In this paper we investigate some families of decision problems associated with a restricted class of Post canonical forms, specifically, those defined over one-letter alphabets whose productions have single premises and contain only one variable. For brevity sake, we call any such form an RPCF (Restricted Post Canonical Form). Constructive proofs are given which show, for any prescribed nonrecursive r.e. many-one degree of unsolvability D, the existence of an RPCF whose word problem is of degree D and an RPCF with (...) axiom whose decision problem is also of degree D. Finally, we show that both of these results are best possible in that they do not hold for one-one degrees. (shrink)

A constructive proof is given which shows that every nonrecursive r.e. many-one degree is represented by the family of decision problems for partial implicational propositional calculi whose well-formed formulas contain at most two distinct variable symbols.

A new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀ x∀ yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

David Hunt has recently developed a new strategy, called the “dispositional omniscience scenario,” or (OOS), which is designed to defeat theological fatalism by showing the compatibility of divine foreknowledge and human (libertarian) free agency. But I argue that Hunt fails to establish his compatibility claim because (DOS) is based on a defective analysis of dispositional belief that is too weak to sustain any divine foreknowledge of future free actions.

In lieu of an abstract, here is a brief excerpt of the content:EMBODIED HUMAN AGENTS INHABITING A MATERIAL WORLD? CHARLES T. HUGHES Chapman University Orange, California I. /n;troduction HE CONCEPT of a "logically possible world" has roven useful in the investigation of issues within many ranches of philosophy, including the philosophy of religion.1 Since this paper includes an analysis of one "possible worlds" objection to Christian theism, based upon the problem of evil, it will prove useful to preface my discussion (...) with a paragraph which summarizes some of the ideas important to the logic of possible worlds. Each logically possible world contains a complete set of compossible facts, which includes the past, present, and future of its objects, properties, and relations. Those complete sets of compossible facts, each of which describes the way the actual world could have been, have in common the truth of all logically necessary propositions (for example, 2 +2 = 4, "Circles cannot be squared "), and the falsity of all logically impossible propositions (for example, 2 +6 = 7, " Circles can be squared"), but differ in their truth values with respect to at least some logically contingent propositions (for example, "Elephants exist" is true in one or more possible worlds but false in all others, and "Unicorns exist " is true in one or more possible worlds but false in all others). An impossible world includes an affirmation of the falsity of at least one necessary proposition (for example,,.._,(2 +2 = 4)), and/or affirms the truth of at least one logical1 See Alvin Plantinga, The Nature of Necessity (Oxford: Clarendon Press, 1974), chapters IX and X. 389 390 CHARLES T. HUGHES ly impossible proposition (for example, " Logically necessary propositions are false in all possible worlds ").2 The actual world, which is our world, is that complete set of compossible propositions which describe the way things actually are. The actual world, therefore, differs from other possible worlds in that all of its contingent propositions correspond to the past, present, and future of all of the objects, properties, and relations which actually obtain. There is an on-going debate about the nature and ontological status of possible worlds and their objects.3 The purpose of this paper is to analyze and reject a particular "possible worlds" objection to Christian theism, based upon the problem of evil, which is not frequently discussed in philosophical literature. The problem is this : If God must actualize a possible world in order to provide human agents with the initial stage of existence they require to develop freely as rational, moral, and religious beings, who choose their own destinies, why did God not create them as incorporeal agents within an immaterial world which would, by its very nature, eliminate the physical dimension of evil and suffering which plagues the actual world? For it seems, prima facie, that God could have realized his purposes by actualizing an immaterial world, populated with human-like incorporeal agents, thereby avoiding all of the terrible evil and suffering made possible by physicality. Therefore, the existence of the actual physical world, with all its evil and suffering, is evidence against the existence of God.4 In order to show that the previously stated objection from evil could pose a genuine threat to theism, I will first identify some of my important assumptions in this paper, and then outline particular Christian ideas to determine how the " possible worlds " 2 For an argument that some impossible propositions are not self-contradictory, see Linda Zagzebski, " What If the Impossible Had Been Actual," in Michael D. Beatty, ed., Christian Theism aml the Problems of Philosophy (Notre Dame: University of Notre Dame Press, 1990), pp. 165-183. a Compare, for example, the positions of D. M. Armstrong, A Combinatorial Theory of Possibility (Cambridge: Cambridge University Press, 1989) and D. Lewis, On the Plurality of Worlds (Oxford: Blackwell, 1986). 4 Some interesting discussion of this type of problem can be found in Austin Farrer, Love Almighty and Ills Unlimited (London: Collins, 1962), pp. 69-76. EMBODIED HUMAN AGENTS 391 objection to Christian theism could arise in the first place. After that, I will discuss the nature of the " possible worlds " objection and how, in principle, it might be rebutted by theists who possess a great amount of... (shrink)