How to address initial criticisms of modal logic, in particular “I can imagine a world where [necessary truth] is false?”

Question

Whether you ultimately agree with it or not, I feel that a study of modal logic, including David Lewis's (and others') notion of possible words is a valuable facet of a proper understanding of logic and metaphysics. However when I attempt to explain necessary truths (where one cannot imagine a world in which they would be false) I am often met with some form of the following criticism:

I can imagine universes in which the axioms of logic and mathematics don't hold, and who are you to tell me what I can and cannot imagine?

Of course, the premise of a necessary truth is that this isn't actually possible, and therefore the person making this statement is mistaken (assuming the example is a necessary truth). With enough time I am usually able to walk the person I'm talking to through why this is the case, but there is serious resistance on the part of many people being introduced to this concept. I understand where they are coming from, but I don't have a strong rebuttal to it other than "Oh really?" which is, of course, not satisfactory.

I most often see this criticism from people early on in their exposure to Philosophy as a formal discipline, so please target your responses for someone with limited knowledge of the discipline.

Answer 1

Firstly, it's worth drawing out the distinction between what is conceivable and what is possible. It's an open question whether it can be useful to view one as a guide to the other, so make it clear that the ability of the human mind to dream up weird and fantastic scenarios doesn't mean that what they describe is something that is in the realm of possibility. Possible worlds might well include some pretty crazy things, but in order to make the notion of what is possible clear, you need to split it apart from what people can imagine.

Secondly, your example student is having a problem with taking Logic and Mathematics to be systems of paradigmatic necessary truths. This seems quite legitimate for someone new to philosophy, since public exposure to these things is generally purely formal. Some Philosophy of Language can help to unsettle some of the prior intuitions that your students might have already established.

Kripke's examples in Naming and Necessity about a posteriori necessity seem to suggest that there can be many conflated notions being bundled into natural intuitions about necessity. It's necessary that Hesperus is identical to Phosphorus, because the two names are of "the same" planet, and identity is by definition reflexive. But it's not analytic or a priori that 'Hesperus', being the evening star, refers to the same thing as 'Phosphorus', the morning star.

If people are thinking about logic and mathematics purely syntactically, they'll generally stick solely to the latter as their conception of necessity. You can use this as an opportunity to make the Use/Mention distinction. Then you can use some examples of descriptions with identical referents, like Water and H20, and embedding them in a particular theoretical context, in this case, molecular chemistry.

Answer 2

I would say that we can divide criticism to logic as pre-logic and post-logic.

In pre-logic criticisms, people don't quite understand logic and are resistant to it already. Often they have only that stereotypical notion that logic turns a person into a non-sensitive robot. All their contact with logic was superficial and at a distance.

In post-logic criticisms, someone knows logic well enough to get close to its boundaries. Then that person can start to discern where logic can or cannot go, and even deal with that kind of statement that "I can imagine such and such".

Probably the person you're refering to is in the first category, but using ideas that are dealt with by people in the second one.

The approach I try in those cases is usually acknowledge what the person is arguing, but saying that those things can only be really dealt with after you have some good grasp of logic. And to do so, first you have to get within its boundaries, and within its boundaries, necessary truths are... necessary. :)

Besides, when a person argues like that, they are usually trying to make a point by using logic - at least in some sense. They are trying to make you conclude, by the premise of their imagination, that there can be such a universe.

Logic is a wonderful system that we stumbled upon - maybe by inventing it, maybe by discovering it - that for some reason works amazingly well. And I think that is enough for us to give it a good credit, even if afterwards we question its limits.