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1.
In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction
of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach
has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives
they possess. It yields a new classification of logics expanding Leibniz hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it
has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which
is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the
semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples
of fuzzy logics and suggests new directions for research in the field. 相似文献
2.
Dag Normann 《Archive for Mathematical Logic》2002,41(8):765-788
We use the theory of domains with totality to construct some logics generalizing ω-logic and β-logic and we prove a completenes
theorem for these logics. The key application is E-logic, the logic related to the functional 3
E. We prove a compactness theorem for sets of sentences semicomputable in 3
E.
Received: 21 January 1998 / Published online: 2 September 2002 相似文献
3.
4.
Bounded commutative residuated ℓ-monoids are a generalization of algebras of propositional logics such as BL-algebras, i.e. algebraic counterparts of the basic fuzzy logic (and hence consequently MV-algebras, i.e. algebras of the Łukasiewicz infinite valued logic) and Heyting algebras, i.e. algebras of the intuitionistic
logic. Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single
individual variable occurs. We introduce and study monadic residuated ℓ-monoids as a generalization of monadic MV-algebras.
Jiří Rachůnek was supported by the Council of Czech Goverment MSM 6198959214. 相似文献
5.
Sergey Pavlov 《Logica Universalis》2011,5(2):319-325
The logic with independent truth and falsehood operators TFL is proposed. In TFL(→) standard truth-conditions for the implication
are adopted. Nevertheless the laws of classical logic are not valid. In this language more then 107 different binary connectives can be defined. So this logic can be treated as universal logic relatively to the class of sentential
logics. 相似文献
6.
L. L. Maksimova 《Algebra and Logic》2006,45(1):49-66
We look at the interplay between the projective Beth property in non-classical logics and interpolation. Previously, we proved
that in positive logics as well as in superintuitionistic and modal ones, the projective Beth property PB2 follows from Craig's interpolation property and implies the restricted interpolation property IPR. Here, we show that IPR and PB2 are equivalent in positive logics, and also in extensions of the superintuitionistic logic KC and of the modal logic Grz.2.
Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President)
and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.
__________
Translated from Algebra i Logika, Vol. 45, No. 1, pp. 85–113, January–February, 2006. 相似文献
7.
J. Sakalauskaitė 《Lithuanian Mathematical Journal》2007,47(3):266-276
In this paper, we consider branching time temporal logic CT L with epistemic modalities for knowledge (belief) and with awareness operators. These logics involve the discrete-time linear
temporal logic operators “next” and “until” with the branching temporal logic operator “on all paths”. In addition, the temporal
logic of knowledge (belief) contains an indexed set of unary modal operators “agent i knows” (“agent i believes”). In a language of these logics, there are awareness operators. For these logics, we present sequent calculi with
a restricted cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness for
these calculi are proved.
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 328–340, July–September, 2007. 相似文献
8.
Dov M. Gabbay 《Logica Universalis》2011,5(2):255-318
In this paper we show that some versions of Dung’s abstract argumentation frames are equivalent to classical propositional
logic. In fact, Dung’s attack relation is none other than the generalised Peirce–Quine dagger connective of classical logic
which can generate the other connectives ?, ù, ú, ?{\neg, \wedge, \vee, \to} of classical logic. After establishing the above correspondence we offer variations of the Dung argumentation frames in parallel
to variations of classical logic, such as resource logics, predicate logic, etc., etc., and create resource argumentation
frames, predicate argumentation frames, etc., etc. We also offer the notion of logic proof as a geometrical walk along the
nodes of a Dung network and thus we are able to offer a geometrical abstraction of the notion of inference based argumentation.
Thus our paper is also a contribution to the question: 相似文献
9.
Michael Pinsker 《Algebra Universalis》2005,54(2):129-148
We first determine the maximal clones on a set X of infinite regular cardinality κ which contain all permutations but not all unary functions, extending a result of Heindorf’s
for countably infinite X. If κ is countably infinite or weakly compact, this yields a list of all maximal clones containing the permutations, since
in that case the maximal clones above the unary functions are known. We then generalize a result of Gavrilov’s to obtain on
all infinite X a list of all maximal submonoids of the monoid of unary functions which contain the permutations.
Received January 8, 2004; accepted in final form December 22, 2004. 相似文献
10.
11.
We engage a study of nonmodal linear logic which takes times ⊗ and the linear conditional ⊸ to be the basic connectives instead
of times and linear negation ()⊥ as in Girard's approach. This difference enables us to obtain a very large subsystem of linear logic (called positive linear
logic) without an involutionary negation (if the law of double negation is removed from linear logic in Girard's formulation,
the resulting subsystem is extremely limited). Our approach enables us to obtain several natural models for various subsystems
of linear logic, including a generic model for the so-called minimal linear logic. In particular, it is seen that these models
arise spontaneously in the transition from set theory to multiset theory. We also construct a model of full (nonmodal) linear
logic that is generic relative to any model of positive linear logic. However, the problem of constructing a generic model
for positive linear logic remains open. Bibliography: 2 titles.
Published inZapiski Nauchnykh Seminarov POMI, Vol. 220, 1995, pp. 23–35. Original 相似文献
12.
This is a sequel of the work done on (strongly) monotonically monolithic spaces and their generalizations. We introduce the
notion of monotonically κ-monolithic space for any infinite cardinal κ and present the relevant results. We show, among other things, that any σ-product of monotonically κ-monolithic spaces is monotonically κ-monolithic for any infinite cardinal κ; besides, it is consistent that any strongly monotonically ω-monolithic space with caliber ω
1 is second countable. We also study (strong) monotone κ-monolithicity in linearly ordered spaces and subspaces of ordinals. 相似文献
13.
Vladimir V. Tkachuk 《Central European Journal of Mathematics》2012,10(2):456-465
Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ
+ if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ
+. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ
+ for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ
+ for compact spaces. 相似文献
14.
Simon Thomas 《Israel Journal of Mathematics》1998,103(1):93-109
We prove that the automorphism tower of every infinite centreless groupG of cardinality κ terminates in less than (2κ)+ steps. We also show that it is consistent withZFC that the automorphism tower of every infinite centreless groupG of regular cardinality κ actually terminates in less than 2κ steps.
Research partially supported by NSF Grants. 相似文献
15.
Petr Cintula 《Archive for Mathematical Logic》2003,42(5):449-468
The ŁΠ and logics were introduced by Godo, Esteva and Montagna. These logics extend many other known propositional and predicate logics,
including the three mainly investigated ones (G?del, product and Łukasiewicz logic).
The aim of this paper is to show some advances in this field. We will see further reduction of the axiomatic systems for both
logics. Then we will see many other logics contained in the ŁΠ family of logics (namely logics induced by the continuous finitely
constructed t-norms and Takeuti and Titani's fuzzy predicate logic).
Received: 1 October 2000 / Revised version: 27 March 2002 /
Published online: 5 November 2002
Partial support of the grant No. A103004/00 of the Grant agency of the Academy of Sciences of the Czech Republic is acknowledged.
Key words or phrases: Fuzzy logic – Łukasiewicz logic – Product logic 相似文献
16.
V. B. Shehtman 《Proceedings of the Steklov Institute of Mathematics》2011,274(1):317-325
The paper gives an overview of new results on two-dimensional modal logics of special type, “Segerberg squares.” They are
defined as usual squares of modal logics with additional connectives corresponding to the diagonal symmetry and two projections
onto the diagonal. In many cases these logics are finitely axiomatizable, complete and have the finite model property. Segerberg
squares are interpreted in the classical predicate logic. 相似文献
17.
This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of
scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which
do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered”
posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size
κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ0 the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem.
The authors warmly thank Uri Abraham for his many useful suggestions and comments. Mirna Džamonja thanks EPSRC for their support
on an EPSRC Advanced Fellowship. 相似文献
18.
We introduce an extension, indexed by a partially ordered set P and cardinal numbers κ,λ, denoted by (κ,<λ)⇝P, of the classical relation (κ,n,λ)→ρ in infinite combinatorics. By definition, (κ,n,λ)→ρ holds if every map F: [κ]
n
→[κ]<λ
has a ρ-element free set. For example, Kuratowski’s Free Set Theorem states that (κ,n,λ)→n+1 holds iff κ ≥ λ
+n
, where λ
+n
denotes the n-th cardinal successor of an infinite cardinal λ. By using the (κ,<λ)⇝P framework, we present a self-contained proof of the first author’s result that (λ
+n
,n,λ)→n+2, for each infinite cardinal λ and each positive integer n, which solves a problem stated in the 1985 monograph of Erdős, Hajnal, Máté, and Rado. Furthermore, by using an order-dimension
estimate established in 1971 by Hajnal and Spencer, we prove the relation $(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1}
{2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor }
$(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1}
{2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor }
, for every infinite cardinal λ and all positive integers n and r with 2≤r<n. For example, (ℵ210,4,ℵ0)→32,768. Other order-dimension estimates yield relations such as (ℵ109,4,ℵ0) → 257 (using an estimate by Füredi and Kahn) and (ℵ7,4,ℵ0)→10 (using an exact estimate by Dushnik). 相似文献
19.
This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in terms of an extension property and a syntactical metarule. This is the final part of the investigation started and developed in the papers (Cintula and Noguera in Arch Math Logic 49(4):417–446, 2010; Arch Math Logic 53(3):353–372, 2016). 相似文献
20.
Ladislav Bican 《Annali dell'Universita di Ferrara》2005,51(1):61-67
Sunto LetG andH be abstract classes of modules. The classH is said to have theG-property if to each infinite cardinal λ there exists a cardinal κ>λ such that for everyF∈H with |F|≥κ and every its submoduleK with |F/K|≤λ there exists a submoduleL ofK such thatF/L/teG and |F/L|<κ. This condition is stronger than the condition (P) requiringL≠0 instead of |F/L|<κ, which was introduced and investigated in [8]. In this note we are going to study the relations of this more general condition
to the existence of precovers with respect to some classes of modules. As an application we obtain some sufficient conditions
for the existence of σ-torsionfree precovers related to a given hereditary torsion theory σ for the categoryR-mod. This result is closely related to and in some sense extends that of [5].
The research has been partially supported by the Grant Agency of the Czech Republic, grant #GAČR 201/03/0937 and also by the
institutional grant MSM 113 200 007. 相似文献