Modal logic: place of the operator in a formalization

Question

Consider the following sentence:

1) Nothing can be cause of itself.

What would be the correct formalization? $\forall x \Box \neg Cxx$, or rather $\Box\forall x \neg Cxx$? And what would be the reason?

Answer 1

Here we are in the actual world. Naked, unmodalized, quantifiers run over things that exist here, making claims about things in the actual world.

So $\forall x \Box Fx$ says everything that exists here is such that it is necessarily $F$ -- it is $F$ here and $F$ in every world.

While $\Box\forall x Fx$ says that necessarily everything is $F$ -- i.e. in every possible world everything (not just the things which also exist here in the actual world) is $F$.

(Example: Some nominalists seem to think that there could be abstract objects, but believe that none in fact exist in the actual world. So, crudely with "$F$" meaning "concrete" they believe the first claim -- everything that actually exists is concrete and is concrete in every other world in which it exists (it is essentially concrete). But they don't believe that necessarily everything is concrete.)

Similarly, $\forall x \Box \neg Cxx$ is true if everything here is such that it necessarily doesn't cause itself.

By contrast, $\Box\forall x \neg Cxx$ says that in any world at all, you won't find things which cause themselves.

The second inituitively seems stronger: for it seems we can wonder about whether there can be self-causing objects (call them "gods" if you like) which don't exist in this world but do in other worlds. They aren't ruled out by the first claim, but are by the second.

Which way should we interpret "Nothing can be cause of itself"? Is this saying that, of anything that actually exists, it is such as to be (necessarily) non-self-causing? Or is it saying that that the possibility of self-causers is ruled out not just here but across all possible worlds? Only context, and the use to which the claim is made, will tell, I suppose.

Answer 2

The difference between the two statements is the distinction between necessity de re and de dicto. A discussion is in Chapter 4 of Melvin Fitting's First-Order Modal Logic.

Do you want to distinguish between these two?

• $\Box\forall x\neg Cxx$: It is necessarily true that nothing is the cause of itself (de dicto, the claim is necessarily true)

• $\forall x\Box\neg Cxx$: All things have the essential property of not being the cause of themselves (de re, all things have a necessary property)

If you want to, then you pick one of the two sentences. Otherwise the sentences are interchangeable.

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The question then naturally arises of what distinction may be drawn between the two sentences.

@PeterSmith opens his excellent post with a clear statement of an important assumption: that "quantifiers run over things that exist here." This is known as the varying domains, or actualist quantification assumption. The constant domain or possibilist quantification assumption, on the other hand, holds that quantifiers run over things that exist either here or elsewhere.

Suppose there is a thing $t$ to be found in some world other than this, but accessible from this. Further suppose that $Ctt$. If we claim

$$ \forall x \Box \neg Cxx $$

under the actualist quantification assumption, the truth of $Ctt$ is of no consequence, because $x$ does not take the value $t$. Under the possibilist quantification assumption, however, $\forall x \Box \neg Cxx$ is false.

Contrast this with $\Box \forall x \neg Cxx$. Suppose we still have $t$ such that $Ctt$ in some accessible world. Then $\Box \forall x \neg Cxx$ is false for both actualists and possibilists.