James Cain

The Doctrine of the Trinity and the Logic of Relative Identity

The doctrine of the Trinity says that there is just one God and three distinct divine persons, each of whom is God. This would seem to imply that there are three divine persons, each a different person the other persons but the same God as the other persons. If we accept what I believe is the most popular account of identity current among logicians then we must hold that this apparent consequence is contradictory. We see this as follows (it will suffice to consider just the relation of Father and Son): logicians generally treat relativized identity expressions of the form ‘is the same A as’ (here ‘ A ’ stands in for a term which relativizes the identity) as being analysable in terms of absolute (or unrelativized) identity according to the following equivalence schema, (E): (E) a is the same A as b if and only if a is identical to b and a is an A and b is an A. The view under consideration affirms the following three sentences: (1) The Father and the Son are persons. (2) The Father is not the same person as the Son. (3) The Father is the same God as the Son.

On the weak Kleene scheme in Kripke's theory of truth

It is well known that the following features hold of AR + Tunder the strong Kleene scheme, regardless of the way the language is Godel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 21? fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n I A(n) is true in the minimal fixed point}, where A(x) is a formula of AR + T) are precisely the HI1 sets. 4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the A sets. 5. The closure ordinal for Kripkes construction of the minimal fixed point is cl 1 . In contrast, we show that under the weak Kleene scheme, depending on the way the Godel numbering is

Free will and the problem of evil

According to the free-will defence, the exercise of free will by creatures is of such value that God is willing to allow the existence of evil which comes from the misuse of free will. A well-known objection holds that the exercise of free will is compatible with determinism and thus, if God exists, God could have predetermined exactly how the will would be exercised; God could even have predetermined that free will would be exercised sinlessly. Thus, it is held, the free-will defence cannot be used as a partial account of why God should have allowed evil to exist. I investigate this objection using Kripkes apparatus for treating modalities and natural kinds to explore the nature of the incompatibilism required by the free-will defence. I show why the objection fails even if the standard arguments for compatibilism are acceptable. This is because the modality involved in the incompatibilism needed by the free-will defence differs from the modality involved in the compatibilism that is supported by standard compatibilist arguments. Finally, an argument is sketched for a variety of incompatibilism of the kind needed by the free-will defence. The topic of free will along with the attendant problem of compatibilism looms large in the philosophy of religion, as it does in metaphysics in general. Yet there are special features of the compatibilist question that must be properly understood to obtain a clear view of the issues when addressing the problem of evil and the free-will defence. These special features arise because there are closely related, but logically distinct, notions of compatibility, and the notion of compatibility that is especially pertinent with regard to the free-will defence is largely ignored in the general literature on the free-will question. As we shall see, standard philosophical arguments that have some force regarding one form of compatibilism may have little bearing on another. Consequently, certain con- siderations concerning compatibilism that are standardly discussed outside the philosophy of religion cannot simply be imported into discussions of the problem of evil. For example, one motivation for developing a compatibilist position is to address a sceptical worry that if science were to show that determinism is true, that might require us to give up the claim that we have free will and moral

The Theory of Computability Developed in Terms of Satisfaction

The notion of computability is developed through the study of the behavior of a set of languages interpreted over the natural numbers which con- tain their own fully defined satisfaction predicate and whose only other vocab- ulary is limited to 0 ,i ndividual variables, the successor function, the identity relation and operators for disjunction, conjunction, and existential quantifica- tion.

Arithmetic with Satisfaction

A language in which we can express arithmetic and which contains its own satisfaction predicate (in the style of Kripkes theory of truth) can be formulated using just two nonlogical primitives: � (the successor function) and Sat (a satisfaction predicate).