I spoke with Dr. Alex Malpass a bit about his debate with Dr. William Lane Craig (WLC) and the charge of a modal scope fallacy. I am going to naively attempt to explain the argument that Dr. Malpass and WLC were having in regards to their debate based upon what he has told me:

Given:

1. ∀xφ(P(x))

2. φ∀x(P(x))

This is recognized by both Dr. Malpass and WLC as not a universally valid inference that is not truth preserving from 1 to 2.

(1.) is read as something like: for all of x + some modal operator + some statement about x.

∀x = for all of x

φ as some model operator like “possible” or “necessary”

P(x) some sentence about x

Example: “∀p□(Ksp→Bsp) would be for all of p it is necessary that if s knows p then s believes p.”

(2.) is read as something like: it is necessary for all of x that + some model operator like “possible” or “necessary”+ some sentence about x.

Example: □∀p(Ksp→Bsp) would be: it is necessary for all of p that if s knows p that s believes p.

1 to 2 is not truth preserving as that is a modal scope fallacy. 2 does not hold true as there are knowledge which do not require belief for a necessary precondition for knowledge and there is no metaphysical necessity for it to be the case that knowledge requires belief so changing the placement of the modal operator from 1 to 2 does not preserve the truth of the universal inference from 1.

Malpass argued to WLC that given:

3. ∀xF(Count(x))

4. F∀x(Count(x))

Is truth preserving and that WLC had not shown it to not be truth preserving in the simple future (F) case while WLC was arguing that:

5. ∀xFP(Count(x))

6. FP∀x(Count(x))

is not truth preserving in the future perfect (FP) case.

Malpass argues that WLC has not shown 3 and 4 to be invalid. He argues while it is an invalid inference (1 and 2) the truth preservation still can hold as you can have a “truth-preserving instance of an invalid inference”. He argues that 5 and 6 are not truth preserving in the future perfect case, but WLC didn’t establish that 3 and 4 are not truth preserving. WLC would have to actually show 3 and 4 are not truth preserving rather than just assert 1 and 2 as a modal scope fallacy as Malpass acknowledges the rule itself is invaild, but still truth preserving in the future case. The difference here is:

The future perfect tense is in the form of S + will/shall + have + past participle such as “I will have counted” or a combination of “it will be” and “it was”, while the simple future tense is in the form of S + am/is/are/will/shall + verb + some statement such as “I will run to the store tomorrow” or “I am going to go to the store”.

In the context of Dr. Malpass and WLC’s debate dealing with countable infinities (a set that has a 1 to 1 correspondence to the naturals (Bijection to N)).

“I will count n” (future)

“I will have counted n (future perfect)” or “It will be that it was that I counted n” (future perfect)

Note: “I will count” is a potential infinity, while “It will be that it was that I counted n” is an actual infinity which is the crux of the argument of comparing trying to compare a potential future infinity to an actual past infinity in the discussion between WLC and Malpass.

“I will necessarily count n” and “It is necessary that I count n” are not the same thing just like “If S knows p then it is necessary that p is true” (Ksp → □p) is not the same thing as saying “It is necessary that if S knows p that p is true”(□Ksp → p).

But putting modality aside let’s assume in the future perfect tense (It will be that it was that I counted n) that you start to count the natural numbers (N) and never stop counting. For every n ∈ N (number as an element of N) at some point you will have counted n, but you will never count all of N. You will never have a time when you complete the counting and therefore it is true that you will never have counted every number.

However, in the simple future tense (I will count n) there exists a time where you will count n and in the present (Now, at the start of the count) you will count all the numbers.

So with (3) have “for all of x, it will be that (I count x)” and (4) you have “it will be for all of x (I count x) and both are true in the simple future tense and therefore truth preserving.

With (5) you have “for all of x, it will be that it was that (I counted x) and (6) you have “it will be that it was that for all of x (I counted x). This is not truth preserving as it is true that for every x that you will have counted it, but not true that you will eventually count every x just like you will never count all of n ∈ N.

Malpass argues that WLC is right in arguing that given 1 and 2 that the future perfect case here is not truth preserving in 5 and 6…but that WLC fails to recognize that it is truth preserving in the simple future case of 3 and 4 and perhaps even fails to see the subtle difference between them.

(Thanks to Dr. Josh Rasmussen and Dr. Alex Malpass for giving this a cursory look over for any glaring mistakes)

William PiiI do agree that some operators can commute under pleasant circumstances. For example, let A and B be 3×3-matrices. For arbitrary A and B, it is quite rare that AB=BA, but ABx=BAx, whenever x is in the kernel of (AB-BA). However, to use this equality, one has to show x is in the specified kernel. Some college algebra books have called such a situation a “conditional equality” due to the necessary condition.

I like the use of the natural numbers ℕ here, as it highlights future perfect tense. Indeed, one definition I have seen for a set to be “finite” is to be in bijective correspondence with a bounded subset of ℕ. Consequently, the counting must be completed, yielding the perfect tense.

Countably infinite, on the other hand, is to correspond bijectively to ℕ itself, which is unbounded. Thus, the enumeration would never reach a termination state, i.e. the perfect tense.

Considering the future tense, “∀xF(Count(x))” would be “for all x, it will be the case that x will be counted”. Also, “F∀x(Count(x))” would be “it will be the case that for all x, x is counted”. If I am understanding F operator correctly, ∀xF(Count(x)) is true as for each distinct x, there is a future time where the enumeration will reach x. On the other hand, F∀x(Count(x)) would be true if there is a future time when ∀x(Count(x)) is true, i.e. when the enumeration has reached all x. I think F∀x(Count(x)) is false since there is no fixed point in time when ∀x(Count(x)) is true.

I am amused that this philosophical issue also an issue with commuting operators.