: We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are well-known to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear; and thus, the whole situation typically becomes much easier to understand. KEY WORDS: Galois connection, closure operation, interior operation, polarity, axiality CLASSIFICATION: Primary: 06A15, 06--01, 06A06 Secondary: 54-01, 54B99, 54H99, 68F05 0. INTRODUCTION Mathematicians are familiar with the following situation: there are two "worlds" and t...

This paper shows that classical logic is inappropriate for hypothetical reasoning and develops an alternative logic for this purpose. The paper focuses on a form of hypothetical reasoning which appears computationally tractable. Specifically, Horn-clause logic is augmented with rules, called embedded implications, which can hypothetically add atomic formulas to a rulebase. By introducing the notion of rulebase independence, we show that these rules can express hypothetical queries which classical logic cannot; and by adopting methods from modal logic, we show these rules to be intuitionistic. In particular, they form a subset of intuitionistic logic having semantic properties similar to those of Horn-clause logic. This report is an expanded version of a paper published in the Proceedings of the Seventh National Conference on Artificial Intelligence, St. Paul, Minnesota, August 21--26 1988, American Association for Artificial Intelligence (AAAI). 1 Introduction Several researchers...

This paper aims to help to elucidate some questions on the duality between the intuitionistic and the paraconsistent paradigms of thought, proposing some new classes of anti-intuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is shown here that anti-intuitionistic logics are paraconsistent, and in particular we develop a first anti-intuitionistic hierarchy starting with Johansson 's dual calculus and ending up with Godel's three-valued dual calculus, showing that no calculus of this hierarchy allows the introduction of an internal implication symbol. Comparing these anti-intuitionistic logics with well-known paraconsistent calculi, we prove that they do not coincide with any of these. On the other hand, by dualizing the hierarchy of the paracomplete (or maximal weakly intuitionistic) many-valued )n## we show that the anti-intuitionistic hierarchy (I )n## obtained from (I )n## does coincide with the hierarchy of the many-valued paraconsistent logics (P )n## . Fundamental properties of our method are investigated, and we also discuss some questions on the duality between the intuitionistic and the paraconsistent paradigms, including the problem of self-duality. We argue that questions of duality quite naturally require refutative systems (which we call elenctic systems) as well as the usual demonstrative systems (which we call deictic systems), and multiple-conclusion logics are used as an appropriate environment to deal with them.

. The paper introduces a notion of context parchment. The notion is illustrated by several examples. It is shown, that every logical context parchment generates a context institution. Morphisms between context parchments are introduced, thus yielding a category of context parchments. The use of universal constructions in the category of context parchments, for modular construction of logics is discussed and illustrated by examples. 1 Introduction Institutions, were introduced to provide an "abstract model theory for specification and programming"---quoting from the title of [8]. The model-theoretic view of logic, advocated by institutions, seems to be very natural in computer science applications, considering the fact, that our main concern is to specify, create, and reason about concrete objects---such as programs or VLSI chips. Context institutions (cf. [13]), enrich the structure of institutions by adding notions such as contexts, and substitutions, retaining at the same time the...

notions, such as `constructive proof', `arbitrary number-theoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifier-free statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 1887--1963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifier-free expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...

. The paper introduces a notion of context parchment. The notion is illustrated by several examples. It is shown, that every logical context parchment generates a context institution. Morphisms between context parchments are introduced, thus yielding a category of context parchments. The use of universal constructions in the category of context parchments, for modular construction of logics is discussed and illustrated by examples. 1 Introduction Institutions, provide an "abstract model theory for specification and programming"--- quoting from the title of [8]. The model-theoretic view of logic, advocated by institutions, seems to be very natural in computer science applications, considering the fact, that our main concern is to specify, create, and reason about concrete objects---such as programs or VLSI chips. Context institutions (cf. [13]), enrich the structure of institutions by adding notions such as contexts, and substitutions, retaining at the same time the modeltheoretic f...

We define process algebras with a generalised operation P for choice. For every infinite cardinal , we prove that the algebra of transition trees with branching degree ! is free in the class of process algebras in which P is defined for all subsets with a cardinality ! . We explain how the expressions of a fragment of the specification language CRL may be used to denote elements of our process algebras. In particular, we explain how choice quantifiers may be used to denote infinite sums. We show that choice quantifiers can simulate both the existential and the universal quantifiers of first-order logic, while the input prefix mechanism can only simulate the universal quantifier. 1

We introduce a propositional logic ICL, which adds to intuitionistic logic elements of classical reasoning without collapsing it into classical logic. This logic includes a new constant for false, which augments false in intuitionistic logic and in minimal logic. The new constant requires a simple-yet-significant modification of intuitionistic logic both semantically and proof-theoretically. We define a Kripke-style semantics as well as a topological space interpretation in which the new constant is given a precise denotation. We define a sequent calculus and prove cut-elimination. We then formulate a natural deduction proof system with a term calculus, one that gives a direct, computational interpretation of contraction. This calculus shows that ICL is fully capable of typing programming language control constructs such as call/cc while maintaining intuitionistic implication as a genuine connective.

We introduce a propositional logic ICL, which adds to intuitionistic logic elements of classical reasoning without collapsing it into classical logic. This logic includes a new constant for false, which augments false in intuitionistic logic and in minimal logic. We define Kripke models for ICL and show how they translate to several other forms of semantics. We define a sequent calculus and prove cut-elimination. We then formulate a natural deduction proof system with a term calculus that gives a direct, computational interpretation of contraction. This calculus shows that ICL is fully capable of typing programming language control operators such as call/cc while maintaining intuitionistic implication as a genuine connective. 1