Clear Language, Clear Mind

“World A: Contains nothing but a single being.
World B: Contains nothing but a single being, that is a different being than the one in World A.

Do you have any objection to either of these worlds? Each seems entirely possible.

Unless you can come up with some sort of objection to THEM, we’ve got a proof. They contain nothing in common, so there cannot be any necessary being.” Smullyan-esque, source.

Where “world” means possible world. This is an interesting counter-example because it goes through even if people find the idea of an empty world contradictory/meaningless. The only part that I find disagreeable is the unnecessary (heh) “cannot” in the last sentence but I am extremely careful with modalities.

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modal-logic-formalization

Intro and types

I’ve been paying very close attention as of late to a special type of discourse: Namely, about what is possible and what is impossible. This study has led me to be very careful about my language use when speaking of such things because there are multiple types of possibilities: Logical, epistemic, physical, metaphysical, practical, technological etc. I have even created a modified version of modal logic that can handle multiple types of possibilities.ⁱ Logical possibility we ought to call L-possible, epistemic possibility we ought to call E-possible etc.

Modal fallacy

And before that I discovered the modal fallacy, which occurs when people confuse the scope of the possibility used. It may be about a single proposition or an entire implication.ⁱⁱ

Versions: Hypothetical and absolute

And then I discovered that even a single type (and pay close attention to the words used) of possibility is used in multiple ways. Let’s call these versions. There is the absolute version and then there is the hypothetical version. I did not invest these terms; Liebniz did.ⁱⁱⁱ

Since I have already written of the aforementioned let me skip them and proceed on defining absolute and hypothetical modaly. Absolute modality is the one I’ve always been talking about and hypothetical is the one that others often talk about, which confuses matters a lot, and ultimately ends up wasting a lot of time.^iv

Definitions

But that is not clear enough, so let me define the first. A proposition is absolutely necessary iff the negation is a contradiction (which has the form [p∧¬p]). A hypothetical impossibility is a proposition which if added to a set of propositions would result in a contradiction. This is the kind of impossibility that we’re talking about when making reductio arguments: “If something, then some contradiction, but that it impossible, so something can’t be true.” Yes, it can in the absolute sense. We ought not to confuse them.

In a later article I attacked a hypothetical impossibility for being an absolute impossibility.^v

The value of the hypothetical impossibility term?

I ask now what value we have of this term. What need do we get covered by accepting this term into our collection of words? None but confusions as far as I can tell. We might as well stop called the hypothetical impossibility for an impossibility at all, and then while we’re at it, we should be very careful in our usage of the necessarily-operator when writing conditionals, so we don’t commit the modal fallacy. It doesn’t matter if we call it ‘must’, ‘cannot’ ‘has to be’ or something else. We must be very clear in our language about this matter, for if anything is certain (meant non-literally), it is that the plain English language is not at all good enough for handling modalities. Clarity is the way forward.

ihttp://deleet.dk/2009/02/09/flere-slags-muligheder-i-en-s%C3%A6tning/

iihttp://deleet.dk/2009/01/07/the-modal-fallacy/

iiiIt is discussed here: http://maverickphilosopher.powerblogs.com/posts/1159490720.shtml but originally from here: http://www.class.uidaho.edu/mickelsen/texts/Leibniz%20-%20Correspondence.htm

ivhttp://www.freeratio.org/showthread.php?t=264125

vhttp://deleet.dk/2009/04/07/does-a-sound-lpoe-establish-that-god-is-impossible/

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:

Argument a

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:

1′. P

2′. P→ □Q

3′. □Q

4′. □Q→S

5′. S

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.

Argument b

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.

References

An awesome source by professor Norman Swartz (Indiana University Ph.D., 1971 (History and Philosophy of Science)):

http://www.sfu.ca/philosophy/swartz/modal_fallacy.htm

Internet Encyclopedia of Philosophy about the modal fallacy in divine foreknowledge.

http://www.iep.utm.edu/f/foreknow.htm#section6

The Fallacy Files about modal fallacies in general. There is a subpage about the specific fallacy I have in mind.

http://www.fallacyfiles.org/modalfal.html

Stanford Encyclopedia of Philosophy about modal logic. They mention the fallacy just before the section on deontic logic.

http://plato.stanford.edu/entries/logic-modal/