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  • Articles  (4,898)
  • Springer  (4,898)
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  • 1995-1999  (2,461)
  • 1990-1994  (2,437)
  • Philosophy  (4,898)
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  • Articles  (4,898)
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  • 1
    Electronic Resource
    Electronic Resource
    Definite descriptions (1990)
    Springer
    Studia logica 49 (1990), S. 87-104 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Three views on definite descriptions are summarized and discussed, including that of P. F. Strawson in which reference failure results in lack of truth value. When reference failure is allowed, a problem arises concerning Universal Instantiation. Van Fraassen solves the problem by the use of supervaluations, preserving as well such theorems as ‘a=a’, and ‘Fa ∨ ∼Fa’, even when the term ‘a’ fails to refer. In the present paper a form of relevant, quasi-analytic implication is set out which allows reference failure to infect even ‘a=a’ and ‘Fa ∨ ∼ Fa’ with lack of truth-value. Reference failure causes lack of truth-value in a subwff to spread throughout any wff built up by the classical connectives. As a result none of the classical firstorder axiom schemes remain as axiom schemes in the system presented.
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    URL: http://dx.doi.org/10.1007/BF00401556
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  • 2
    Electronic Resource
    Electronic Resource
    Semantic holism (1990)
    Springer
    Studia logica 49 (1990), S. 67-82 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract A bivalent valuation is snt iff sound (standard PC inference rules take truths only into truths) and non-trivial (not all wffs are assigned the same truth value). Such a valuation is normal iff classically correct for each connective. Carnap knew that there were non-normal snt valuations of PC, and that the gap they revealed between syntax and semantics could be “jumped” as follows. Let VAL snt be the set of snt valuations, and VAL nrm be the set of normal ones. The bottom row in the table for the wedge ‘∨’ is not semantically determined by VAL snt, but if one deletes from VAL snt all those valuations that are not classically correct at the aforementioned row, one jumps straights to VAL nrm and thus to classical semantics. The conjecture we call semantic holism claims that the same thing happens for any semantic indeterminacy in any row in the table of any connective of PC, i.e., to remove it is to jump straight to classical semantics. We show (i) why semantic holism is plausible and (ii) why it is nevertheless false. And (iii) we pose a series of questions concerning the number of possible steps or jumps between the indeterminate semantics given by VAL snt and classical semantics given by VAL nrm.
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    URL: http://dx.doi.org/10.1007/BF00401554
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  • 3
    Electronic Resource
    Electronic Resource
    On a complexity-based way of constructivizing the recursive functions (1990)
    Springer
    Studia logica 49 (1990), S. 133-149 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Let g E(m, n)=o mean that n is the Gödel-number of the shortest derivation from E of an equation of the form ϕ(m)=k. Hao Wang suggests that the condition for general recursiveness ∀m∃n(g E(m, n)=o) can be proved constructively if one can find a speedfunction ϕ s s, with ϕ s(m) bounding the number of steps for getting a value of ϕ(m), such that ∀m∃n≦ϕ s(m) s.t. g E(m, n)=o. This idea, he thinks, yields a constructivist notion of an effectively computable function, one that doesn't get us into a vicious circle since we intuitively know, to begin with, that certain proofs are constructive and certain functions effectively computable. This paper gives a broad ‘possibility’ proof for the existence of such classes of effectively computable functions, with Wang's idea of effective computability generalized along a number of dimensions.
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    URL: http://dx.doi.org/10.1007/BF00401559
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  • 4
    Electronic Resource
    Electronic Resource
    Every world can see a reflexive world (1990)
    Springer
    Studia logica 49 (1990), S. 175-181 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Let ℭ be the class of frames satisfying the condition $$\forall x\exists y(Ry \wedge yRy)$$ (“every world can see a reflexive world”). LetKMT be the system obtained by adding to the minimal normal modal systemK the axiom $$M((Lp_1 \supset p_1 ) \wedge ... \wedge (Lp_n \supset p_n ))$$ for eachn ⩾ 1. The main results proved are: (1)KMT is characterized by ℭ. (2)KMT has the finite model property. (3) There are frames forKMT which are not in ℭ, but allfinite frames forKMT are in ℭ. (4)KMT is decidable. (5)KMT is not finitely axiomatizable. (6) The class of all frames forKMT is not definable by any formula of first-order logic. (7) 〈W, R〉 is a frame forKMT iff for everyx εW, the worlds thatx can see form a sub-frame which is not finitely colourable.
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    URL: http://dx.doi.org/10.1007/BF00935597
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  • 5
    Electronic Resource
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    Call for papers (1990)
    Springer
    Studia logica 49 (1990), S. 288-288 
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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/BF00935605
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  • 6
    Electronic Resource
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    Deontic logic and possible worlds semantics: A historical sketch (1990)
    Springer
    Studia logica 49 (1990), S. 273-282 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper describes and compares the first step in modern semantic theory for deontic logic which appeared in works of Stig Kanger, Jaakko Hintikka, Richard Montague and Saul Kripke in late 50s and early 60s. Moreover, some further developments as well as systematizations are also noted.
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    URL: http://dx.doi.org/10.1007/BF00935603
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  • 7
    Electronic Resource
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    Axiomatization of the De Morgan type rules (1990)
    Springer
    Studia logica 49 (1990), S. 333-343 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract In Section 1 we show that the De Morgan type rules (= sequential rules in L(∨, ⌝) which remain correct if ∧ and ∨ are interchanged) are finitely based. Section 2 contains a similar result for L(→). These results are essentially based on special properties of some equational theories.
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  • 8
    Electronic Resource
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    Temporal necessity and the conditional (1990)
    Springer
    Studia logica 49 (1990), S. 345-363 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract In [12] Richmond Thomason and Anil Gupta investigate a semantics for conditional logic that combines the ideas of [8] and [9] with a branching time model of tense logic. The resulting branching time semantics for the conditional is intended to capture the logical relationship between temporal necessity and the conditional. The central principle of this logical relationship is Past Predominance, according to which past similarities and differences take priority over future similarities and differences in determining the comparative similarity of alternative possible histories with respect to a given present moment. In this paper I will use ordinary possible worlds semantics (i.e. Kripke frames) to solve the completeness problem for a system of logic that combines conditional logic with temporal necessity in the context of Past Predominance. Branching time models turn out not to be necessary for the articulation of Past Predominance, and this means that one can axiomatize Past Predominance without first having to solve a much more difficult problem: the completeness problem for the logic of temporal necessity in the context of branching time. Thomason and Gupta argue in [12] that in addition to Past Predominance, temporal necessity and the conditional are logically related, by what have become known as the Edelberg Inferences, whose apparent validity motivates the very complicated theory presented at the end of [12]. I will conclude this paper by examining how the Edelberg inferences would be incorporated into the possible worlds based system presented in the earlier sections of this paper.
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  • 9
    Electronic Resource
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    Logical equations and admissible rules of inference with parameters in modal provability logics (1990)
    Springer
    Studia logica 49 (1990), S. 215-239 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper concerns modal logics of provability — Gödel-Löb systemGL and Solovay logicS — the smallest and the greatest representation of arithmetical theories in propositional logic respectively. We prove that the decision problem for admissibility of rules (with or without parameters) inGL andS is decidable. Then we get a positive solution to Friedman's problem forGL andS. We also show that A. V. Kuznetsov's problem of the existence of finite basis for admissible rules forGL andS has a negative solution. Afterwards we give an algorithm deciding the solvability of logical equations inGL andS and constructing some solutions.
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    URL: http://dx.doi.org/10.1007/BF00935600
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  • 10
    Electronic Resource
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    Klemens Szaniawski (1925–1990) (1990)
    Springer
    Studia logica 49 (1990), S. I 
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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/BF00935594
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  • 11
    Electronic Resource
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    Verisimilitude based on concept analysis (1990)
    Springer
    Studia logica 49 (1990), S. 307-320 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract In the paper ordering relations for comparison of verisimilitude of theories are introduced and discussed. The relations refer to semantic analysis of the results of theories, in particular to analysis of concepts the theories deal with.
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  • 12
    Electronic Resource
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    On simplicity of formulas (1990)
    Springer
    Studia logica 49 (1990), S. 401-419 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Simple formula should contain only few quantifiers. In the paper the methods to estimate quantity and quality of quantifiers needed to express a sentence equivalent to given one.
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    URL: http://dx.doi.org/10.1007/BF00370372
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  • 13
    Electronic Resource
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    Categorial grammars determined from linguistic data by unification (1990)
    Springer
    Studia logica 49 (1990), S. 431-454 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract We provide an algorithm for determining a categorial grammar from linguistic data that essentially uses unification of type-schemes assigned to atoms. The algorithm presented here extends an earlier one restricted to rigid categorial grammars, introduced in [4] and [5], by admitting non-rigid outputs. The key innovation is the notion of an optimal unifier, a natural generalization of that of a most general unifier.
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    URL: http://dx.doi.org/10.1007/BF00370157
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  • 14
    Electronic Resource
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    Models for relevant modal logics (1990)
    Springer
    Studia logica 49 (1990), S. 501-514 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Semantics are given for modal extensions of relevant logics based on the kind of frames introduced in [7]. By means of a simple recipe we may obtain from a class FRM (L) of unreduced frames characterising a (non-modal) logic L, frame-classes FRM □ (L.M) characterising conjunctively regular modal extensions L.M of L. By displaying an incompleteness phenomenon, it is shown how the recipe fails when reduced frames are under consideration.
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    URL: http://dx.doi.org/10.1007/BF00370161
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  • 15
    Electronic Resource
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    Conditionals and monotonic belief revisions: the success postulate (1990)
    Springer
    Studia logica 49 (1990), S. 557-566 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract One of the main applications of the logic of theory change is to the epistemic analysis of conditionals via the so-called Ramsey test. In the first part of the present note this test is studied in the “limiting case” where the theory being revised is inconsistent, and it is shown that this case manifests an intrinsic incompatibility between the Ramsey test and the AGM postulate of “success”. The paper then analyses the use of the postulate of success, and a weakening of it, generating axioms of conditional logic via the test, and it is shown that for certain purposes both success and weak success are quite superfluous. This suggests the proposal of abandoning both success and weak success entirely, thus permitting retention of the postulate of “preservation” discarded by Gärdenfors.
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    URL: http://dx.doi.org/10.1007/BF00370165
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  • 16
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    Books received (1990)
    Springer
    Studia logica 49 (1990), S. 601-611 
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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/BF00370169
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  • 17
    Electronic Resource
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    Some independence results in interpretability logic (1991)
    Springer
    Studia logica 50 (1991), S. 29-38 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract A Kripke-style semantics developed by de Jongh and Veltman is used to investigate relations between several extensions of interpretability logic, IL.
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  • 18
    Electronic Resource
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    Rosser orderings and free variables (1991)
    Springer
    Studia logica 50 (1991), S. 71-80 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract It is shown that for arithmetical interpretations that may include free variables it is not the Guaspari-Solovay system R that is arithmetically complete, but their system R −. This result is then applied to obtain the nonvalidity of some rules under arithmetical interpretations including free variables, and to show that some principles concerning Rosser orderings with free variables cannot be decided, even if one restricts oneself to “usual” proof predicates.
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  • 19
    Electronic Resource
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    Some unifying fixed point principles (1991)
    Springer
    Studia logica 50 (1991), S. 129-141 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This article is written for both the general mathematican and the specialist in mathematical logic. No prior knowledge of metamathematics, recursion theory or combinatory logic is presupposed, although this paper deals with quite general abstractions of standard results in those three areas. Our purpose is to show how some apparently diverse results in these areas can be derived from a common construction. In Section 1 we consider five classical fixed point arguments (or rather, generalizations of them) which we present as problems that the reader might enjoy trying to solve. Solutions are given at the end of the section. In Section 2 we show how all these solutions can be obtained as special cases of a single fixed point theorem. In Section 3 we consider another generalization of the five fixed point results of Section 1 and show that this is of the same strength as that of Section 2. In Section 4 we show some curious strengthenings of results of Section 3 which we believe to be of some interest on their own accounts.
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  • 20
    Electronic Resource
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    Three-membered domains for aristotle's syllogistic (1991)
    Springer
    Studia logica 50 (1991), S. 181-187 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The paper shows that for any invalid polysyllogism there is a procedure for constructing a model with a domain with exactly three members and an interpretation that assigns non-empty, non-universal subsets of the domain to terms such that the model invalidates the polysyllogism.
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    URL: http://dx.doi.org/10.1007/BF00370181
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  • 21
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    Logical feedback (1991)
    Springer
    Studia logica 50 (1991), S. 225-239 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Just as non-well-founded sets extend the usual sets of ZF, so do root reflexive propositional formulas extends the usual class of Boolean expressions. Though infinitary, these formulas are generated by finite patterns. They possess transition functions instead of truth values and have applications in electric circuit theory.
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    URL: http://dx.doi.org/10.1007/BF00370184
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  • 22
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    On structural completeness of implicational logics (1991)
    Springer
    Studia logica 50 (1991), S. 275-297 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules.
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  • 23
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    Books received (1991)
    Springer
    Studia logica 50 (1991), S. 351-357 
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    Topics: Mathematics , Philosophy
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  • 24
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    Introduction (1991)
    Springer
    Studia logica 50 (1991), S. 365-374 
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  • 25
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    A deduction theorem schema for deductive systems of propositional logics (1991)
    Springer
    Studia logica 50 (1991), S. 385-390 
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    Topics: Mathematics , Philosophy
    Notes: Abstract We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) ⊑ L and p, A(p, q) ⊢s q.
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  • 26
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    Construction of monadic three-valued Łukasiewicz algebras (1991)
    Springer
    Studia logica 50 (1991), S. 473-483 
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    Topics: Mathematics , Philosophy
    Notes: Abstract The notion of monadic three-valued Łukasiewicz algebras was introduced by L. Monteiro ([12], [14]) as a generalization of monadic Boolean algebras. A. Monteiro ([9], [10]) and later L. Monteiro and L. Gonzalez Coppola [17] obtained a method for the construction of a three-valued Łukasiewicz algebra from a monadic Boolea algebra. In this note we give the construction of a monadic three-valued Łukasiewicz algebra from a Boolean algebra B where we have defined two quantification operations ∃ and ∃* such that ∃∀*x=∀*∃x (where ∀*x=-∃*-x). In this case we shall say that ∃ and ∃* commutes. If B is finite and ∃ is an existential quantifier over B, we shall show how to obtain all the existential quantifiers ∃* which commute with ∃. Taking into account R. Mayet [3] we also construct a monadic three-valued Łukasiewicz algebra from a monadic Boolean algebra B and a monadic ideal I of B. The most essential results of the present paper will be submitted to the XXXIX Annual Meeting of the Unión Matemática Argentina (October 1989, Rosario, Argentina).
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  • 27
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    Semantics for relevance logic with identity (1992)
    Springer
    Studia logica 51 (1992), S. 1-20 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract Models are constructed for a variety of systems of quantified relevance logic with identity. Models are given for systems with different principles governing the transitivity of identity and substitution, and the relative merits of these principles are discussed. The models in this paper are all extensions of the semantics of Fine's “Semantics for Quantified Relevance Logic” (Journal of Philosophical Logic 17 (1988)).
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  • 28
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    Partial monotonicity and a new version of the Ramsey test (1992)
    Springer
    Studia logica 51 (1992), S. 21-47 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract We introduce two new belief revision axioms: partial monotonicity and consequence correctness. We show that partial monotonicity is consistent with but independent of the full set of axioms for a Gärdenfors belief revision sytem. In contrast to the Gärdenfors inconsistency results for certain monotonicity principles, we use partial monotonicity to inform a consistent formalization of the Ramsey test within a belief revision system extended by a conditional operator. We take this to be a technical dissolution of the well-known Gärdenfors dilemma. In addition, we present the consequential correctness axiom as a new measure of minimal revision in terms of the deductive core of a proposition whose support we wish to excise. We survey several syntactic and semantic belief revision systems and evaluate them according to both the Gärdenfors axioms and our new axioms. Furthermore, our algebraic characterization of semantic revision systems provides a useful technical device for analysis and comparison, which we illustrate with several new proofs. Finally, we have a new inconsistency result, which is dual to the Gärdenfors inconsistency results. Any elementary belief revision system that is consequentially correct must violate the Gärdenfors axiom of strong boundedness (K*8), which we characterize as yet another monotonicity condition.
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  • 29
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    Lambek calculus with restricted contraction and expansion (1992)
    Springer
    Studia logica 51 (1992), S. 125-143 
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    ISSN: 1572-8730
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    Topics: Mathematics , Philosophy
    Notes: Abstract This paper deals with some strengthenings of the non-directional product-free Lambek calculus by means of additional structural rules. In fact, the rules contraction and expansion are restricted to basic types. For each of the presented systems the usual proof-theoretic notions are discussed, some new concepts especially designed for these calculi are introduced reflecting their intermediate position between the weaker and the stronger sequent-systems.
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  • 30
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    Quantified extensions of canonical propositional intermediate logics (1992)
    Springer
    Studia logica 51 (1992), S. 195-214 
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    Topics: Mathematics , Philosophy
    Notes: Abstract The quantified extension of a canonical prepositional intermediate logic is complete with respect to the generalization of Kripke semantics taking into consideration set-valued functors defined on a category.
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  • 31
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    Revision algebra semantics for conditional logic (1992)
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    Studia logica 51 (1992), S. 279-316 
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    Notes: Abstract The properties of belief revision operators are known to have an informal semantics which relates them to the axioms of conditional logic. The purpose of this paper is to make this connection precise via the model theory of conditional logic. A semantics for conditional logic is presented, which is expressed in terms of algebraic models constructed ultimately out of revision operators. In addition, it is shown that each algebraic model determines both a revision operator and a logic, that are related by virtue of the stable Ramsey test.
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    Action incompleteness (1992)
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    Studia logica 51 (1992), S. 533-550 
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    Notes: Abstract The author has previously introduced an operatorδ into dynamic logic which takes formulae to terms; the suggested reading ofδ A was “the bringing about of A” or “the seeing to it that A”. After criticism from S. K. Thomason and T. J. Surendonk the author now presents an improved version of his theory. The crucial feature is the introduction of an operatorOK taking terms to formulae; the suggested reading of OKα is “α always terminates”.
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    Normal bimodal logics of ability and action (1992)
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    Studia logica 51 (1992), S. 519-532 
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    Notes: Abstract The basic bimodal systemK/K can be interpreted as an analysis of the logic of ability developed in [1]. Where in [1] we would express the claimI can bring it about that P using the formula , with its non-normal operator , we will now use the formula . Here is a normal alethic possibilitation operator. is a normal necessitation operator, but it is independent of , and not subject to an alethic interpretation. Rather, is interpreted to meanI bring it about that P. The result is a simplification and clarification of a combined logic of ability and action like that in [2], but employing only normal operators. A number of extensions of the basic systemK/K are constructed, first by strengthening the two normal sublogics independently and then by linking the two sublogics via axiom schemata involving both operators. The result is a series of increasingly strong systems which more and more adequately fulfill our expectations for a satisfactory logic of action and ability.
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    Selective bibliography in the logic of action (1992)
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    Studia logica 51 (1992), S. 579-589 
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    On Łukasiewicz-Moisil algebras of fuzzy sets (1993)
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    Studia logica 52 (1993), S. 95-111 
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    Notes: Abstract The set (X, J) of fuzzy subsetsf:X→J of a setX can be equipped with a structure of ϑ-valued Łukasiewicz-Moisil algebra, where ϑ is the order type of the totally ordered setJ. Conversely, every Łukasiewicz-Moisil algebra — and in particular every Post algebra — is isomorphic to a subalgebra of an algebra of the form (X, J), whereJ has an order type $$\bar \theta $$ ⩾ϑ. The first result of this paper is a characterization of those ϑ-valued Łukasiewicz-Moisil algebras which are isomorphic to an algebra of the form (X, J) (Theorem 1). Then we prove that (X, J) is a Post algebra if and only if the setJ is dually well-ordered (Theorem 2) and we give a characterization of those ϑ-valued Post algebras with are isomorphic to an algebra of the form (X, J) (Theorem 3 and Proposition 2).
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    The simple substitution property of the intermediate propositional logics on finite slices (1993)
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    Studia logica 52 (1993), S. 41-62 
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    Notes: Abstract The simple substitution property provides a systematic and easy method for proving a theorem by an axiomatic way. The notion of the property was introduced in Hosoi [4] but without a definite name and he showed three examples of the axioms with the property. Later, the property was given it's name as above in Sasaki [7]. Our main result here is that the necessary and sufficient condition for a logicL on a finite slice to have the simple substitution property is thatL is finite. Here the necessity part is essentially new, for the sufficiency part has been proved in Hosoi and Sasaki [5]. Also the proof of sufficiency part is improved here. For logics on the ω-th slice, the condition for them to have the simple substitution property is not yet known. We abbreviate the simple substitution property asSSP.
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    Some results on the Kripke sheaf semantics for super-intuitionistic predicate logics (1993)
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    Studia logica 52 (1993), S. 73-94 
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    Notes: Abstract Some properties of Kripke-sheaf semantics for super-intuitionistic predicate logics are shown. The concept ofp-morphisms between Kripke sheaves is introduced. It is shown that if there exists ap-morphism from a Kripke sheaf κ1 into κ2 then the logic characterized by κ1 is contained in the logic characterized by κ2. Examples of Kripke-sheaf complete and finitely axiomatizable super-intuitionistic (and intermediate) predicate logics each of which is Kripke-frame incomplete are given. A correction to the author's previous paper “Kripke bundles for intermediate predicate logics and Kripke frames for intuitionistic modal logics” (Studia Logica, 49(1990), pp. 289–306 ) is stated.
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    A study of intermediate propositional logics on the third slice (1993)
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    Studia logica 52 (1993), S. 15-21 
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    Notes: Abstract The intermediate logics have been classified into slices (cf. Hosoi [1]), but the detailed structure of slices has been studied only for the first two slices (cf. Hosoi and Ono [2]). In order to study the structure of slices, we give a method of a finer classification of slices & n (n ≥ 3). Here we treat only the third slice as an example, but the method can be extended to other slices in an obvious way. It is proved that each subslice contains continuum of logics. A characterization of logics in each subslice is given in terms of the form of models.
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    A reasoning method for a paraconsistent logic (1993)
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    Studia logica 52 (1993), S. 281-289 
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    Notes: Abstract A proof method for automation of reasoning in a paraconsistent logic, the calculus C1* of da Costa, is presented. The method is analytical, using a specially designed tableau system. Actually two tableau systems were created. A first one, with a small number of rules in order to be mathematically convenient, is used to prove the soundness and the completeness of the method. The other one, which is equivalent to the former, is a system of derived rules designed to enhance computational efficiency. A prototype based on this second system was effectively implemented.
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    How to bereally contraction free (1993)
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    Studia logica 52 (1993), S. 381-391 
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    Notes: Abstract A logic is said to becontraction free if the rule fromA → (A →B) toA →B is not truth preserving. It is well known that a logic has to be contraction free for it to support a non-trivial naïve theory of sets or of truth. What is not so well known is that if there isanother contracting implication expressible in the language, the logic still cannot support such a naïve theory. A logic is said to berobustly contraction free if there is no such operator expressible in its language. We show that a large class of finitely valued logics are each not robustly contraction free, and demonstrate that some other contraction free logics fail to be robustly contraction free. Finally, the sublogics of Łω (with the standard connectives) are shown to be robustly contraction free.
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    Automated theorem proving for non-classical logics (1993)
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    Studia logica 52 (1993), S. 489-489 
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    An intensional epistemic logic (1993)
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    Studia logica 52 (1993), S. 259-280 
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    Keywords: Leibniz's Substitution Principle ; Intension and Extension ; Non-classical Logic Programming ; Quantified Epistemic logic ; Variable Domain Possible-worlds semantics ; Skolemization Theorem for Modal logic ; Horn Theories ; Equational Programs ; Paramodulation ; SLD-Resolution and Many-sorted Unification
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    Notes: Abstract One of the fundamental properties inclassical equational reasoning isLeibniz's principle of substitution. Unfortunately, this propertydoes not hold instandard epistemic logic. Furthermore,Herbrand's lifting theorem which isessential to thecompleteness ofresolution andParamodulation in theclassical first order logic (FOL), turns out to be invalid in standard epistemic logic. In particular, unlike classical logic, there is no skolemization normal form for standard epistemic logic. To solve these problems, we introduce anintensional epistemic logic, based on avariation of Kripke's possible-worlds semantics that need not have a constant domain. We show how a weaker notion of substitution through indexed terms can retain the Herbrand theorem. We prove how the logic can yield a satisfibility preserving skolemization form. In particular, we present an intensional principle for unifing indexed terms. Finally, we describe asound andcomplete inference system for a Horn subset of the logic withequality, based onepistemic SLD-resolution.
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    An essay on resolution logics (1993)
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    Studia logica 52 (1993), S. 309-322 
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    Notes: Abstract This paper discusses the resolution principle in the context of non-classical logics.
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    The relative complexity of analytic tableaux and SL-resolution (1993)
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    Studia logica 52 (1993), S. 323-337 
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    Notes: Abstract In this paper we describe an improvement of Smullyan's analytic tableau method for the propositional calculus-Improved Parent Clash Restricted (IPCR) tableau-and show that it is equivalent to SL-resolution in complexity.
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    Satisfiability on hypergraphs (1993)
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    Studia logica 52 (1993), S. 393-404 
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    Notes: Abstract In [4] R.Cowen considers a generalization of the resolution rule for hypergraphs and introduces a notion of satisfiability of families of sets of vertices via 2-colorings piercing elements of such families. He shows, for finite hypergraphs with no one-element edges that if the empty set is a consequence ofA by the resolution rule, thenA is not satisfiable. Alas the converse is true for a restricted class of hypergraphs only, and need not to be true in the general case. In this paper we show that weakening slightly the notion of satisfiability, we get the equivalence of unsatisfiability and the derivability of the empty set for any hypergraph. Moreover, we show the compactness property of hypergraph satisfiability (in the weaker sense) and state its equivalence to BPI, i.e. to the statement that in every Boolean algebra there exists an ultrafilter.
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    Sentential constants in systems near R (1993)
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    Studia logica 52 (1993), S. 443-455 
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    Notes: Abstract An Ackermann constant is a formula of sentential logic built up from the sentential constant t by closing under connectives. It is known that there are only finitely many non-equivalent Ackermann constants in the relevant logic R. In this paper it is shown that the most natural systems close to R but weaker than it-in particular the non-distributive system LR and the modalised system NR-allow infinitely many Ackermann constants to be distinguished. The argument in each case proceeds by construction of an algebraic model, infinite in the case of LR and of arbitrary finite size in the case of NR. The search for these models was aided by the computer program MaGIC (Matrix Generator for Implication Connectives) developed by the author at the Australian National University.
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    Inductive inference in the limit for first-order sentences (1993)
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    Studia logica 52 (1993), S. 491-517 
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    Notes: Abstract The paper investigates learning functions for first order languages. Several types of convergence and identification in the limit are defined. Positive and negative results on learning problems are presented throughout the paper.
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    Syntactical results on the arithmetical completeness of modal logic (1993)
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    Studia logica 52 (1993), S. 549-564 
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    Notes: Abstract In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS ϕ of a modal sequentS, which allow the transformation of PA-proofs ofS ϕ into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of the form □p i,m i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic.
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    On a property of BCK-identities (1994)
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    Studia logica 53 (1994), S. 227-234 
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    Notes: Abstract A BCK-algebra is an algebra in which the terms are generated by a set of variables, 1, and an arrow. We mean by aBCK-identity an equation valid in all BCK-algebras. In this paper using a syntactic method we show that for two termss andt, if neithers=1 nort=1 is a BCK-identity, ands=t is a BCK-identity, then the rightmost variables of the two terms are identical. This theorem was conjectured firstly in [5], and then in [3]. As a corollary of this theorem, we derive that the BCK-algebras do not form a variety, which was originally proved algebraically by Wroński ([4]). To prove the main theorem, we use a Gentzen-type logical system for the BCK-algebras, introduced by Komori, which consists of the identity axiom, the right and the left introduction rules of the implication, the exchange rule, the weakening rule and the cut. As noted in [2], the cut-elimination theorem holds for this system.
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    A gentzen system for conditional logic (1994)
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    Studia logica 53 (1994), S. 243-257 
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    Notes: Abstract Conditional logic is the deductive system 〈 $$\mathcal{L}$$ , ⊨〉 where $$\mathcal{L}$$ is the set of propositional connectives {∧, ∨,′} and ⊨ is the structural finitary consequence relation on the absolutely free algebra $$F_{m_\mathcal{L} } $$ that preserves degrees of truth over the structure of truth values 〈C, ≤〉. HereC is the non-commutative regular extension of the 2-element Boolean algebra to 3 truth values {t, u, f}, andf〈u〈t. In this paper we give a Gentzen type axiomatization for conditional logic.
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    Non-axiomatizability of Grice's implicature (1994)
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    Studia logica 53 (1994), S. 343-349 
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    Notes: Abstract The aim of this paper is to test Grice's theory of conversational implication [1], so-calledimplicature, by putting it into operation in the simplest possible formal language, that is, by constructing an adequate zero-order (sentential) logic. We are going to give a recursive formal description of Grice's maxims and show that the description cannot be made finite.
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    Perfect MV-algebras are categorically equivalent to abelianl-groups (1994)
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    Studia logica 53 (1994), S. 417-432 
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    Notes: Abstract In this paper we prove that the category of abelianl-groups is equivalent to the category of perfect MV-algebras. Furthermore, we give a finite equational axiomatization of the variety generated by perfect MV-algebras.
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    Cecylia Rauszer (1994)
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    Studia logica 53 (1994), S. 467-471 
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    A finite analog to the löwenheim-skolem theorem (1994)
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    Studia logica 53 (1994), S. 503-532 
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    Notes: Abstract The traditional model theory of first-order logic assumes that the interpretation of a formula can be given without reference to its deductive context. This paper investigates an interpretation which depends on a formula's location within a derivation. The key step is to drop the assumption that all quantified variables must have the same range and to require only that the ranges of variables in a derivation must be related in such way as to preserve the soundness of the inference rules. With each (consistent) derivation there is associated a “Buridan-Volpin (orBV) structure” [M, {r(x)}] which is simply a Tarski structureM for the language and a map giving the ranger(x) of each variablex in the derivation. IfLK* is (approximately) the classical sequent calculusLK of Gentzen from which the structural contraction rules have been dropped, then our main result reads: If a set of first-ordered formulas Γ has a Tarski modelM, then from any normal derivationD inLK* of Γ ⇒ Δ can be constructed aBV modelM D=[M, {r(x)}] of Γ where each ranger(x) is finite.
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    Helena Rasiowa (1995)
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    Studia logica 54 (1995), S. 1-2 
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    Books received (1995)
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    Studia logica 54 (1995), S. 129-137 
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    The relative consistency of system RRC* and some of its extensions (1994)
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    Studia logica 53 (1994), S. 351-360 
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    Notes: Abstract We present a relative consistency proof for second order systemRRC* and for certain important extensions of this system. The proof proceeds as follows: we prove first the equiconsistency of the strongest of such extensions (viz., systemH RRC*+(∃/CP**)) with second order systemT λ * . Now, N. Cocchiarella has shown thatT λ * is relatively consistent to systemT*+Ext; clearly, it follows thatH RRC*+(∃/CP**) is relatively consistent toT*+E xt. As an immediate consequence, the relative consistency ofRRC* and the other extensions also follows, being all of them subsystems ofH RRC*+(∃/CP**).
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    A note on the ω-incompleteness formalization (1994)
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    Studia logica 53 (1994), S. 389-396 
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    Notes: Abstract The paper studies two formal schemes related to ω-completeness. LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where α(x) is a formula with the only free variable x): (a) (for anyn) (⊢ T α(n)), (b) ⊢ T ∀ x Pr T (−α(x)−) and (c) ⊢ T ∀xα(x) (the notational conventions are those of Smoryński [3]). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) ⇒ (c) and (a) ⇒ (b), where α ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) ⇒ (c) is equivalent to ¬Cons T and 2. the schema corresponding to (a) ⇒ (b) is not consistent with 1-CON T. The former result follows from a simple adaptation of the ω-incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma.
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    On the canonicity of Sahlqvist identities (1994)
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    Studia logica 53 (1994), S. 473-491 
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    Notes: Abstract We give a simple proof of the canonicity of Sahlqvist identities, using methods that were introduced in a paper by Jónsson and Tarski in 1951.
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    A complete minimal logic of the propositional contents of thought (1995)
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    Studia logica 54 (1995), S. 391-410 
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    Notes: Abstract Our purpose is to formulate a complete logic of propositions that takes into account the fact that propositions are both senses provided with truth values and contents of conceptual thoughts. In our formalization, propositions are more complex entities than simple functions from possible worlds into truth values. They have a structure of constituents (a content) in addition to truth conditions. The formalization is adequate for the purposes of the logic of speech acts. It imposes a stronger criterion of propositional identity than strict equivalence. Two propositions P and Q are identical if and only if, for any illocutionary force F, it is not possible to perform with success a speech act of the form F(P) without also performing with success a speech act of the form F(Q). Unlike hyperintensional logic, our logic of propositions is compatible with the classical Boolean laws of propositional identity such as the symmetry and the associativity of conjunction and the reduction of double negation.
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    Tableaus for many-valued modal logic (1995)
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    Studia logica 55 (1995), S. 63-87 
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    Notes: Abstract We continue a series of papers on a family of many-valued modal logics, a family whose Kripke semantics involves many-valued accessibility relations. Earlier papers in the series presented a motivation in terms of a multiple-expert semantics. They also proved completeness of sequent calculus formulations for the logics, formulations using a cut rule in an essential way. In this paper a novel cut-free tableau formulation is presented, and its completeness is proved.
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    Averaging the truth-value in Łukasiewicz logic (1995)
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    Studia logica 55 (1995), S. 113-127 
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    Notes: Abstract Chang's MV algebras are the algebras of the infinite-valued sentential calculus of Łukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of ‘average degree of truth’ of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.
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    Undefinability of propositional quantifiers in the modal system S4 (1995)
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    Studia logica 55 (1995), S. 259-271 
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    Keywords: Primary 03B45, 06E25 ; Secondary 18C10
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    Notes: Abstract We show that (contrary to the parallel case of intuitionistic logic, see [7], [4]) there does not exist a translation fromS42 (the propositional modal systemS4 enriched with propositional quantifiers) intoS4 that preserves provability and reduces to identity for Boolean connectives and □.
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    Algorithmic problems concerning first-order definability of modal formulas on the class of all finite frames (1995)
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    Studia logica 55 (1995), S. 421-448 
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    Notes: Abstract The main result is that is no effective algorithmic answer to the question:how to recognize whether arbitrary modal formula has a first-order equivalent on the class of finite frames. Besides, two known problems are solved: it is proved algorithmic undecidability of finite frame consequence between modal formulas; the difference between global and local variants of first-order definability of modal formulas on the class of transitive frames is shown.
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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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    The abstract variable-binding calculus (1995)
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    Studia logica 55 (1995), S. 129-179 
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    Notes: Abstract Theabstract variable binding calculus (VB-calculus) provides a formal frame-work encompassing such diverse variable-binding phenomena as lambda abstraction, Riemann integration, existential and universal quantification (in both classical and nonclassical logic), and various notions of generalized quantification that have been studied in abstract model theory. All axioms of the VB-calculus are in the form of equations, but like the lambda calculus it is not a true equational theory since substitution of terms for variables is restricted. A similar problem with the standard formalism of the first-order predicate logic led to the development of the theory of cylindric and polyadic Boolean algebras. We take the same course here and introduce the variety of polyadic VB-algebras as a pure equational form of the VB-calculus. In one of the main results of the paper we show that every locally finite polyadic VB-algebra of infinite dimension is isomorphic to a functional polyadic VB-algebra that is obtained from a model of the VB-calculus by a natural coordinatization process. This theorem is a generalization of the functional representation theorem for polyadic Boolean algebras given by P. Halmos. As an application of this theorem we present a strong completeness theorem for the VB-calculus. More precisely, we prove that, for every VB-theory T that is obtained by adjoining new equations to the axioms of the VB-calculus, there exists a model D such that ⊢T s=t iff ⊨D s=t. This result specializes to a completeness theorem for a number of familiar systems that can be formalized as VB-calculi. For example, the lambda calculus, the classical first-order predicate calculus, the theory of the generalized quantifierexists uncountably many and a fragment of Riemann integration.
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    Positive modal logic (1995)
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    Studia logica 55 (1995), S. 301-317 
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    Notes: Abstract We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply ∧, ∨, □, ◊. The postulates (and theorems) are all deducibility statements ϕ ⊢ ψ. The only postulates that might not be obvious are $$\diamondsuit \varphi \wedge \square \psi \vdash \diamondsuit (\varphi \wedge \psi )\square (\varphi \vee \psi ) \vdash \square \varphi \vee \diamondsuit \psi $$ . It is shown thatK + is complete with respect to the usual Kripke-style semantics. The proof is by way of a Henkin-style construction, with “possible worlds” being taken to be prime theories. The construction has the somewhat unusual feature of using at an intermediate stage disjoint pairs consisting of a theory and a “counter-theory”, the counter-theory replacing the role of negation in the standard construction. Extension to other modal logics is discussed, as well as a representation theorem for the corresponding modal algebras. We also discuss proof-theoretic arguments.
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    From the editors (1996)
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    Studia logica 57 (1996), S. 3-4 
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    Two approaches to the formalisation of defeasible deontic reasoning (1996)
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    Studia logica 57 (1996), S. 73-90 
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    Keywords: deontic reasoning ; defeasible reasoning ; moral dilemmas
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    Notes: Abstract This paper compares two ways of formalising defeasible deontic reasoning, both based on the view that the issues of conflicting obligations and moral dilemmas should be dealt with from the perspective of nonmonotonic reasoning. The first way is developing a special nonmonotonic logic for deontic statements. This method turns out to have some limitations, for which reason another approach is recommended, viz. combining an already existing nonmonotonic logic with a deontic logic. As an example of this method the language of Reiter's default logic is extended to include modal expressions, after which the argumentation framework in default logic of [20, 22] is used to give a plausible logical analysis of moral dilemmas and prima facie obligations.
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    Deontic database constraints, violation and recovery (1996)
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    Studia logica 57 (1996), S. 139-165 
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    Keywords: deontic logic ; database integrity ; constraints ; constraint violation ; recovery
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    Notes: Abstract The paper discusses the potential value of a deontic approach to database specification. More specifically, some different types of integrity constraints are considered and a distinction is drawn between necessary (“hard”) and deontic (“soft”) constraints. Databases are compared with other normative systems. A deontic logic for database specification is proposed and the problems of how to react to, and of how to correct, or repair, a situation which arises through norm violation are discussed in the context of this logic. The limitations of the proposed logic and possible modifications and extensions of it are analysed.
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    Doing well enough: Toward a logic for common-sense morality (1996)
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    Studia logica 57 (1996), S. 167-192 
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    Keywords: deontic logic ; supererogation ; ordering semantics ; indifference ; optimality
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    Notes: Abstract On the traditional deontic framework, what is required (what morality demands) and what is optimal (what morality recommends) can't be distinguished and hence they can't both be represented. Although the morally optional can be represented, the supererogatory (exceeding morality's demands), one of its proper subclasses, cannot be. The morally indifferent, another proper subclass of the optional-one obviously disjoint from the supererogatory-is also not representable. Ditto for the permissibly suboptimal and the morally significant. Finally, the minimum that morality allows finds no place in the traditional scheme. With a focus on the question, “What would constitute a hospitable logical neighborhood for the concept of supererogation?”, I present and motivate an enriched logical and semantic framework for representing all these concepts of common sense morality.
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    The finite model property for BCI and related systems (1996)
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    Studia logica 57 (1996), S. 303-323 
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    Keywords: finite models ; substructural logics ; the Lambek calculus ; 03B40 ; 03B60
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    Notes: Abstract We prove the finite model property (fmp) for BCI and BCI with additive conjunction, which answers some open questions in Meyer and Ono [11]. We also obtain similar results for some restricted versions of these systems in the style of the Lambek calculus [10, 3]. The key tool is the method of barriers which was earlier introduced by the author to prove fmp for the product-free Lambek calculus [2] and the commutative product-free Lambek calculus [4].
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    Equivalential and algebraizable logics (1996)
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    Studia logica 57 (1996), S. 419-436 
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    Keywords: algebraizable logics ; equivalential logics ; implicative logics ; protoalgebraic logics
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    Notes: Abstract The notion of an algebraizable logic in the sense of Blok and Pigozzi [3] is generalized to that of a possibly infinitely algebraizable, for short, p.i.-algebraizable logic by admitting infinite sets of equivalence formulas and defining equations. An example of the new class is given. Many ideas of this paper have been present in [3] and [4]. By a consequent matrix semantics approach the theory of algebraizable and p.i.-algebraizable logics is developed in a different way. It is related to the theory of equivalential logics in the sense of Prucnal and Wroński [18], and it is extended to nonfinitary logics. The main result states that a logic is algebraizable (p.i.-algebraizable) iff it is finitely equivalential (equivalential) and the truth predicate in the reduced matrix models is equationally definable.
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    A Completeness Proof for a Logic with an Alternative Necessity Operator (1997)
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    Studia logica 58 (1997), S. 99-112 
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    Keywords: knowledge logic ; contingency ; completeness ; complexity
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    Notes: Abstract We show the completeness of a Hilbert-style system LK defined by M. Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information. The completeness proof uses a variant of Makinson's canonical model construction. Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete.
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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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    Towards a Rough Mereology-Based Logic for Approximate Solution Synthesis. Part 1 (1997)
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    Studia logica 58 (1997), S. 143-184 
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    Keywords: approximate reasoning ; synthesis of complex objects ; rough sets ; mereology ; rough mereology
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    Notes: Abstract We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as schemes constructed in order to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely.
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    Axiomatising first-order temporal logic: Until and since over linear time (1996)
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    Studia logica 57 (1996), S. 279-302 
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    Keywords: temporal logic ; axiomatization ; first-order temporal/modal logics
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    Notes: Abstract We present an axiomatisation for the first-order temporal logic with connectives Until and Since over the class of all linear flows of time. Completeness of the axiom system is proved. We also add a few axioms to find a sound and complete axiomatisation for the first order temporal logic of Until and Since over rational numbers time.
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    Game logic and its applications I (1996)
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    Studia logica 57 (1996), S. 325-354 
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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 logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II.
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    Helena Rasiowa in memoriam (1996)
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    Studia logica 57 (1996), S. 459-459 
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    An Elementary Proof of Chang's Completeness Theorem for the Infinite-valued Calculus of Lukasiewicz (1997)
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    Studia logica 58 (1997), S. 79-97 
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    Keywords: MV algebra ; completeness of the Lukasiewicz calculus ; infinite-valued logic
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    Notes: Abstract The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.
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    Simulation and Transfer Results in Modal Logic – A Survey (1997)
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    Studia logica 59 (1997), S. 149-177 
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    Keywords: modal logic ; transfer ; interpretation ; tense logic ; intuitionistic logic
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    Notes: Abstract This papers gives a survey of recent results about simulations of one class of modal logics by another class and of the transfer of properties of modal logics under extensions of the underlying modal language. We discuss: the transfer from normal polymodal logics to their fusions, the transfer from normal modal logics to their extensions by adding the universal modality, and the transfer from normal monomodal logics to minimal tense extensions. Likewise, we discuss simulations of normal polymodal logics by normal monomodal logics, of nominals and the difference operator by normal operators, of monotonic monomodal logics by normal bimodal logics, of polyadic normal modal logics by polymodal normal modal logics, and of intuitionistic modal logics by normal bimodal logics.
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    Knowledge on Treelike Spaces (1997)
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    Studia logica 59 (1997), S. 271-301 
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    Keywords: bimodal logics ; logics of knowledge ; branching time ; persistence ; convexity
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    Notes: Abstract This paper presents a bimodal logic for reasoning about knowledge during knowledge acquisitions. One of the modalities represents (effort during) non-deterministic time and the other represents knowledge. The semantics of this logic are tree-like spaces which are a generalization of semantics used for modeling branching time and historical necessity. A finite system of axiom schemes is shown to be canonically complete for the formentioned spaces. A characterization of the satisfaction relation implies the small model property and decidability for this system.
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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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    Linear Lambda-Terms and Natural Deduction (1998)
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    Studia logica 60 (1998), S. 209-231 
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    Elementary Extensions of External Classes in a Nonstandard Universe (1998)
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    Studia logica 60 (1998), S. 253-273 
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    Keywords: nonstandard set theory ; inner subuniverses ; constructibility ; iterated elementary extensions
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    Notes: Abstract In continuation of our study of HST, Hrbaček set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st-∈-language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe. We show that, given a standard cardinal κ, any set R ⊑ *κ generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "external" class L[S(R)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in S(R). We prove that under some mild saturation-like requirements for R the class L[S(R)] models a certain κ-version of HST including the principle of κ+-saturation; moreover, in this case L[S(R′)] is an elementary extension of L[S(R)] in the st-∈-language whenever sets R ⊑ R′ satisfy the requirements.
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    URL: http://dx.doi.org/10.1023/A:1005064032270
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    Preservation Theorems Without Continuum Hypothesis (1998)
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    Studia logica 60 (1998), S. 343-355 
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    ISSN: 1572-8730
    Keywords: Horn sentence ; reduced product ; reduced power ; ultrapower ; and ultraproduct
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Many results concerning the equivalence between a syntactic form of formulas and a model theoretic conditions are proven directly without using any form of a continuum hypothesis. In particular, it is demonstrated that any reduced product sentence is equivalent to a Horn sentence. Moreover, in any first order language without equality one now has that a reduced product sentence is equivalent to a Horn sentence and any sentence is equivalent to a Boolean combination of Horn sentences.
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    URL: http://dx.doi.org/10.1023/A:1005028511181
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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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    ISSN: 1572-8730
    Keywords: Modal logic ; definability ; irreflexivity rule ; characterization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    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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    URL: http://dx.doi.org/10.1023/A:1005081328928
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    The Writing of the MV-algebras (1998)
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    Studia logica 61 (1998), S. 3-6 
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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/10.1023/A:1005069911813
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    Books Received: Jürgen Mittelstrass, Enzyklopädie Philosophie und Wissenschaftstheorie (1998)
    Springer
    Studia logica 60 (1998), S. 441-448 
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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/10.1023/A:1005041713907
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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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    ISSN: 1572-8730
    Keywords: BCK-algebra ; commutative BCK-algebra ; poset with difference ; D-poset ; MV-algebra ; quantum MV-algebra ; orthoalgebra
    Source: Springer Online Journal Archives 1860-2000
    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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    URL: http://dx.doi.org/10.1023/A:1005089629837
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    Commodious Axiomatization of Quantifiers in Multiple-Valued Logic (1998)
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    Studia logica 61 (1998), S. 101-121 
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    ISSN: 1572-8730
    Keywords: many-valued logic ; infinitely-valued logic ; distribution quantifiers ; distributive lattices ; semantic tableaux ; sequent calculi
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract We provide tools for a concise axiomatization of a broad class of quantifiers in many-valued logic, so-called distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem for finite distributive lattices are used to derive tableau-style axiomatizations of distribution quantifiers.
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    URL: http://dx.doi.org/10.1023/A:1005086415447
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    On Some Kripke Complete and Kripke Incomplete Intermediate Predicate Logics (1998)
    Springer
    Studia logica 61 (1998), S. 281-292 
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    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The Kripke-completeness and incompleteness of some intermediate predicate logics is established. In particular, we obtain a Kripke-incomplete logic (H* +A+D+K) where H* is the intuitionistic predicate calculus, A is a disjunction-free propositional formula, D = ∀x(P(x) V Q) ⊃ ∀xP(x) V Q, K = ¬¬∀x(P(x) V ¬P(x)) (the negative answer to a question of T. Shimura).
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    URL: http://dx.doi.org/10.1023/A:1005029515564
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    Subdirectly Irreducible Modal Algebras and Initial Frames (1999)
    Springer
    Studia logica 62 (1999), S. 269-282 
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    ISSN: 1572-8730
    Keywords: duality theory in modal logic ; modal algebras ; generated frames ; subdirect irreducibility
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The duality between general frames and modal algebras allows to transfer a problem about the relational (Kripke) semantics into algebraic terms, and conversely. We here deal with the conjecture: the modal algebra A is subdirectly irreducible (s.i.) if and only if the dual frame A* is generated. We show that it is false in general, and that it becomes true under some mild assumptions, which include the finite case and the case of K4. We also prove that a Kripke frame F is generated if and only if the dual algebra F* is s.i. The technical result is that A is s.i. when the set of points which generate the dual frame A* is not of zero measure.
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    URL: http://dx.doi.org/10.1023/A:1026407804348
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    Monotone Majorizable Functionals (1999)
    Springer
    Studia logica 62 (1999), S. 283-289 
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    ISSN: 1572-8730
    Keywords: Monotone functionals ; monotone majorizable functionals ; hereditarily majorizable functionals ; simply typed lambda-calculus ; extensionality ; Dialectica interpretation
    Source: Springer Online Journal Archives 1860-2000
    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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    URL: http://dx.doi.org/10.1023/A:1026459821186
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    From Brouwerian Counter Examples to the Creating Subject (1999)
    Springer
    Studia logica 62 (1999), S. 305-314 
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    ISSN: 1572-8730
    Keywords: intuitionism ; Brouwerian counter examples ; continuum ; creating subject
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    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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    URL: http://dx.doi.org/10.1023/A:1026411905257
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    Preface (1999)
    Springer
    Studia logica 62 (1999), S. 117-120 
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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/10.1023/A:1026442515466
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    Marginalia on Sequent Calculi (1999)
    Springer
    Studia logica 62 (1999), S. 291-303 
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    ISSN: 1572-8730
    Keywords: sequent calculus ; normal deduction ; cutfree deduction ; complete discharge convention
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    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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    URL: http://dx.doi.org/10.1023/A:1026413320413
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    A Translation of Intuitionistic Predicate Logic into Basic Predicate Logic (1999)
    Springer
    Studia logica 62 (1999), S. 341-352 
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    ISSN: 1572-8730
    Keywords: Basic Predicate Logic ; Intuitionistic Predicate Logic ; translation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    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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    URL: http://dx.doi.org/10.1023/A:1005196310070
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