Title:

For some proposition and so many possible worlds

In this thesis, I deal with the notions of a condition holding for some proposition and a proposition being true in a certain number of possible worlds. These notions are called propositional quantifiers and numerical modalizers respectively. In each chapter, I attempt to dispose of a system. A system consists of: a language; axioms and rules of inference; and an interpretation. To dispose of a system is to prove its decidability and its consistency and completeness for the given interpretation. I shall, in passing, make applications to de definability, translatability and other topics. In Chapter 1, I consider the system 5SQ. Its language is that of 55 with Q as a fresh unary operator. Its axioms and rules of inference are those for 85 plus the following special axiomschemes for Q: 1) QA::>MA 2) ( 2) Q A ::> L (A::> B) V L (A ::> B) 3) L (A == B) ::> (Q A ::> Q B) 4) Q A ::> L Q A. 'QA' is interpreted as 'A is true in exactly one possible world.' I dispose of the system by showing that every formula in it is equivalent to one in normal form. In Chapter 2 I consider the system S5n (n for numerical modalizer). Its language is that of S5 but with the unary operators Qk for each nonnegative integer k. Its axioms and rules are those of S5 plus the following special axiomschemes for Qk: 1) Qk A>~Q1 A, 1˂K 2) Qk A ≡ Vki≡oQi (A˄ B) ˄ QkI (A ˄ ~ B) 3) L (A ≡ B) ˂ (Qk A˂ Qk B) 4) Qk A ˂ L Qk A 5) Qo A ≡ L ~ A, 1 ≥ 0, k ≥ 1 ‘Qk A’ is interpreted as 'A is true in exactly k possible worlds.' I dispose of this system by generalising the normal forms of S5Q. In the chapters 35, I consider three systems which result from adding propositional quantifiers to S5. The first two systems, S5╥+ and S5╥, contain the usual axioms and rules for quantifiers. The first contains, in addition, the axiomscheme G = (Ǝ P) (P ˄ (R) (R > L (P > R))). The last, S5╥, results from S5╥ by restricting the Scheme of Specification, viz., (P) A (P) > A (B), B free for P in A (P), to formulas B of the propositional calculus. To interpret these systems we must specify which propositions the variable P ranges over. For St╥, we merely require that if p and q are propositions, then (not p) and (p or q) are also propositions. For S5╥+, we also require that each possible world be describable i.e. that there be a proposition which is true in that world alone. And for S5╥, we require not that each possible world be describable but that there be a proposition which is true in just those possible worlds which are describable. Again, we dispose of the systems by normal forms. This requires that we eliminate quantifiers and nested occurrences of L by adding new symbols to the language. For S5╥+, the operators Qk suffice. For S5╥, the operators Qk suffice. For S5╥, we also require the constant g and a fresh unary operator N. For S5╥, even greater additions are required. In the last two chapters, 6 and 7, I turn to systems which have essentially the same language as S5╥. However, ‘Qk A' is now interpreted as 'A is true in exactly k possible worlds accessible from the given world.' Different conditions on R, the relation of accessibility, lead to different axioms. In chapter 6 I consider the conditions of reflexivity, symmetry and transitivity, and in Chapter 7 the conditions of being a partial, convergent, total or dense order. I prove consistency and completeness by the method of maximally consistent systems. The method can yield decidability results, but I do not go into the matter. I have, as a rule, not given acknowledgements for wellestablished results or terminology. The main references are at the end of each chapter. Fuller references are in the bibliography.
