A Modal-Ontological Argument and Leibniz’s View of Possible Worlds Nino B. Cocchiarella April 22, 2018 Abstract We critically discuss an ontological argument that purports to prove not only that God, or a God-like being, exists, but in addition that God’s existence is necessary. This requires turning to a modal logic, S5 in par- ticular, in which the argument is presented. We explain why the argument fails. We then attempt a second version in which one of its premises is strengthened. That attempt also fails because of its use of the Carnap- Barcan formula in a context in which that formula is not valid. A third is presented as well using the proper name ‘God’as a singular term, but it too fails for the same reason, though in a later section we show how this last argument can be validated under a re-interpretation of the quanti…ers of the background logic. In our later sections, we explain what is wrong with the original …rst premise as a representation of what Leibniz meant by the consistency of God’s existence, speci…cally as God’s existence in a possible world. Possible worlds exist only as ideas in God’smind, and the consistency of God’s existence cannot be God’s existence in a possi- ble world. Realism regarding possible worlds must be rejected. Only our world is real, the result of an ontological act of creation. We also explain in a related matter why according to Leibniz, Boethius, Aquinas and other medieval philosophers, God’somniscience does not imply fatalism. The classical ontological arguments of Anselm and Descartes were attempts to prove that God exists. Leibniz went a step further and tried to prove not only that God exists, but also that God exists necessarily. An argument for such a conclusion would need to be framed within a modal logic. Leibniz, unfortunately did not have the tools of modern logic, including especially those of modern modal logic. Today, we have a number of di¤erent modal logics, but we will restrict our discussion here to …rst-order S5 with identity. We will …rst consider a modal argument proposed by Charles Pigden, who began with a premise of Leibniz’s, namely that God’s existence is consistent, and which Leibniz tried to defend in a separate argument.1 Pigden accepted this premise Copyright c 2018 Nino B. Cocchiarella 1 Pigdon described his argument in a 1997 posting in the (now defunct) online discussion group Russell-l. 1 and represented it in modal logic as God’sexistence being possible, except that instead of using the name ‘God’as a singular term, Pigdon assumed that a God- like being is possible, in symbols ( x)G(x), where the predicate G is read as ‘is God-like’.2 We discuss Pigdon’sargument9 below and explain why it does not work in quite the way it was intended. We also construct two related arguments, one with ‘God’as a singular term; but these arguments fail as well. In section 4 we explain why Pigdon’s…rst premise is fallacious and does not really represent what Leibniz meant by the consistency of God’sexistence. In section 5, we give a di¤erent interpretation of the quanti…ers so as to cover abstract and divine entities as transcendent objects, and show how an argument with ‘God’ as a singular term can be validated. Finally, we also examine a fatalist argument based on a fallacy similar to the fallacy of representing Leibniz’s view of the consistency of God’sexistence as God’sexistence in a possible world. 1 A Modal-Ontological Argument The modal argument Pigdon proposed was not just to prove that God exists, but in addition to prove that God necessarily exist. As just expressed, the claim, at least in modal logic as we will see, is really a conjunction of two sub-claims, namely (1) that God exists, and (2) that God’sexistence is necessary. In English we just say ‘God necessarily exists’or ‘Necessarily God exists’,and that seems to su¢ ce for both parts of the claim. But that is because we make an implicit assumption in how we use a name like ‘God’,speci…cally that the name denotes the same entity through di¤erent times and di¤erent possible worlds. That implicit assumption requires a certain formula in modal logic stipulating that the name is a “rigid designator”, which to add as a premise, would simply beg the question at issue.3 Pigdon avoids this issue by not using the name ‘God’, but by using the predicate G for being God-like. This change, however, may mislead us in how the conclusion of the argument is to be formally represented. As we explain below, Pigdon’s original argument is valid in …rst-order S5, but it does not prove the necessary existence of a God-like being; that is, again to emphasize the implicit conjunction involved, it does not prove (1) the existence of a God-like being and (2) that that being’sexistence is necessary. It was this sort of conclusion that was what Leibniz wanted to prove, and which was what Pigdon claimed he was attempting to defend. As indicated, the logical framework that we start with is standard …rst- order modal logic S5 with identity. Semantically, the …rst-order quanti…ers are assumed to quantify only over the objects that exist in a given possible world (of a system of possible worlds). We also assume that di¤erent possible worlds will in general consist of di¤erent objects that exist in those worlds, i.e., there is no one …xed domain of objects. Pigden’s modal ontological argument is as 2 Pigdon explained his premise as representing Leibniz’sview in a later posting of 11-22-97 to the Russell-l group. 3 The stipulation would be ( x)(God = x), which in modal logic clearly implies that God exists. 9 2 follows: (1) Possibly there exists a God-like thing: ( x)G(x). (2) Necessarily, if there exists a God-like thing,9 then necessarily there exists a God-like thing, in symbols [( x)G(x) ( x)G(x)]: 9 ! 9 Therefore, necessarily there exists a God-like thing, i.e., conclusion: ( x)G(x).4 9 We understand Pigden’s …rst premise here, i.e., premise (1), as stating in formal symbolic terms (of the object language) that a God-like being is possible, or in more metaphysical terms that a God-like being exists in some possible world. This is how Pigdon represented Leibniz’s view that the existence of God is consistent. We will take up a philosophical discussion of this claim— or really the related claim that the existence of God is possible— later in section 5. Premise (2) has a di¤erent problematic, which we will take up in section 2 below. For now, let us note that the conclusion does not show that there exists (in our world) a necessary God-like being, i.e., ( x) G(x), where the existential 9 quanti…er comes before and not after. Nor does the conclusion exclude the possibility of polytheism, i.e., the existence of many God-like beings in any given possible world. And it allows that any God-like being that exists in one possible world need not also exist in any other possible world, and hence that none need necessarily exist. These are all de…ciencies of the argument— at least as a reconstruction of Leibniz’sview. Note that even if we modify Pigden’sargument by replacing the clause “there exists a God-like thing” in (1) and (2) by “there exists exactly one God-like thing”, so as to get monotheism and exclude polytheism, all that follows is that in each possible world there exists exactly one God-like thing— where the God-like thing that exists in one world need not necessarily be the same as the God-like thing that exists in any other possible world, and hence that none need necessarily exist. Now let us be clear: what Leibniz claimed, and what the argument does not show, is that there exists a God-like being that necessarily exists, i.e., in symbols ( x)[G(x) ( y)(x = y)], which requires a free logic, i.e., a logic free of existential9 presuppositions.^ 9 5 The point of requiring ( y)(x = y) (and hence 9 4 Proof: because [( x)G(x) ( x)G(x)] 9 ! 9 [( x)G(x) ( x)G(x)] is provable in S5, then by premises (1) and (2) and modus ponens,! 9 ( x)!G(x) follows.9 But because ( x)G(x) ( x)G(x) is also provable in S5 9 9 ! 9 (and in fact is an instance of the distinctive axiom schema of S5), the conclusion ( x)G(x) then follows. 9 5 In free logic ( y)(x = y) is not provable with y free. In standard …rst-order modal logic, on the other hand,9 ( y)(x = y) is provable, and therefore by modal generalization, so is 9 ( y)(x = y), and consequently, by exstential generalization, so is ( x)( y)(x = y). In “free”9 logic neither universal instantiation nor existential generalization9 are9 valid without quali…cation, speci…cally the quali…cation that the singular term we are instantiating to, or generalizing from, is a “rigid designator”. 3 a free logic) as part of the condition that is necessary is that di¤erent worlds will in general consist of di¤erent objects, so that it is logically possible that there is an existent being x in our world that in some possible world does not exist (in that world) but nevertheless falls under the concept G in that world, i.e., G(x) is true in that other world even though x does not exist in that world.