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A Primer On Galois Connections
 York Academy of Science
, 1992
"... : 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 wellknown to most mathematicians who deal with order theory; they seem to be less known to to ..."
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Cited by 29 (3 self)
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: 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 wellknown 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, 0601, 06A06 Secondary: 5401, 54B99, 54H99, 68F05 0. INTRODUCTION Mathematicians are familiar with the following situation: there are two "worlds" and t...
A Logic for Hypothetical Reasoning
 Department of Computer Science, Rutgers University
, 1988
"... 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, Hornclause logic is augmented with rules, called embedde ..."
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Cited by 15 (9 self)
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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, Hornclause 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 Hornclause 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 2126 1988, American Association for Artificial Intelligence (AAAI). 1 Introduction Several researchers...
Knowledge theoretic properties of topological spaces
 Knowledge Representation and Uncertainty, Logic at Work, volume 808 of Lecture Notes in Artificial Intelligence
, 1994
"... We study the topological models of a logic of knowledge for topological reasoning, introduced by Larry Moss and Rohit Parikh ([1992]). Among our results is a solution of a conjecture by the formentioned authors, finite satisfiability property and decidability for the theory of topological models. 1 ..."
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Cited by 9 (1 self)
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We study the topological models of a logic of knowledge for topological reasoning, introduced by Larry Moss and Rohit Parikh ([1992]). Among our results is a solution of a conjecture by the formentioned authors, finite satisfiability property and decidability for the theory of topological models. 1
Propositional quantification in the topological semantics for S4
 Notre Dame Journal of Formal Logic
, 1997
"... quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S ..."
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Cited by 4 (1 self)
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quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub S4πt, isstrictly weaker than its Kripkean counterpart. I prove here that secondorder arithmetic can be recursively embedded in S4πt. Inthe course of the investigation, I also sketch a proof of Fine’s and Kripke’s results that the Kripkean system S4π+ is recursively isomorphic to secondorder logic. 1Introduction One way to extend a propositional logic to a language with propositional quantifiers is to begin with a semantics for the logic; extract from the semantics a notion of a proposition; and interpret the quantifiers as ranging over the propositions. Thus, Fine [4] extends the Kripke semantics for modal logics to propositionally quantified systems S5π+, S4π+, S4.2π+, and such: given a Kripke frame, the quantifiers range over all sets of possible worlds. S5π+ is decidable ([4] and Kaplan [14]). In later unpublished work, Fine and Kripke independently showed that S4π+, S4.2π+, K4π+, Tπ+, Kπ+, and Bπ+ and others are recursively isomorphic to full secondorder classical logic. (Fine informs me that he later proved this stronger result. Kripke informs me that he too proved this stronger result in the early 1970s. A proof of this result occurs in Kaminski and Tiomkin [13], who use techniques similar to those used in Kremer [16] and to those used below. These techniques do not apply to S4.3π+. But according to Kaminski and Tiomkin, work of Gurevich and Shelah ([9], [10], and [39]) implies that secondorder arithmetic is interpretable in S4.3π+ and furthermore that, under
AntiIntuitionism and Paraconsistency
 URL = http://www.cle.unicamp.br/eprints/vol 3,n 1,2003.html
, 2003
"... 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 antiintuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is s ..."
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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 antiintuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is shown here that antiintuitionistic logics are paraconsistent, and in particular we develop a first antiintuitionistic hierarchy starting with Johansson 's dual calculus and ending up with Godel's threevalued dual calculus, showing that no calculus of this hierarchy allows the introduction of an internal implication symbol. Comparing these antiintuitionistic logics with wellknown 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) manyvalued )n## we show that the antiintuitionistic hierarchy (I )n## obtained from (I )n## does coincide with the hierarchy of the manyvalued 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 selfduality. 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 multipleconclusion logics are used as an appropriate environment to deal with them.
Context Parchments
"... . 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 uni ..."
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. 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 modeltheoretic 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 objectssuch 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...
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providi ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree 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 18871963 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 quantifierfree 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 ...
Presenting Context Institutions: Context Parchments
"... . 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 uni ..."
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. 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 modeltheoretic 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 objectssuch 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...
Intuitionistic Control Logic
, 2012
"... 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 simpley ..."
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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 simpleyetsignificant modification of intuitionistic logic both semantically and prooftheoretically. We define a Kripkestyle 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 cutelimination. 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.
An Intuitionistic Logic for Sequential Control
"... 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 ..."
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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 cutelimination. 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