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  • 1
    Electronic Resource
    Electronic Resource
    On the {↔, ∼} -reduct of the intuitionistic consequence operation (1981)
    Springer
    Studia logica 40 (1981), S. 55-66 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract The intuitionistic consequence operation restricted to the language with ↔ (equivalence) and ∼ (negation) as the only connectives is axiomatized by means of a finite set of sequential rules of inference.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01837555
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  • 2
    Electronic Resource
    Electronic Resource
    On fragments of Medvedev's logic (1981)
    Springer
    Studia logica 40 (1981), S. 39-54 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectivesΦ such that $$\{ \to , \vee , \urcorner \} \not \subseteq \Phi \subseteq \{ \to , \wedge , \urcorner \} $$ theΦ-fragment ofMV equals theΦ fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: is an intermediate logic based on the axiom (⌝a→b∨c) →(⌉a→b)∨(⌝a → c) separable?
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01837554
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  • 3
    Electronic Resource
    Electronic Resource
    Individualistic formal approach to deontic logic (1981)
    Springer
    Studia logica 40 (1981), S. 99-102 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Some people approve of certain general rules of behavior, or some concrete cases. The others disapprove of or are indifferent to them. In this paper I suggest an axiom system which formalizes the use of these utterances. It may be considered as a special (“individualistic”) approach to deontic logic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01874702
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  • 4
    Electronic Resource
    Electronic Resource
    The lattice of strengthenings of a strongly finite consequence operation (1981)
    Springer
    Studia logica 40 (1981), S. 177-193 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract First, we prove that the lattice of all structural strengthenings of a given strongly finite consequence operation is both atomic and coatomic, it has finitely many atoms and coatoms, each coatom is strongly finite but atoms are not of this kind — we settle this by constructing a suitable counterexample. Second, we deal with the notions of hereditary: algebraicness, strong finitisticity and finite approximability of a strongly finite consequence operation. Third, we formulate some conditions which tell us when the lattice of all structural strengthenings of a given strongly finite consequence operation is finite, and subsequently we give some applications of them.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01874707
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  • 5
    Electronic Resource
    Electronic Resource
    2-Element matrices (1981)
    Springer
    Studia logica 40 (1981), S. 315-353 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Sections 1, 2 and 3 contain the main result, the strong finite axiomatizability of all 2-valued matrices. Since non-strongly finitely axiomatizable 3-element matrices are easily constructed the result reveals once again the gap between 2-valued and multiple-valued logic. Sec. 2 deals with the basic cases which include the important F i ∞ from Post's classification. The procedure in Sec. 3 reduces the general problem to these cases. Sec. 4 is a study of basic algebraic properties of 2-element algebras. In particular, we show that equational completeness is equivalent to the Stone-property and that each 2-element algebra generates a minimal quasivariety. The results of Sec. 4 will be applied in Sec. 5 to maximality questions and to a matrix free characterization of 2-valued consequences in the lattice of structural consequences in any language. Sec. 6 takes a look at related axiomatization. problems for finite algebras and matrices. We study the notion of a propositional consequence with equality and, among other things, present explicit axiomatizations of 2-valued consequences with equality.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00401653
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  • 6
    Electronic Resource
    Electronic Resource
    Distributive lattices with a dual homomorphic operation. II (1981)
    Springer
    Studia logica 40 (1981), S. 391-404 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract An Ockham lattice is defined to be a distributive lattice with 0 and 1 which is equipped with a dual homomorphic operation. In this paper we prove: (1) The lattice of all equational classes of Ockham lattices is isomorphic to a lattice of easily described first-order theories and is uncountable, (2) every such equational class is generated by its finite members. In the proof of (2) a characterization of orderings of ω with respect to which the successor function is decreasing is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00401657
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  • 7
    Electronic Resource
    Electronic Resource
    On the relation provable equivalence and on partitions in effectively inseparable sets (1981)
    Springer
    Studia logica 40 (1981), S. 29-37 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract We generalize a well-knownSmullyan's result, by showing that any two sets of the kindC a = {x/⊢ x↔a} andC b = {x/⊢ x↔b} are effectively inseparable (if I ⌿ ↔b). Then we investigate logical and recursive consequences of this fact (see Introduction).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01837553
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  • 8
    Electronic Resource
    Electronic Resource
    On some ascending chains of brouwerian modal logics (1981)
    Springer
    Studia logica 40 (1981), S. 75-87 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper specifies classes of framesmaximally omnitemporally characteristic for Thomas' normal modal logicT 2 + and for each logic in the ascending chain of Segerberg logics investigated by Segerberg and Hughes and Cresswell. It is shown that distinct a,scending chains of “generalized” Segerberg logics can be constructed from eachT n + logic (n ⩾ 2). The set containing allT n + and Segerberg logics can be totally- (linearly-) ordered but not well-ordered by the inclusion relation. The order type of this ordered setΓ isω *(ω + 1). Throughout the paper my approach is fundamentally semantical.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01837557
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  • 9
    Electronic Resource
    Electronic Resource
    A propositional logic with explicit fixed points (1981)
    Springer
    Studia logica 40 (1981), S. 155-175 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper studies a propositional logic which is obtained by interpreting implication as formal provability. It is also the logic of finite irreflexive Kripke Models. A Kripke Model completeness theorem is given and several completeness theorems for interpretations into Provability Logic and Peano Arithmetic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01874706
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  • 10
    Electronic Resource
    Electronic Resource
    The degrees of maximality of the intuitionistic propositional logic and of some of its fragments (1981)
    Springer
    Studia logica 40 (1981), S. 195-198 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Professor Ryszard Wójcicki once asked whether the degree of maximality of the consequence operationC determined by the theorems of the intuitionistic propositional logic and the detachment rule for the implication connective is equal to $$2^{2^\aleph 0} $$ ? The aim of the present paper is to give the affirmative answer to the question. More exactly, it is proved here that the degree of maximality ofC Ψ — theΨ — fragment ofC, is equal to $$2^{2^\aleph 0} $$ , for every such that → εΨ.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01874708
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