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Little Theories
 Automated DeductionCADE11, volume 607 of Lecture Notes in Computer Science
, 1992
"... In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable wa ..."
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Cited by 52 (16 self)
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In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.
Modal Deduction in SecondOrder Logic and Set Theory
 Journal of Logic and Computation
, 1997
"... We investigate modal deduction through translation into standard logic and set theory. Derivability in the minimal modal logic is captured precisely by translation into a weak, computationally attractive set theory \Omega\Gamma This approach is shown equivalent to working with standard firstorder t ..."
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Cited by 11 (7 self)
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We investigate modal deduction through translation into standard logic and set theory. Derivability in the minimal modal logic is captured precisely by translation into a weak, computationally attractive set theory \Omega\Gamma This approach is shown equivalent to working with standard firstorder translations of modal formulas in a theory of general frames. Next, deduction in a more powerful secondorder logic of general frames is shown equivalent with settheoretic derivability in an `admissible variant' of \Omega\Gamma Our methods are mainly modeltheoretic and settheoretic, and they admit extension to richer languages than that of basic modal logic. 1 Introduction We are interested in analyzing general deduction for modal formulae [4]. The standard systems used for this purpose are the socalled "minimal modal logic" K or, if one wants to work over general frames (as we do), the system K s obtained from K adding a substitution rule. We do not consider special purpose calculi for ...
Modal Logic and nonwellfounded Set Theory: translation, bisimulation, interpolation.
, 1998
"... ..."
Compiling Dyadic FirstOrder Specifications into Map Algebra
"... Two techniques are designed for eliminating quantifiers from an existentially quantified conjunction of dyadic literals, in terms of the operators... , ∩, and... of the TarskiChinGivant formalism of relations. The use of such techniques is illustrated through increasingly challenging examp ..."
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Cited by 3 (3 self)
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Two techniques are designed for eliminating quantifiers from an existentially quantified conjunction of dyadic literals, in terms of the operators... , &cap;, and... of the TarskiChinGivant formalism of relations. The use of such techniques is illustrated through increasingly challenging examples, and their algorithmic complexity is assessed.
Logical Aspects of Quantum (Non)Individuality
, 2008
"... In this paper I consider some logical and mathematical aspects of the discussion of the identity and individuality of quantum entities. I shall point out that for some aspects of the discussion, the logical basis cannot be put aside; on the contrary, it leads us to unavoidable conclusions which may ..."
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Cited by 2 (2 self)
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In this paper I consider some logical and mathematical aspects of the discussion of the identity and individuality of quantum entities. I shall point out that for some aspects of the discussion, the logical basis cannot be put aside; on the contrary, it leads us to unavoidable conclusions which may have consequences in how we articulate certain concepts related to quantum theory. Behind the discussion, there is a general argument which suggests the possibility of a metaphysics of nonindividuals, based on a reasonable interpretation of quantum basic entities. I close the paper with a suggestion that consists in emphasizing that quanta should be referred to by the cardinalities of the collections to which they belong, for which an adequate mathematical framework seems to be possible. “The subjectpredicate logic to which we are accustomed depends for its convenience upon the fact that at the usual temperature of the earth there are approximately permanent ‘things’. This would not be true at the temperature of the sun, and is only roughly true at the temperature to which we are accustomed. ” (Russell [1957]) 1
Threevariable statements of setpairing
 Theoretical Computer Science
"... The approach to algebraic specifications of set theories proposed by Tarski and Givant inspires current research aimed at taking advantage of the purely equational nature of the resulting formulations for enhanced automation of reasoning on aggregates of various kinds: sets, bags, hypersets, etc. Th ..."
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Cited by 1 (0 self)
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The approach to algebraic specifications of set theories proposed by Tarski and Givant inspires current research aimed at taking advantage of the purely equational nature of the resulting formulations for enhanced automation of reasoning on aggregates of various kinds: sets, bags, hypersets, etc. The viability of the said approach rests upon the possibility to form ordered pairs and to decompose them by means of conjugated projections. Ordered pairs can be conceived of in many ways: along with the most classic one, several other pairing functions are examined, which can be preferred to it when either the axiomatic assumptions are too weak to enable pairing formation à la Kuratowski, or they are strong enough to make the specification of conjugated projections particularly simple, and their formal properties easy to check within the calculus of binary relations.
Compiling Dyadic FirstOrder Specications into Map Algebra
"... Two techniques are designed for eliminating quanti ers from an existentially quanti ed conjunction of dyadic literals, in terms of the operators , \, and of the TarskiChinGivant formalism of relations. ..."
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Two techniques are designed for eliminating quanti ers from an existentially quanti ed conjunction of dyadic literals, in terms of the operators , \, and of the TarskiChinGivant formalism of relations.
ON SKOLEMISING ZERMELO’S SET THEORY
, 2009
"... Abstract. We give a Skolemised presentation of Zermelo’s set theory (with notations for comprehension, powerset, etc.) and show that this presentation is conservative w.r.t. the usual one (where sets are introduced by existential axioms). Conservativity is achieved by an explicit deskolemisation pro ..."
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Abstract. We give a Skolemised presentation of Zermelo’s set theory (with notations for comprehension, powerset, etc.) and show that this presentation is conservative w.r.t. the usual one (where sets are introduced by existential axioms). Conservativity is achieved by an explicit deskolemisation procedure that transforms terms and formulæ of the extended language into provably equivalent formulæ of the core language of set theory. Finally we show that the notation {t(x)  x ∈ u} (‘the set of all t(x) where x ranges over u’) is also definable in this framework, which proves that the weak form of replacement which is needed to define syntactic constructs such as (settheoretic) λabstraction and infinitary Cartesian product does not need Fraenkel and Skolem’s replacement scheme to be justified. §1. Introduction. Set theory [2, 3] is traditionally presented with a very economical firstorder language whose atomic formulæ are built from two binary predicate symbols = and ∈ and whose underlying term algebra is reduced to variables—the language provides no constant or function symbol. Although convenient in the perspective of a modeltheoretic study, the language
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"... Remarks on the theory of quasisets Abstract. Quasiset theory has been proposed as a means of handling collections of indiscernible objects. Although the most direct application of the theory is quantum physics, it can be seen per se as a nonclassical logic (a nonreflexive logic). In this paper w ..."
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Remarks on the theory of quasisets Abstract. Quasiset theory has been proposed as a means of handling collections of indiscernible objects. Although the most direct application of the theory is quantum physics, it can be seen per se as a nonclassical logic (a nonreflexive logic). In this paper we revise and correct some aspects of quasiset theory as presented in [11], so as to avoid some misunderstandings and possible misinterpretations about the results achieved by the theory. Some further ideas with regard to quantum field theory are also advanced in this paper.