A Comparison

Abstract

represents adverbs within higher-order modal logic. This paper will show some of the comparative advantages of the former over the latter. Despite these advantages, the extensional formalisation has some fatal shortcomings: the impossibility of handling negation and of representing the functional dependence of the modifier upon the modified entity. I will show how by introducing projection functions the idea of the adverb living on the verb can be better captured. The semantic I will introduce for such adverbial modifiers differs from the traditional ones. When the denotation of the verb on which the adverb depends is empty, the valuation of the whole sentence will be undefined. I will also show how this new valuation scheme is more suitable for expressing negation in natural languages. Finally, I will show how this valuation scheme can be extended to become compatible with the partial logic built up by Ebbingaus. their The purpose of this paper is mainly expository, presenting the relevant logics of Anderson and Belnap in the context of a wider theory encompassing many weaker logics as well as the relevant logics, some modal logics including S4, a fuzzy logic and classical truth-functional logic. The system given here is a formulation of the weak relevant logic DW in the style of Lemmon's version of orthodox logic [1], with two innovations. The first of these is to use two ways of combining assumptions: an extensional way to go with & and v, and an intensional way to go with -> and ~. This apparatus makes it possible to obtain a great range of logics, sustaining very different motivations, by varying the structural rules while keeping the logical rules constant. From such a viewpoint the connectives of these logics can be seen as meaning the same despite the differences regarding what entails what, these differences being fundamentally structural. The second innovation is to introduce the combinators (or at least some combinator-like objects) into the logical syntax as special "bunches of assumptions". These allow many logics to be defined in an extremely natural way within the single deductive system DW; hence the "unified theory" of the title. systems maximally accessible to with in a one-term in logic. MST MSTQ MSTR for multisets with rational multiset multiset x). Multisets in which with infinite multiplicities are This paper presents higher-level sequent-systems for intuitionistic analogues of S5 and S4. As in [1], rules for modal constants involve sequents of level 2, i.e. sequents having collections of ordinary sequents of level 1 on the left and right of the turnstile. Starting from a canonical higher-level sequent formulation of S5, the restriction of sequents of level 2 to those with the single-conclusion property produces S4, without changing anything else. A similar restriction on sequents of level 1 produces Heyting S5, and if this restriction is made on sequents of both level 1 and 2, we obtain Heyting S4. The paper contains a brief discussion of Kripke-style models for the intuitionistic propositional modal logics in question. theories of geometry, from Euclidean and non-Euclidean to semi-Euclidean, enumerative, finite, and differential to semi-Riemannian, Gaussian infinitesimal, non-Legendrian, non-Archimedean and even synergetic geometry, and the much greater formal strength of arithmetic to date, this situation is hardly surprising. My objective in this brief paper will be to clarify the main lines of development in this field in a mini-course style while aiming to help uncover the elements of a theory of geometry comparable to PA or reasons for the lack thereof. To do this, we will first consider Tarski's axiomatizations of Euclidean geometry [1] and Heyting's axiomatizations of projective geometry [2]. trees and forests have been studied by many logicians from various points of view. H. Friedman's theory of tree-embedding, Leeb's theory of jungles, Buchholz's new theory of ordinal notations and Takeuti's theory of ordinal diagrams are considered important examples of such theories. We give close relationships among these theories from the viewpoint of our theory of quasi-ordinal diagrams (which was defined in [3]). These relationships