If this world is the only possible world, what does that mean for validity in logical arguments?

Question

Validity is usually defined in terms of possible worlds. More specifically, an argument with premise set S and conclusion C is said to be valid if and only if there is no possible world with all the members of S true but C false. But, what if this world, the actual world, is the only possible world, as I and many others believe? Then all sorts of arguments that are intuitively invalid would be valid. In fact, all arguments with a true conclusion would be valid, since there is no other world where the conclusion is false. So, does this mean that actualism implies that validity is "trivialized" in this way? And have any philosophy books or papers argued for this analysis?

Answer 1

Firstly, I would say that understanding validity in terms of possible worlds is only one way of doing so, and it wouldn't really be correct to say it is the usual way. There are about a dozen different ways of characterising validity: however that is a subject for another question. You are asking about how validity relates to 'actualism', which is the position that the way the world actually is, is the only possible way it could be.

The two don't conflict, or at least not in any simple way. It helps to focus on what we mean by 'possible'. To say of something that it is possible or impossible may be to say one of many different things. E.g., we might say that something is physically possible to mean that its existence or occurrence would not violate any known laws of nature. That something is epistemically possible means that its truth would not contradict anything we know. That something is legally possible (or permissible) means that it does not conflict with any laws or regulations. Some philosophers also make use of the concept of metaphysical possibility, which relates to theories of what is necessarily, as opposed to contingently, true. And we have logical possibility, which is perhaps the weakest kind, and says of something that it does not conflict with our understanding of logic, or more concretely, with some system of logic to which we subscribe. There are also other kinds of possibility.

To hold that the actual world is the only world that exists does not conflict with other kinds of possibility. It would not be inconsistent to form descriptions of worlds that are not actual, as we do in fiction, and to say of such worlds that what holds there does not conflict with logic. Such worlds would then be logically possible, but not actual. It would also not be inconsistent to say of the actual world that some proposition is epistemically possible, meaning that it might be true for all we know. Indeed, one of the typical uses of logic is to examine the space of what is epistemically possible in order to deduce what is actually true.

The upshot for validity is that we can still choose to think of validity as there being no possible world in which the premises of an argument hold and the conclusion does not, but we need to understand that the range of possible worlds extends to all logically possible worlds, and not merely those that exist. In fact, actualists often do refer to possible worlds, because they are a useful device for thinking about logic, but they understand them as sets of descriptions, or as sets of states of affairs, but without ontological commitment.