The much mentioned the modal fallacy is not a fallacy (that is, is a valid inference rule) if one accepts an exotic view about modalities and necessities that is logically implied by a particular understanding of infallible knowledge and a knower.

### Infallible knowledge

Some people seem to think that some known things are false and thus the need for a term like infallible knowledge, for that kind of knowledge that cannot be of false things. However that term “infallible knowledge” (and it’s under-term “infallible foreknowledge”) is subject to some interpretation. Is it best understood as:

A. If something is known, then it is necessarily true.

Or?:

B. Necessarily, if something is known, then it is true.

Or equivalently, in terms of “cannot” instead of “necessarily”:

A. If something is known, then it cannot be false.

B. It cannot be false that, if something is known, then it is true.^{1}

I contend that the second interpretation, (B), is the best. However suppose that one accepts the first, (A).

### The assumption of the existence of a foreknower

Now let’s assume that there is someone that knows everything (which is the case), the knower. He posses infallible knowledge á la (A). Now we can work out the implications.

The foreknower exists and knows everything (that is the case):

1. There exists at least one person and that for all propositions, that a proposition is the case logically implies that that person knows that proposition.

(∃x)(∀P)(P⇒Kx(P))

Whatever is known is necessarily the case (A):

2. For all propositions and for all persons, that a person knows a proposition logically implies that that proposition is necessarily true.

(∀P)(∀x)(Kx(P)⇒□P)

Thus, every proposition that is the case is necessarily the case:

Thus, 3. For all propositions, that a proposition is the case logically implies that it is necessarily the case.

⊢ (∀P)(P⇒□P) [from 1, 2, HS]

Thus, everything that is logically possible is the case:

Thus, 4. For all propositions, that a proposition is logically possible logically implies that it is the case.

⊢ (∀P)(◊P⇒P) [from 3, others]^{2}

Thus, everything that is logically possible is necessarily the case:

Thus, 5. For all propositions, that a proposition is logically possible logically implies that it is necessarily the case.

⊢ (∀P)(◊P⇒□P) [from 3, 4, HS]

This is called modal collapse. The acceptance of that all possibilities are necessarily the case.

Thus, the modal fallacy is no longer a fallacy:

Thus, 6. For all propositions (P) and for all propositions (Q), that a proposition (P) is the case, and that that proposition (P) logically implies a proposition (Q), logically implies that that proposition (Q) is necessarily the case.

⊢ (∀P)(∀Q)(P∧(P⇒Q)⇒□Q) [from 3]^{3}

And so we can validly infer from a proposition being the case and that that proposition logically implies some other proposition to that that other proposition is necessarily the case.

### Notes

1Or “cannot be not-true” to avoid relying on monoalethism (and the principle of bivalence) which means that truth bearers only have a single truth value.

For all I see, 3 is false because you get p->Np from premises that are not logically true, such as the existence of a foreknower.

4 seems false as well: instead of ‘logically possible’, it should say just ‘possible’; there may be propositions that are logically possible but mathematically or metaphysically impossible; it doesn’t follow that these are true.

I wrote down the inferences above, so it’s not too difficult to follow. You can challenge any of them you want, but they seem valid to me. I just went through them again. I had to because I wrote this post 3 years ago! I suppose I could spend an hour or so proving it with proof trees, but I rather not.

I don’t understand what you mean by “metaphysical possibility” or “mathematical possibility”. The first seems like nonsense, and the second seems to be the same as logical possibility, which makes the second name redundant.