124.221 Metaphysics Propositional Modal Logic Modal logic is the logic of necessity, possibility and other related notions. We introduce two new sentential operators ◊ and ☐, to be understood as ‘Possibly’ and ‘Necessarily’ respectively. (Sentential operators are devices that take sentences to form new sentences: thus, in adding ◊ and ☐ to standard propositional logic we can form such sentences as ☐P, ◊(P ∧ Q), P ∧ ☐Q, ☐◊P.) We use these to formalize a number of locutions of English. If P is the sentence ‘Pigs can fly’, then ◊P formalizes such English sentences as ‘Pigs might have flown’ ‘Pigs might have been able to fly’ ‘It could have been the case that pigs fly’, ‘It’s possible for pigs to have flown’. Similarly if Q is the sentence ‘1+1=2’ then ☐Q formalizes such sentences as ‘1+1 has to be 2’ ‘1+1 is necessarily 2’ ‘1+1 couldn’t have failed to be 2’. In understanding possibility and necessity we employ the device of possible worlds. A possible world is a way that the world, i.e. everything, might have been. (‘Possible world’ is thus a bit of a misnomer; ‘possible universe’ would have been better.) There is a great deal of controversy over what possible worlds are: whether they really exist, or whether they are just things that we somehow construct in the imagination; however we do not need to get into this here. (Those interested might have a look at David Lewis’s book On the Plurality of Worlds; he thinks that they really exist.) What is important for our purposes is that they provide a way of understanding talk of possibility and necessity. Roughly, sentence that is necessarily true is one that is true in every possible world, and a sentence that is possibly true is one that is true in at least one possible world. Before making this idea more precise, let us pause to mention a use of possibility talk which cannot be accommodated in the possible worlds framework; and some other uses which, despite initial appearances to the contrary, can. First, the use which cannot be accommodated. Suppose I say to you, out of the blue, ‘Is 329 x 747 equal to 254,763?’ Unless you are remarkably good at mental arithmetic, you are likely to say something like ‘I don’t know; it’s possible that it is, but I’d have to work it out’. However, as you’ll find if you do work it out, 329 x 747 is in fact equal to 245,763. But if that is true in this world, then it is true in any possible world. So when you said it was possible that it was equal to 254,763, your utterance cannot be understood in possible worlds terms. In fact this kind of possibility, sometimes called ‘epistemic possibility’ since it concerns what one does not know is not the case, is not at all easy to analyse, and there is no agreed account of it. We now turn to a set of possibility claims that can be understood in the possible worlds framework, despite initial appearances to the contrary. Suppose we say that it’s not possible for someone to travel faster than light, or that, given their anatomy, it’s not possible for pigs to fly, or that we can’t possibly go to the cinema at 1 o’clock on Wednesday since we have an important metaphysics class. In each of these cases we are not asserting that there is no possible world in which someone goes faster than light, or that pigs fly, or in which we do go to the cinema at 1 o’clock on Wednesday. But we can still understand these claims in possible world terms; we2simply need to restrict the set of possible worlds that we are considering. Thus, in the first case, there is no possible world in which the actual laws of physics hold in which someone travels faster than light; in the second, there is no possible world in which the actual laws of physics hold, and pigs have their actual anatomy, and pigs fly; in the third, there is no possible world in which my obligation to go to classes is as it actually is, and I live up to those obligations, and I go to the cinema at 1 o’clock on Wednesday. (This last example might need some fine tuning; but I trust that the idea is clear.) We now turn to ask how this talk of possible worlds might be formalized. What we have said so far might suggest that we can think of the modal operators as quantifiers that range over possible worlds: ☐P is true just in case ∀w (P is true in w); ◊P is true just in case ∃w (P is true in w). Unfortunately things are not quite so simple. In understanding the modal operators in this way we have prevented ourselves from saying various things that we might well want to say. The point becomes clear once we come to iterated (repeated) modalities. Suppose we want say that whilst a certain state of affairs isn’t possible, it could have been possible. In short: it is possible that it is possible. That seems to make some sense: it seems plausible that ◊◊P is weaker that ◊P. How can we spell this out in terms of possible world? The standard way is this: rather than just saying that ◊P is true just in case there is a world in which P is true, say that it is true just in case there a world which we can get to from this world, in which P is true. Then we can make sense of the idea that it is possible that it is possible that P without it being possible that P as follows: whilst from this possible world we can’t get to another world in which P, we can get to a world in which possibly P; and from that world we can get to another world in which P. What we need for this then, is the idea of there being some worlds that you can get to directly from this world, and others that you cannot; or, in the jargon, an accessibility relation. Certain worlds are accessible from others. We can then state all this a little more generally and formally. ☐P is true at a world w1 iff every world which is accessible from w1 is a world in which P is true; similarly ◊P is true at w1 just in case there exists a world w2 which is accessible from w1 in