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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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: 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
 In Proc. of AAAI88
, 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 embed ..."
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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 ruZebuse independence, it is shown that these rules can express hypothetical queries which classical logic cannot. By adopting methods from modal logic, these rules are then shown to be intuitionistic. In particular, they form a subset of intuitionistic logic having semantic properties similar to those of Hornclause logic. 1
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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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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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.
On the Priorean temporal logic with ‘around now’ over the real line
"... We consider the temporal language with the Priorean operators G and H expressing that a formula is true at all future times and all past times, plus an operator ✷ expressing that a formula is true throughout some open interval containing the evaluation time (i.e., it is true ‘around now’). We show t ..."
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We consider the temporal language with the Priorean operators G and H expressing that a formula is true at all future times and all past times, plus an operator ✷ expressing that a formula is true throughout some open interval containing the evaluation time (i.e., it is true ‘around now’). We show that the logic of real numbers time in this language is finitely axiomatisable, answering an implicit question of Shehtman (1993). We also show that the logic has PSPACEcomplete complexity, but is not Kripke complete and has no strongly complete axiomatisation.
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...
On the Expressiveness of the Choice Quantifier
 ANNALS OF PURE AND APPLIED LOGIC
"... 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 exp ..."
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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 firstorder logic, while the input prefix mechanism can only simulate the universal quantifier.
DEGREES AND REVERSE MATHEMATICS
, 2011
"... We investigate the complexity of mathematical problems from two perspectives: Medvedev degrees and reverse mathematics. In the Medvedev degrees, we calculate the complexity of its firstorder theory, and we also calculate the complexities of the firstorder theories of several related structures. We ..."
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We investigate the complexity of mathematical problems from two perspectives: Medvedev degrees and reverse mathematics. In the Medvedev degrees, we calculate the complexity of its firstorder theory, and we also calculate the complexities of the firstorder theories of several related structures. We characterize the joinirreducible Medvedev degrees and deduce several consequences for the interpretation of propositional logic in the Medvedev degrees. We equate the size of chains of Medvedev degrees with the size of chains of sets of reals under ⊆. In reverse mathematics, we analyze the strength of classical theorems of finite graph theory generalized to the countable. In particular, we consider Menger’s theorem, Birkhoff’s theorem, and unfriendly partitions. BIOGRAPHICAL SKETCH Paul was born on February 28, 1983 in Richland, Washington during the final episode Goodbye, Farewell and Amen of the popular television series M*A*S*H.
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.