This is a translation of my earlier article on the subject. Link.
The modal fallacy
By Emil Kirkegaard, Deleet.dk
This fallacy is rather common among persons who are not well versed in logic (especially modal logic). Consider these two not logically identical sentences:
I) If there exists at least one subject S that knows which outcome U situation F will have at time t1, then outcome U will necessarily happen at time t1.
II) Necessarily, if there exists at least one subject S that knows which outcome U situation F will have at time t1, then outcome U will happen at time t1.
An unlucky property of the natural language is that it does not distinguish between these two sentences and that one normally almost always uses (I) if one is not aware of the difference. This problem is actual in Danish and English and maybe other languages as well. What happens is that ‘necessary(-ily)’ gets misplaced. It gets placed in the consequence of an implication but in reality it speaks (or should speak) of the entire implication. Logically the sentences can be formalized like this:
I’) P→ □Q
II’) □(P→ Q)
Now the difference should be clear and it should also be clear that (I) is false and that (2) does not support what one normally believes that it supports. Let us consider two arguments where the first is very common among young atheists who are not well versed in logic:
If God knows which outcome situation F will have at time t1, then the situation will necessarily have the outcome he knows it will have. If the situation necessarily will have the outcome, then all humans who are involved in situation F have no free will in F. Moreover, if God knows the outcome of all situations, then no man has a free will.
The argument is a little complicated, let us just look at the first part of it:
1. God knows which outcome situation F will have at time t1. (premise or hypothesis)
2. If God knows which outcome situation F will have at time t1, then the situation will necessarily have that outcome. (premise)
3. The situation will necessarily have that outcome. (1, 2)
4. If the situation will necessarily have that outcome, then no human in the situation F has a free will.
5. No human in situation F has a free will. (3, 4).
[snip a bit about the Danish language not having a future case]. Note that ‘necessarily’ will typically not be placed in the start of a sentence in natural language like one does in philosophy to reduce ambiguity (cf. (3)). The argument (a) can me formalized like this:
2′. P→ □Q
The argument is valid but the problem is that (2) is false. Defenders of the argument or similar will typically argue (2) by noting that if someone knows something, then it is necessarily true, because otherwise they would not know it. This is also false in this specific formulation. That whatever one believes is true is a necessary condition for knowledge (cf. JTB+) but from here it does not follow that it is necessarily true.
Remember that a necessary truth is true in all worlds and therefore it follows that if a person knows something then that something is true in all logically possible world. But this is false because there is a logically possible world where Earth is flat but still I know that Earth is round.
Recall sentence (I) and (II) from earlier. The analogue sentences for knowledge are these:
I”) If someone knows p, then p is necessarily true.
II”) Necessarily, if someone knows p. then p is true.
(I”) is false and (II”) is true. But for argument (a) to be sound, then it is required that (I”) is true. If we substitute (I”) with (II”), then the argument is no longer valid because □Q doesn’t follow from (1) and (2).
Let’s consider another argument.
If my mother knows which education I will choose after high school, then I will necessarily choose that education. My mother knows which education I will choose after high school, therefore I will necessarily choose that education.
The we can again spot the problem where ‘necessarily’ is misplaced. It is true when it speaks of the entire implication but false then it only speaks of the consequence. If this argument was sound then it would prove that I cannot change my mind, but this is false.
An awesome source by professor Norman Swartz (Indiana University Ph.D., 1971 (History and Philosophy of Science)):
Internet Encyclopedia of Philosophy about the modal fallacy in divine foreknowledge.
The Fallacy Files about modal fallacies in general. There is a subpage about the specific fallacy I have in mind.
Stanford Encyclopedia of Philosophy about modal logic. They mention the fallacy just before the section on deontic logic.