Results for 'Propositional calculus'

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  1.  3
    A Finite Hilbert‐Style Axiomatization of the Implication‐Less Fragment of the Intuitionistic Propositional Calculus.Jordi Rebagliato & Ventura Verdú - 1994 - Mathematical Logic Quarterly 40 (1):61-68.
    In this paper we obtain a finite Hilbert-style axiomatization of the implicationless fragment of the intuitionistic propositional calculus. As a consequence we obtain finite axiomatizations of all structural closure operators on the algebra of {–}-formulas containing this fragment.
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  2.  12
    A Cut-Free Gentzen Formulation of Basic Propositional Calculus.Kentaro Kikuchi & Katsumi Sasaki - 2003 - Journal of Logic, Language and Information 12 (2):213-225.
    We introduce a Gentzen style formulation of Basic Propositional Calculus(BPC), the logic that is interpreted in Kripke models similarly tointuitionistic logic except that the accessibility relation of eachmodel is not necessarily reflexive. The formulation is presented as adual-context style system, in which the left hand side of a sequent isdivided into two parts. Giving an interpretation of the sequents inKripke models, we show the soundness and completeness of the system withrespect to the class of Kripke models. The cut-elimination (...)
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  3.  3
    Simulating Non-Prenex Cuts in Quantified Propositional Calculus.Emil Jerábek & Phuong Nguyen - 2011 - Mathematical Logic Quarterly 57 (5):524-532.
    We show that the quantified propositional proof systems Gi are polynomially equivalent to their restricted versions that require all cut formulas to be prenex Σqi . Previously this was known only for the treelike systems G*i. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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  4. On Interpreting the S5 Propositional Calculus: An Essay in Philosophical Logic.Michael J. Carroll - 1976 - Dissertation, University of Iowa
    Discusses alternative interpretations of the modal operators, for the modal propositional logic S5.
     
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  5. Propositional Calculus.P. H. Nidditch - 1962 - New York: Dover Publications.
  6. Tables for the Propositional Calculus (Logico-Mathematical Brain).René[from old catalog] Calvache - 1966 - Miami, Fla..
     
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  7.  9
    Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic.Michael Kaminski & Nissim Francez - 2014 - 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.  4
    Basic Propositional Calculus I.Mohammad Ardeshir & Wim Ruitenburg - 1998 - Mathematical Logic Quarterly 44 (3):317-343.
    We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E1, a theory axiomatized by T → ⊥. The intersection CPC ∩ E1 is axiomatizable by the Principle of the Excluded Middle A V ∨ ⌝A. If (...)
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  9.  6
    The Lambek Calculus Extended with Intuitionistic Propositional Logic.Michael Kaminski & Nissim Francez - 2016 - Studia Logica 104 (5):1051-1082.
    We present sound and complete semantics and a sequent calculus for the Lambek calculus extended with intuitionistic propositional logic.
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  10.  7
    Valuation Semantics for Intuitionic Propositional Calculus and Some of its Subcalculi.Andréa Lopari? - 2011 - Principia: An International Journal of Epistemology 14 (1):125-33.
    In this paper, we present valuation semantics for the Propositional Intuitionistic Calculus (also called Heyting Calculus) and three important subcalculi: the Implicative, the Positive and the Minimal Calculus (also known as Kolmogoroff or Johansson Calculus). Algorithms based in our definitions yields decision methods for these calculi. DOI:10.5007/1808-1711.2010v14n1p125.
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  11. On the Period of Sequences (an(P)) in Intuitionistic Propositional Calculus.Wim Ruitenburg - 1984 - Journal of Symbolic Logic 49 (3):892 - 899.
    In classical propositional calculus for each proposition A(p) the following holds: $\vdash A(p) \leftrightarrow A^3(p)$ . In this paper we consider what remains of this in the intuitionistic case. It turns out that for each proposition A(p) the following holds: there is an n ∈ N such that $\vdash A^n(p) \leftrightarrow A^{n + 2}(p)$ . As a byproduct of the proof we give some theorems which may be useful elsewhere in propositional calculus.
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  12.  1
    Problems of Substitution and Admissibility in the Modal System Grz and in Intuitionistic Propositional Calculus.V. V. Rybakov - 1990 - Annals of Pure and Applied Logic 50 (1):71-106.
    Questions connected with the admissibility of rules of inference and the solvability of the substitution problem for modal and intuitionistic logic are considered in an algebraic framework. The main result is the decidability of the universal theory of the free modal algebra imageω extended in signature by adding constants for free generators. As corollaries we obtain: there exists an algorithm for the recognition of admissibility of rules with parameters in the modal system Grz, the substitution problem for Grz and for (...)
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  13.  11
    A Finite Model Theorem for the Propositional Μ-Calculus.Dexter Kozen - 1988 - Studia Logica 47 (3):233 - 241.
    We prove a finite model theorem and infinitary completeness result for the propositional -calculus. The construction establishes a link between finite model theorems for propositional program logics and the theory of well-quasi-orders.
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  14.  23
    An Alternative Propositional Calculus for Application to Empirical Sciences.Paul Weingartner - 2010 - Studia Logica 95 (1/2):233 - 257.
    The purpose of the paper is to show that by cleaning Classical Logic (CL) from redundancies (irrelevances) and uninformative complexities in the consequence class and from too strong assumptions (of CL) one can avoid most of the paradoxes coming up when CL is applied to empirical sciences including physics. This kind of cleaning of CL has been done successfully by distinguishing two types of theorems of CL by two criteria. One criterion (RC) forbids such theorems in which parts of the (...)
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  15.  11
    Basic Propositional Calculus II. Interpolation.Mohammad Ardeshir & Wim Ruitenburg - 2001 - Archive for Mathematical Logic 40 (5):349-364.
    Let ℒ and ? be propositional languages over Basic Propositional Calculus, and ℳ = ℒ∩?. Weprove two different but interrelated interpolation theorems. First, suppose that Π is a sequent theory over ℒ, and Σ∪ {C⇒C′} is a set of sequents over ?, such that Π,Σ⊢C⇒C′. Then there is a sequent theory Φ over ℳ such that Π⊢Φ and Φ, Σ⊢C⇒C′. Second, let A be a formula over ℒ, and C 1, C 2 be formulas over ?, such (...)
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  16.  46
    Is Propositional Calculus Categorical?Jaroslav Peregrin - manuscript
    According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done within first-order (...)
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  17.  28
    Three Contributions to the Two-Valued Propositional Calculus.Stanisław Jaśkowski - 1975 - Studia Logica 34 (1):121 - 132.
    Three chapters contain the results independent of each other. In the first chapter I present a set of axioms for the propositional calculus which are shorter than the ones known so far, in the second one I give a method of defining all ternary connectives, in the third one, I prove that the probability of propositional functions is preserved under reversible substitutions.
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  18.  15
    Independence of Two Nice Sets of Axioms for the Propositional Calculus.T. Thacher Robinson - 1968 - Journal of Symbolic Logic 33 (2):265-270.
    Kanger [4] gives a set of twelve axioms for the classical propositional Calculus which, together with modus ponens and substitution, have the following nice properties: (0.1) Each axiom contains $\supset$ , and no axiom contains more than two different connectives. (0.2) Deletions of certain of the axioms yield the intuitionistic, minimal, and classical refutability1 subsystems of propositional calculus. (0.3) Each of these four systems of axioms has the separation property: that if a theorem is provable in (...)
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  19.  20
    A Proof of Axiomatizability of Łukasiewicz's Three-Valued Implicational Propositional Calculus.T. Prucnal - 1967 - Studia Logica 20 (1):144-144.
    LetL 3 c be the smallest set of propositional formulas, which containsCpCqpCCCpqCrqCCqpCrpCCCpqCCqrqCCCpqppand is closed with respect to substitution and detachment. Let $\mathfrak{M}_3^c $ be Łukasiewicz’s three-valued implicational matrix defined as follows:cxy=min (1,1−x+y), where $x,y \in \{ 0,\tfrac{1}{2},1\}$ . In this paper the following theorem is proved: $$L_3^c = E( \mathfrak{M}_3^c )$$ The idea used in the proof is derived from Asser’s proof of completeness of the two-valued propositional calculus. The proof given here is based on the Pogorzelski’s (...)
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  20.  19
    A Note on the Completeness of Kozen's Axiomatisation of the Propositional Μ-Calculus.Igor Walukiewicz - 1996 - Bulletin of Symbolic Logic 2 (3):349-366.
    The propositional μ -calculus is an extension of the modal system K with a least fixpoint operator. Kozen posed a question about completeness of the axiomatisation of the logic which is a small extension of the axiomatisation of the modal system K. It is shown that this axiomatisation is complete.
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  21.  18
    A Note on the System of Propositional Calculus with Primitive Rule of Extensionality.K. Hałkowska - 1967 - Studia Logica 20 (1):150-150.
    The present paper deals with a systemS of propositional calculus, conjunction, equivalence and falsum being its primitive terms.The only primitive rule inS is the rule of extensionality defined by the scheme: $\frac{{E\alpha \beta ,\Phi (\alpha )}}{{\Phi (\beta )}}$.
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  22.  19
    Quantum Logic and the Classical Propositional Calculus.Othman Qasim Malhas - 1987 - Journal of Symbolic Logic 52 (3):834-841.
    In much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.
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  23.  6
    Rudimentary Kripke Models for the Intuitionistic Propositional Calculus.Kosta Došen - 1993 - Annals of Pure and Applied Logic 62 (1):21-49.
    It is shown that the intuitionistic propositional calculus is sound and complete with respect to Kripke-style models that are not quasi-ordered. These models, called rudimentary Kripke models, differ from the ordinary intuitionistic Kripke models by making fewer assumptions about the underlying frames, but have the same conditions for valuations. However, since accessibility between points in the frames need not be reflexive, we have to assume, besides the usual intuitionistic heredity, the converse of heredity, which says that if a (...)
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  24.  4
    Description of All Functions Definable by Formulæ of the 2nd Order Intuitionistic Propositional Calculus on Some Linear Heyting Algebras.Dimitri Pataraia - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):457-483.
    Explicit description of maps definable by formulæ of the second order intuitionistic propositional calculus is given on two classes of linear Heyting algebras?the dense ones and the ones which possess successors. As a consequence, it is shown that over these classes every formula is equivalent to a quantifier free formula in the dense case, and to a formula with quantifiers confined to the applications of the successor in the second case.
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  25.  3
    Sets, Classes and the Propositional Calculus.E. Lopez-Escobar - 2005 - Manuscrito 28 (2):417-448.
    The propositional calculus AoC, “Algebra of Classes”,and the extended propositional calculus EAC, “Extended Algebra ofClasses” are introduced in this paper. They are extensions, by additionalpropositional functions which are not invariant under the biconditional,of the corresponding classical propositional systems. Theirorigin lies in an analysis, motivated by Cantor’s concept of the cardinalnumbers, of A. P. Morse’s impredicative, polysynthetic set theory.
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  26.  5
    Interpolants, Cut Elimination and Flow Graphs for the Propositional Calculus.Alessandra Carbone - 1997 - Annals of Pure and Applied Logic 83 (3):249-299.
    We analyse the structure of propositional proofs in the sequent calculus focusing on the well-known procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing the flow of formulas in it . We show some general facts about logical graphs such as acyclicity of cut-free proofs and acyclicity of contraction-free proofs , and we (...)
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  27.  20
    Formalization of Functionally Complete Propositional Calculus with the Functor of Implication as the Only Primitive Term.Czesław Lejewski - 1989 - Studia Logica 48 (4):479 - 494.
    The most difficult problem that Leniewski came across in constructing his system of the foundations of mathematics was the problem of defining definitions, as he used to put it. He solved it to his satisfaction only when he had completed the formalization of his protothetic and ontology. By formalization of a deductive system one ought to understand in this context the statement, as precise and unambiguous as possible, of the conditions an expression has to satisfy if it is added to (...)
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  28.  13
    The Calculus of Higher-Level Rules, Propositional Quantification, and the Foundational Approach to Proof-Theoretic Harmony.Peter Schroeder-Heister - 2014 - Studia Logica 102 (6):1185-1216.
    We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational account of proof-theoretic harmony. With every set of introduction rules (...)
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  29.  12
    Examining Fragments of the Quantified Propositional Calculus.Steven Perron - 2008 - Journal of Symbolic Logic 73 (3):1051-1080.
    When restricted to proving $\Sigma _{i}^{q}$ formulas, the quantified propositional proof system $G_{i}^{\ast}$ is closely related to the $\Sigma _{i}^{b}$ theorems of Buss's theory $S_{2}^{i}$ . Namely, $G_{i}^{\ast}$ has polynomial-size proofs of the translations of theorems of $S_{2}^{i}$ , and $S_{2}^{i}$ proves that $G_{i}^{\ast}$ is sound. However, little is known about $G_{i}^{\ast}$ when proving more complex formulas. In this paper, we prove a witnessing theorem for $G_{i}^{\ast}$ similar in style to the KPT witnessing theorem for $T_{2}^{i}$ . This witnessing (...)
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  30.  22
    Metalogic of Intuitionistic Propositional Calculus.Alex Citkin - 2010 - Notre Dame Journal of Formal Logic 51 (4):485-502.
    With each superintuitionistic propositional logic L with a disjunction property we associate a set of modal logics the assertoric fragment of which is L . Each formula of these modal logics is interdeducible with a formula representing a set of rules admissible in L . The smallest of these logics contains only formulas representing derivable in L rules while the greatest one contains formulas corresponding to all admissible in L rules. The algebraic semantic for these logics is described.
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  31. Extracting the Resolution Algorithm From a Completeness Proof for the Propositional Calculus.Robert Constable & Wojciech Moczydłowski - 2009 - Annals of Pure and Applied Logic 161 (3):337-348.
    We prove constructively that for any propositional formula in Conjunctive Normal Form, we can either find a satisfying assignment of true and false to its variables, or a refutation of showing that it is unsatisfiable. This refutation is a resolution proof of ¬. From the formalization of our proof in Coq, we extract Robinson’s famous resolution algorithm as a Haskell program correct by construction. The account is an example of the genre of highly readable formalized mathematics.
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  32.  39
    A Propositional Calculus with Denumerable Matrix.Michael Dummett - 1959 - Journal of Symbolic Logic 24 (2):97-106.
  33. On Formulas of One Variable in Intuitionistic Propositional Calculus.Iwao Nishimura - 1960 - Journal of Symbolic Logic 25 (4):327-331.
  34. Completeness of Intuitionistic Propositional Calculus.Harvey Friedman - manuscript
    An assignment is a function f that assigns subsets of N to some atoms. Then f is extended to f* which sends every formula A of HPC to a subset of S(A).
     
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  35. The Metatheory of the Classical Propositional Calculus is Not Axiomatizable.Ian Mason - 1985 - Journal of Symbolic Logic 50 (2):451-457.
  36.  59
    Propositional Calculus for Contradictory Deductive Systems.Stanisław Jaśkowski - 1969 - Studia Logica 24 (1):143 - 160.
  37. Peirce's Axioms for Propositional Calculus.A. N. Prior - 1958 - Journal of Symbolic Logic 23 (2):135-136.
  38.  21
    A Formalization of the Propositional Calculus of H-B Logic.Cecylia Rauszer - 1974 - Studia Logica 33 (1):23 - 34.
  39.  23
    Quantified Propositional Calculus and a Second-Order Theory for NC1.Stephen Cook & Tsuyoshi Morioka - 2005 - Archive for Mathematical Logic 44 (6):711-749.
  40.  5
    On the Discussive Conjunction in the Propositional Calculus for Inconsistent Deductive Systems.Stanisław Jaśkowski - 1999 - Logic and Logical Philosophy 7:57.
  41.  16
    The Separation Theorem of Intuitionist Propositional Calculus.Alfred Horn - 1962 - Journal of Symbolic Logic 27 (4):391-399.
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  42.  17
    A Propositional Calculus for Inconsistent Deductive Systems.Stanisław Jaśkowski - 1999 - Logic and Logical Philosophy 7:35.
  43.  13
    Notes on the Axiomatics of the Propositional Calculus.C. A. Meredith & A. N. Prior - 1963 - Notre Dame Journal of Formal Logic 4 (3):171-187.
  44.  16
    The Finite Model Property and Subsystems of Classical Propositional Calculus.Ronald Harrop - 1959 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 5 (1-2):29-32.
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  45.  41
    Epistemological Remarks on the Propositional Calculus.Karl Britton - 1935 - Analysis 3 (4):57 - 63.
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  46.  17
    Remarks on Discussive Propositional Calculus.Tomasz Furmanowski - 1975 - Studia Logica 34 (1):39 - 43.
  47.  12
    iH-Propositional Calculus.A. Figallo Jr, G. Ramón & S. Saad - 2006 - Bulletin of the Section of Logic 35 (4):157-162.
  48.  16
    An Alternative Formalisation of Sobociński's Three-Valued Implicational Propositional Calculus.Alan Rose - 1956 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 2 (10-15):166-172.
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  49.  23
    A Formalisation, Using Non‐Standard Rules, of A 5‐Valued Propositional Calculus.Alan Rose† - 1987 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (2):187-192.
  50.  17
    A Formalisation of Post'sm-Valued Propositional Calculus with Variable Functors.Alan Rose - 1965 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 11 (3):221-226.
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