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By
Rasiowa, Helena
6 Citations
Post algebras of order ω^{+} as a semantic foundation for ω^{+}valued predicate calculi were examined in [5]. In this paper Post spaces of order ω^{+} being a modification of Post spaces of order n≥2 (cf. Traczyk [8], Dwinger [1], Rasiowa [6]) are introduced and Post fields of order ω^{+} are defined. A representation theorem for Post algebras of order ω^{+} as Post fields of sets is proved. Moreover necessary and sufficient conditions for the existence of representations preserving a given set of infinite joins and infinite meets are established and applied to LindenbaumTarski algebras of elementary theories based on ω^{+}valued predicate calculi in order to obtain a topological characterization of open theories.
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By
Alchourrón, Carlos E.; Makinson, David
108 Citations
This paper is concerned with formal aspects of the logic of theory change, and in particular with the process of shrinking or contracting a theory to eliminate a proposition. It continues work in the area by the authors and Peter Gärdenfors. The paper defines a notion of “safe contraction” of a set of propositions, shows that it satisfies the Gärdenfors postulates for contraction and thus can be represented as a partial meet contraction, and studies its properties both in general and under various natural constraints.
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By
Smullyan, Raymond M.
2 Citations
Selfreferential sentences have played a key role in Tarski's proof [9] of the nondefinibility of arithmetic truth within arithmetic and Gödel's proof [2] of the incompleteness of Peano Arithmetic. In this article we consider some new methods of achieving selfreference in a uniform manner.
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By
Goldblatt, Robert
10 Citations
A foundational algebra (
$$\mathfrak{B}$$
, f, λ) consists of a hemimorphism f on a Boolean algebra
$$\mathfrak{B}$$
with a greatest solution λ to the condition α⩽f(x). The quasivariety of foundational algebras has a decidable equational theory, and generates the same variety as the complex algebras of structures (X, R), where f is given by Rimages and λ is the nonwellfounded part of binary relation R.
The corresponding results hold for algebras satisfying λ=0, with respect to complex algebras of wellfounded binary relations. These algebras, however, generate the variety of all (
$$\mathfrak{B}$$
,f) with f a hemimorphism on
$$\mathfrak{B}$$
).
Admitting a second hemimorphism corresponding to the transitive closure of R allows foundational algebras to be equationally defined, in a way that gives a refined analysis of the notion of diagonalisable algebra.
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By
Benthem, Johan
10 Citations
Contemporary historians of logic tend to credit Bernard Bolzano with the invention of the semantic notion, of consequence, a full century before Tarski. Nevertheless, Bolzano's work played no significant rôle in the genesis of modern logical semantics. The purpose of this paper is to point out three highly original, and still quite relevant themes in Bolzano's work, being a systematic study of possible types of inference, of consistency, as well as their metatheory. There are certain analogies with Tarski's concerns here, although the main thrust seems to be different, both philosophically and technically. Thus, if only obliquely, we also provide some additional historical perspective on Tarski's achievement.
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By
Czelakowski, Janusz
22 Citations
The first known statements of the deduction theorems for the firstorder predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas α and β, βεC(X∪{{a}}) iff P(α, β) AC(X). [P(α, β) denotes the set of formulas which result by the simultaneous substitution of α for p and β for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the joinsemilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian.
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By
Czelakowski, Janusz; Malinowski, Grzegorz
7 Citations
The aim of the article is to outline the historical background and the present state of the methodology of deductive systems invented by Alfred Tarski in the thirties. Key notions of Tarski's methodology are presented and discussed through, the recent development of the original concepts and ideas.
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By
McCarty, Willard; Hood, Patricia
Conclusion
Given the current state of word processing and the hardware for which it has been designed, WordPerfect seems to be among the best of what is available. It manifests a reasonable balance between the demands of the novice, who will appreciate the simplicity of its user interface, and those of the experienced person, who will be more interested in the power and flexibility of the tool. This simplicity has, however, been achieved at the cost of a certain amount of rigidity. We wonder, then, how long the package can remain current with respect to the evolution of word processing. At the same time, we realize that not every academic user of word processing will care about the evolution of this genre. For those who don't our criticisms are mostly irrelevant, although anyone interested in this package would do well to compare it to something on the level of Nota Bene or T3.
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By
Mijajlović, Žarko
3 Citations
In paper [5] it was shown that a great part of model theory of logic with the generalized quantifier Qx = “there exist uncountably many x” is reducible to the model theory of first order logic with an extra binary relation symbol. In this paper we consider when the quantifier Qx can be “syntactically” defined in a first order theory T. That problem was raised by Kosta Došen when he asked if the quantifier Qx can be eliminated in Peano arithmetic. We answer that question fully in this paper.
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By
McKee, T. A.
2 Citations
A simple propositional operator is introduced which generalizes pairwise equivalence and occurs widely in mathematics. Attention is focused on a replacement theorem for this notion of generalized equivalence and its use in producing further generalized equivalences.
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By
Jankowski, Andrzej W.; Zawadowski, Marek
For a complete Heyting lattice ℒ, we define a category Etale (ℒ). We show that the category Etale (ℒ) is equivalent to the category of the sheaves over ℒ, Sh(ℒ), hence also with ℒvalued sets, see [2], [1]. The category Etale(ℒ) is a generalization of the category Etale (X), see [1], where X is a topological space.
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By
Giambrone, Steve
[10] offers two (cutfree) subscripted Gentzen systems, G_{2}T_{+} and G_{2}R_{+}, which are claimed to be equivalent in an appropriate sense to the positive relevant logics T_{+} and R_{+}, respectively. In this paper we show that that claim is false. We also show that the argument in [10] for the further claim that cut and/or modus ponens is admissible in two other subscripted Gentzen systems, G_{1}T_{+} and G_{1}R_{+}, is unsound.
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By
Czelakowski, Janusz
8 Citations
With each sentential logic C, identified with a structural consequence operation in a sentential language, the class Matr^{*} (C) of factorial matrices which validate C is associated. The paper, which is a continuation of [2], concerns the connection between the purely syntactic property imposed on C, referred to as Maehara Interpolation Property (MIP), and three diagrammatic properties of the class Matr* (C): the Amalgamation Property (AP), the (deductive) Filter Extension Property (FEP) and Injections Transferable (IT). The main theorem of the paper (Theorem 2.2) is analogous to the Wroński's result for equational classes of algebras [13]. It reads that for a large class of logics the conjunction of (AP) and (FEP) is equivalent to (IT) and that the latter property is equivalent to (MIP).
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By
Goranko, Valentin
13 Citations
This paper deals with, prepositional calculi with strong negation (Nlogics) in which the Craig interpolation theorem holds. Nlogics are defined to be axiomatic strengthenings of the intuitionistic calculus enriched with a unary connective called strong negation. There exists continuum of Nlogics, but the Craig interpolation theorem holds only in 14 of them.
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By
Ono, Hiroakira
6 Citations
In this paper, a semantics for predicate logics without the contraction rule will be investigated and the completeness theorem will be proved. Moreover, it will be found out that our semantics has a close connection with Bethtype semantics.
By
Wasilewska, Anita
We introduce here and investigate the notion of an alternative tree of decomposition. We show (Theorem 5) a general method of finding out all nonalternative trees of the alternative tree determined by a diagram of decomposition.
By
FattorosiBarnaba, M.; Caro, F.
47 Citations
We study a modal system ¯T, that extends the classical (prepositional) modal system T and whose language is provided with modal operators M inn (nεN) to be interpreted, in the usual kripkean semantics, as “there are more than n accessible worlds such that...”. We find reasonable axioms for ¯T and we prove for it completeness, compactness and decidability theorems.
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By
Wasilewska, Anita
We use the algebraic theory of programs as in Blikle [2], Mazurkiewicz [5] in order to show that the difference between programs with and without recursion is of the same kind as that between cut free Gentzen type formalizations of predicate and prepositional logics.
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By
Deutsch, Harry
2 Citations
A modified filtrations argument is used to prove that the relevant logic S of [2] is decidable.
By
Jankowski, Andrzej W.
1 Citations
This paper is a continuation of investigations on Galois connections from [1], [3], [10]. It is a continuation of [2]. We have shown many results that link properties of a given closure space with that of the dual space. For example: for every ωdisjunctive closure space X the dual closure space is topological iff the base of X generated by this dual space consists of the ωprime sets in X (Theorem 2). Moreover the characterizations of the satisfiability relation for classical logic are shown. Roughly speaking our main result here is the following: a satisfiability relation in a logic L with, a countable language is a fragment of the classical one iff the compactness theorem for L holds (Theorems 3–8).
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By
Turner, Raymond
5 Citations
By the term ‘nominalization’ I mean any process which transforms a predicate or predicate phrase into a noun or noun phrase, e.g. ‘feminine’ is transformed into ‘feminity’. I call these derivative nouns abstract singular terms. Our aim is to provide a modeltheoretic interpretation for a formal language which admits the occurrence of such abstract singular terms.
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By
Ursini, Aldo
We make use of a Theorem of BurrisMcKenzie to prove that the only decidable variety of diagonalizable algebras is that defined by ‘τ0=1’. Any variety containing an algebra in which τ0≠1 is hereditarily undecidable. Moreover, any variety of intuitionistic diagonalizable algebras is undecidable.
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By
Orłowska, Ewa
33 Citations
In the paper we define a class of languages for representation o knowledge in those application areas when a complete information about a domain is not available. In the languages we introduce modal operators determined by accessibility relations depending on parameters.
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By
Takano, Mitio
3 Citations
A structure A for the language L, which is the firstorder language (without equality) whose only nonlogical symbol is the binary predicate symbol ɛ, is called a quasi ɛstruoture iff (a) the universe A of A consists of sets and (b) aɛb is true in A ↔ (∃[p) a = {p } & p ε b] for every a and b in A, where a(b) is the name of a (b). A quasi ɛstructure A is called an ɛstructure iff (c) {p } ε A whenever p ε a ε A. Then a closed formula σ in L is derivable from Leśniewski's axiom ∀x, y[x ɛy ↔∃u (u ɛ x)∧∀u; v(u, v ɛ x→ uɛv)∧∀u(u ɛ x→ u ɛ y)] (from the axiom ∀x, y(xɛ y → x ɛ x)∧∀x, y, z(x ɛ y ɛ z → y ɛx ɛ z)) iff σ is true in every ɛstructure (in every quasi ɛstructure).
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By
Zawadowski, Marek
This paper is a continuation of the investigation from [13]. The main theorem states that the general and the existential quantifiers are (χ, λreducible in some Grothendieck toposes. Using this result and Theorems 4.1, 4.2 [13] we get the downward SkolemLöwenheim theorem for semantics in these toposes.
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By
Jankowski, Andrzej W.
1 Citations
The main result of this paper is the following theorem: a closure space X has an 〈α, δ, Q〉regular base of the power
$$\mathfrak{n}$$
iff X is Qembeddable in
$$B_{\alpha ,\delta }^\mathfrak{n} $$
It is a generalization of the following theorems:
(i)
Stone representation theorem for distributive lattices (α = 0, δ = ω, Q = ω),
(ii)
universality of the Alexandroff's cube for T_{0}topological spaces (α = ω, δ = ∞, Q = 0),
(iii)
universality of the closure space of filters in the lattice of all subsets for 〈α, δ〉closure spaces (Q = 0).
By this theorem we obtain some characterizations of the closure space
$$F_\mathfrak{m} $$
given by the consequence operator for the classical propositional calculus over a formalized language of the zero order with the set of propositional variables of the power
$$\mathfrak{m}$$
. In particular we prove that a countable closure space X is embeddable with finite disjunctions preserved into F_{ω} iff X is a consistent closure space satisfying the compactness theorem and X contains a 〈0, ω〉base consisting of ωprime sets.
This paper is a continuation of [7], [2] and [3].
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By
Vakarelov, Dimiter
4 Citations
The main results of the paper are the following: For each monadic prepositional formula φ which is classically true but not intuitionistically so, there is a continuum of intuitionistic monotone modal logics L such that L+φ is inconsistent.
There exists a consistent intuitionistic monotone modal logic L such that for any formula φ of the kind mentioned above the logic L+φ is inconsistent.
There exist at least countably many maximal intuitionistic monotone modal logics.
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By
Došen, Kosta
30 Citations
This paper, a sequel to “Models for normal intuitionistic modal logics” by M. Božić and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of S4 and S5. For these prepositional logics Kripkestyle models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown how the holding of formulae characteristic for particular logics is equivalent to conditions for the relations of the models. Modalities in these logics are also investigated.
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By
Jankowski, Andrzej W.
5 Citations
In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T_{0}closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a “generalized Alexandroff's cube” that is universal for T_{0}closure spaces. By this theorem we obtain the following characterization of the consequence operator of the classical logic: If ℒ is a countable set and C: P(ℒ) → P(ℒ) is a closure operator on X, then C satisfies the compactness theorem iff the closure space 〈ℒ,C〉 is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.
We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2^{ω} there exists a subset X′ of irrationals and a subset X″ of the Cantor's set such that X is both a continuous image of X′ and a continuous image of X″.
We assume the reader is familiar with notions in [5].
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By
Becker, Lawrence C.
1 Citations
Summary
Quibbles aside,Notebook II is a solid program for people who need a database manager for large fields of text. It is efficient, reliable, well documented, and easy to use.WordStar users may well already know about Pro/Tem Software's other programs,Footnote andBibliography.
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