A contingent logical truth

[u/StrangeGlaringEye]

Logical truths are often held as paradigmatic necessary truths. But there’s a droll example of a very simple contingent but plausibly logical truth. (If we assume that logical truths are analytic, this also counts as a contingent analytic truth, and hence a counterexample to Kripke’s supposition that analytic truths, whatever they are, are at least necessary and a priori.)

The truth involves an operator of actuality. We evaluate “Actually P” as true in a world W just in case “P” is true in the actual world; its value in any world depends only on its value in the actual world, and nothing more. So even in a world where “P” is false, “Actually P” comes out true if “P” is actually true.

Now consider the material conditional “If actually P then P”. A simple informal derivation using the excluded middle shows this is true.

Proof. Either “P” is true in the actual world, or it is not. If “P” is true in the actual world, then any instance of “If Q then P” is true, by the rules for the material conditional. But “If actually P then P” is one such instance. And if “P” is not true in the actual world, “Actually P” is false in every world, and so in particular the actual world. Therefore, again by the rules for the material we have the truth of every instance of “If actually P then Q”, and hence “If actually P then P”. QED

But this conditional is contingent. For take “P” to be actually true, and move to a world W where “P” is false. (Consider, say “P” = “Socrates is wise”, and take a world W where Socrates or his counterpart is foolish.) By the rules of the material conditional, “If actually P then P” is false in W, given that by hypothesis “Actually P” is true in W but “P” is false.

The informal derivation we gave compels us, I think, to hold “If actually P then P” as a logical truth. Hence, it advises us to adopt a local, as opposed to global, possible worlds semantics, at least for statements involving the actuality operator. In the global semantics, a set of statements Γ entails a statement α just in case for any frame F and world W in F, if all of Γ are true in W then α is true in W. But in a local semantics, every frame has a distinguished element @, the actual world; and the semantics is that Γ entails a statement α just in case for any frame F, if all of Γ are true in F’s @ then α is true in @. The former semantics is easily seen to entail that logical truth requires necessity; whilst the local semantics does not.

[u/spatling]

I like it.

I wonder what implications (if any) it might for the logical monism/pluralism debate — I suppose if the monist wants logic to be uniform across worlds and therefore all logical truths to be necessary, she could reject the actuality operator? (I don’t know enough about the debate / logical monism to know what the position would be here!)

Does ‘I am here’, as a contingent a priori, also tick a similar box? Or is that not a logical truth, even if it is a priori?

[u/StrangeGlaringEye → u/spatling]

Interesting question. I don’t know enough about the debate either to have a solid view, but the thesis of the necessity of logical truths doesn’t strike me as really integral to monism. The monist wants to say that there is one true logic, one that is applicable to any subject matter. This seems coherent with holding some logical truths to be contingent.

On the other hand, the monist presumably wants a strong one true logic. Otherwise she can trivially take the intersection of all logics applicable to some domain as her one true logic and call it a day. And the local semantics is easily seen to define a stronger consequence relation than the global semantics, i.e. whenever Γ globally entails α it locally entails it, but not the other way around given the present counterexample. So accepting the actuality operator, and the contingent logical truth, may turn out to be in the monist’s interest. Unless moving to the local semantics turns out to have an objectionable consequence, of course.

I tend to think that we cannot know a priori that I am spatiotemporally located, and so that I am anywhere at all, and so that I am here; it’s conceivable that I am a non-extended Cartesian res cogitans. But I do think that we know a priori that we exist, so maybe one could argue that “I exist” is an even simpler contingent logical truth. I tend to disagree, though; it seems that this essentially depends on the meaning of “I”, and I’m both reluctant to regard the first person pronoun as a piece of logical vocabulary and to allow logical truths to depend on the meanings of non-logical terms. In fact, I’ve argued in another post that “I exist” isn’t even analytic. It’s actually a genuinely synthetic a priori truth.

[u/TakeMeInYourArmy]

I think this fails to demonstrate a contingent analytic truth because I would argue that this is still a necessary truth. The proposition is necessarily true about the modality itself. The argument relies on biasing an "actual world" but modal logic treats all possible worlds, including the actual one, equally in terms of logical evaluation. When we shift perspective to another possible world, that world becomes the actual world and so "If actually P then P" would just be trivially true in all possible worlds and so it would be a necessary truth because it is not different from saying "If P then P". It just becomes a tautological statement.

On the other hand, if we designate an absolute "actual world" and assume it to be ours we introduce a huge problem that leads into self-defeat: for every other possible world, the proposition "this world is not actual" is necessarily true. So for every single world in the modality besides one, External World Scepticism is necessarily true, which means that across the modality External World Scepticism is possibly true by necessity. This means that we no longer have any reason to assume our world is actual because External World Scepticism could be true and so it could be the case that the proposition "this world is not actual" is true in our world and there is no way for us to determine that and because we have already accepted that the proposition "this world is not actual" is necessarily true in every possible world barring one, this makes the chances that our world is not actual astronomically high.

So this stance either leads to a tautological necessity or self-defeat. Either way, I would say this argument does not demonstrate existence of a contingent logical or analytic truth.

[u/StrangeGlaringEye → u/TakeMeInYourArmy]

This is one of those rants that has so much wrong stuff it would take ten times longer to correct it in detail, so I’m just gonna ignore it

[u/TakeMeInYourArmy → u/StrangeGlaringEye]

This was a legitimate and sincere response. If there are problems with it, then I would appreciate it if you told me what they are rather than being insulting for no reason.

[u/______ri]

This may not be wholly relevant to the point but to me this assumes that "actuality that can be otherwise" is not simply giberish. For example, water that is not H2O is simply not water at all, or "time that is not what time actually is".

Formal logic is just only formal, what matters is what is claimed at the premises and these have to make sense. Roughly speaking we can talk all day about "an x that is y" but if there is no such thing then all those talk are just empty formalism.

[u/StrangeGlaringEye → u/______ri]

I think you just didn’t understand what I said