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1.
In this paper we prove that thek-ary fragment of transitive closure logic is not contained in the extension of the (k–1)-ary fragment of partial fixed point logic by all (2k–1)-ary generalized quantifiers. As a consequence, the arity hierarchies of all the familiar forms of fixed point logic are strict simultaneously with respect to the arity of the induction predicates and the arity of generalized quantifiers.Although it is known that our theorem cannot be extended to the sublogic deterministic transitive closure logic, we show that an extension is possible when we close this logic under congruence.Supported by a grant from the University of Helsinki. This research was initiated while he was a Junior Researcher at the Academy of FinlandThis article was processed by the author using the LATEX style filepljourlm from Springer-Verlag. 相似文献
2.
Th. Leontiadis 《Applicable analysis》2013,92(1-4):225-230
In this paper we shall prove that the extension Î of a generalized (or full) Daniell-integral I over a non distributive lattice Ê with values in a weakly б-distributive 1-group G has all the fundamental limiting properties of the Lebesgue integral, i.e. the properties stated i n the monotone convergence theorem, Fatou's-lemma and Lebesgue's dominated convergence theorem. 相似文献
3.
Thomas Macaulay Ferguson 《Logica Universalis》2014,8(2):165-191
This paper offers a framework for extending Arnon Avron and Iddo Lev’s non-deterministic semantics to quantified predicate logic with the intent of resolving several problems and limitations of Avron and Anna Zamansky’s approach. By employing a broadly Fregean picture of logic, the framework described in this paper has the benefits of permitting quantifiers more general than Walter Carnielli’s distribution quantifiers and yielding a well-behaved model theory. This approach is purely objectual and yields the semantical equivalence of both α-equivalent formulae and formulae differing only by codenotative terms. Finally, we make a brief excursion into non-deterministic model theory, proving a strong ?o?’ Theorem and compactness for all finitely-valued, non-deterministic logics whose quantifiers have intensions describable in a first order metalanguage. 相似文献
4.
In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic (FNL). We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition. 相似文献
5.
Joseph Sgro 《Israel Journal of Mathematics》1976,25(3-4):249-272
In this paper we formulate a first order theory of continuous functions on product topologies via generalized quantifiers.
We present an axiom system for continuous functions on product topologies and prove a completeness theorem for them with respect
to topological models. We also show that if a theory has a topological model which satisfies the Hausdorff separation axiom,
then it has a 0-dimensional, normal topological model. We conclude by obtaining an axiomatization for topological algebraic
structures, e.g. topological groups, proving a completeness theorem for the analogue with countable conjunctions and disjunctions,
and presenting counterexamples to interpolation and definability. 相似文献
6.
Masanao Ozawa 《Annals of Pure and Applied Logic》2021,172(4):102938
In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws. To justify the new assignment, we prove the Transfer Principle, showing that this assignment of a truth value to every bounded ZFC theorem has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula. We study the most general class of truth value assignments and obtain necessary and sufficient conditions for them to satisfy the Transfer Principle, to satisfy De Morgan's Laws, and to satisfy both. For the class of assignments with polynomially definable logical operations, we determine exactly 36 assignments that satisfy the Transfer Principle and exactly 6 assignments that satisfy both the Transfer Principle and De Morgan's Laws. 相似文献
7.
H. Imhof 《Archive for Mathematical Logic》1999,38(8):491-513
The capability of logical systems to express their own satisfaction relation is a key issue in mathematical logic. Our notion
of self definability is based on encodings of pairs of the type (structure, formula) into single structures wherein the two components can be
clearly distinguished. Hence, the ambiguity between structures and formulas, forming the basis for many classical results,
is avoided. We restrict ourselves to countable, regular, logics over finite vocabularies. Our main theorem states that self
definability, in this framework, is equivalent to the existence of complete problems under quantifier free reductions. Whereas
this holds true for arbitrary structures, we focus on examples from Finite Model Theory. Here, the theorem sheds a new light
on nesting hierarchies for certain generalized quantifiers. They can be interpreted as failure of self definability in the
according extensions of first order logic. As a further application we study the possibility of the existence of recursive
logics for PTIME. We restate a result of Dawar concluding from recursive logics to complete problems. We show that for the model checking
Turing machines associated with a recursive logic, it makes no difference whether or not they may use built in clocks.
Received: 7 February 1997 相似文献
8.
G. V. Mikaelyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2018,53(3):147-152
In this paper we introduce generalized characteristics for meromorphic in the halfplane functions, and generalize the Levin’s formula and the first fundamental theorem for Tsuji’s characteristics. 相似文献
9.
Tin-Yau Tam 《Linear and Multilinear Algebra》2013,61(1-3):87-113
The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered. 相似文献
10.
Harvey Friedman 《Israel Journal of Mathematics》1973,14(2):205-212
We prove that the Beth definability theorem fails for a comprehensive class of first-order logics with cardinality quantifiers.
In particular, we give a counterexample to Beth’s theorem forL(Q), which is finitary first-order logic (with identity) augmented with the quantifier “there exists uncountably many”.
This research was partially supported by NSF GP29254. 相似文献
11.
Makoto Tatsuta Ken-etsu Fujita Hiroshi Nakano 《Annals of Pure and Applied Logic》2010,161(11):1390-1399
This paper shows that the inhabitation problem in the lambda calculus with negation, product, polymorphic, and existential types is decidable, where the inhabitation problem asks whether there exists some term that belongs to a given type. In order to do that, this paper proves the decidability of the provability in the logical system defined from the second-order natural deduction by removing implication and disjunction. This is proved by showing the quantifier elimination theorem and reducing the problem to the provability in propositional logic. The magic formulas are used for quantifier elimination such that they replace quantifiers. As a byproduct, this paper also shows the second-order witness theorem which states that a quantifier followed by negation can be replaced by a witness obtained only from the formula. As a corollary of the main results, this paper also shows Glivenko’s theorem, Double Negation Shift, and conservativity for antecedent-empty sequents between the logical system and its classical version. 相似文献
12.
《Annals of Pure and Applied Logic》2023,174(4):103231
In this paper we study Littlewood's Tauberian theorem from a proof theoretic perspective. We first use the Dialectica interpretation to produce an equivalent, finitary formulation of the theorem, and then carry out an analysis of Wielandt's proof to extract concrete witnessing terms. We argue that our finitization can be viewed as a generalized Tauberian remainder theorem, and we instantiate it to produce two concrete remainder theorems as a corollary, in terms of rates of convergence and rates metastability, respectively. We rederive the standard remainder estimate for Littlewood's theorem as a special case of the former. 相似文献
13.
《Applied Mathematics Letters》2003,16(1):97-103
In this paper, we first prove a new fixed-point theorem from which the Kakutani's fixed-point theorem in locally convex topological vector spaces is immediately extended to H-spaces. Then, we establish a new existence theorem of equilibrium for generalized games in H-spaces, by applying our fixed-point theorem. 相似文献
14.
Alexandra Bruasse-Bac 《Comptes Rendus Mathematique》2002,334(2):93-96
We define a purely relational model of second order linear logic. In the absence of any notion of coherence, we will especially concentrate on establishing a normal form theorem that will give rise to the interpretation of the second order quantifiers. To cite this article: A. Bruasse-Bac, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 93–96 相似文献
15.
San‐Min Wang 《Mathematical Logic Quarterly》2013,59(4-5):268-273
In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members. 相似文献
16.
《Annals of Pure and Applied Logic》2023,174(5):103245
This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of Allwein and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic. 相似文献
17.
Hongjun Zhou Bin Zhao 《Fuzzy Sets and Systems》2012,187(1):33-57
Bosbach and Rie?an states on residuated lattices both are generalizations of probability measures on Boolean algebras. Recently, two types of generalized Bosbach states on residuated lattices were introduced by Georgescu and Mure?an through replacing the standard MV-algebra in the original definition with arbitrary residuated lattices as codomains. However, several interesting problems there remain still open. The first part of the present paper gives positive answers to these open problems. It is proved that every generalized Bosbach state of type II is of type I and the similarity Cauchy completion of a residuated lattice endowed with an order-preserving generalized Bosbach state of type I is unique up to homomorphisms preserving similarities, where the codomain of the type I state is assumed to be Cauchy-complete. Consequently, many existing results about generalized Bosbach states can be further strengthened. The second part of the paper introduces the notion of relative negation (with respect to a given element, called relative element) in residuated lattices, and then many issues with the canonical negation such as Glivenko property, semi-divisibility, generalized Rie?an state of residuated lattices can be extended to the context of such relative negations. In particular, several necessary and sufficient conditions for the set of all relatively regular elements of a residuated lattice to be special residuated lattices are given, of which one extends the well-known Glivenko theorem, and it is also proved that relatively generalized Rie?an states vanishing at the relative element are uniquely determined by their restrictions on the MV-algebra consisting of all relatively regular elements when the domain of the states is relatively semi-divisible and the codomain is involutive. 相似文献
18.
Martin Lück 《Annals of Pure and Applied Logic》2018,169(9):928-969
In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(~) of Väänänen's first-order team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion.As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-preserving elimination of modalities and quantifiers from formulas. 相似文献
19.
In this paper we characterize convex games by means of Owen's multilinear extension and the marginal worth vectors associated with even or odd permutations. Therefore we have obtained a refinement of the classic theorem; Shapley (1971), Ichiishi (1981) in order to characterize the convexity of a game by its marginal worth vectors. We also give new expressions for the marginal worth vectors in relation to unanimity coordinates and the first partial derivatives of Owen's multilinear extension. A sufficient condition for the convexity is given and also one application to the integer part of a convex game. 相似文献
20.
Abir Nour 《Mathematical Logic Quarterly》1999,45(4):457-466
In order to modelize the reasoning of an intelligent agent represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this papers, we consider logic systems called L′T without this kind of constants but limited to the case where T is a finite poset. We prove a weak deduction theorem. We introduce also an algebraic semantics using Hey ting algebra with operators. To prove the completeness theorem of the L′T system with respect to the algebraic semantics, we use the method of H. Rasiowa and R. Sikorski for first order logic. In the propositional case, a corollary allows us to assert that it is decidable to know “if a propositional formula is valid”. We study also certain relations between the L′T logic and the intuitionistic and classical logics. 相似文献