LOGIC If possibly p, then necessarily possibly p If possibly necessarily p, then necessarily p which are provable in S5 and are characteristic features of that system The relative merits of S4 and S5 as systems of modal logic remain a matter of debate today, and not only among logicians Some philosophers of religion, for instance, have argued that if it is possible that a necessary being (i.e God) exists, then a necessary being does exist This involves a tacit appeal to the second of the S5 theses listed above There are a number of parallels between modal operators and the quantifiers of predicate logic The interdefinability of ‘necessary’ and ‘possible’ parallels the interdefinability of ‘all’ and ‘some’ Just as ‘For all x, Fx’ entails ‘Fa’, so ‘Necessarily p’ entails ‘p’, and just as ‘Fa’ entails ‘For some x, Fx’, so ‘p’ entails ‘possibly p’ There are laws of distribution in modal logic that are the analogues of those in quantification theory: thus it is necessary that p and q if and only if it is both necessary that p and necessary that q, and it is possible that p or q if and only if it is either possible that p or it is possible that q Because of this, if we introduce quantification into modal logic, and use modal operators and quantifiers together, we have a system that resembles double quantification In quantified modal logic it is important to mark the order in which the operators and quantifiers are placed It is easily seen that ‘For all x, x is possibly F’ is not the same as ‘It is possible that for all x, x is F’: in a fair lottery, everyone has a chance of being the winner, but there is no chance that everyone is the winner Likewise we must distinguish between ‘There is something that necessarily Ös’ and ‘Necessarily, there is something that Ös’ It is true that of necessity there is someone than whom no one is more obese However, that person is not necessarily so obese: it is perfectly possible for him to slim and cease to be a champion fatty Sentences in which the modal operator precedes the quantifier (as in the second of each of the two pairs above) were called in the Middle Ages modals de dicto, and sentences in which the quantifier comes earlier (as in the first of each of the two pairs above) were called modals de re These terms have been revived by modern modal logicians to make very similar distinctions Despite the parallels between modal logic and quantification theory there is also an important difference, once we introduce into the system the notion of identity In the technical term introduced by Quine, modal 118