Infallible knowledge, the modal fallacy and modal collapse

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.

2This follows like this: I. □P⇔¬◊¬P (definition of ◊). II. Thus, P⇒¬◊¬P. [I, 3, Equi., HS] Thus, ◊¬P⇒¬P. [II, CS, DN] Thus, III. ◊P⇒P. [II, Substitution of ¬P for P]

3This follow like this: P∧(P⇒Q)⇒Q is just MP, and from 3 it follows that any proposition that is the case is necessarily the case.