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Journal articles on the topic 'Propositional logic'

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Published: 4 June 2021

Last updated: 22 June 2024

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1

WHITEN, BILL. "A SIMPLE ALGORITHM FOR DEDUCTION." ANZIAM Journal 51, no. 1 (July 2009): 102–22. http://dx.doi.org/10.1017/s1446181109000352.

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AbstractIt is shown that a simple deduction engine can be developed for a propositional logic that follows the normal rules of classical logic in symbolic form, but the description of what is known about a proposition uses two numeric state variables that conveniently describe unknown and inconsistent, as well as true and false. Partly true and partly false can be included in deductions. The multi-valued logic is easily understood as the state variables relate directly to true and false. The deduction engine provides a convenient standard method for handling multiple or complicated logical relations. It is particularly convenient when the deduction can start with different propositions being given initial values of true or false. It extends Horn clause based deduction for propositional logic to arbitrary clauses. The logic system used has potential applications in many areas. A comparison with propositional logic makes the paper self-contained.

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2

Al-Khowarizmi, Al-Khowarizmi, Asrar Aspia Manurung, and Mulkan Azhari. "Design of an Application to Calculate Student Grades in Learning Logic Informatics Propositional Calculus Material." Hanif Journal of Information Systems 1, no. 1 (August 24, 2023): 18–25. http://dx.doi.org/10.56211/hanif.v1i1.7.

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Propositional calculus is a method used to calculate the truth value of a proposition. Propositional calculus is commonly studied by various students, from high school to college. Propositional calculus studies the truth value (True/False) of a proposition. The proposition is then processed into the steps of working on the proposition according to the order of operator priority, which is packaged into truth tables and logical expressions. The proposition work done is not easy, but consumes a lot of time and writing media if done manually. Errors in doing proposition work result in inaccurate truth values obtained. Therefore, the author is interested in making an application to calculate student grades using the propositional calculus method that can be used to overcome the problem of working on propositions manually, so that it can produce accurate truth values, which can also be used as a learning medium and media for students to calculate and see the results of their grades manually.

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3

Bedregal, Benjamín René Callejas, and Anderson Paiva Cruz. "Propositional Logic as a Propositional Fuzzy Logic." Electronic Notes in Theoretical Computer Science 143 (January 2006): 5–12. http://dx.doi.org/10.1016/j.entcs.2005.05.023.

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4

Citkin, Alex. "Deductive systems with unified multiple-conclusion rules." Logical Investigations 26, no. 2 (December 13, 2020): 87–105. http://dx.doi.org/10.21146/2074-1472-2020-26-2-87-105.

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Our goal is to develop a syntactical apparatus for propositional logics in which the accepted and rejected propositions have the same status and are being treated in the same way. The suggested approach is based on the ideas of Ƚukasiewicz used for the classical logic and in addition, it includes the use of multiple conclusion rules. A special attention is paid to the logics in which each proposition is either accepted or rejected.

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5

FRITZ, PETER. "LOGICS FOR PROPOSITIONAL CONTINGENTISM." Review of Symbolic Logic 10, no. 2 (March 20, 2017): 203–36. http://dx.doi.org/10.1017/s1755020317000028.

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AbstractRobert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete.

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6

Gehrke, Mai, Carol Walker, and Elbert Walker. "A Mathematical Setting for Fuzzy Logics." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 05, no. 03 (June 1997): 223–38. http://dx.doi.org/10.1142/s021848859700021x.

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The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and -x=1-x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as three-valued logic. The propositional logic obtained when the algebra of truth values is the set {(a, b)|a≤ b and a,b∈[0,1]} of subintervals of the unit interval with component-wise operations, is propositional interval-valued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by finite algebras, it follows that there are finite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are discussed for both of these logics.

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7

O'Hearn, Peter W., and David J. Pym. "The Logic of Bunched Implications." Bulletin of Symbolic Logic 5, no. 2 (June 1999): 215–44. http://dx.doi.org/10.2307/421090.

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AbstractWe introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers and which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels.

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8

Gärdenfors, Peter. "Propositional logic based on the dynamics of belief." Journal of Symbolic Logic 50, no. 2 (June 1985): 390–94. http://dx.doi.org/10.2307/2274226.

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In this article propositions will be identified with a certain kind of changes of belief. The intended interpretation is that a proposition is characterised by the change it would induce if added to a state of belief. Propositions will thus be defined as functions from states of belief to states of belief. A set of postulates concerning the properties and existence of propositions will be formulated. A proposition will be said to be a tautology iff it is the identity function on states of belief. The main result is that the logic determined by the set of postulates is intuitionistic propositional logic.The basic epistemic concept is that of a belief model, which is defined as a pair 〈, 〉, where is a nonempty set and is a class of functions from to . The elements in will be called states of belief and they will be denoted K, K′,…. A discussion of the epistemological interpretation of the states of belief can be found in Gärdenfors [2]. Here, no assumptions about the structure of the elements in will be made.The elements in will be called propositions, and A, B, C, … will be used as variables over . Functions from states of belief to states of belief can be characterised as epistemic inputs. The intended interpretation of the functions in is that they correspond to changes of belief where the new evidence is accepted as “certain” or “known” in the resulting state of belief. This means that not all functions defined on can properly be called propositions.

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9

MAHER, MICHAEL J. "Propositional defeasible logic has linear complexity." Theory and Practice of Logic Programming 1, no. 6 (November 2001): 691–711. http://dx.doi.org/10.1017/s1471068401001168.

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Defeasible logic is a rule-based nonmonotonic logic, with both strict and defeasible rules, and a priority relation on rules. We show that inference in the propositional form of the logic can be performed in linear time. This contrasts markedly with most other propositional nonmonotonic logics, in which inference is intractable.

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10

DZIK, WOJCIECH, and PIOTR WOJTYLAK. "UNIFICATION IN SUPERINTUITIONISTIC PREDICATE LOGICS AND ITS APPLICATIONS." Review of Symbolic Logic 12, no. 1 (December 3, 2018): 37–61. http://dx.doi.org/10.1017/s1755020318000011.

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AbstractWe introduce unification in first-order logic. In propositional logic, unification was introduced by S. Ghilardi, see Ghilardi (1997, 1999, 2000). He successfully applied it in solving systematically the problem of admissibility of inference rules in intuitionistic and transitive modal propositional logics. Here we focus on superintuitionistic predicate logics and apply unification to some old and new problems: definability of disjunction and existential quantifier, disjunction and existential quantifier under implication, admissible rules, a basis for the passive rules, (almost) structural completeness, etc. For this aim we apply modified specific notions, introduced in propositional logic by Ghilardi, such as projective formulas, projective unifiers, etc.Unification in predicate logic seems to be harder than in the propositional case. Any definition of the key concept of substitution for predicate variables must take care of individual variables. We allow adding new free individual variables by substitutions (contrary to Pogorzelski & Prucnal (1975)). Moreover, since predicate logic is not as close to algebra as propositional logic, direct application of useful algebraic notions of finitely presented algebras, projective algebras, etc., is not possible.

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11

Grossi, Davide, Emiliano Lorini, and Francois Schwarzentruber. "The Ceteris Paribus Structure of Logics of Game Forms." Journal of Artificial Intelligence Research 53 (May 27, 2015): 91–126. http://dx.doi.org/10.1613/jair.4666.

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The article introduces a ceteris paribus modal logic, called CP, interpreted on the equivalence classes induced by finite sets of propositional atoms. This logic is studied and then used to embed three logics of strategic interaction, namely atemporal STIT, the coalition logic of propositional control (CL−PC) and the starless fragment of the dynamic logic of propositional assignments (DL−PA). The embeddings highlight a common ceteris paribus structure underpinning the key operators of all these apparently very different logics and show, we argue, remarkable similarities behind some of the most influential formalisms for reasoning about strategic interaction

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12

KREMER, PHILIP. "COMPLETENESS OF SECOND-ORDER PROPOSITIONAL S4 AND H IN TOPOLOGICAL SEMANTICS." Review of Symbolic Logic 11, no. 3 (September 2018): 507–18. http://dx.doi.org/10.1017/s1755020318000229.

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AbstractWe add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

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13

Tzouvaras, Athanassios. "Propositional superposition logic." Logic Journal of the IGPL 26, no. 1 (November 23, 2017): 149–90. http://dx.doi.org/10.1093/jigpal/jzx054.

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14

Savinov, A. A. "Fuzzy propositional logic." Fuzzy Sets and Systems 60, no. 1 (November 1993): 9–17. http://dx.doi.org/10.1016/0165-0114(93)90284-o.

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15

Gries, David, and Fred B. Schneider. "Equational propositional logic." Information Processing Letters 53, no. 3 (February 1995): 145–52. http://dx.doi.org/10.1016/0020-0190(94)00198-8.

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16

Wölfl, Stefan. "Propositional Q-Logic." Journal of Philosophical Logic 31, no. 5 (October 2002): 387–414. http://dx.doi.org/10.1023/a:1020163602542.

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17

Dyrkolbotn, Sjur, and Michał Walicki. "Propositional discourse logic." Synthese 191, no. 5 (May 30, 2013): 863–99. http://dx.doi.org/10.1007/s11229-013-0297-x.

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18

Fairtlough, Matt, and Michael Mendler. "Propositional Lax Logic." Information and Computation 137, no. 1 (August 1997): 1–33. http://dx.doi.org/10.1006/inco.1997.2627.

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19

Kamide, Norihiro. "Relating first-order monadic omega-logic, propositional linear-time temporal logic, propositional generalized definitional reflection logic and propositional infinitary logic." Journal of Logic and Computation 27, no. 7 (February 27, 2017): 2271–301. http://dx.doi.org/10.1093/logcom/exx006.

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20

Aghaei, Mojtaba, and Mohammad Ardeshir. "A Bounded Translation of Intuitionistic Propositional Logic into Basic Propositional Logic." MLQ 46, no. 2 (May 2000): 199–206. http://dx.doi.org/10.1002/(sici)1521-3870(200005)46:2<199::aid-malq199>3.0.co;2-b.

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21

MALINK, MARKO, and ANUBAV VASUDEVAN. "THE PERIPATETIC PROGRAM IN CATEGORICAL LOGIC: LEIBNIZ ON PROPOSITIONAL TERMS." Review of Symbolic Logic 13, no. 1 (November 6, 2018): 141–205. http://dx.doi.org/10.1017/s1755020318000266.

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AbstractGreek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$.

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22

Zantema, Hans, and Jan Friso Groote. "Transforming equality logic to propositional logic." Electronic Notes in Theoretical Computer Science 86, no. 1 (May 2003): 162–73. http://dx.doi.org/10.1016/s1571-0661(04)80661-3.

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23

Kakas, Antonis C., Paolo Mancarella, and Francesca Toni. "On Argumentation Logic and Propositional Logic." Studia Logica 106, no. 2 (July 19, 2017): 237–79. http://dx.doi.org/10.1007/s11225-017-9736-x.

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24

Aranda, Víctor. "Completud débil y Post completud en la escuela de Hilbert." Humanities Journal of Valparaiso, no. 14 (December 29, 2019): 449. http://dx.doi.org/10.22370/rhv2019iss14pp449-466.

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The aim of this paper is to clarify why propositional logic is Post complete and its weak completeness was almost unnoticed by Hilbert and Bernays, while first-order logic is Post incomplete and its weak completeness was seen as an open problem by Hilbert and Ackermman. Thus, I will compare propositional and first-order logic in the Prinzipien der Mathematik, Bernays’s second Habilitationsschrift and the Grundzüge der Theoretischen Logik. The so called “arithmetical interpretation”, the conjunctive and disjunctive normal forms and the soundness of the propositional rules of inference deserve special emphasis.

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25

PASEAU, ALEXANDER. "CAPTURING CONSEQUENCE." Review of Symbolic Logic 12, no. 2 (March 4, 2019): 271–95. http://dx.doi.org/10.1017/s1755020318000291.

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AbstractFirst-order formalisations are often preferred to propositional ones because they are thought to underwrite the validity of more arguments. We compare and contrast the ability of some well-known logics—these two in particular—to formally capture valid and invalid arguments. We show that there is a precise and important sense in which first-order logic does not improve on propositional logic in this respect. We also prove some generalisations and related results of philosophical interest. The rest of the article investigates the results’ philosophical significance. A first moral is that the correct way to state the oft-cited superiority of first-order logic vis-à-vis propositional logic is more nuanced than often thought. The second moral concerns semantic theory; the third logic’s use as a tool for discovery. A fourth and final moral is that second-order logic’s transcendence of first-order logic is greater than first-order logic’s transcendence of propositional logic.

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26

Aravantinos, V., R. Caferra, and N. Peltier. "Decidability and Undecidability Results for Propositional Schemata." Journal of Artificial Intelligence Research 40 (March 22, 2011): 599–656. http://dx.doi.org/10.1613/jair.3351.

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We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions and iterated connectives ranging over intervals parameterized by arithmetic variables. The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called bound-linear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability.

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27

Pathak, Surendra Raj, and Raj Narayan Yadav. "A study on Contact Algebra by Browerian Logic." Amrit Research Journal 1, no. 1 (September 17, 2020): 78–85. http://dx.doi.org/10.3126/arj.v1i1.32458.

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There is a close relation between Boolean logic or two-valued logic and an electric di-contact algebra. Two-valued logic is concerned with propositions which are either true or false and which can be combined in various ways. Similarly, the switches of circuits are activated by contacts which, open or closed, can be combined in analogous ways. But there are positions which are not two-valued - a generalisation of truth values of a proposition leads to an n-valued logic. It is then natural to raise the query whether it is possible to generalise the notion of switching contact analogous to the generalisation of truth value of a proposition. If it is so, does there exist an isomorphism between propositional algebra in n-valued logic and a structure in switching circuits based on contact values? The solution of the problems leads to a new algebra. Here we have reviewed this contact algebra by Browerian logic.

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28

McKay, C. G. "A consistent prepositional logic without any finite models." Journal of Symbolic Logic 50, no. 1 (March 1985): 38–41. http://dx.doi.org/10.2307/2273785.

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Some propositional logics (e.g. the classical system) can be characterized by a finite model, while others (e.g. Heyting's) which have the finite model property (FMP) can be characterized by an infinite set of finite models. Still others (e.g. certain extensions of Heyting's logic) which lack the FMP can only be characterized by a set of models, at least one of which is infinite. Yet all these logics admit finite models even though they may not be characterized by them. (For example, they all admit the 2-element Boolean algebra as a model in the sense that all their theorems are valid on that algebra when the propositional connectives are interpreted in the usual manner.) The object of the present paper is to give a (not too artificial) example of a propositional logic which is consistent and which admits only infinite models. It therefore lacks the FMP in a very strong sense. Such a propositional logic, I shall call hyperinfinite. The existence of hyperinfinite logics was already plausible from a result in abstract algebra which says that there are varieties of algebras of which the only finite element is the trivial algebra (see [3]).I wish to thank Professor A. S. Troelstra, Amsterdam, for comments on an early version of this paper. The constructive criticism of two anonymous referees has also been useful.The hyperinfinite propositional logic to be described is obtained from Positive Logic—the negative-free part of Heyting's logic—by adjoining certain axioms which govern the use of a unary modal connective.

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29

Brast-McKie, Benjamin. "Identity and Aboutness." Journal of Philosophical Logic 50, no. 6 (October 25, 2021): 1471–503. http://dx.doi.org/10.1007/s10992-021-09612-w.

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AbstractThis paper develops a theory of propositional identity which distinguishes necessarily equivalent propositions that differ in subject-matter. Rather than forming a Boolean lattice as in extensional and intensional semantic theories, the space of propositions forms a non-interlaced bilattice. After motivating a departure from tradition by way of a number of plausible principles for subject-matter, I will provide a Finean state semantics for a novel theory of propositions, presenting arguments against the convexity and nonvacuity constraints which Fine (Journal of Philosophical Logic, 4545, 199–226 13, 14, 15) introduces. I will then move to compare the resulting logic of propositional identity (PI1) with Correia’s (The Review of Symbolic Logic, 9, 103–122 9) logic of generalised identity (GI), as well as the first degree fragment of Angell’s (2) logic of analytic containment (AC). The paper concludes by extending PI1 to include axioms and rules for a subject-matter operator, providing a much broader theory of subject-matter than the principles with which I will begin.

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30

van Alten, C. J. "The finite model property for knotted extensions of propositional linear logic." Journal of Symbolic Logic 70, no. 1 (March 2005): 84–98. http://dx.doi.org/10.2178/jsl/1107298511.

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AbstractThe logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: . It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such logic has the finite model property with respect to its algebraic semantics and hence that the logic is decidable.

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31

GORANKO, VALENTIN, and ANTTI KUUSISTO. "LOGICS FOR PROPOSITIONAL DETERMINACY AND INDEPENDENCE." Review of Symbolic Logic 11, no. 3 (April 2, 2018): 470–506. http://dx.doi.org/10.1017/s1755020317000272.

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AbstractThis paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic${\cal D}$and Propositional Independence Logic${\cal I}$are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics${{\cal L}_D}$and${{\cal L}_{\,I\,}}$, based on Kripke semantics, and propose them as alternatives for${\cal D}$and${\cal I}$, respectively. We analyse the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We argue that${{\cal L}_D}$and${{\cal L}_{\,I\,}}$naturally resolve a range of interpretational problems that arise in${\cal D}$and${\cal I}$. We also obtain sound and complete axiomatizations for${{\cal L}_D}$and${{\cal L}_{\,I\,}}$.

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32

Punčochář, Vít, and Igor Sedlár. "Inquisitive Propositional Dynamic Logic." Journal of Logic, Language and Information 30, no. 1 (January 8, 2021): 91–116. http://dx.doi.org/10.1007/s10849-020-09326-3.

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33

Al-Odhari, Adel Mohammed. "Features of propositional logic." Pure Mathematical Sciences 10, no. 1 (2021): 35–44. http://dx.doi.org/10.12988/pms.2021.91275.

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34

Correia, Fabrice. "Propositional Logic of Essence." Journal of Philosophical Logic 29, no. 3 (June 2000): 295–313. http://dx.doi.org/10.1023/a:1004796309066.

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35

Japaridze, Giorgi. "Propositional computability logic I." ACM Transactions on Computational Logic 7, no. 2 (April 2006): 302–30. http://dx.doi.org/10.1145/1131313.1131318.

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36

Japaridze, Giorgi. "Propositional computability logic II." ACM Transactions on Computational Logic 7, no. 2 (April 2006): 331–62. http://dx.doi.org/10.1145/1131313.1131319.

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37

Caridroit, Thomas, Sébastien Konieczny, and Pierre Marquis. "Contraction in propositional logic." International Journal of Approximate Reasoning 80 (January 2017): 428–42. http://dx.doi.org/10.1016/j.ijar.2016.06.010.

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38

Shiver, Anthony. "Propositional Logic Card Games." Teaching Philosophy 36, no. 1 (2013): 51–58. http://dx.doi.org/10.5840/teachphil20133614.

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39

Baratella, Stefano. "Continuous propositional modal logic." Journal of Applied Non-Classical Logics 28, no. 4 (May 11, 2018): 297–312. http://dx.doi.org/10.1080/11663081.2018.1468677.

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40

Prevost, N., R. E. Jennings, L. Jorgenson, and F. D. Fracehia. "Visualization in propositional logic." IEEE Computer Graphics and Applications 16, no. 2 (March 1996): 6–8. http://dx.doi.org/10.1109/38.486673.

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41

Azizi-Sultan, Ahmad-Saher. "Constrained Pseudo-Propositional Logic." Logica Universalis 14, no. 4 (November 5, 2020): 523–35. http://dx.doi.org/10.1007/s11787-020-00266-x.

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42

Castañeda, Hector-Neri. "Leibniz's complete propositional logic." Topoi 9, no. 1 (March 1990): 15–28. http://dx.doi.org/10.1007/bf00147626.

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43

Ren, Xuanzhi. "Fullness and Decidability in Continuous Propositional Logic." Mathematics 10, no. 23 (November 25, 2022): 4455. http://dx.doi.org/10.3390/math10234455.

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In this paper we consider general continuous propositional logics and prove some basic properties about them. First, we characterize full systems of continuous connectives of the form {¬,,f} where f is a unary connective. We also show that, in contrast to the classical propositional logic, a full system of continuous propositional logic cannot contain only one continuous connective. We then construct a closed full system of continuous connectives without any constants. Such a system does not have any tautologies. For the rest of the paper we consider the standard continuous propositional logic as defined by Yaacov, I.B and Usvyatsov, A. We show that Strong Compactness and Craig Interpolation fail for this logic, but approximated versions of Strong Compactness and Craig Interpolation hold true. In the last part of the paper, we introduce various notions of satisfiability, falsifiability, tautology, and fallacy, and show that they are either NP-complete or co-NP-complete.

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44

Onoprienko, A. A. "Topological Models of Propositional Logic of Problems and Propositions." Moscow University Mathematics Bulletin 77, no. 5 (October 2022): 236–41. http://dx.doi.org/10.3103/s0027132222050059.

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45

Antonelli, G. Aldo, and Richmond H. Thomason. "Representability in second-order propositional poly-modal logic." Journal of Symbolic Logic 67, no. 3 (September 2002): 1039–54. http://dx.doi.org/10.2178/jsl/1190150147.

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AbstractA propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.

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Ramos, Jaime, João Rasga, and Cristina Sernadas. "Schema Complexity in Propositional-Based Logics." Mathematics 9, no. 21 (October 21, 2021): 2671. http://dx.doi.org/10.3390/math9212671.

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The essential structure of derivations is used as a tool for measuring the complexity of schema consequences in propositional-based logics. Our schema derivations allow the use of schema lemmas and this is reflected on the schema complexity. In particular, the number of times a schema lemma is used in a derivation is not relevant. We also address the application of metatheorems and compare the complexity of a schema derivation after eliminating the metatheorem and before doing so. As illustrations, we consider a propositional modal logic presented by a Hilbert calculus and an intuitionist propositional logic presented by a Gentzen calculus. For the former, we discuss the use of the metatheorem of deduction and its elimination, and for the latter, we analyze the cut and its elimination. Furthermore, we capitalize on the result for the cut elimination for intuitionistic logic, to obtain a similar result for Nelson’s logic via a language translation.

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Iliukhina, N. A. "ABOUT PROJECTIONS OFTHE MENTAL STRUCTURE INTO LANGUAGE AND SPEECH (on the example ofthe concept-proposition)." Voprosy Kognitivnoy Lingvistiki, no. 1 (2021): 70–79. http://dx.doi.org/10.20916/1812-3228-2021-1-70-79.

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The author examines, using the example of a concept-proposition, multiple manifestations of the connection between a mental unit and units of a language system, argues that the principle of structuring knowledge, characteristic of a proposition, is found in the semantic structure of a sentence, a word-formation nest, in the logic of lexical metonymy, in the logic of transferring definitions, as well as in the ability of a noun to represent knowledge of a propositional nature in speech, concludes that there is a deep commonality of mental and linguistic activity. The semantic structure of the sentence and the structure of the syntactic proposition are isomorphic to the structure of the mental proposition. The vectors of the transfer of the definition from one term of the sentence to another, as well as to the designation of the entire situation, often have propositional logic and are closed by the framework of one sentence. Propositional logic is observed in many word-building nests, especially consistent with the structure of nests organized by a polyactant verb. Among the verbal derivatives in the nest, the percentage of lexemes that name the components of the corresponding situation is significant, in some cases - all the main components of the situation. At the lexico-semantic level, the projection of a proposition on the phenomenon of metonymy is described. Among the models of transference, a variety is highlighted, called propositional metonymy. It includes transfers of the name, the vectors of which (shift of the focus of attention) reflect the structure of the proposition. Another manifestation of the connection between a proposition and linguistic units, considered in the article, is the facts of using a single noun in speech of any lexical and grammatical semantics to represent all situations of any structural complexity. The realized perspective of the research (from the mental unit to linguistic units and processes that reveal with it a certain isomorphism in the logic of categorization and conceptualization of knowledge) allows us to reveal an important line of interaction between the mental and linguistic levels.

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Kremer, Philip. "On the complexity of propositional quantification in intuitionistic logic." Journal of Symbolic Logic 62, no. 2 (June 1997): 529–44. http://dx.doi.org/10.2307/2275545.

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AbstractWe define a propositionally quantified intuitionistic logic Hπ+ by a natural extension of Kripke's semantics for propositional intuitionistic logic. We then show that Hπ+ is recursively isomorphic to full second order classical logic. Hπ+ is the intuitionistic analogue of the modal systems S5π+, S4π+, S4.2π+, K4π+, Tπ+, Kπ+ and Bπ+, studied by Fine.

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Devesas Campos, Marco, and Marcelo Fiore. "Classical logic with Mendler induction." Journal of Logic and Computation 30, no. 1 (January 2020): 77–106. http://dx.doi.org/10.1093/logcom/exaa004.

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Abstract We investigate (co-) induction in classical logic under the propositions-as-types paradigm, considering propositional, second-order and (co-) inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-) iterator and show that it is strongly normalizing. We prove this using a reducibility argument.

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Liu Honglan, Zhang Dezheng, Hao Weidong, and Gao Sihua. "Operations on Propositions and Relations between Propositions in Probabilistic Propositional Logic." International Journal of Advancements in Computing Technology 4, no. 3 (February 29, 2012): 18–25. http://dx.doi.org/10.4156/ijact.vol4.issue3.3.

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