Search results for 'propositional logic' (try it on Scholar)

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  1.  12
    Theodore Hailperin & Ontologically Neutral Logic (2001). Kenneth Harris and Branden Fitelson/Comments on Some Completeness Theorems of Urquhart and Méndez & Salto 51–55 Dominic Gregory/Completeness and Decidability Results for Some Propositional Modal Logics Containing “Actu. [REVIEW] Journal of Philosophical Logic 30:617-618.
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  2.  44
    Claes Strannegård, Simon Ulfsbäcker, David Hedqvist & Tommy Gärling (2010). Reasoning Processes in Propositional Logic. Journal of Logic, Language and Information 19 (3):283-314.
    We conducted a computer-based psychological experiment in which a random mix of 40 tautologies and 40 non-tautologies were presented to the participants, who were asked to determine which ones of the formulas were tautologies. The participants were eight university students in computer science who had received tuition in propositional logic. The formulas appeared one by one, a time-limit of 45 s applied to each formula and no aids were allowed. For each formula we recorded the proportion of the (...)
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  3.  22
    Konrad Zdanowski (2009). On Second Order Intuitionistic Propositional Logic Without a Universal Quantifier. Journal of Symbolic Logic 74 (1):157-167.
    We examine second order intuitionistic propositional logic, IPC². Let $F_\exists $ be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in $F_\exists $ that is, for φ € $F_\exists $ φ is a classical tautology if and only if ¬¬φ is a tautology of IPC². We show that for each sentence φ € $F_\exists $ (without free variables), φ is a classical tautology if and only if φ is an intuitionistic tautology. As (...)
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  4.  3
    Kosta Dosen & Zoran Petric (2012). Isomorphic Formulae in Classical Propositional Logic. Mathematical Logic Quarterly 58 (1):5-17.
    Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.
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  5.  3
    K. Ishii (2003). New Sequent Calculi for Visser's Formal Propositional Logic. Mathematical Logic Quarterly 49 (5):525.
    Two cut-free sequent calculi which are conservative extensions of Visser's Formal Propositional Logic are introduced. These satisfy a kind of subformula property and by this property the interpolation theorem for FPL are proved. These are analogies to Aghaei-Ardeshir's calculi for Visser's Basic Propositional Logic.
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  6. Xingxing He, Jun Liu, Yang Xu, Luis Martínez & Da Ruan (2012). On Α-Satisfiability and its Α-Lock Resolution in a Finite Lattice-Valued Propositional Logic. Logic Journal of the Igpl 20 (3):579-588.
    Automated reasoning issues are addressed for a finite lattice-valued propositional logic LnP(X) with truth-values in a finite lattice-valued logical algebraic structure—lattice implication algebra. We investigate extended strategies and rules from classical logic to LnP(X) to simplify the procedure in the semantic level for testing the satisfiability of formulas in LnP(X) at a certain truth-value level α (α-satisfiability) while keeping the role of truth constant formula played in LnP(X). We propose a lock resolution method at a certain truth-value (...)
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  7.  7
    Michael Kaminski & Nissim Francez (2014). Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic. Studia Logica 102 (3):479-497.
    We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.
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  8.  3
    Michael Kaminski & Nissim Francez (forthcoming). The Lambek Calculus Extended with Intuitionistic Propositional Logic. Studia Logica:1-32.
    We present sound and complete semantics and a sequent calculus for the Lambek calculus extended with intuitionistic propositional logic.
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  9.  41
    Kevin C. Klement, Propositional Logic. Internet Encyclopedia of Philosophy.
    Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study (...)
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  10.  71
    Mauro Ferrari, Camillo Fiorentini & Guido Fiorino (2004). A Secondary Semantics for Second Order Intuitionistic Propositional Logic. Mathematical Logic Quarterly 50 (2):202-210.
    In this paper we propose a Kripke-style semantics for second order intuitionistic propositional logic and we provide a semantical proof of the disjunction and the explicit definability property. Moreover, we provide a tableau calculus which is sound and complete with respect to such a semantics.
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  11.  3
    R. Alonderis (2013). A Proof-Search Procedure for Intuitionistic Propositional Logic. Archive for Mathematical Logic 52 (7-8):759-778.
    A sequent root-first proof-search procedure for intuitionistic propositional logic is presented. The procedure is obtained from modified intuitionistic multi-succedent and classical sequent calculi, making use of Glivenko’s Theorem. We prove that a sequent is derivable in a standard intuitionistic multi-succedent calculus if and only if the corresponding prefixed-sequent is derivable in the procedure.
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  12.  4
    M. Alizadeh & M. Ardeshir (2004). On the Linear Lindenbaum Algebra of Basic Propositional Logic. Mathematical Logic Quarterly 50 (1):65.
    We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra.
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  13.  11
    Dimitar P. Guelev (1999). A Propositional Dynamic Logic with Qualitative Probabilities. Journal of Philosophical Logic 28 (6):575-604.
    This paper presents an w-completeness theorem for a new propositional probabilistic logic, namely, the dynamic propositional logic of qualitative probabilities (DQP), which has been introduced by the author as a dynamic extension of the logic of qualitative probabilities (Q P) introduced by Segerberg.
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  14.  7
    Stefano Cavagnetto (2009). Some Applications of Propositional Logic to Cellular Automata. Mathematical Logic Quarterly 55 (6):605-616.
    In this paper we give a new proof of Richardson's theorem [31]: a global function G[MATHEMATICAL DOUBLE-STRUCK CAPITAL A] of a cellular automaton [MATHEMATICAL DOUBLE-STRUCK CAPITAL A] is injective if and only if the inverse of G[MATHEMATICAL DOUBLE-STRUCK CAPITAL A] is a global function of a cellular automaton. Moreover, we show a way how to construct the inverse cellular automaton using the method of feasible interpolation from [20]. We also solve two problems regarding complexity of cellular automata formulated by Durand (...)
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  15.  35
    Balder ten Cate (2006). Expressivity of Second Order Propositional Modal Logic. Journal of Philosophical Logic 35 (2):209-223.
    We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and (...)
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  16.  8
    Joanna Golińska-Pilarek & Taneli Huuskonen (2016). Non-Fregean Propositional Logic with Quantifiers. Notre Dame Journal of Formal Logic 57 (2):249-279.
    We study the non-Fregean propositional logic with propositional quantifiers, denoted by $\mathsf{SCI}_{\mathsf{Q}}$. We prove that $\mathsf{SCI}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in $\mathsf{SCI}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of $\mathsf{SCI}_{\mathsf{Q}}$-sentences. Finally, we present a translation of $\mathsf{SCI}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove (...)
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  17.  17
    Vítězslav Švejdar (2003). On the Polynomial-Space Completeness of Intuitionistic Propositional Logic. Archive for Mathematical Logic 42 (7):711-716.
    We present an alternative, purely semantical and relatively simple, proof of the Statman's result that both intuitionistic propositional logic and its implicational fragment are PSPACE-complete.
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  18.  16
    John Cantwell (2015). An Expressivist Bilateral Meaning-is-Use Analysis of Classical Propositional Logic. Journal of Logic, Language and Information 24 (1):27-51.
    The connectives of classical propositional logic are given an analysis in terms of necessary and sufficient conditions of acceptance and rejection, i.e. the connectives are analyzed within an expressivist bilateral meaning-is-use framework. It is explained how such a framework differs from standard inferentialist frameworks and it is argued that it is better suited to address the particular issues raised by the expressivist thesis that the meaning of a sentence is determined by the mental state that it is conventionally (...)
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  19.  9
    René David & Marek Zaionc (2009). Counting Proofs in Propositional Logic. Archive for Mathematical Logic 48 (2):185-199.
    We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite.
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  20.  23
    Morten H. Sørensen & Paweł Urzyczyn (2010). A Syntactic Embedding of Predicate Logic Into Second-Order Propositional Logic. Notre Dame Journal of Formal Logic 51 (4):457-473.
    We give a syntactic translation from first-order intuitionistic predicate logic into second-order intuitionistic propositional logic IPC2. The translation covers the full set of logical connectives ∧, ∨, →, ⊥, ∀, and ∃, extending our previous work, which studied the significantly simpler case of the universal-implicational fragment of predicate logic. As corollaries of our approach, we obtain simple proofs of nondefinability of ∃ from the propositional connectives and nondefinability of ∀ from ∃ in the second-order intuitionistic (...)
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  21.  25
    Peter Roeper & Hugues Leblanc (1999). Absolute Probability Functions for Intuitionistic Propositional Logic. Journal of Philosophical Logic 28 (3):223-234.
    Provided here is a characterisation of absolute probability functions for intuitionistic (propositional) logic L, i.e. a set of constraints on the unary functions P from the statements of L to the reals, which insures that (i) if a statement A of L is provable in L, then P(A) = 1 for every P, L's axiomatisation being thus sound in the probabilistic sense, and (ii) if P(A) = 1 for every P, then A is provable in L, L's axiomatisation (...)
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  22.  4
    Ioana Leuştean (2006). Non-Commutative Łukasiewicz Propositional Logic. Archive for Mathematical Logic 45 (2):191-213.
    The non-commutative counterpart of the well-known Łukasiewicz propositional logic is developed, in strong connection with the algebraic theory of psMV-algebras. An extension by a new unary logical connective is also considered and a stronger completeness result is proved for this system.
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  23.  19
    A. D. Yashin (1999). Irreflexive Modality in the Intuitionistic Propositional Logic and Novikov Completeness. Journal of Philosophical Logic 28 (2):175-197.
    A. Kuznetsov considered a logic which extended intuitionistic propositional logic by adding a notion of 'irreflexive modality'. We describe an extension of Kuznetsov's logic having the following properties: (a) it is the unique maximal conservative (over intuitionistic propositional logic) extension of Kuznetsov's logic; (b) it determines a new unary logical connective w.r.t. Novikov's approach, i.e., there is no explicit expression within the system for the additional connective; (c) it is axiomatizable by means of (...)
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  24.  5
    Diderik Batens (2013). Propositional Logic Extended with a Pedagogically Useful Relevant Implication. Logic and Logical Philosophy.
    First and foremost, this paper concerns the combination of classical propositional logic with a relevant implication. The proposed combination is simple and transparent from a proof theoretic point of view and at the same time extremely useful for relating formal logic to natural language sentences. A specific system will be presented and studied, also from a semantic point of view. The last sections of the paper contain more general considerations on combining classical propositional logic with (...)
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  25.  3
    M. Aghaei & M. Ardeshir (2000). A Bounded Translation of Intuitionistic Propositional Logic Into Basic Propositional Logic. Mathematical Logic Quarterly 46 (2):195-206.
    In this paper we prove a bounded translation of intuitionistic propositional logic into basic propositional logic. Our new theorem, compared with the translation theorem in [1], has the advantage that it gives an effective bound on the translation, depending on the complexity of formulas.
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  26.  38
    Michael J. Carroll (1976). On Interpreting the S5 Propositional Calculus: An Essay in Philosophical Logic. Dissertation, University of Iowa
    Discusses alternative interpretations of the modal operators, for the modal propositional logic S5.
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  27.  1
    Karl Dürr (1951/1980). The Propositional Logic of Boethius. Greenwood Press.
    The text of the treatise “The Propositional Logic of Boethius” was finished in 1939. Prof. Jan Lukasiewicz wished at that time to issue it in the second volume of “Collectanea Logica”; as a result of political events, he was not able to carry out his plan. In 1938, I published an article in “Erkenntnis” entitled “AUS- sagenlogik im Mittelalter”; this article included the contents of a paper which I read to the International Congress for the Unity of Science (...)
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  28. Yehoshua Sagiv (1979). An Algorithm for Inferring Multivalued Dependencies That Works Also for a Subclass of Propositional Logic. Dept. Of Computer Science, University of Illinois at Urbana-Champaign.
     
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  29. Guo‐Jun Wang & Yan‐Hong She (2006). A Topological Characterization of Consistency of Logic Theories in Propositional Logic. Mathematical Logic Quarterly 52 (5):470-477.
    The main purpose of this note is to characterize consistency of logic theories in propositional logic by means of topological concept. Based on the concepts of truth degree of formulas and similarity degree between formulas the concept of logic metric space has been proposed by the first author. It is proved in this note that a closed logic theory Γ is consistent if and only if it contains no interior point in the logic metric (...)
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  30.  11
    Jan Krajíček (1995). Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press.
    This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then (...)
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  31.  44
    Marcello D'Agostino & Luciano Floridi (2009). The Enduring Scandal of Deduction: Is Propositional Logic Really Uninformative? Synthese 167 (2):271 - 315.
    Deductive inference is usually regarded as being "tautological" or "analytical": the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by (...)
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  32.  60
    Fabrice Correia (2000). Propositional Logic of Essence. Journal of Philosophical Logic 29 (3):295-313.
    This paper presents a propositional version of Kit Fine's (quantified) logic for essentialist statements, provides it with a semantics, and proves the former adequate (i.e. sound and complete) with respect to the latter.
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  33.  27
    Brian Hill & Francesca Poggiolesi (2010). A Contraction-Free and Cut-Free Sequent Calculus for Propositional Dynamic Logic. Studia Logica 94 (1):47 - 72.
    In this paper we present a sequent calculus for propositional dynamic logic built using an enriched version of the tree-hypersequent method and including an infinitary rule for the iteration operator. We prove that this sequent calculus is theoremwise equivalent to the corresponding Hilbert-style system, and that it is contraction-free and cut-free. All results are proved in a purely syntactic way.
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  34.  13
    Ernst Zimmermann (2002). A Predicate Logical Extension of a Subintuitionistic Propositional Logic. Studia Logica 72 (3):401-410.
    We develop a predicate logical extension of a subintuitionistic propositional logic. Therefore a Hilbert type calculus and a Kripke type model are given. The propositional logic is formulated to axiomatize the idea of strategic weakening of Kripke''s semantic for intuitionistic logic: dropping the semantical condition of heredity or persistence leads to a nonmonotonic model. On the syntactic side this leads to a certain restriction imposed on the deduction theorem. By means of a Henkin argument strong (...)
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  35.  13
    Giorgi Japaridze (2000). The Propositional Logic of Elementary Tasks. Notre Dame Journal of Formal Logic 41 (2):171-183.
    The paper introduces a semantics for the language of propositional additive-multiplicative linear logic. It understands formulas as tasks that are to be accomplished by an agent (machine, robot) working as a slave for its master (user, environment). This semantics can claim to be a formalization of the resource philosophy associated with linear logic when resources are understood as agents accomplishing tasks. I axiomatically define a decidable logic TSKp and prove its soundness and completeness with respect to (...)
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  36.  32
    Mojtaba Aghaei & Mohammad Ardeshir (2001). Gentzen-Style Axiomatizations for Some Conservative Extensions of Basic Propositional Logic. Studia Logica 68 (2):263-285.
    We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.
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  37.  43
    Xuefeng Wen (2007). A Propositional Logic with Relative Identity Connective and a Partial Solution to the Paradox of Analysis. Studia Logica 85 (2):251 - 260.
    We construct a a system PLRI which is the classical propositional logic supplied with a ternary construction , interpreted as the intensional identity of statements and in the context . PLRI is a refinement of Roman Suszko’s sentential calculus with identity (SCI) whose identity connective is a binary one. We provide a Hilbert-style axiomatization of this logic and prove its soundness and completeness with respect to some algebraic models. We also show that PLRI can be used to (...)
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  38.  13
    Merlijn Sevenster (2006). On the Computational Consequences of Independence in Propositional Logic. Synthese 149 (2):257 - 283.
    Sandu and Pietarinen [Partiality and Games: Propositional Logic. Logic J. IGPL 9 (2001) 101] study independence friendly propositional logics. That is, traditional propositional logic extended by means of syntax that allow connectives to be independent of each other, although the one may be subordinate to the other. Sandu and Pietarinen observe that the IF propositional logics have exotic properties, like functional completeness for three-valued functions. In this paper we focus on one of their (...)
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  39.  12
    Rosalie Iemhoff (2001). On the Admissible Rules of Intuitionistic Propositional Logic. Journal of Symbolic Logic 66 (1):281-294.
    We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conjecture by de Jongh and Visser is proved. We also present a proof system for the admissible rules, and give semantic criteria for admissibility.
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  40.  1
    G. Sandu & A. Pietarinen (2001). Partiality and Games: Propositional Logic. Logic Journal of the IGPL 9 (1):101-121.
    We study partiality in propositional logics containing formulas with either undefined or over-defined truth-values. Undefined values are created by adding a four-place connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally complete for all partial functions. Transjunction is seen to be motivated from a game-theoretic perspective, emerging from a two-stage extensive form semantic game of imperfect information between two players. This game-theoretic approach yields an interpretation where partiality is generated (...)
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  41.  14
    John T. Kearns (1997). Propositional Logic of Supposition and Assertion. Notre Dame Journal of Formal Logic 38 (3):325-349.
    This presentation of a system of propositional logic is a foundational paper for systems of illocutionary logic. The language contains the illocutionary force operators '' for assertion and ' ' for supposition. Sentences occurring in proofs of the deductive system must be prefixed with one of these operators, and rules of take account of the forces of the sentences. Two kinds of semantic conditions are investigated; familiar truth conditions and commitment conditions. Accepting a statement A or rejecting (...)
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  42.  32
    Richard Zach (1999). Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic. Bulletin of Symbolic Logic 5 (3):331-366.
    Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, (...)
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  43.  13
    Ahti-Veikko Pietarinen (2001). Propositional Logic of Imperfect Information: Foundations and Applications. Notre Dame Journal of Formal Logic 42 (4):193-210.
    I will show that the semantic structure of a new imperfect-information propositional logic can be described in terms of extensive forms of semantic games. I will discuss some ensuing properties of these games such as imperfect recall, informational consistency, and team playing. Finally, I will suggest a couple of applications that arise in physics, and most notably in quantum theory and quantum logics.
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  44. Wojciech Dzik, Jouni Järvinen & Michiro Kondo (2010). Intuitionistic Propositional Logic with Galois Connections. Logic Journal of the IGPL 18 (6):837-858.
    In this work, an intuitionistic propositional logic with a Galois connection is introduced. In addition to the intuitionistic logic axioms and inference rule of modus ponens, the logic contains only two rules of inference mimicking the performance of Galois connections. Both Kripke-style and algebraic semantics are presented for IntGC, and IntGC is proved to be complete with respect to both of these semantics. We show that IntGC has the finite model property and is decidable, but Glivenko's (...)
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  45. Albert Visser (2002). Substitutions of Σ10-Sentences: Explorations Between Intuitionistic Propositional Logic and Intuitionistic Arithmetic. Annals of Pure and Applied Logic 114 (1-3):227-271.
    This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional (...) . Here NNIL is the class of propositional formulas with no nestings of implications to the left . The identical embedding of IPC -derivability into a consequence relation has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL -preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA. (shrink)
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  46.  5
    D. Skvortsov (1997). Non-Axiomatizable Second Order Intuitionistic Propositional Logic. Annals of Pure and Applied Logic 86 (1):33-46.
    The second order intuitionistic propositional logic characterized by the class of all “principal” Kripke frames is non-recursively axiomatizable, as well as any logic of a class of principal Kripke frames containing every finite frame.
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  47.  11
    Theodore Hailperin (1984). Boole's Abandoned Propositional Logic. History and Philosophy of Logic 5 (1):39-48.
    The approach used by Boole in Mathematical analysis of logic to develop propositional logic was based on the idea of ?cases? or ?conjunctures of circumstances?. But this was dropped in Laws of thought in favor of one which Boole considered to be more satisfactory, that of using the notion of ?time for which a proposition is true?. We show that, when suitable clarifications and corrections are made, the earlier approach? which accords with modern logic in eschewing (...)
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  48.  1
    Lex Hendriks (2000). Doing Logic by Computer: Interpolation in Fragments of Intuitionistic Propositional Logic. Annals of Pure and Applied Logic 104 (1-3):97-112.
    In this paper we study the interpolation property in fragments of intuitionistic and propositional logic, using both proof theoretic and semantic techniques. We will also sketch some computational methods, based on the semantical techniques introduced, to obtain counterexamples in fragment where interpolation does not hold.
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  49.  1
    Sven Hartmann & Sebastian Link (2008). Characterising Nested Database Dependencies by Fragments of Propositional Logic. Annals of Pure and Applied Logic 152 (1):84-106.
    We extend the earlier results on the equivalence between the Boolean and the multivalued dependencies in relational databases and fragments of the Boolean propositional logic. It is shown that these equivalences are still valid for the databases that store complex data elements obtained from the recursive nesting of record, list, set and multiset constructors. The major proof argument utilises properties of Brouwerian algebras.The equivalences have several consequences. Firstly, they provide new insights into databases that are not in first (...)
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  50. Albert Visser (2002). Substitutions of< I> Σ_< Sub> 1< Sup> 0-Sentences: Explorations Between Intuitionistic Propositional Logic and Intuitionistic Arithmetic. [REVIEW] Annals of Pure and Applied Logic 114 (1):227-271.
    This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional (...) . Here NNIL is the class of propositional formulas with no nestings of implications to the left . The identical embedding of IPC -derivability into a consequence relation has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL -preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA. (shrink)
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