.1) □(If Kp, then p)

But it is important to note that this is as far as we can go. There is no way to go from 3.1 to 3:

□(If Kp, then p), therefore (If Kp, then □p)

So premise 3 does not follow from 3.1. Moreover, I say that 3.1 is actually the correct logical form of: ‘If God knows that you will do x, then it is necessary that you will do x.’

Admittedly, the word ‘necessarily’ is in the consequent in the sentence, and that seems to count against my claim. But then we systematically leave it there when we express both de re (of the thing) and de dicto (of the word) modalities, which should have it in different places. This means we fail to distinguish between the scope of the modality in natural language. Getting the scope of the modality right will solve the problem.

Quine’s example in Word and Object (p120) is that about cyclists being necessarily two-legged (and mathematicians being necessarily rational). To adapt his example, we would say:

If x is a cyclist, then it is necessary that he has two legs.

This sentence also has the word ‘necessary’ in the consequent, when it should be prefixing the whole conditional. It expresses only that under the description of the word cyclist, x has two legs. It is possible that x falls and gets one of his legs somehow cut off, and then x would not have two legs. It is not a necessary truth about x that he has two legs, only a necessary requirement for being a cyclist. So it is necessary de dicto that x has two legs, but not necessary de re. If we speak carefully, we would say:

It is necessary that (if x is a cyclist, then x has two legs)

The above formulation is compatible with the fact that x could fall off his bike and lose a leg, because he would then stop being a cyclist. Neither him being a cyclist nor having two legs is necessary; what is necessary is the connection between being a cyclist and having two legs.

This shows that we regularly fail to state the correct logical form when expressing de dicto modal claims. Sometimes, even though the word ‘necessarily’, etc, is in the consequent, it should be prefixing the whole conditional. And I say that premise 3 is one of these cases.

So if 3.1 was used in place, it would say that it is necessary that if God knows you will do x, then you will do x, just like it is necessary that if x is a cyclist, then he has two legs. Just like with the cyclist example, you do not have to do x (and he could fall off his bike). x doesn’t have to have two-legs, its just that it is necessary that if he is a cyclist then he does. You don’t have to do x, it just that it is necessary that if God knows that you will, then you will. In each case, the conditional is necessary, meaning that the one condition is never true without the other, but the other can be false. If it is false, then the antecedent condition would be false too.

One way of putting this is that it God knows contingent truths, like that you will do x. It is necessary that he knows them, but he only knows them if they are true. It is contingent that you will x, so it is contingent that p is true, and thus God might not know it. All that is necessary is that if it is true, he knows it; and if he knows it, it is true.

If we plug 3.1 in place of 3 in our argument from above, it stops us being able to move from it being true that p, to it being necessary that p. We needed that to get our contradiction, so we have blocked the contradiction. Thus foreknowledge and freedom are compatible.

Conclusion.

So I gave an argument for the incompatibility between divine foreknowledge and freedom, making it as strong as possible, which showed logically that the two concepts lead to a contradiction. However, I suggested the the logical form of the third premise was incorrect, allowing us to keep all the strong assumptions and show that no contradiction is forthcoming.

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9 thoughts on “The Compatibility of Omniscience and Freedom”

  1. I agree with the above analysis. I agree that 3.1 is the correct logical form and that 3 isn’t because 3 commits what I have read termed as the “modal muddle”. However, it seems to me that 3.1 takes the form of a(x then y). Could one, then, use the law of distribution like in high school algebra, in order to derive (ax then ay) just like one can derive (ax+ay) from a(x+y) using the law of distribution? In other words, can we get (ax then ay) from a(x then y) and can we then make a logically sound argument that omniscience and freedom are logically incompatible?

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    1. Yes, you can move from □(If Kp, then p) to (If □Kp, then □p). So it says that if it is necessary that God knows that p, then it is necessary that p. And maybe it is necessary that God knows p, for all p? I don’t think so. If p is contingent, then the fact that God knows that p is also contingent. For example, God knows I am drinking coffee, but it is contingent that I am drinking coffee – I could have made tea instead. So because it could have been that it was false that I am drinking coffee, it follows that it could have been that it was false that God knows I am drinking coffee. Therefore, it is not necessary that God knows that I am drinking coffee; rather it is contingent that God knows this. So what we got with your application of the law of distribution was that if it is necessary that God knows it, then it it necessary. But it is only ever true that God necessarily knows p if p is necessary. Remember, we have Kp iff p, meaning that God knowing that p and p being true are equivalent. So they always have the same modal status. This means you cannot go from a contingent truth, filter it through God’s knowledge and make it necessary at the end.

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  2. IF someone could out my mistakes below, I will be able to understand this better.

    I believe that me, and many others, get the argument in the wrong way because we associate “necessary” with the following definition:

    DEF: Let P be a proposition. Saying that P is *necessary*, means that P can be logically deduced without any premises. Example, all tautologies are necessary.

    Based on this…I will try to illustrate the confusion that (at least) I have.

    1) It is necessary that (if p, then q). (de dicto)

    That would mean that the conditional (if p, then q) can be logically deduced without premises. A way to prove that the conditional is necessary would be to introduce the supposition p, and deductively reach q. Then, supposition elimination gives (if p, then q). All without premises.

    2) if p, then necessarily q. (de re)

    This one would mean that if p is true, then q can be logically deduced without premises. It sounds contradictory on the one hand because…we are saying IF P, THEN….(which essentially gives P the role of a premise)…and at the same time that “q can be logically deduced WITHOUT PREMISES”) which says that nothing plays the role of a premise.

    On the other hand, if we are asked to prove “if p, then necessarily q” IT SEEMS TO ME that it is exactly the whole prove we did for case 1, that is: introduce the supposition p…..then show that deductive reasoning leads necessarily to q.

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    1. I guess…the confusion in (2) is because of interpreting it based on two possible ways of phrasing it:

      2) if p, then necessarily q.
      In this phrasing I think I considered it same as “if p, then q”, and using the word “necessarily” simply emphasises the meaning of a conditional.

      2.1) if p, then q is necessary.
      In this phrasing I considered it like something different from “if p, then q” and called it contradictory based on the “definition of necessary” that I specified above.

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  3. “And maybe it is necessary that God knows p, for all p? I don’t think so.” – Alex

    Is the following equivalent:

    “And maybe it is necessary that God is omniscient? I dont think so.” — Alex.1

    ?

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  4. I think I am now able to put into words the reason why i think they are incompatible.

    You concluded:

    [1] It is necessary that: IF God foreknows x, then x.

    if you ask God today: will Juan eat cereal tomorrow? and He says: I know “Juan will not eat cereal tomorrow”.

    Then there is no way that I can go against his knowledge.There is no way that I can eat cereal. So it does limit my freedom. Either I eat cereal and he did not know…or He knows and it is not possible for me to eat it.

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  5. Hi Juan. In the basic example, God knows that you will do p tomorrow, but you don’t. This means that you will do p tomorrow, and that you will do it necessarily follows from God knowing it. This doesn’t mean that you will necessarily do it. Your doing p is still contingent. You might not do it, even though you will in fact do it.

    In your example, the only difference seems to be that God has told you that you will do p. I don’t think that this means that you know that you will do p. God telling you is insufficient to know the future. Sometimes God foretells things in the bible (like the destruction of Nineveh) which end up not happening. Medieval theologians generally considered these to be elliptical conditional predictions (‘I will smite Nineveh unless it repents’, etc). So even if God tells you something will happen, this doesn’t mean that it will.

    Anyway, let’s say that we ignore this, and imagine that God telling you that you will do p means that you know that you will do p. Well, if you know that you will do p, then you will do p, because truth is a condition of knowledge (you can’t know something false). So in this scenario, you will do p.

    You claim that it is impossible that you don’t do p in this scenario. And you not doing p is incompatible with you knowing that you will do p. But, and this is all that matters, it being true that you will do p is not incompatible with it being possible that you do not p. (You will do p) and (it is possible that you will not do p) are logically compatible, there is no contradiction between them both being true. Adding that God knows whatever, or even that you do, makes no difference.

    The intuition that you are expressing is deeply felt by a lot of people, but when you go through the logic carefully, it is fallacious.

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  6. I would like to say at the outset of this response that my intention is not to refute you. In other words, I hope that you don’t take this as me being stubbornly attached to my own understanding, unwilling to change. Instead, my intention is to give you genuine feedback and explain how my current understanding is preventing me to see things the way you do. I believe that…with the feedback…you may be better equipped to attack my misunderstanding (and as you said…the misunderstanding of many others).

    If you ask me… my guess, as to WHAT might be preventing me to clearly understand your point, is miscommunication. Specifically, having different meanings for certain terms. Particularly, the term “possible”. Most likely, you are using it with a formal definition, perhaps rooted in modal logic (of which I know virtually nothing about). I am thinking about it perhaps differently, rooted in probability. I elaborate below.

    Without further ado…let me start with what you rightly identified as “…and this is all that matters…”

    [1] “it being true that you will do x is not incompatible with it being possible that you do not x. ”

    To check if I understand you….I will provide the following example to illustrate what I think you mean. I would like you to tell me if I am getting your point.

    ————EXAMPLE——————————————–
    Lets assume that tomorrow I will go ice-skating. Given this, we can use the excluded middle to state:

    — (Juan will either fall during his ice-skating activity) OR (He will not).

    It is clear that ONE AND ONLY ONE will happen because they are logical-complements. However, simply because ONE will happen, doesn’t imply/mean that the other was impossible all along (This is [1]).

    At this point, Juan and all confused people would ask: Why is this the case? Can you elaborate?

    I think the explanation you would provide would be based on what the term “possible” means….and at least I would go about it like this:

    If we are asked TODAY to provide the probability of occurrence for each of those outcomes (fall, not-fall) we know that one will have probability p and the other (1-p)…again, because they are complements of each other. Also, AND THIS IS THE IMPORTANT PART:

    [2] p NEED NOT be 0 nor 1, it could be strictly between…so that (1-p) is also NOT 0 nor 1.
    In such a case, we would answer TODAY, that both outcomes have a non-zero probability of occurring = i.e. that neither is certain = i.e. that neither is impossible = i.e. that both are possible.

    To emphasise, even though ONE WILL IN FACT HAPPEN….say “fall will happen”…since neither need be certain…it is compatible to say “not-falling was possible”. <—An instance of [1]

    ^—THIS is my current way to illustrate how I understand the term "possible". and to illustrate what I think you meant by [1].
    —————————————————————————

    If you agree that my understanding is compatible with yours….then I have a counterargument that applies to god foreknowledge.

    First: Formally define "possible", "impossible", "certain".

    My definitions for "possible" and "impossible" are founded on the concept of probability for complementary outcomes. Specifically, the term "possible" applies to a statement A when its probability of occurrence p≠0. Small Theorem_1, when p≠1 (i.e. not certainty), then "possible" also applies to its complement ~A because (1-p)≠ 0. The term "impossible" applies to a statement B whenever its probability p = 0. In that case, its compliment ~B is not only possible but more specifically "certain" because its probability is (1-p) = (1-0) = 1.

    Second: Probabilities can be affected by knowledge.

    For instance, what is the unconditional probability that you have a decease D vs. what is the probability that you have decease D given that you took a 95% accurate test and it was positive for the decease . These two probabilities are both asking about the outcome ("You have decease D")…but they need not be equal (HINT: apply Bayes Theorem). Therefore, Knowledge (of the test being positive) is capable of affecting probabilities….thus (given my definition of "possible") Knowledge may change possibilities. (I am clearly considering Knowledge as a subset of Probability-Conditions)

    So now…going back to [1]

    1) "it being true that you will do x is not incompatible with it being possible that you do not x. "

    Again, I think you said it in the sense that even though ONE WILL IN FACT HAPPEN… "do x"… it need not be that its probability was 1 (certainty/necessity) to begin with…therefore it need not be that its complement "do not x" was impossible to begin with (i.e. possibility is allowed). I understand this… But I THINK this is the case because of the whole "it need not be" alternative …and I THINK we say it because of uncondional probabilities…like saying "no condition so far requires probabilities to be 0 or 1 exactly"

    BUT if there is a condition…in which "do x" has conditional probability of p=1, then the whole "it need not be" fails…and IT DOES NEED BE that "do not x" is impossible because (1-p) would be 0.

    Isn't "God Knows" such a condition?

    EXTREMELY SORRY FOR HOW LONG THIS IS….Sometimes I don't know how to be concise and clear….the problem is that I may still be unclear.

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  7. Before you read…the never ending comment above…I think…this topic might be awesome if you discuss it in a video with Ozzy, Matt, etc. I think it would be really interesting. Consider it!

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