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  • Articles  (3,526)
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  • 1
    Electronic Resource
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    Reduced products of logical matrices (1980)
    Springer
    Studia logica 39 (1980), S. 19-43 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The class Matr(C) of all matrices for a prepositional logic (ℒ, C) is investigated. The paper contains general results with no special reference to particular logics. The main theorem (Th. (5.1)) which gives the algebraic characterization of the class Matr(C) states the following. Assume C to be the consequence operation on a prepositional language induced by a class K of matrices. Let m be a regular cardinal not less than the cardinality of C. Then Matr (C) is the least class of matrices containing K and closed under m-reduced products, submatrices, matrix homomorphisms, and matrix homomorphic counter-images.
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    URL: http://dx.doi.org/10.1007/BF00373095
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  • 2
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    Fifth International Wittgenstein Symposium (1980)
    Springer
    Studia logica 39 (1980), S. 99-99 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
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    URL: http://dx.doi.org/10.1007/BF00373099
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  • 3
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    Some kinds of modal completeness (1980)
    Springer
    Studia logica 39 (1980), S. 125-141 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract In the modal literature various notions of “completeness” have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity and possession of the finite model property — and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability and preservation under ultrafilter extensions are introduced and studied as well.
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    URL: http://dx.doi.org/10.1007/BF00370316
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  • 4
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    Omega-consistency and the diamond (1980)
    Springer
    Studia logica 39 (1980), S. 237-243 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract G is the result of adjoining the schema □ (□qA→A)→□qA to K; the axioms of G* are the theorems of G and the instances of the schema □qA→A and the sole rule of G* is modus ponens. A sentence is ω-provable if it is provable in P(eano) A(rithmetic) by one application of the ω-rule; equivalently, if its negation is ω-inconsistent in PA. Let ω-Bew(x) be the natural formalization of the notion of ω-provability. For any modal sentence A and function ϕ mapping sentence letters to sentences of PA, inductively define A ωϕ by: p ωϕ = ϕ(p) (p a sentence letter); ⊥ωϕ= ⊥; (A→B)suωϕ}= (A ωϕ→Bωϕ); and (□qA)ωϕ= ω-Bew(⌜A ωϕ⌝)(⌜S⌝) is the numeral for the Gödel number of the sentence S). Then, applying techniques of Solovay (Israel Journal of Mathematics 25, pp. 287–304), we prove that for every modal sentence A,⊢ G A iff for all ϕ, ⊢ PA A ωϕ; and for every modal sentence A, ⊢ G* A iff for all ϕ, A ωϕ is true.
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  • 5
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    Interpolation in loop-free logic (1980)
    Springer
    Studia logica 39 (1980), S. 297-310 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Model-theoretic methods are used to extend Craig's Interpolation Theorem to the loop-free portion of Pratt's dynamic logic of programs with simple assignments.
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  • 6
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    An example of strongly finite consequence operation with 2ℵ0 standard strengthenings (1980)
    Springer
    Studia logica 39 (1980), S. 375-379 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Using ideas from Murskii [3], Tokarz [4] and Wroński [7] we construct some strongly finite consequence operation having 2%0 standard strengthenings. In this way we give the affirmative answer to the following question, stated in Tokarz [4]: are there strongly finite logics with the degree of maximality greater than ℵ0?
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  • 7
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    Finite axiomatization for some intermediate logics (1980)
    Springer
    Studia logica 39 (1980), S. 415-423 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract LetN. be the set of all natural numbers (except zero), and letD n * = {k ∈N ∶k|n} ∪ {0} wherek¦n if and only ifn=k.x f or somex∈N. Then, an ordered setD n * = 〈D n * , ⩽ n , wherex⩽ ny iffx¦y for anyx, y∈D n * , can easily be seen to be a pseudo-boolean algebra. In [5], V.A. Jankov has proved that the class of algebras {D n * ∶n∈B}, whereB =,{k ∈N∶ ⌉ $$\mathop \exists \limits_{n \in N} $$ (n 〉 1 ≧n 2 k)is finitely axiomatizable. The present paper aims at showing that the class of all algebras {D n * ∶n∈B} is also finitely axiomatizable. First, we prove that an intermediate logic defined as follows: $$LD = Cn(INT \cup \{ p_3 \vee [p_3 \to (p_1 \to p_2 ) \vee (p_2 \to p_1 )]\} )$$ finitely approximatizable. Then, defining, after Kripke, a model as a non-empty ordered setH = 〈K, ⩽〉, and making use of the set of formulas true in this model, we show that any finite strongly compact pseudo-boolean algebra ℬ is identical with. the set of formulas true in the Kripke modelH B = 〈P(ℬ), ⊂〉 (whereP(ℬ) stands for the family of all prime filters in the algebra ℬ). Furthermore, the concept of a structure of divisors is defined, and the structure is shown to beH D n * = 〈P (D n * ), ⊂〉for anyn∈N. Finally, it is proved that for any strongly compact pseudo-boolean algebraU satisfying the axiomp 3∨ [p 3→(p1→p2)∨(p2→p1)] there is a structure of divisorsD * n such that it is possible to define a strong homomorphism froomiH D n * ontoH D U . Exploiting, among others, this property, it turns out to be relatively easy to show that $$LD = \mathop \cap \limits_{n \in N} E(\mathfrak{D}_n^* )$$ .
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  • 8
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    On the {↔, ∼} -reduct of the intuitionistic consequence operation (1981)
    Springer
    Studia logica 40 (1981), S. 55-66 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract The intuitionistic consequence operation restricted to the language with ↔ (equivalence) and ∼ (negation) as the only connectives is axiomatized by means of a finite set of sequential rules of inference.
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  • 9
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    On fragments of Medvedev's logic (1981)
    Springer
    Studia logica 40 (1981), S. 39-54 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectivesΦ such that $$\{ \to , \vee , \urcorner \} \not \subseteq \Phi \subseteq \{ \to , \wedge , \urcorner \} $$ theΦ-fragment ofMV equals theΦ fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: is an intermediate logic based on the axiom (⌝a→b∨c) →(⌉a→b)∨(⌝a → c) separable?
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  • 10
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    Individualistic formal approach to deontic logic (1981)
    Springer
    Studia logica 40 (1981), S. 99-102 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Some people approve of certain general rules of behavior, or some concrete cases. The others disapprove of or are indifferent to them. In this paper I suggest an axiom system which formalizes the use of these utterances. It may be considered as a special (“individualistic”) approach to deontic logic.
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  • 11
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    A logic for propositions with indefinite truth values (1982)
    Springer
    Studia logica 41 (1982), S. 197-226 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract In the first part of this paper a logic is defined for propositions whose probability of being true may not be known. A speaker's beliefs about which propositions are true are still interesting in this case. The meaning of propositions is determined by the consequences of asserting them: in this logic there are debates which incur certain costs for the protagonists. The second part of the paper describes the mathematics of the resulting logic which displays several novel features.
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  • 12
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    The logic of ability, freedom and responsibility (1982)
    Springer
    Studia logica 41 (1982), S. 227-248 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract The aim of this paper is to offer a rigorous explication of statements ascribing ability to agents and to develop the logic of such statements. A world is said to be feasible iff it is compatible with the actual past-and-present. W is a P-world iff W is feasible and P is true in W (where P is a proposition). P is a sufficient condition for Q iff every P world is a Q world. P is a necessary condition for Q iff Q is a sufficient condition forP. Each individual property S is shown to generate a rule for an agent X. X heeds S iff X makes all his future choices in accordance with S. (Note that X may heed S and yet fail to have it). S is a P-strategy for X iff X's heeding S together with P is a necessary and sufficient condition for X to have S. (P-strategies are thus rules which X is able to implement on the proviso P).Provisional opportunity: X has the opportunity to A provided P iff there is an S such that S is a P-strategy for X and X's implementing S is a sufficient condition for X's doing A. P is etiologically complete iff for every event E which P reports P also reports an etiological ancestry of E, and P is true. Categorical opportunity: X has the opportunity to A iff there is a P such that P is etiologically complete and X has the opportunity to A provided P. For X to have the ability to A there must not only be an appropriate strategy, but X must have a command of that strategy. X steadfastly intends A iff X intends A at every future moment at which his doing A is not yet inevitable. X has a command of S w.r.t. A and P iff X's steadfastly intending A together with P is a sufficient condition for X to implement S. Provisional ability: X can A provided P iff there is an S such that S is a P-strategy for X, X's implementing S is a sufficient condition for X's doing A, and X has a command of S w.r.t. A and P. Categorical ability: X can A iff there is a P such that P is etiologically complete and X can A provided P. X is free w.r.t. to A iff X can A and X can non- A. X is free iff there is an A such that X is free w.r.t. A.
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  • 13
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    A formal ontology of situations (1982)
    Springer
    Studia logica 41 (1982), S. 381-413 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract A generalized Wittgensteinian semantics for propositional languages is presented, based on a lattice of elementary situations. Of these, maximal ones are possible worlds, constituting a logical space; minimal ones are logical atoms, partitioned into its dimensions. A verifier of a proposition α is an elementary situation such that if real it makes α true. The reference (or objective) of a proposition is a situation, which is the set of all its minimal verifiers. (Maximal ones constitute its locus.) Situations are shown to form a Boolean algebra, and the Boolean set algebra of loci is its representation. Wittgenstein's is a special case, admitting binary dimensions only.
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  • 14
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    An extension of the deontic calculus DSC1 (1983)
    Springer
    Studia logica 42 (1983), S. 129-137 
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    Topics: Mathematics , Philosophy
    Notes: Abstract The chief aim of the paper is to extend the calculusDSC 1 (see [4]) in such a way as to satisfy all the requirements listed in [4] as well as a further stipulation — called ‘the principle of uninvolvement’ — to the effect that neither deontic compatibility nor deontic incompatibility of codes (see [2]) should be presupposed in deontic logic.
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  • 15
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    Subject-predicate calculus free from existential import (1983)
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    Studia logica 42 (1983), S. 209-221 
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    Topics: Mathematics , Philosophy
    Notes: Abstract Two subject-predicate calculi with equality,SP = and its extensionUSP′ =, are presented as systems of natural deduction. Both the calculi are systems of free logic. Their presentation is preceded by an intuitive motivation. It is shown that Aristotle's syllogistics without the laws of identitySaP andSiP is definable withinSP =, and that the first-order predicate logic is definable withinUSP′ =.
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  • 16
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    On the complexity-relativized strong reducibilites (1983)
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    Studia logica 42 (1983), S. 259-267 
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    Topics: Mathematics , Philosophy
    Notes: Abstract This paper discusses refinements of the natural ordering of them-degrees (1-degrees) of strong recursive reducibility classes. Such refinements are obtained by posing complexity conditions on the reduction function. The discussion uses the axiomatic complexity theory and is hence very general. As the main result it is proved that if the complexity measure is required to be linearly bounded (and space-like), then a natural class of refinements forms a lattice with respect to a natural ordering upon them.
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  • 17
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    Logical operations and iterated infinitely deep languages (1983)
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    Studia logica 42 (1983), S. 243-249 
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    Topics: Mathematics , Philosophy
    Notes: Abstract We discuss an abstract notion of a logical operation and corresponding logics. It is shown that if all the logical operations considered are implicitely definable in a logic ℒ*, then the same holds also for the logic obtained from these operations. As an application we show that certain iterated forms of infinitely deep languages are implicitely definable in game quantifier languages. We consider also relations between structures and show that Karttunen's characterization of elementary equivalence for the ordinary infinitely deep languages can be generalized to hold for the iterated infinitely deep languages. An early version of this work was presented in the Abstracts Section of ICM '78.
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  • 18
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    A decision procedure for the systemE (of entailment). I. (1983)
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    Studia logica 42 (1983), S. 139-164 
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    Topics: Mathematics , Philosophy
    Notes: Abstract A system of natural deduction is presented whose set of theses is identical with that of the systemE of entailment. For that system a decision procedure is described proving the decidability ofE.
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  • 19
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    On some proof theoretical properties of the modal logic GL (1983)
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    Studia logica 42 (1983), S. 453-459 
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    Notes: Abstract This paper deals with the system of modal logicGL, in particular with a formulation of it in terms of sequents. We prove some proof theoretical properties ofGL that allow to get the cut-elimination theorem according to Gentzen's procedure, that is, by double induction on grade and rank.
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  • 20
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    Completeness theorems for some intermediate predicate calculi (1983)
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    Studia logica 42 (1983), S. 431-441 
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    Notes: Abstract We give completeness results — with respect to Kripke's semantic — for the negation-free intermediate predicate calculi: (1) $$\begin{gathered} BD = positive predicate calculus PQ + B:(\alpha \to \beta )v(\beta \to \alpha ) \hfill \\ + D:\forall x\left( {a\left( x \right)v\beta } \right) \to \forall xav\beta \hfill \\ \end{gathered}$$ (2) $$T_n D = PQ + T_n :\left( {a_0 \to a_1 } \right)v \ldots v\left( {a_n \to a_{n + 1} } \right) + D\left( {n \geqslant 0} \right)$$ and the superintuitionistic predicate calculus: (3) $$B^1 DH_2^ \urcorner = BD + intuitionistic negation + H_2^ \urcorner : \urcorner \forall xa \to \exists x \urcorner a.$$ The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3]. For a general account on negation-free intermediate predicate calculi — see Casari-Minari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].
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  • 21
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    Ayda Ignez Arruda (1936–1983) (1984)
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    Studia logica 43 (1984), S. 1-2 
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  • 22
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    On the meaning of indexicals (1983)
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    Studia logica 42 (1983), S. 285-291 
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    Notes: Abstract The approach adopted in the paper is based on the theory known as Montague grammar. Accepting, in general, that theory — especially in its modified version, which is due to Thomason and Kaplan — the author points out certain inadequacy in its treatment of the meaning of some indexical expressions and suggests some modification of its theoretical framework in order to avoid that shortcoming. It is claimed that to do justice to the meaning of so-called indefinite indexicals (such as “we”, “you”, “now”) two kinds of dependence of their semantic values upon the context of use must be taken into account — a semantic (or lexical) and a pragmatic (or extralexical) one.
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  • 23
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    Paraconsistent logic and model theory (1984)
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    Studia logica 43 (1984), S. 17-32 
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    Notes: Abstract The object of this paper is to show how one is able to construct a paraconsistent theory of models that reflects much of the classical one. In other words the aim is to demonstrate that there is a very smooth and natural transition from the model theory of classical logic to that of certain categories of paraconsistent logic. To this end we take an extension of da Costa'sC 1 = (obtained by adding the axiom ⌝⌝A ↔A) and prove for it results which correspond to many major classical model theories, taken from Shoenfield [5]. In particular we prove counterparts of the theorems of Łoś-Tarski and Chang-Łoś-Suszko, Craig-Robinson and the Beth definability theorem.
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    On the relevant systemsP andP * and some related systems (1984)
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    Studia logica 43 (1984), S. 33-49 
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    Notes: Abstract In this paper we study the systemsP andP * (see Arruda and da Costa,O paradoxo de Curry-Moh Shaw-Kwei, Boletim da Sociedade Matemātica de São Paulo 18 (1966)) and some related systems. In the last section, we prove that certain set theories havingP andP * as their underlying logics are non-trivial.
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  • 25
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    Zeno's paradoxes and temporal becoming in dialectical atomism (1984)
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    Studia logica 43 (1984), S. 169-180 
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    Notes: Abstract The homogeneity of time (i.e. the fact that there are no privileged moments) underlies a fundamental symmetry relating to the energy conservation law. On the other hand the obvious asymmetry between past and future, expressed by the metaphor of the “arrow of time” or “flow of time” accounts for the irreversibility of what happens. One takes this for granted but the conceptual tension it creates against the background of time's presumed homogeneity calls for an explanation of temporal becoming. Here, it is approached with the help of a claim to the effect that the instant (moment) itself has a structure isomorphic to that of time as a whole. Then the asymmetry of past and future in regard to temporal becoming is associated with the internal structure of the very moment, and not with external relations between different moments of time. In this paper ideas of ancient atomism and contemporary dialectics are brought together. It is for the sake of a contrast to what is known as logical atomism that I choose to call this view dialectical atomism. The latter admits dialectical contradictions and, so far as the logical status of contradictions is concerned, bears reference to paraconsistent logics. In the paper there is an outline of a method of converting any consistent axiomatic formal system into a paraconsistent theory.
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    Correspondence as an intertheory relation (1983)
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    Studia logica 42 (1983), S. 363-371 
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    Notes: Abstract In this paper we give the gist of our reconstructed notion of (limiting case) correspondence. Our notion is very general, so that it should be applicable to all the cases in which a correspondence has been said to exist in actual science.
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    On the eliminative explanation of social theories (1983)
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    Studia logica 42 (1983), S. 331-345 
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    Notes: Abstract The paper discusses eliminative explanation in which a (social) successor theory correctively explains and, as a consequence, eliminates its predecessor theory. Technical concepts and results from general logic are applied to the explication of corrective explanation, especially to the notion of framework translation that it involves.
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    On the axiomatization of finiteK-frames (1983)
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    Studia logica 42 (1983), S. 383-388 
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    Notes: Abstract We find a short way to construct a formula which axiomatizes a given finite frame of the modal logicK, in the sense that for each finite frameA, we construct a formula ωA which holds in those and only those frames in which every formula true inA holds. To obtain this result we find, for each finite model $$\mathfrak{A}$$ and each natural numbern, a formula ω $$\mathfrak{A}$$ which holds in those and only those models in which every formula true in $$\mathfrak{A}$$ , and involving the firstn propositional letters, holds.
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    The representation of Takeuti's $$\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $$ -operator (1983)
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    Studia logica 42 (1983), S. 407-415 
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    Notes: Abstract Gaisi Takeuti has recently proposed a new operation on orthomodular latticesL, $$\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $$ :P(L)»L. The properties of $$\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $$ suggest that the value of $$\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $$ (A) (A) $$ \subseteq $$ L) corresponds to the degree in which the elements ofA behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular latticesL and the existence of two-valued homomorphisms onL.
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    The preservation of coherence (1984)
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    Studia logica 43 (1984), S. 89-106 
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    Notes: Abstract It is argued that the preservation of truth by an inference relation is of little interest when premiss sets are contradictory. The notion of a level of coherence is introduced and the utility of modal logics in the semantic representation of sets of higher coherence levels is noted. It is shown that this representative role cannot be transferred to first order logic via frame theory since the modal formulae expressing coherence level restrictions are not first order definable. Finally, an inference relation, calledyielding, is introduced which is intermediate between the coherence preservingforcing relation introduced elsewhere by the authors and the coherence destroying, inference relation of classical logic.
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    Aristotle's thesis in consistent and inconsistent logics (1984)
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    Studia logica 43 (1984), S. 107-116 
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    Notes: Abstract A typical theorem of conaexive logics is Aristotle's Thesis(A), ∼(A→∼A).A cannot be added to classical logic without producing a trivial (Post-inconsistent) logic, so connexive logics typically give up one or more of the classical properties of conjunction, e.g.(A & B)→A, and are thereby able to achieve not only nontriviality, but also (negation) consistency. To date, semantical modellings forA have been unintuitive. One task of this paper is to give a more intuitive modelling forA in consistent logics. In addition, while inconsistent but nontrivial theories, and inconsistent nontrivial logics employing prepositional constants (for which the rule of uniform substitution US fails), have both been studied extensively within the paraconsistent programme, inconsistent nontrivial logics (closed under US) do not seem to have been. This paper gives sufficient conditions for a logic containingA to be inconsistent, and then shows that there is a class of inconsistent nontrivial logics all containingA. A second semantical modelling forA in such logics is given. Finally, some informal remarks about the kind of modellingA seems to require are made.
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    Trees and diagrams of decomposition (1985)
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    Studia logica 44 (1985), S. 139-158 
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    Notes: Abstract We introduce here and investigate the notion of an alternative tree of decomposition. We show (Theorem 5) a general method of finding out all non-alternative trees of the alternative tree determined by a diagram of decomposition.
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    Graded modalities. I (1985)
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    Studia logica 44 (1985), S. 197-221 
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    Notes: Abstract 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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    Sentential logics and Maehara Interpolation Property (1985)
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    Studia logica 44 (1985), S. 265-283 
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    Notes: Abstract 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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    Key notions of Tarski's methodology of deductive systems (1985)
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    Studia logica 44 (1985), S. 321-351 
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    Notes: Abstract 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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    On the logic of theory change: Safe contraction (1985)
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    Studia logica 44 (1985), S. 405-422 
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    Notes: Abstract 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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    Errata (1986)
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    Studia logica 45 (1986), S. 2-2 
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    Craig's interpolation theorem for the intuitionistic logic and its extensions—A semantical approach (1986)
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    Studia logica 45 (1986), S. 19-33 
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    Notes: Abstract A semantical proof of Craig's interpolation theorem for the intuitionistic predicate logic and some intermediate prepositional logics will be given. Our proof is an extension of Henkin's method developed in [4]. It will clarify the relation between the interpolation theorem and Robinson's consistency theorem for these logics and will enable us to give a uniform way of proving the interpolation theorem for them.
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    Interpolation and amalgamation properties in varieties of equivalential algebras (1986)
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    Studia logica 45 (1986), S. 35-38 
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    Notes: Abstract Important positive as well as negative results on interpolation property in fragments of the intuitionistic propositional logic (INT) were obtained by J. I. Zucker in [6]. He proved that the interpolation theorem holds in purely implicational fragment of INT. He also gave an example of a fragment of INT for which interpolation fails. This fragment is determined by the constant falsum (⊥), well known connectives: implication (→) and conjunction (∧), and by a ternary connective δ defined as follows: δ (p, q, r)= df (p∧q)∨(ℸp∧r). Extending this result of J. I. Zucker, G. R. Renardel de Lavalette proved in [5] that there are continuously many fragments of INT without the interpolation property. This paper is meant to continue the research mentioned above. To be more precise, its aim is to answer questions concerning interpolation and amalgamation properties in varieties of equivalential algebras, particularly in the variety determined by the purely equivalential fragment of INT.
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    Three theories of nominalized predicates (1985)
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    Studia logica 44 (1985), S. 165-186 
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    Notes: Abstract 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 model-theoretic interpretation for a formal language which admits the occurrence of such abstract singular terms.
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    Decision problems for classes of diagonalizable algebras (1985)
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    Studia logica 44 (1985), S. 87-89 
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    Notes: Abstract We make use of a Theorem of Burris-McKenzie 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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    Galois structures (1985)
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    Studia logica 44 (1985), S. 109-124 
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    Notes: Abstract 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 ω-disjunctive closure space X the dual closure space is topological iff the base of X generated by this dual space consists of the ω-prime 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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    Semantical analysis of predicate logics without the contraction rule (1985)
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    Studia logica 44 (1985), S. 187-196 
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    Notes: Abstract 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 Beth-type semantics.
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    Sheaves over Heyting lattices (1985)
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    Studia logica 44 (1985), S. 237-256 
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    Notes: Abstract 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 ℒ-valued 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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    An algebraic study of well-foundedness (1985)
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    Studia logica 44 (1985), S. 423-437 
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    Notes: Abstract 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 quasi-variety 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 R-images and λ is the non-wellfounded 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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    Erratum (1985)
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    Studia logica 44 (1985), S. 446-446 
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    A cut-free gentzen-type system for the logic of the weak law of excluded middle (1986)
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    Studia logica 45 (1986), S. 39-53 
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    Notes: Abstract The logic of the weak law of excluded middleKC p is obtained by adding the formula ℸA ∨ ℸ ℸA as an axiom scheme to Heyting's intuitionistic logicH p . A cut-free sequent calculus for this logic is given. As the consequences of the cut-elimination theorem, we get the decidability of the propositional part of this calculus, its separability, equality of the negationless fragments ofKC p andH p , interpolation theorems and so on. From the proof-theoretical point of view, the formulation presented in this paper makes clearer the relations betweenKC p ,H p , and the classical logic. In the end, an interpretation of classical propositional logic in the propositional part ofKC p is given.
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    Logics of some kripke frames connected with Medvedev notion of informational types (1986)
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    Studia logica 45 (1986), S. 101-118 
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    Notes: Abstract Intermediate prepositional logics we consider here describe the setI(Ω) of regular informational types introduced by Yu. T. Medvedev [7]. He showed thatI(Ω) is a Heyting algebra. This algebra gives rise to the “logic of infinite problems” from [13] denoted here asLM 1. Some other definitions of negation inI(Ω) lead to logicsLM n (n ⩽ ω). We study inclusions between these and other systems, proveLM n to be non-finitely axiomatizable (n ⩽ ω) and recursively axiomatizable (n 〈 ω). We also show that formulas in one variable do not separateLM ∞ from Heyting's logicH, andLM n (n 〈 ω) from Scott's logic (H+S).
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    Why objects exist but events occur (1986)
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    Studia logica 45 (1986), S. 371-375 
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    Notes: Abstract I distinguish between sentences like (1) Last Thursday we drove from Wellington to Waikanae and (2) Last Thursday my copy of Aspects of the Theory of Syntax remained on my bookshelf. Sentence (2) has the subinterval property. If it is true at an interval t it is true at every subinterval of t. (1) lacks this property. (1) reports an event. (2) reports a state. Events do not have the subinterval property but states do have it, and so do objects. If something is a linguist at an interval t then that person is a linguist at all subintervals of t. I argue that ‘exists’ applies to things which have the subdinterval property, and ‘occurs’ applies to things which lack it.
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    On decidable consequence operators (1986)
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    Studia logica 45 (1986), S. 415-424 
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    Notes: Abstract The main theorem says that a consequence operator is an effective part of the consequence operator for the classical prepositional calculus iff it is a consequence operator for a logic satisfying the compactness theorem, and in which every finitely axiomatizable theory is decidable.
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    Countably-categorical Boolean algebras with distinguished ideals (1987)
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    Studia logica 46 (1987), S. 121-135 
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    Notes: Abstract In the paper all countable Boolean algebras with m distinguished. ideals having countably-categorical elementary theory are described and constructed. From the obtained characterization it follows that all countably-categorical elementary theories of Boolean algebras with distinguished ideals are finite-axiomatizable, decidable and, consequently, their countable models are strongly constructivizable.
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    First-order fuzzy logic (1987)
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    Studia logica 46 (1987), S. 87-109 
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    Notes: Abstract This paper is an attempt to develop the many-valued first-order fuzzy logic. The set of its truth, values is supposed to be either a finite chain or the interval 〈0, 1〉 of reals. These are special cases of a residuated lattice 〈L, ∨, ∧, ⊗, →, 1, 0〉. It has been previously proved that the fuzzy propositional logic based on the same sets of truth values is semantically complete. In this paper the syntax and semantics of the first-order fuzzy logic is developed. Except for the basic connectives and quantifiers, its language may contain also additional n-ary connectives and quantifiers. Many propositions analogous to those in the classical logic are proved. The notion of the fuzzy theory in the first-order fuzzy logic is introduced and its canonical model is constructed. Finally, the extensions of Gödel's completeness theorems are proved which confirm that the first-order fuzzy logic is also semantically complete.
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    Semi-Post algebras (1987)
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    Studia logica 46 (1987), S. 149-160 
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    Notes: Abstract In this paper, semi-Post algebras are introduced and investigated. The generalized Post algebras are subcases of semi-Post algebras. The so called primitive Post constants constitute an arbitrary partially ordered set, not necessarily connected as in the case of the generalized Post algebras examined in [3]. By this generalization, semi-Post products can be defined. It is also shown that the class of all semi-Post algebras is closed under these products and that every semi-Post algebra is a semi-Post product of some generalized Post algebras.
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    Studia logica 46 (1987), S. 205-207 
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    Relevant implication and the weak deduction theorem (1987)
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    Studia logica 46 (1987), S. 239-245 
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    Notes: Abstract It is shown that the implicational fragment of Anderson and Belnap's R, i.e. Church's weak implicational calculus, is not uniquely characterized by MP (modus ponens), US (uniform substitution), and WDT (Church's weak deduction theorem). It is also shown that no unique logic is characterized by these, but that the addition of further rules results in the implicational fragment of R. A similar result for E is mentioned.
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    W-algebras which are Boolean products of members of SR[1] and CW-algebras (1987)
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    Studia logica 46 (1987), S. 265-274 
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    Notes: Abstract We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.
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    Studia logica 46 (1987), S. 279-282 
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    Axiomatizable classes with strong homomorphisms (1987)
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    Studia logica 46 (1987), S. 113-120 
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    Notes: Abstract In the paper A. I. Malcev's problem on the characterization of axioms for classes with strong homomorphisms is being solved.
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    No non-trivial quasivariety of BCK-algebras has decidable first order theory (1987)
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    Studia logica 46 (1987), S. 343-345 
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    Notes: Abstract Using the semantic embedding technique the theorem announced by the title is proved.
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    A simplification of a completeness proof of Guaspari and Solovay (1987)
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    Studia logica 46 (1987), S. 187-192 
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    Notes: Abstract The modal completeness proofs of Guaspari and Solovay (1979) for their systems R and R − are improved and the relationship between R and R − is clarified.
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    Gaifman's theorem on categorial grammars revisited (1988)
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    Studia logica 47 (1988), S. 23-33 
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    Notes: Abstract The equivalence of (classical) categorial grammars and context-free grammars, proved by Gaifman [4], is a very basic result of the theory of formal grammars (an essentially equivalent result is known as the Greibach normal form theorem [1], [14]). We analyse the contents of Gaifman's theorem within the framework of structure and type transformations. We give a new proof of this theorem which relies on the algebra of phrase structures and exhibit a possibility to justify the key construction used in Gaifman's proof by means of the Lambek calculus of syntactic types [15].
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    Remarks on a survey article on many valued logic by A. Urquhart (1987)
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    Studia logica 46 (1987), S. 275-278 
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    Infinitary prepositional normal modal logic (1987)
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    Studia logica 46 (1987), S. 291-309 
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    Notes: Abstract A logic with normal modal operators and countable infinite conjunctions and disjunctions is introduced. A Hilbert's style axiomatization is proved complete for this logic, as well as for countable sublogics and subtheories. It is also shown that the logic has the interpolation property.
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    Variations on the Ramsey test: More triviality results (1987)
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    Studia logica 46 (1987), S. 321-327 
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    Notes: Abstract The purpose of this note is to formulate some weaker versions of the so called Ramsey test that do not entail the following unacceptable consequence If A and C are already accepted in K, then “if A, then C” is also accepted in K. and to show that these versions still lead to the same triviality result when combined with a preservation criterion.
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    A weak intuitionistic propositional logic with purely constructive implication (1987)
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    Studia logica 46 (1987), S. 371-382 
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    Notes: Abstract We introduce subsystems WLJ and SI of the intuitionistic propositional logic LJ, by weakening the intuitionistic implication. These systems are justifiable by purely constructive semantics. Then the intuitionistic implication with full strength is definable in the second order versions of these systems. We give a relationship between SI and a weak modal system WM. In Appendix the Kripke-type model theory for WM is given.
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    Modal operators with probabilistic interpretations, I (1987)
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    Studia logica 46 (1987), S. 383-393 
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    Notes: Abstract We present a class of normal modal calculi PFD, whose syntax is endowed with operators M r (and their dual ones, L r), one for each r ε [0,1]: if a is α sentence, M rα is to he read “the probability that a is true is strictly greater than r” and to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model. Every such a model is a kripkean model, enriched by a family of regular (see below) probability evaluations with range in a fixed finite subset F of [0,1]: there is one such a function for every world w, P F(w,-), and this allows to evaluate M ra as true in the world w iff p F(w, α) 〉 r. For every fixed F as before, suitable axioms and rules are displayed, so that the resulting system P FD is complete and compact with respect to the class of all the F-restricted probabilistic kripkean models.
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    Normal predicative logics with graded modalities (1988)
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    Studia logica 47 (1988), S. 11-22 
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    Notes: Abstract In this work we extend results from [4], [3] and [2] about propositional calculi with graded modalities to the predicative level. Our semantic is based on Kripke models with a single domain of interpretation for all the worlds. Therefore the axiomatic system will need a suitable generalization of the Barcan formula. We haven't considered semantics with world-relative domains because they don't present any new difficulties with respect to classical case. Our language will have, as in [1], constant and function symbols, but they will have a rigid interpretation. In this instance the terms will be considered “rigid designators”, that is, their interpretations will be the same in all possible worlds (cfr. [5]). Nevertheless we will not consider languages with identity: in this way we avoid the problems of identity in modal contexts. On the other hand, we think that in that context would arise essentially the same problems which occur in classical predicative modal logic. Finally we will consider only denumerable languages because the extension to non denumerable ones isn't conceptually problematic.
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    The adequacy condition as a definition of elementary interpretation (1988)
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    Studia logica 47 (1988), S. 57-69 
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    Notes: Abstract A definition of elementary interpretation, equivalent (up to isomorphisms) to the ones of [3] and [4], is given. The defining condition, used here, seems to confirm that intuitions agree with the choice of the class of elementary interpretations, which was done in [3].
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    Graded modalities. III (the completeness and compactness of S40) (1988)
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    Studia logica 47 (1988), S. 99-110 
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    Notes: Abstract We go on along the trend of [2] and [1], giving an axiomatization of S4 0 and proving its completeness and compactness with respect to the usual reflexive and transitive Kripke models. To reach this results, we use techniques from [1], with suitable adaptations to our specific case.
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    On finite linear intermediate predicate logics (1988)
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    Studia logica 47 (1988), S. 391-399 
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    Notes: Abstract An intermediate predicate logicS + n (n〉0) is introduced and investigated. First, a sequent calculusGS n is introduced, which is shown to be equivalent toS + n and for which the cut elimination theorem holds. In § 2, it will be shown thatS + n is characterized by the class of all linear Kripke frames of the heightn.
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    Alpha-conversion, conditions on variables and categorical logic (1989)
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    Studia logica 48 (1989), S. 319-360 
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    Notes: Abstract We present the paradigm of categories-as-syntax. We briefly recall the even stronger paradigm categories-as-machine-language which led from λ-calculus to categorical combinators viewed as basic instructions of the Categorical Abstract Machine. We extend the categorical combinators so as to describe the proof theory of first order logic and higher order logic. We do not prove new results: the use of indexed categories and the description of quantifiers as adjoints goes back to Lawvere and has been developed in detail in works of R. Seely. We rather propose a syntactic, equational presentation of those ideas. We sketch the (quasi-equational) categorical structures for dependent types, following ideas of J. Cartmell (contextual categories). All these theories of categorical combinators, together with the translations from λ-calculi into them, are introduced smoothly, thanks to the systematic use of - an abstract variable-free notation for λ-calculus, going back to N. De Bruijn, - a sequent formulation of the natural deduction.
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    Jerzy Słupecki (1904–1987): Life and work (1989)
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    Studia logica 48 (1989), S. 401-411 
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    Sequent-systems and groupoid models. II (1989)
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    Studia logica 48 (1989), S. 41-65 
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    Notes: Abstract The purpose of this paper is to connect the proof theory and the model theory of a family of prepositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related to BCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hilbert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics. Below is the sequel to the first part of the paper, which appeared in the previous issue of this journal (vol. 47 (1988), pp. 353–386). The first part contained sections on sequent-systems and Hilbert-formulations, and here is the third section on groupoid models. This second part is meant to be read in conjunction with the first part.
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    Intuitive semantics for some three-valued logics connected with information, contrariety and subcontrariety (1989)
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    Studia logica 48 (1989), S. 565-575 
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    Notes: Abstract Four known three-valued logics are formulated axiomatically and several completeness theorems with respect to nonstandard intuitive semantics, connected with the notions of information, contrariety and subcontrariety is given.
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    On comparison of theories by their contents (1989)
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    Studia logica 48 (1989), S. 617-622 
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    Notes: Abstract Popper's definition of verisimilitude was criticized for its paradoxical consequences in the case of false theories. The aim of this paper is to show that paradoxes disappear if the falsity content of a theory is defined with help of dCn or Cn −1.
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    Pre-Announcement International Symposium on ‘structures in mathematical theories’ (1989)
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    Studia logica 48 (1989), S. I 
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    First-order modal theories (1980)
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    Studia logica 39 (1980), S. 159-202 
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    Notes: Abstract This paper is part of a general programme of developing and investigating particular first-order modal theories. In the paper, a modal theory of propositions is constructed under the assumption that there are genuinely singular propositions, ie. ones that contain individuals as constituents. Various results on decidability, axiomatizability and definability are established.
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    A modal sequent calculus for a fragment of arithmetic (1980)
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    Studia logica 39 (1980), S. 245-256 
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    Notes: Abstract Global properties of canonical derivability predicates (the standard example is Pr() in Peano Arithmetic) are studied here by means of a suitable propositional modal logic GL. A whole book [1] has appeared on GL and we refer to it for more information and a bibliography on GL. Here we propose a sequent calculus for GL and, by exhibiting a good proof procedure, prove that such calculus admits the elimination of cuts. Most of standard results on GL are then easy consequences: completeness, decidability, finite model property, interpolation and the fixed point theorem.
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    On the size of refutation Kripke models for some linear modal and tense logics (1980)
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    Studia logica 39 (1980), S. 325-333 
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    Notes: Abstract LetL be any modal or tense logic with the finite model property. For eachm, definer L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr L (m) for some linear modal and tense logicsL.
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    Intuitionistic uniformity principles for propositions and some applications (1980)
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    Studia logica 39 (1980), S. 361-369 
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    Notes: Abstract This note deals with the prepositional uniformity principlep-UP: ∧p ∈ Κ ∨x ∈N A (p, x) → ∨x ∈N ∧p ∈ Κ A (p, x) (Ω species of all propositions) in intuitionistic mathematics.p-UP is implied by WC and KS. But there are interestingp-UP-cases which require weak KS resp. WC only. UP for number species follows fromp-UP by extended bar-induction (ranging over propositions) and suitable weak continuity. As corollaries we have the disjunction property and the existential definability w.r.t. concrete objects. Other consequences are: there is no non-trivial countable partition ofΩ;id is the only injective function fromΩ toΩ; there are no many-place injective prepositional functions; card (Ω) is incomparable with the cardinality of all metric spaces containing at least three elements.
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    On constructive functions ranging over propositions (1980)
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    Studia logica 39 (1980), S. 371-374 
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    Notes: Abstract It is shown that there is no constructive extensional truth-value mapping from the speciesP of all propositions into known constructive structures ≠ P.
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    λ-Normal forms in an intensional logic for English (1980)
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    Studia logica 39 (1980), S. 311-324 
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    Notes: Abstract Montague [7] translates English into a tensed intensional logic, an extension of the typed λ-calculus. We prove that each translation reduces to a formula without λ-applications, unique to within change of bound variable. The proof has two main steps. We first prove that translations of English phrases have the special property that arguments to functions are modally closed. We then show that formulas in which arguments are modally closed have a unique fully reduced λ-normal form. As a corollary, translations of English phrases are contained in a simply defined proper subclass of the formulas of the intensional logic.
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    The undecidability of the first-order theory of diagonalizable algebras (1980)
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    Studia logica 39 (1980), S. 355-359 
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    Notes: Abstract The undecidability of the first-order theory of diagonalizable algebras is shown here.
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    Quantified modal logic: Non-normal worlds and propositional attitudes (1982)
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    Studia logica 41 (1982), S. 41-65 
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    Notes: Abstract One way to obtain a comprehensive semantics for various systems of modal logic is to use a general notion of non-normal world. In the present article, a general notion of modal system is considered together with a semantic framework provided by such a general notion of non-normal world. Methodologically, the main purpose of this paper is to provide a logical framework for the study of various modalities, notably prepositional attitudes. Some specific systems are studied together with semantics using non-normal worlds of different kinds.
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    Books received (1982)
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    Studia logica 41 (1982), S. 83-90 
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    The semantics of Hoare's Iteration Rule (1982)
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    Studia logica 41 (1982), S. 141-158 
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    Notes: Abstract Hoare's Iteration Rule is a principle of reasoning that is used to derive correctness assertions about the effects of implementing a while-command. We show that the propositional modal logic of this type of command is axiomatised by Hoare's rule in conjunction with two additional axioms. The proof also establishes decidability of the logic. The paper concludes with a discussion of the relationship between the logic of “while” and Segerberg's axiomatisation of propositional dynamic logic.
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    A propositional fragment of Leśniewski's ontology and its formulation by the tableau method (1982)
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    Studia logica 41 (1982), S. 181-195 
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    Notes: Abstract The propositional fragment L 1 of Leśniewski's ontology is the smallest class (of formulas) containing besides all the instances of tautology the formulas of the forms: ɛ(a, b) ⊃ ɛ(a, a), ɛ(a, b) ∧ ɛ(b,).⊃ ɛ(a, c) and ɛ(a, b) ∧ ɛ(b, c). ⊃ ɛ(b, a) being closed under detachment. The purpose of this paper is to furnish another more constructive proof than that given earlier by one of us for: Theorem A is provable in L 1 iff TA is a thesis of first-order predicate logic with equality, where T is a translation of the formulas of L 1 into those of first-order predicate logic with equality such that Tɛ(a, b) = FblxFax (Russeltian-type definite description), TA ∨ B = TA ∨ TB, T ∼ A = ∼TA, etc. For the proof of this theorem use is made of a tableau method based upon the following reduction rules: $$\begin{gathered} \frac{{G\left[ {A \vee B} \right]}}{{G\left[ {A \vee B{\text{\_}}} \right] \vee \sim A|G|[A \vee B\_] \vee \sim B,}}{\text{ }}\frac{{G[\varepsilon (a,b)\_]}}{{G[\varepsilon (a,b)\_] \vee \sim \varepsilon (a,a),}} \hfill \\ \frac{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_]}}{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_], \vee \sim \varepsilon (a,c),}}{\text{ }}\frac{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_]}}{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_] \vee \sim \varepsilon (b,a),}} \hfill \\ \end{gathered} $$ where F[A +] (G[A−]) means that A occurs in F[A +] (G[A−]) as its positive (negative) part in accordance with the definition given by Schütte.
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    A deontic logic of action (1982)
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    Studia logica 41 (1982), S. 269-282 
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    Notes: Abstract The formal language studied in this paper contains two categories of expressions, terms and formulas. Terms express events, formulas propositions. There are infinitely many atomic terms and complex terms are made up by Boolean operations. Where α and β are terms the atomic formulas have the form α=β (α is the same as β), Forb α (α is forbidden) and Perm α (α is permitted). The formulae are truth functional combinations of these. An algebraic and a model theoretic account of validity are given and an axiomatic system is provided for which they are characteristic. The ‘closure principle’, that what is not forbidden is permitted is shown to hold at the level of outcomes but not at the level of events. In the two final sections some other operators are considered and a semantics in terms of action games.
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    Meinongian type theory and its applications (1982)
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    Studia logica 41 (1982), S. 297-307 
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    Notes: Abstract In this paper I propose a fundamental modification of standard type theory, produce a new kind of type theoretic language, and couch in this language a comprehensive theory of abstract individuals and abstract properties and relations of every type. I then suggest how to employ the theory to solve the four following philosophical problems: (A) the identification and ontological status of Frege's Senses; (B) the deviant behavior of terms in propositional attitude contexts; (C) the non-identity of necessarily equivalent propositions, and (D) the “paradox” of analysis. We can roughly describe these solutions as follows: (A) the senses of English names and descriptions which denote individuals will be modelled as abstract individuals; the senses of English relation denoting expressions of a given type will be modelled as abstract relations of that type. (B) Inside de dicto attitude contexts, these English expressions denote (the abstract objects which serve as) their senses. (C) Relations and propositions will not be identified with their extensions, nor with functions or sets of any kind. They will be taken as primitive, and precise “being” and identity conditions will be proposed consistent with the view that necessarily equivalent relations and propositions may be distinct. (D) With the modelling described in (A), the expressions “being a brother” and “being a male sibling” may both denote the same property, though (the abstract properties which serve as) their senses may differ. Just as Frege predicts, “being a brother just is being a male sibling” is an informative identity statement because the terms flanking the identity sign have the same denotation, though distinct senses.
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    What is quantum logic? (1982)
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    Studia logica 41 (1982), S. 311-316 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The paper describes in detail the procedure of identification of the inner language and an inner logico of a physical theory. The procedure is a generalization of the original ideas of J. von Neuman and G. Birkhoff about quantum logic.
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    The lattice of strengthenings of a strongly finite consequence operation (1981)
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    Studia logica 40 (1981), S. 177-193 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract First, we prove that the lattice of all structural strengthenings of a given strongly finite consequence operation is both atomic and coatomic, it has finitely many atoms and coatoms, each coatom is strongly finite but atoms are not of this kind — we settle this by constructing a suitable counterexample. Second, we deal with the notions of hereditary: algebraicness, strong finitisticity and finite approximability of a strongly finite consequence operation. Third, we formulate some conditions which tell us when the lattice of all structural strengthenings of a given strongly finite consequence operation is finite, and subsequently we give some applications of them.
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    2-Element matrices (1981)
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    Studia logica 40 (1981), S. 315-353 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Sections 1, 2 and 3 contain the main result, the strong finite axiomatizability of all 2-valued matrices. Since non-strongly finitely axiomatizable 3-element matrices are easily constructed the result reveals once again the gap between 2-valued and multiple-valued logic. Sec. 2 deals with the basic cases which include the important F i ∞ from Post's classification. The procedure in Sec. 3 reduces the general problem to these cases. Sec. 4 is a study of basic algebraic properties of 2-element algebras. In particular, we show that equational completeness is equivalent to the Stone-property and that each 2-element algebra generates a minimal quasivariety. The results of Sec. 4 will be applied in Sec. 5 to maximality questions and to a matrix free characterization of 2-valued consequences in the lattice of structural consequences in any language. Sec. 6 takes a look at related axiomatization. problems for finite algebras and matrices. We study the notion of a propositional consequence with equality and, among other things, present explicit axiomatizations of 2-valued consequences with equality.
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    Distributive lattices with a dual homomorphic operation. II (1981)
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    Studia logica 40 (1981), S. 391-404 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract An Ockham lattice is defined to be a distributive lattice with 0 and 1 which is equipped with a dual homomorphic operation. In this paper we prove: (1) The lattice of all equational classes of Ockham lattices is isomorphic to a lattice of easily described first-order theories and is uncountable, (2) every such equational class is generated by its finite members. In the proof of (2) a characterization of orderings of ω with respect to which the successor function is decreasing is given.
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    Tadeusz Kotarbiński (1886–1981) (1982)
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    Studia logica 41 (1982), S. 1-2 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
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    Preface (1983)
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    Studia logica 42 (1983), S. 117-118 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
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    The strong proof from hypotheses and conditionals: Some theorems of deduction for relevant systems (1983)
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    Studia logica 42 (1983), S. 165-171 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The aim of this paper is to present a modified version of the notion of strong proof from hypotheses (definition D2), and to give three deduction theorems for the relevant logicsR (theoremsT1, andT2) andE (theoremT3).
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    The logic of algebraic rules as a generalization of equational logic (1983)
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    Studia logica 42 (1983), S. 251-257 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract In this paper we start an investigation of a logic called the logic of algebraic rules. The relation of derivability of this logic is defined on universal closures of special disjunctions of equations extending the relation of derivability of the usual equational logic. The paper contains some simple theorems and examples given in justification for the introduction of our logic. A number of open questions is posed.
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    Logical semantics as an empirical science (1983)
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    Studia logica 42 (1983), S. 299-313 
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    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Exact philosophy consists of various disciplines scattered and separated. Formal semantics and philosophy of science are good examples of two such disciplines. The aim of this paper is to show that there is possible to find some integrating bridge topics between the two fields, and to show how insights from the one are illuminating and suggestive in the other.
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    The logic of viewpoints (1983)
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    Studia logica 42 (1983), S. 187-196 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract In this paper a propositional logic of viewpoints is presented. The language of this logic consists of the usual modal operatorsL (of necessity) andM (of possibility) as well as of two new operatorsA andR. The intuitive interpretations ofA andR are “from all viewpoints” and “from some viewpoint”, respectively. Semantically the language is interpreted by using Kripke models augmented with sets of “viewpoints” and with a new alternativeness relation for the operatorA. Truth values of formulas are evaluated with respect to a world and a viewpoint. Various axiomatizations of the logic of viewpoints are presented and proved complete. Finally, some applications are given.
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    Cardinalities of proper ideals in some lattices of strengthenings of the intuitionistic propositional logic (1983)
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    Studia logica 42 (1983), S. 173-177 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract We prove that each proper ideal in the lattice of axiomatic, resp. standard strengthenings of the intuitionistic propositional logic is of cardinality 2ℵ0. But, each proper ideal in the lattice of structural strengthenings of the intuitionistic propositional logic is of cardinality 22ℵ0. As a corollary we have that each of these three lattices has no atoms.
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