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  • Articles  (3,096)
  • Springer  (3,096)
  • National Academy of Sciences
  • 1995-1999  (2,461)
  • 1975-1979  (635)
  • Philosophy  (3,096)
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  • Articles  (3,096)
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  • 1
    Electronic Resource
    Electronic Resource
    Remarks on discussive propositional calculus (1975)
    Springer
    Studia logica 34 (1975), S. 39-43 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02314422
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  • 2
    Electronic Resource
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    On the Jaśkowski's method of suppositions (1975)
    Springer
    Studia logica 34 (1975), S. 187-200 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02123386
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  • 3
    Electronic Resource
    Electronic Resource
    Some basic theorems of qualitative probability (1975)
    Springer
    Studia logica 34 (1975), S. 257-264 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02125228
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  • 4
    Electronic Resource
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    Notes on the Rasiowa-Sikorski lemma (1975)
    Springer
    Studia logica 34 (1975), S. 265-268 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper aims at formulating a condition neccssary and sufficient for the existing of a prime filter preserving enumerable infinite joins and meets in a distributive lattice.
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    URL: http://dx.doi.org/10.1007/BF02125229
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  • 5
    Electronic Resource
    Electronic Resource
    Eliminability in a cardinal (1976)
    Springer
    Studia logica 35 (1976), S. 71-89 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02120953
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  • 6
    Electronic Resource
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    On the autological character of diagonalizable algebras (1976)
    Springer
    Studia logica 35 (1976), S. 327-333 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Summary Let ℱ be the first order theory of diagonalizable algebras. We define a bijection ϕ from the atomic formulas of ℱ (identities) to the open formulas of ℱ. ϕ is an algebraic analogous of ≒. We prove that ϕ, ϕ-1 preserve the validity.
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  • 7
    Electronic Resource
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    A very strong intuitionistic theory (1976)
    Springer
    Studia logica 35 (1976), S. 377-385 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02123404
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  • 8
    Electronic Resource
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    The gentzen style axiomatization of 433-1433-1433-1logic (1976)
    Springer
    Studia logica 35 (1976), S. 433-445 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02123408
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  • 9
    Electronic Resource
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    A Kripke-style semantics for R-Mingle using a binary accessibility relation (1976)
    Springer
    Studia logica 35 (1976), S. 163-172 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02120878
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  • 10
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    Some results concerning finite model separability of propositional calculi (1976)
    Springer
    Studia logica 35 (1976), S. 179-189 
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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/BF02120880
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  • 11
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    Corrigenda (1977)
    Springer
    Studia logica 36 (1977), S. 240-240 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02121271
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  • 12
    Electronic Resource
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    The Leśniewski-Curry theory of syntactical categories and the categorially open functors (1976)
    Springer
    Studia logica 35 (1976), S. 191-201 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02120881
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  • 13
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    Reviews (1976)
    Springer
    Studia logica 35 (1976), S. 203-211 
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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/BF02120882
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  • 14
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    Hyperintensional logic (1975)
    Springer
    Studia logica 34 (1975), S. 25-38 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02314421
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  • 15
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    Systems of Leśniewski's ontology with the functor of weak inclusion as the only primitive term (1977)
    Springer
    Studia logica 36 (1977), S. 323-349 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02120669
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  • 16
    Electronic Resource
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    Quantifiers in ontology (1977)
    Springer
    Studia logica 36 (1977), S. 301-307 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper is a reaction to G. Küng's and J. T. Canty's ‘Substitutional Quantification and Leśniewskian quantifiers'Theoria 36 (1970), 165–182. I reject their arguments that quantifiers in Ontology cannot be referentially interpreted but I grant that there is what can be called objectual — referential interpretation of quantifiers and that because of the unrestricted quantification in Ontology the quantifiers in Ontology should not be given a so-called objectual-referential interpretation. I explain why I am in agreement with Küng and Canty's recommendation that Ontology's quantifiers not be substitutionally interpreted even if Leśniewski intended them to be so interpreted. A notion of an interpretation which is referential but yet which does not interpret ∃ as an assertor of existence of objects in a domain is developed. It is then shown that a first order version of Ontology is satisfied by those special kind of referential interpretations which read ∃ as ‘Something’ as epposed to ‘Something existing’.
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  • 17
    Electronic Resource
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    Some preliminary suggestions for the mirroring of non-metaphysical modalities in Leśniewski's ontology (1977)
    Springer
    Studia logica 36 (1977), S. 363-376 
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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/BF02120671
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  • 18
    Electronic Resource
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    Semantics for finitary predicate calculi (1976)
    Springer
    Studia logica 35 (1976), S. 227-241 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02282485
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  • 19
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    Projective and inductive generation of abstract logics (1976)
    Springer
    Studia logica 35 (1976), S. 249-255 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract An abstract logic 〈A, C〉 consists of a finitary algebraA and a closure systemC onA. C induces two other closure systems onA, C P andC I, by projective and inductive generation respectively. The various relations amongC, C P andC I are determined. The special case thatC is the standard equational closure system on monadic terms is studied in detail. The behavior of Boolean logics with respect to projective and inductive generation is determined.
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  • 20
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    A correction to the paper “On expandability of models of Peano arithmetic. I” (1977)
    Springer
    Studia logica 36 (1977), S. 237-237 
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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/BF02121269
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  • 21
    Electronic Resource
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    The uniqueness of the fixed-point in every diagonalizable algebra (1976)
    Springer
    Studia logica 35 (1976), S. 335-343 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Summary It is well known that, in Peano arithmetic, there exists a formulaTheor (x) which numerates the set of theorems. By Gödel's and Löb's results, we have that Therefore, the considered «equations» admit, up to provable equivalence, only one solution. In this paper we prove (Corollary 1), that, in general, ifP (x) is an arbitrary formula built fromTheor (x), then the fixed-point ofP (x) (which exists by the diagonalization lemma) is unique up to provable equivalence. This result is settled referring to the concept of diagonalizable algebra (see Introduction).
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  • 22
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    A new way of normalizing intuitionist propositional logic (1976)
    Springer
    Studia logica 35 (1976), S. 387-408 
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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/BF02123405
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  • 23
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    On finitely based consequence operations (1976)
    Springer
    Studia logica 35 (1976), S. 453-458 
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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/BF02123410
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  • 24
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    Applications of Kripke models to Heyting-Brouwer logic (1977)
    Springer
    Studia logica 36 (1977), S. 61-71 
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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/BF02121115
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  • 25
    Electronic Resource
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    Possible worlds and many truth values (1978)
    Springer
    Studia logica 37 (1978), S. 195-204 
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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/BF02124804
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  • 26
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    An axiomatization of the modal theory of the veiled recession frame (1979)
    Springer
    Studia logica 38 (1979), S. 37-47 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The veiled recession frame has served several times in the literature to provide examples of modal logics failing to have certain desirable properties. Makinson [4] was the first to use it in his presentation of a modal logic without the finite model property. Thomason [5] constructed a (rather complicated) logic whose Kripke frames have an accessibility relation which is reflexive and transitive, but which is satisfied by the (non-transitive) veiled recession frame, and hence incomplete. In Van Benthem [2] the frame was an essential tool to find simple examples of incomplete logics, axiomatized by a formula in two proposition letters of degree 2, or by a formula in one proposition letter of degree 4 (the degree of a modal formula ϕ is the maximal number of nested occurrences of the necessity operator in ϕ). In [3] we showed that the modal logic determined by the veiled recession frame is incomplete, and besides that, is an immediate predecessor of classical logic (or, more precisely, the modal logic axiomatized by the formula p↔□p), and hence is a logic, maximal among the incomplete ones. Considering the importance of the modal logic determined by the veiled recession frame, it seems worthwhile to ask for an axiomatization, and in particular, to answer the question if it is finitely axiomatizable. In the present paper we find a finite axiomatization of the logic, and in fact, a rather simple one consisting of formulas in at most two proposition letters and of degree at most three.
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  • 27
    Electronic Resource
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    On defining necessity in terms of entailment (1979)
    Springer
    Studia logica 38 (1979), S. 95-104 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract In their book Entailment, Anderson and Belnap investigate the consequences of defining ‘Lp’ (it is necessary that p) in system E as (p→p)→p. Since not all theorems are equivalent in E, this raises the question of whether there are reasonable alternative definitions of necessity in E. In this paper, it is shown that a definition of necessity in E satisfies the conditions {⊢ E Lp→p, ⊢EL(p→q)→(Lp→Lq), ⊢E p→Lp} if and only if its has the form C 1→.C2→ ...→. Cn→p, where each C iis equivalent in E to either p→p or ((p→p)→p)→p.
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    URL: http://dx.doi.org/10.1007/BF00370435
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  • 28
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    On analysing relevance constructively (1979)
    Springer
    Studia logica 38 (1979), S. 171-185 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper lays out a program for analysing relevance constructively. It begins with a summary of results concerning the system C of Pottinger [197a] which has entailment, relevant implication, S4 strict implication, and intuitionist implication among its connectives. A full working out of the motivation for C will require formal analysis of informal concepts derived from the usual explanation of the meanings of the constants of intuitionist propositional logic. Formal machinery which should be adequate for the proof theoretic side of the analysis of the nonmodal part of C is described in detail, and the direction in which semantical results are to be sought is indicated.
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  • 29
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    An extension of the Łukasiewicz logic to the modal logic of quantum mechanics (1978)
    Springer
    Studia logica 37 (1978), S. 149-155 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract An attempt is made to include the axioms of Mackey for probabilities of experiments in quantum mechanics into the calculusŁ x0 of Łukasiewicz. The obtained calculusŁQ contains an additional modal signQ and four modal rules of inference. The propositionQx is read “x is confirmed”. The most specific rule of inference may be read: for comparable observations implication is equivalent to confirmation of material implication. The semantic truth ofŁQ is established by the interpretation with the help of physical objects obeying to the rules of quantum mechanics. The embedding of the usual quantum propositional logic inŁQ is accomplished.
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  • 30
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    Foundations of Probability theory, statistical inference, and statistical theories of science (1978)
    Springer
    Studia logica 37 (1978), S. 213-219 
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    URL: http://dx.doi.org/10.1007/BF02124806
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  • 31
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    Editorial note (1979)
    Springer
    Studia logica 38 (1979), S. 315-316 
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    URL: http://dx.doi.org/10.1007/BF00405389
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  • 32
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    On two problems of Harvey Friedman (1979)
    Springer
    Studia logica 38 (1979), S. 247-262 
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    Topics: Mathematics , Philosophy
    Notes: Abstract The paper considers certain properties of intermediate and moda propositional logics. The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule ∼x→y∨z/(∼x→y)∨(∼x→z). The second part includes a proof of the theorem ensuring existence of a greatest structurally complete intermediate logic having the disjunction property. This theorem confirms H. Friedman's conjecture 41 (cf. [2], problem 41). In the third part the reader will find a criterion which allows us to obtain sets satisfying the conditions of Friedman's problem 42, on the basis of intermediate logics satisfying the conditions of problem 41. Finally, the fourth part contains a proof of a criterion which allows us to obtain modal logics endowed with Hallden's property on the basis of structurally complete intermediate logics having the disjunction property.
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  • 33
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    Semantics for connexive logics. I (1978)
    Springer
    Studia logica 37 (1978), S. 393-412 
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    Topics: Mathematics , Philosophy
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    URL: http://dx.doi.org/10.1007/BF02176171
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  • 34
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    Calculating self-referential statements, I: Explicit calculations (1979)
    Springer
    Studia logica 38 (1979), S. 17-36 
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    Notes: Abstract The proof of the Second Incompleteness Theorem consists essentially of proving the uniqueness and explicit definability of the sentence asserting its own unprovability. This turns out to be a rather general phenomenon: Every instance of self-reference describable in the modal logic of the standard proof predicate obeys a similar uniqueness and explicit definability law. The efficient determination of the explicit definitions of formulae satisfying a given instance of self-reference reduces to a simple algebraic problem-that of solving the corresponding fixed-point equation in the modal logic. We survey techniques for the efficient calculation of such fixed-points.
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  • 35
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    A modal calculus analogous to K4W, based on intuitionistic propositional logic, Iℴ (1979)
    Springer
    Studia logica 38 (1979), S. 297-311 
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    Topics: Mathematics , Philosophy
    Notes: Abstract This paper treats a kind of a modal logic based on the intuitionistic propositional logic which arose from the “provability” predicate in the first order arithmetic. The semantics of this calculus is presented in both a relational and an algebraic way. Completeness theorems, existence of a characteristic model and of a characteristic frame, properties of FMP and FFP and decidability are proved.
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  • 36
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    The concept of a proposition in classical and quantum physics (1979)
    Springer
    Studia logica 38 (1979), S. 337-353 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract A proposition is associated in classical mechanics with a subset of phase space, in quantum logic with a projection in Hilbert space, and in both cases with a 2-valued observable or test. A theoretical statement typically assigns a probability to such a “pure test”. However, since a pure test is an idealization not realizable experimentally, it is necessary — to give such a statement a practical meaning — to describe how it can be approximated by “feasible tests”. This gives rise to a search for a formal representation of feasible tests, which leads via mixed tests (weighted means of pure tests) to vague tests (convex sets of mixed tests). A model is described in which the latter form a “continuous lattice”; the pure and mixed tests are the maximal elements and the feasible tests form a “basis”. Each type of test has its own logic; this is illustrated by the passage from mixed tests to pure tests, which corresponds to the transition from L∞ to classical logic.
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  • 37
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    Relevant Robinson's arithmetic (1979)
    Springer
    Studia logica 38 (1979), S. 407-418 
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    Topics: Mathematics , Philosophy
    Notes: Abstract In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in relevant arithmetic.
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  • 38
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    Alternative semantics for quantified first degree relevant logic (1979)
    Springer
    Studia logica 38 (1979), S. 211-231 
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    Notes: Abstract A system FDQ of first degree entailment with quantification, extending classical quantification logic Q by an entailment connective, is axiomatised, and the choice of axioms defended and also, from another viewpoint, criticised. The system proves to be the equivalent to the first degree part of the quantified entailmental system EQ studied by Anderson and Belnap; accordingly the semantics furnished are alternative to those provided for the first degree of EQ by Belnap. A worlds semantics for FDQ is presented, and the soundness and completeness of FDQ proved, the main work of the paper going into the proof of completeness. The adequacy result is applied to yield, as well as the usual corollaries, weak relevance of FDQ and the fact that FDQ is the common first degree of a wide variety of (constant domain) quantified relevant logics. Finally much unfinished business at the first degree is discussed.
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    Call for papers on paraconsistent logic (1979)
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    Studia logica 38 (1979), S. 313-313 
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    Prepositional attitudes in weak pragmatics (1979)
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    Studia logica 38 (1979), S. 365-374 
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    Notes: Abstract Sentences attributing beliefs, doubts, wants, and the like (propositional attitudes, in Russell's terminology) have posed a major problem for semantics. Recently the pragmatic description of language has become more systematic. I shall discuss the formalization of pragmatics, and propose an analysis of belief attribution that avoids some main problems apparently inherent in the semantic approach.
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    Natural dualities for varieties ofn-valued łukasiewicz algebras (1995)
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    Studia logica 54 (1995), S. 333-370 
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    Keywords: Łukasiewicz algebra ; natural duality ; piggyback duality ; free algebra ; equational base
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    Notes: Abstract Natural dualities are developed for varieties ofn-valued Łukasiewicz algebras with and without negation. These dualities are based on hom-functors, and parallel Stone duality for Boolean algebras. A translation is described which relates the natural dualities to the corresponding restricted Priestley dualities. This enables a unified approach to free algebras to be presented, whence R. Cignoli's characterisations of the finitely generated free algebras are elucidated and new descriptions of arbitrary free algebras obtained. Finally it is shown how dualities for subvarieties encode equational bases. An alternative approach is thereby provided to results of M.E. Adams and R. Cignoli.
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    Joins of minimal quasivarieties (1995)
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    Studia logica 54 (1995), S. 371-389 
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    Keywords: Quasivariety ; lattice of quasivarieties ; distributive (0, 1)-lattice ; nullary operations ; free lattice ; Priestley space ; graph ; Primary: 06D99, 08C15 ; Secondary: 06B25
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    Notes: Abstract LetL(K) denote the lattice (ordered by inclusion) of quasivarieties contained in a quasivarietyK and letD 2 denote the variety of distributive (0, 1)-lattices with 2 additional nullary operations. In the present paperL(D 2) is described. As a consequence, ifM+N stands for the lattice join of the quasivarietiesM andN, then minimal quasivarietiesV 0,V 1, andV 2 are given each of which is generated by a 2-element algebra and such that the latticeL(V 0+V1), though infinite, still admits an easy and nice description (see Figure 2) while the latticeL(V 0+V1+V2), because of its intricate inner structure, does not. In particular, it is shown thatL(V 0+V1+V2) contains as a sublattice the ideal lattice of a free lattice with ω free generators. Each of the quasivarietiesV 0,V 1, andV 2 is generated by a 2-element algebra inD 2.
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    On Jaśkowski's discussive logics (1995)
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    Studia logica 54 (1995), S. 33-60 
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    Notes: Abstract We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.
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    Algebraic study of Sette's maximal paraconsistent logic (1995)
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    Studia logica 54 (1995), S. 89-128 
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    Keywords: Primary — 03B22, 03B53, 03G25 ; Secondary — 03C05, 08B15, 08B26, 08C15 ; paraconsistent logic ; algebraizable deductive system ; equivalent algebraic semantics ; abstract logic ; Smiley model
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    Notes: Abstract The aim of this paper is to study the paraconsistent deductive systemP 1 within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse thatP 1 is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebraS being the unique quasivariety semantics forP 1. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated byS is not equivalent to any algebraizable deductive system. We also show thatP 1 has no algebraic semantics in the sense of Czelakowski. Among other results, we study the variety generated by the algebraS. This enables us to prove in a purely algebraic way that the only proper non-trivial axiomatic extension ofP 1 is the classical deductive systemPC. Throughout the paper we also study those abstract logics which are in a way similar toP 1, and are called hereabstract Sette logics. We obtain for them results similar to those obtained for distributive abstract logics by Font, Verdú and the author.
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    Books received (1995)
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    Studia logica 54 (1995), S. 251-266 
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    Books received (1995)
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    Studia logica 54 (1995), S. 411-424 
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    The logics of orthoalgebras (1995)
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    Studia logica 55 (1995), S. 3-22 
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    Keywords: orthoalgebra ; unsharp orthoalgebra ; D-poset ; partial quantum logic
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    Notes: Abstract The notion of unsharp orthoalgebra is introduced and it is proved that the category of unsharp orthoalgebras is isomorphic to the category of D-posets. A completeness theorem for some partial logics based on unsharp orthoalgebras, orthoalgebras and orthomodular posets is proved.
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    Symmetrical Heyting algebras with a finite order type of operators (1995)
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    Studia logica 55 (1995), S. 89-98 
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    Notes: Abstract The main purpose of this paper is to introduce a class of algebraic structures related to many-valued Łukasiewicz algebras. They are symmetrical Heyting algebras with a set of modal operators indexed by a finite completely symmetric poset. A representation theorem is given for these (not functionally complete) algebras.
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    On variable separation in modal and superintuitionistic logics (1995)
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    Studia logica 55 (1995), S. 99-112 
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    Notes: Abstract In this paper we find an algebraic equivalent of the Hallden property in modal logics, namely, we prove that the Hallden-completeness in any normal modal logic is equivalent to the so-called super-embedding property of a suitable class of modal algebras. The joint embedding property of a class of algebras is equivalent to the Pseudo-Relevance Property. We consider connections of the above-mentioned properties with interpolation and amalgamation. Also an algebraic equivalent of of the principle of variable separation in superintuitionistic logics will be found.
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    Abstract modal logics (1995)
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    Studia logica 55 (1995), S. 273-299 
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    Keywords: Primary ; 03B22, 03B45 ; Secondary ; 03G10, 03G5, 03G25
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    Notes: Abstract In this paper we develop a general framework to deal with abstract logics associated with a given modal logic. In particular we study the abstract logics associated with the weak and strong deductive systems of the normal modal logicK and its intuitionistic version. We also study the abstract logics that satisfy the conditionC +(X)=C(∪ i≤n I n X) and find the modal deductive systems whose abstract logics, in addition to being classical or intuitionistic, satisfy that condition. Finally we study the deductive systems whose abstract logics satisfy, in addition to the already mentioned properties, the property that the operatorC + is classical relative to some new defined operations.
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    Axiomatic extensions of the constructive logic with strong negation and the disjunction property (1995)
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    Studia logica 55 (1995), S. 377-388 
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    Notes: Abstract We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B εV, thenA×B is a homomorphic image of some well-connected algebra ofV. We prove: each varietyV of Nelson algebras with PQWC lies in the fibre σ−1(W) for some varietyW of Heyting algebras having PQWC, for any varietyW of Heyting algebras with PQWC the least and the greatest varieties in σ−1(W) have PQWC, there exist varietiesW of Heyting algebras having PQWC such that σ−1(W) contains infinitely many varieties (of Nelson algebras) with PQWC.
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    Books received (1995)
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    Studia logica 55 (1995), S. 449-451 
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    Semi-DeMorgan algebras (1996)
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    Studia logica 56 (1996), S. 151-183 
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    Keywords: Semi-DeMorgan algebra ; duality
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    Notes: Abstract Semi-DeMorgan algebras are a common generalization of DeMorgan algebras and pseudocomplemented distributive lattices. A duality for them is developed that builds on the Priestley duality for distributive lattices. This duality is then used in several applications. The subdirectly irreducible semi-DeMorgan algebras are characterized. A theory of “partial diagrams” is developed, where properties of algebras are tied to the omission of certain partial diagrams from their duals. This theory is then used to find and give axioms for the largest variety of semi-DeMorgan algebras with the congruence extension property.
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    Directions in generalized quantifier theory (1995)
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    Studia logica 55 (1995), S. 389-419 
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    Notes: Abstract We give a condensed survey of recent research on generalized quantifiers in logic, linguistics and computer science, under the following headings: Logical definability and expressive power, Polyadic quantifiers and linguistic definability, Weak semantics and axiomatizability, Computational semantics, Quantifiers in dynamic settings, Quantifiers and modal logic, Proof theory of generalized quantifiers.
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    Quantum MV algebras (1996)
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    Studia logica 56 (1996), S. 393-417 
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    Keywords: MV algebra ; Multi-valued logic ; quantum MV algebra ; unsharp orthoalgebra ; unsharp quantum mechanics
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    Notes: Abstract We introduce the notion of quantum MV algebra (QMV algebra) as a generalization of MV algebras and we show that the class of all effects of any Hilbert space gives rise to an example of such a structure. We investigate some properties of QMV algebras and we prove that QMV algebras represent non-idempotent extensions of orthomodular lattices.
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    From the editors (1996)
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    Studia logica 57 (1996), S. 3-4 
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    Internal approach to external sets and universes (1996)
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    Studia logica 56 (1996), S. 293-322 
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    Keywords: nonstandard set theory ; partially saturated subuniverses
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    Notes: Abstract In this article
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    A reduction rule for Peirce formula (1996)
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    Studia logica 56 (1996), S. 419-426 
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    Keywords: Curry-Howard isomorphism ; typed lambda-calculus ; classical logic
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    Notes: Abstract A reduction rule is introduced as a transformation of proof figures in implicational classical logic. Proof figures are represented as typed terms in a λ-calculus with a new constant P ((α→β)→α). It is shown that all terms with the same type are equivalent with respect to β-reduction augmented by this P-reduction rule. Hence all the proofs of the same implicational formula are equivalent. It is also shown that strong normalization fails for βP-reduction. Weak normalization is shown for βP-reduction with another reduction rule which simplifies α of ((α → β) → α) → α into an atomic type.
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    Detachment and defeasibility in deontic logic (1996)
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    Studia logica 57 (1996), S. 5-18 
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    Keywords: deontic logic ; conditional obligation ; dyadic deontic logics ; defeasibility
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    Notes: Abstract The purpose of the paper is to present a logical framework that allow to formalize a kind of prima facie duties, defeasible conditional duties, indefeasible conditional duties and actual (indefeasible) duties, as well as to show their logical interconnections.
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    A logic of comparative obligation (1996)
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    Studia logica 57 (1996), S. 117-137 
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    Keywords: deontic logic ; conflicting obligation ; partial ordering ; modal logic ; urgency comparison
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    Notes: Abstract Normal systems of modal logic, interpreted as deontic logics, are unsuitable for a logic of conflicting obligations. By using modal operators based on a more complex semantics, however, we can provide for conflicting obligations, as in [9], which is formally similar to a fragment of the logic of ability later given in [2], Having gone that far, we may find it desirable to be able to express and consider claims about the comparative strengths, or degrees of urgency, of the conflicting obligations under which we stand. This paper, building on the formalism of the logic of ability in [2], provides a complete and decidable system for such a language.
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    Reasoning about actions and obligations in first-order logic (1996)
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    Studia logica 57 (1996), S. 221-237 
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    Keywords: deontic logic ; logic of action
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    Notes: Abstract We describe a new way in which theories about the deontic status of actions can be represented in terms of the standard two-sorted first-order extensional predicate calculus. Some of the resulting formal theories are easy to implement in Prolog; one prototype implementation—R. M. Lee's deontic expert shell DX—is briefly described.
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    Definability of directly indecomposable congruence modular algebras (1996)
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    Studia logica 57 (1996), S. 239-241 
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    Keywords: congruence modular algebras ; directly indecomposable algebras ; central elements
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    Notes: Abstract It is proved that the directly indecomposable algebras in a congruence modular equational class ν form a ∀∃∀ first-order class provided that ν fulfils some two natural assumptions.
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    On maximal intermediate predicate constructive logics (1996)
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    Studia logica 57 (1996), S. 373-408 
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    Keywords: intermediate predicate logic ; nonstandard intermediate predicate logic ; predicate constructive logic ; nonstandard predicate constructive logic ; maximal predicate constructive logic ; maximal nonstandard predicate constructive logic ; smooth predicate constructive logic ; 03B55 ; 03C90
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    Notes: Abstract We extend to the predicate frame a previous characterization of the maximal intermediate propositional constructive logics. This provides a technique to get maximal intermediate predicate constructive logics starting from suitable sets of classically valid predicate formulae we call maximal nonstandard predicate constructive logics. As an example of this technique, we exhibit two maximal intermediate predicate constructive logics, yet leaving open the problem of stating whether the two logics are distinct. Further properties of these logics will be also investigated.
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    A Logic for Information Systems (1997)
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    Studia logica 58 (1997), S. 3-16 
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    Keywords: modal logic ; information system ; complete calculus
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    Notes: Abstract A conception of an information system has been introduced by Pawlak. The study has been continued in works of Pawlak and Orlowska and in works of Vakarelov. They had proposed some basic relations and had constructed a formal system of a modal logic that describes the relations and some of their Boolean combinations. Our work is devoted to a generalization of this approach. A class of relation systems and a complete calculus construction method for these systems are proposed. As a corollary of our main result, our paper contains a solution of a Vakarelov's problem: how to construct a formal system that describes all the Boolean combinations of the basic relations.
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    Fuzzy Logic and Arithmetical Hierarchy, II (1997)
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    Studia logica 58 (1997), S. 129-141 
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    Keywords: fuzzy logic ; many-valued logic ; undecidability
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    Notes: Abstract A very simple many-valued predicate calculus is presented; a completeness theorem is proved and the arithmetical complexity of some notions concerning provability is determined.
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    Some Syntactic Approaches to the Handling of Inconsistent Knowledge Bases: A Comparative Study Part 1: The Flat Case (1997)
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    Studia logica 58 (1997), S. 17-45 
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    Keywords: inconsistency ; argumentation ; nonmonotonic reasoning ; syntax-sensitivity
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    Notes: Abstract This paper presents and discusses several methods for reasoning from inconsistent knowledge bases. A so-called argued consequence relation, taking into account the existence of consistent arguments in favour of a conclusion and the absence of consistent arguments in favour of its contrary, is particularly investigated. Flat knowledge bases, i.e., without any priority between their elements, are studied under different inconsistency-tolerant consequence relations, namely the so-called argumentative, free, universal, existential, cardinality-based, and paraconsistent consequence relations. The syntax-sensitivity of these consequence relations is studied. A companion paper is devoted to the case where priorities exist between the pieces of information in the knowledge base.
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    A Logic for Reasoning about Relative Similarity (1997)
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    Studia logica 58 (1997), S. 185-226 
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    Keywords: rough sets ; entities ; properties ; similarity ; lower and upper approximations ; multi-modal logics ; completeness
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    Notes: Abstract A similarity relation is a reflexive and symmetric binary relation between objects. Similarity is relative: it depends on the set of properties of objects used in determining their similarity or dissimilarity. A multi-modal logical language for reasoning about relative similarities is presented. The modalities correspond semantically to the upper and lower approximations of a set of objects by similarity relations corresponding to all subsets of a given set of properties of objects. A complete deduction system for the language is presented.
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    Game Logic and its Applications II (1997)
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    Studia logica 58 (1997), S. 273-303 
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    Keywords: infinitary predicate KD4 ; common knowledge ; Nash equilibrium ; undecidability on playability
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    Notes: Abstract This paper provides a Genzten style formulation of the game logic framework GLm (0 ≤ m ≤ ω), and proves the cut-elimination theorem for GLm. As its application, we prove the term existence theorem for GLω used in Part I.
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    Labelled Resolution for Classical and Non-classical Logics (1997)
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    Studia logica 59 (1997), S. 179-216 
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    Notes: Abstract Resolution is an effective deduction procedure for classical logic. There is no similar "resolution" system for non-classical logics (though there are various automated deduction systems). The paper presents resolution systems for intuistionistic predicate logic as well as for modal and temporal logics within the framework of labelled deductive systems. Whereas in classical predicate logic resolution is applied to literals, in our system resolution is applied to L(abelled) R(epresentation) S(tructures). Proofs are discovered by a refutation procedure defined on LRSs, that imposes a hierarchy on clause sets of such structures together with an inheritance discipline. This is a form of Theory Resolution. For intuitionistic logic these structures are called I(ntuitionistic) R(epresentation) S(tructures). Their hierarchical structure allows the restriction of unification of individual variables and/or constants without using Skolem functions. This structures must therefore be preserved when we consider other (non-modal) logics. Variations between different logics are captured by fine tuning of the inheritance properties of the hierarchy. For modal and temporal logics IRS's are extended to structures that represent worlds and/or times. This enables us to consider all kinds of combined logics.
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    The Greatest Extension of S4 into which Intuitionistic Logic is Embeddable (1997)
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    Studia logica 59 (1997), S. 345-358 
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    Keywords: modal logic ; intuitionistic logic ; Gödel translation
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    Notes: Abstract This paper gives a characterization of those quasi-normal extensions of the modal system S4 into which intuitionistic propositional logic Int is embeddable by the Gödel translation. It is shown that, as in the normal case, the set of quasi-normal modal companions of Int contains the greatest logic, M*, for which, however, the analog of the Blok-Esakia theorem does not hold. M* is proved to be decidable and Halldén-complete; it has the disjunction property but does not have the finite model property.
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    Kripke Bundle Semantics and C-set Semantics (1997)
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    Studia logica 58 (1997), S. 395-401 
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    Keywords: Kripke bundle semantics ; C-set semantics ; superintuitionistic logics
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    Notes: Abstract Kripke bundle [3] and C-set semantics [1] [2] are known as semantics which generalize standard Kripke semantics. In [3] and in [1], [2] it is shown that Kripke bundle and C-set semantics are stronger than standard Kripke semantics. Also it is true that C-set semantics for superintuitionistic logics is stronger than Kripke bundle semantics [5]. In this paper, we show that Q-S4.1 is not Kripke bundle complete via C-set models. As a corollary we can give a simple proof showing that C-set semantics for modal logics are stronger than Kripke bundle semantics.
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    Strategic Construction of Fitch-style Proofs (1998)
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    Studia logica 60 (1998), S. 45-66 
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    Notes: Abstract Symlog is a system for learning symbolic logic by computer that allows students to interactively construct proofs in Fitch-style natural deduction. On request, Symlog can provide guidance and advice to help a student narrow the gap between goal theorem and premises. To effectively implement this capability, the program was equipped with a theorem prover that constructs proofs using the same methods and techniques the students are being taught. This paper discusses some of the aspects of the theorem prover's design, including its set of proof-construction strategies, its unification algorithm as well as some of the tradeoffs between efficiency and pedagogy.
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    Lambda Calculus and Intuitionistic Linear Logic (1997)
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    Studia logica 59 (1997), S. 417-448 
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    Keywords: linear logic ; Curry-Howard isomorphism ; typed λ-calculi
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    Notes: Abstract The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources. This paper introduces a typed functional language Λ! and a categorical model for it. The terms of Λ! encode a version of natural deduction for Intuitionistic Linear Logic such that linear and non linear assumptions are managed multiplicatively and additively, respectively. Correspondingly, the terms of Λ! are built out of two disjoint sets of variables. Moreover, the λ-abstractions of Λ! bind variables and patterns. The use of two different kinds of variables and the patterns allow a very compact definition of the one-step operational semantics of Λ!, unlike all other extensions of Curry-Howard Isomorphism to Intuitionistic Linear Logic. The language Λ! is Church-Rosser and enjoys both Strong Normalizability and Subject Reduction. The categorical model induces operational equivalences like, for example, a set of extensional equivalences. The paper presents also an untyped version of Λ! and a type assignment for it, using formulas of Linear Logic as types. The type assignment inherits from Λ! all the good computational properties and enjoys also the Principal-Type Property.
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    On the Logic of Acceptance and Rejection (1998)
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    Studia logica 60 (1998), S. 233-251 
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    Keywords: autoepistemic logic
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    Notes: Abstract The logic of acceptance and rejection (AEL2) is a nonmonotonic formalism to represent states of knowledge of an introspective agent making decisions about available information. Though having much in common, AEL2 differs from Moore's autoepistemic logic (AEL) by the fact that the agent not only can accept or reject a given fact, but he/she also has the possibility not to make any decision in case he/she does not have enough knowledge.
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    Characterizations of Negative Definability in Modal Logic (1998)
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    Studia logica 60 (1998), S. 357-386 
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    Keywords: Modal logic ; definability ; irreflexivity rule ; characterization
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    Notes: Abstract Negative definability ([18]) is an alternative way of defining classes of Kripke frames via a modal language, one that enables us, for instance, to define the class of irreflexive frames. Besides a list of closure conditions for negatively definable classes, the paper contains two main theorems. First, a characterization is given of negatively definable classes of (rooted) finite transitive Kripke frames and of such classes defined using both traditional (positive) and negative definitions. Second, we characterize the negatively definable classes of rooted general frames.
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    Program Constructions that are Safe for Bisimulation (1998)
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    Studia logica 60 (1998), S. 311-330 
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    Notes: Abstract It has been known since the seventies that the formulas of modal logic are invariant for bisimulations between possible worlds models — while conversely, all bisimulation-invariant first-order formulas are modally definable. In this paper, we extend this semantic style of analysis from modal formulas to dynamic program operations. We show that the usual regular operations are safe for bisimulation, in the sense that the transition relations of their values respect any given bisimulation for their arguments. Our main result is a complete syntactic characterization of all first-order definable program operations that are safe for bisimulation. This is a semantic functional completeness result for programming, which may be contrasted with the more usual analysis in terms of computational power. The 'Safety Theorem' can be modulated in several ways. We conclude with a list of variants, extensions, and further developments.
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    Modal Deduction in Second-Order Logic and Set Theory - II (1998)
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    Studia logica 60 (1998), S. 387-420 
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    Keywords: Modal Logic ; Modal Deduction ; Translation Methods ; Second-Order Logic ; Set Theory
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    Notes: Abstract In this paper, we generalize the set-theoretic translation method for poly-modal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor such a translation to work with specific cases of extended modal logics.
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    Connections Between BCK-algebras and Difference Posetse (1998)
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    Studia logica 60 (1998), S. 421-439 
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    Keywords: BCK-algebra ; commutative BCK-algebra ; poset with difference ; D-poset ; MV-algebra ; quantum MV-algebra ; orthoalgebra
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    Topics: Mathematics , Philosophy
    Notes: Abstract We discuss the interrelations between BCK-algebras and posets with difference. Applications are given to bounded commutative BCK-algebras, difference posets, MV-algebras, quantum MV-algebras and orthoalgebras.
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    Axiomatizations with Context Rules of Inference in Modal Logic (1998)
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    Studia logica 61 (1998), S. 179-197 
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    Keywords: modal logic ; inference rules ; axiomatizations ; completeness
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    Notes: Abstract A certain type of inference rules in (multi-) modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.
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    Book Reviews (1998)
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    Studia logica 61 (1998), S. 293-310 
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    Curry-Howard Terms for Linear Logic (1998)
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    Studia logica 61 (1998), S. 223-235 
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    Keywords: linear logic ; Curry-Howard terms ; strong normalization
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    Notes: Abstract In this paper we 1. provide a natural deduction system for full first-order linear logic, 2. introduce Curry-Howard-style terms for this version of linear logic, 3. extend the notion of substitution of Curry-Howard terms for term variables, 4. define the reduction rules for the Curry-Howard terms and 5. outline a proof of the strong normalization for the full system of linear logic using a development of Girard's candidates for reducibility, thereby providing an alternative to Girard's proof using proof-nets.
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    Isomorphism and Equational Equivalence of Continuous λ-Models (1998)
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    Studia logica 61 (1998), S. 403-415 
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    Notes: Abstract We will present several results on two types of continuous models of λ-calculus, namely graph models and extensional models. By introducing a variant of Engeler's model construction, we are able to generalize the results of [7] and to give invariants that determine a large family of graph models up to applicative isomorphism. This covers all graph models considered in the litterature so far. We indicate briefly how these invariants may be modified in order to determine extensional models as well. Furthermore, we use our construction to exhibit $$2^{N_0 } $$ graph models that are not equationally equivalent. We indicate once again how the construction passes on to extensional models.
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    Book Reviews (1998)
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    Studia logica 61 (1998), S. 429-448 
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    Problems of Equivalence, Categoricity of Axioms and States Description in Databases (1998)
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    Studia logica 61 (1998), S. 347-366 
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    Keywords: Algebraic logic ; equivalence of data bases ; description of states of data bases
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    Notes: Abstract The paper is devoted to applications of algebraic logic to databases. In databases a query is represented by a formula of first order logic. The same query can be associated with different formulas. Thus, a query is a class of equivalent formulae: equivalence here being similar to that in the transition to the Lindenbaum-Tarski algebra. An algebra of queries is identified with the corresponding algebra of logic. An algebra of replies to the queries is also associated with algebraic logic. These relations lie at the core of the applications. In this paper it is shown how the theory of Halmos (polyadic) algebras (a notion introduced by Halmos as a tool in the algebraization of the first order predicate calculus) is used to create the algebraic model of a relational data base. The model allows us, in particular, to solve the problem of databases equivalence as well as develop a formal algebraic definition of a database's state description. In this paper we use the term "state description" for the logical description of the model. This description is based on the notion of filters in Halmos algebras. When speaking of a state description, we mean the description of a function which realizes the symbols of relations as real relations in the given system of data.
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    Book review (1999)
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    Studia logica 62 (1999), S. 107-116 
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    Instructions for authors (1999)
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    Studia logica 62 (1999), S. 315-315 
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    Bull's Theorem by the Method of Diagrams (1999)
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    Studia logica 62 (1999), S. 163-176 
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    Keywords: propositional modal logics ; finite model property ; completeness property
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    Notes: Abstract We show how to use diagrams in order to obtain straightforward completeness theorems for extensions of K4.3 and a very simple and constructive proof of Bull's theorem: every normal extension of S4.3 has the finite model property.
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    On the Elementarity of Some Classes of Abelian-by-Infinite Groups (1999)
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    Studia logica 62 (1999), S. 201-213 
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    Monotone Majorizable Functionals (1999)
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    Studia logica 62 (1999), S. 283-289 
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    Keywords: Monotone functionals ; monotone majorizable functionals ; hereditarily majorizable functionals ; simply typed lambda-calculus ; extensionality ; Dialectica interpretation
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    Topics: Mathematics , Philosophy
    Notes: Abstract Several properties of monotone functionals (MF) and monotone majorizable functionals (MMF) used in the earlier work by the author and van de Pol are proved. It turns out that the terms of the simply typed lambda-calculus define MF, but adding primitive recursion, and even monotonic primitive recursion changes the situation: already λZ.Z(1 — sg) is not MMF. It is proved that extensionality is not Dialectica-realizable by MMF, and a simple example of a MF which is not hereditarily majorizable is given.
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    From Brouwerian Counter Examples to the Creating Subject (1999)
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    Studia logica 62 (1999), S. 305-314 
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    Keywords: intuitionism ; Brouwerian counter examples ; continuum ; creating subject
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    Notes: Abstract The original Brouwerian counter examples were algorithmic in nature; after the introduction of choice sequences, Brouwer devised a version which did not depend on algorithms. This is the origin of the ‘creating subject’ technique. The method allowed stronger refutations of classical principles. Here it is used to show that ‘negative dense’ subsets of the continuum are indecomposable.
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    Preface (1999)
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    Studia logica 62 (1999), S. 117-120 
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    Marginalia on Sequent Calculi (1999)
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    Studia logica 62 (1999), S. 291-303 
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    Keywords: sequent calculus ; normal deduction ; cutfree deduction ; complete discharge convention
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    Notes: Abstract The paper discusses the relationship between normal natural deductions and cutfree proofs in Gentzen (sequent) calculi in the absence of term labeling. For Gentzen calculi this is the usual version; for natural deduction this is the version under the complete discharge convention, where open assumptions are always discharged as soon as possible. The paper supplements work by Mints, Pinto, Dyckhoff, and Schwichtenberg on the labeled calculi.
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    A Translation of Intuitionistic Predicate Logic into Basic Predicate Logic (1999)
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    Studia logica 62 (1999), S. 341-352 
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    Keywords: Basic Predicate Logic ; Intuitionistic Predicate Logic ; translation
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    Notes: Abstract Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula ϕ in the language {∨, ∧, →, ⊤, ⊥, ∀, ∃}, we associate two sequences of formulas 〈ϕ0,ϕ1,...〉 and 〈ϕ0,ϕ1,...〉 in the same language. We prove that for every sequent ϕ ⇒ ψ, there are natural numbers m, n, such that IQC ⊢ ϕ ⇒ ψ, iff BQC ⊢ ϕn ⇒ ψm. Some applications of this translation are mentioned.
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    Proof-Theoretic Modal PA-Completeness I: A System-Sequent Metric (1999)
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    Studia logica 63 (1999), S. 27-48 
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    Keywords: Proof-Theory ; Probability Logic ; metric between sentences and systems ; characteristic formula of a modal sequent
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    Notes: Abstract This paper is the first of a series of three articles that present the syntactic proof of the PA-completeness of the modal system G, by introducing suitable proof-theoretic objects, which also have an independent interest. We start from the syntactic PA-completeness of modal system GL-LIN, previously obtained in [7], [8], and so we assume to be working on modal sequents S which are GL-LIN-theorems. If S is not a G-theorem we define here a notion of syntactic metric d(S, G): we calculate a canonical characteristic fomula H of S (char(S)) so that ⊢G ∼ H → (∼S) and ⊢GL-LIN ∼ H, and the complexity σ of ∼ H gives the distance d(S, G) of S from G. Then, in order to produce the whole completeness proof as an induction on this d(S, G), we introduce the tree-interpretation of a modal sequent Q into PA, that sends the letters of Q into PA-formulas describing the properties of a GL-LIN-proof P of Q: It is also a d(*, G)-metric linked interpretation, since it will be applied to a proof-tree T of ∼ H with H = char(S) and σ(∼ H) = d(S, G).
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    Complete Infinitary Type Logics (1999)
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    Studia logica 63 (1999), S. 85-119 
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    Keywords: infinitary sequent calculi ; type theory ; Takeuti's conjecture ; logicism
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    Notes: Abstract For each regular cardinal κ, we set up three systems of infinitary type logic, in which the length of the types and the length of the typed syntactical constructs are 〈 κ. For a fixed κ, these three versions are, in the order of increasing strength: the local system Σ(κ), the global system gΣ(κ) (the difference concerns the conditions on eigenvariables) and the τ-system τΣ(κ) (which has anti-selection terms or Hilbertian τ-terms, and no conditions on eigenvariables). A full cut elimination theorem is proved for the local systems, and about the τ-systems we prove that they admit cut-free proofs for sequents in the τ-free language common to the local and global systems. These two results follow from semantic completeness proofs. Thus every sequent provable in a global system has a cut-free proof in the corresponding τ-systems. It is, however, an open question whether the global systems in themselves admit cut elimination.
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    New Intuitionistic Logical Constants and Novikov Completeness (1999)
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    Studia logica 63 (1999), S. 151-180 
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    Keywords: intuitionistic logic ; new logical constants ; Novikov completeness
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    Notes: Abstract Extending the language of the intuitionistic propositional logic Int with additional logical constants, we construct a wide family of extensions of Int with the following properties: (a) every member of this family is a maximal conservative extension of Int; (b) additional constants are independent in each of them.
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    Substructural Implicational Logics Including the Relevant Logic E (1999)
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    Studia logica 63 (1999), S. 181-212 
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    Keywords: substructural logic ; relevant logic ; cut-elimination
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    Topics: Mathematics , Philosophy
    Notes: Abstract We introduce several restricted versions of the structural rules in the implicational fragment of Gentzen's sequent calculus LJ. For example, we permit the applications of a structural rule only if its principal formula is an implication. We investigate cut-eliminability and theorem-equivalence among various combinations of them. The results include new cut-elimination theorems for the implicational fragments of the following logics: relevant logic E, strict implication S4, and their neighbors (e.g., E-W and S4-W); BCI-logic, BCK-logic, relevant logic R, and the intuitionistic logic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005210613711
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  • 98
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    The Finite Model Property for the Implicational Fragment of IPC Without Exchange and Contraction (1999)
    Springer
    Studia logica 63 (1999), S. 213-222 
    Details
    ISSN: 1572-8730
    Keywords: substructural logic ; algebraization ; residuation algebra ; Kripke semantics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The aim of this paper is to show that the implicational fragment BKof the intuitionistic propositional calculus (IPC) without the rules of exchange and contraction has the finite model property with respect to the quasivariety of left residuation algebras (its equivalent algebraic semantics). It follows that the variety generated by all left residuation algebras is generated by the finite left residuation algebras. We also establish that BKhas the finite model property with respect to a class of structures that constitute a Kripke-style relational semantics for it. The results settle a question of Ono and Komori [OK85].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005262630549
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  • 99
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    Definability of Leibniz Equality (1999)
    Springer
    Studia logica 63 (1999), S. 223-243 
    Details
    ISSN: 1572-8730
    Keywords: Leibniz equality ; Universal Horn theory ; Equality-free logic ; Definability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Given a structure for a first-order language L, two objects of its domain can be indiscernible relative to the properties expressible in L, without using the equality symbol, and without actually being the same. It is this relation that interests us in this paper. It is called Leibniz equality. In the paper we study systematically the problem of its definibility mainly for classes of structures that are the models of some equality-free universal Horn class in an infinitary language Lκκ, where κ is an infinite regular cardinal.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005214714620
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  • 100
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    Proof-Theoretic Modal PA-Completeness II: The Syntactic Countermodel (1999)
    Springer
    Studia logica 63 (1999), S. 245-268 
    Details
    ISSN: 1572-8730
    Keywords: Proof-Theory ; Provability Logic ; countermodel of a sequent ; classification of arithmetical interpretations of modal logic
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper is the second part of the syntactic demonstration of the Arithmetical Completeness of the modal system G, the first part of which is presented in [9]. Given a sequent S so that ⊢GL-LIN S, ⊬G S, and given its characteristic formula H = char(S), which expresses the non G-provability of S, we construct a canonical proof-tree T of ~ H in GL-LIN, the height of which is the distance d(S, G) of S from G. T is the syntactic countermodel of S with respect to Gand is a tool of general interest in Provability Logic, that allows some classification in the set of the arithmetical interpretations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005266731458
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