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Summary Propositional logic is the simpler of the two modern classical logics.  It ignores entirely the structure within propositions.  In classical propositional logic, molecular or compound propositions are built up from atomic propositions by means of the connectives, whose meaning is given by their truth tables.  The principle by which the meaning or truth conditions of compound propositions can be recovered by this "building up" process is known as compositionality. This leaf node is a sub-category of classical logic.  As such, non-standard propositional logics are not normally classified in this category—unless a comparison between classical logic and another logic is being drawn or one is reduced to the other—although restrictions of propositional logic in which nothing not a theorem in ordinary propositional logic is a theorem in the restriction do fit here.  Also appropriate here are modest extensions of propositional logic, provided that Boole's three laws of thought are not violated, viz. a proposition is either true or false, not neither, and not both. Meta-theoretical results for propositional logic are also generally classified as "proof theory," "model theory," "mathematical logic," etc.
Key works See below.
Introductions Because of the age of propositional logic there are literally hundreds of introductions to logic which cover this subject reasonably well.   Instructors will have their own favorites. In selecting a book for classroom use, I recommend checking one thing: how much meta-theory is included, so that the book is neither below nor above the level students can handle.
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  1. added 2020-05-26
    The Number of Lines in Frege Proofs with Substitution.Alasdair Urquhart - 1997 - Archive for Mathematical Logic 37 (1):15-19.
    . We prove that for sufficiently large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $n$\end{document}, there are tautologies of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $O$\end{document} that require proofs containing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Omega$\end{document} lines in axiomatic systems of propositional logic based on the rules of substitution and detachment.
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  2. added 2020-03-10
    Partiality and Adjointness in Modal Logic.Wesley H. Holliday - 2014 - In Rajeev Goré, Barteld Kooi & Agi Kurucz (eds.), Advances in Modal Logic, Volume 10. CSLI Publications. pp. 313-332.
    Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual (...)
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  3. added 2020-02-05
    Traits essentiels d'une formalisation adéquate.Gheorghe-Ilie Farte - 2020 - Argumentum. Journal of the Seminar of Discursive Logic, Argumentation Theory and Rhetoric 18 (1):163-174.
    In order to decide whether a discursive product of human reason corresponds or not to the logical order, one must analyze it in terms of syntactic correctness, consistency, and validity. The first step in logical analysis is formalization, that is, the process by which logical forms of thoughts are represented in different formal languages or logical systems. Although each thought can be properly formalized in different ways, the formalization variants are not equally adequate. The adequacy of formalization seems to depend (...)
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  4. added 2020-01-11
    The Accident of Logical Constants.Tristan Grøtvedt Haze - 2020 - Thought: A Journal of Philosophy 9 (1):34-42.
    Work on the nature and scope of formal logic has focused unduly on the distinction between logical and extra-logical vocabulary; which argument forms a logical theory countenances depends not only on its stock of logical terms, but also on its range of grammatical categories and modes of composition. Furthermore, there is a sense in which logical terms are unnecessary. Alexandra Zinke has recently pointed out that propositional logic can be done without logical terms. By defining a logical-term-free language with the (...)
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  5. added 2020-01-06
    Oskari Kuusela, Wittgenstein on Logic as the Method of Philosophy: Re‐Examining the Roots and Development of Analytic Philosophy . Xi + 297, £55.00 Hb. [REVIEW]Alessio Persichetti - 2019 - Philosophical Investigations 42 (4):424-427.
  6. added 2019-07-25
    The Immersion Method II (Logic & Malcolm X).Virgil W. Brower - 2012 - Inside Higher Ed, May 3.
  7. added 2019-07-04
    Choice-Free Stone Duality.Nick Bezhanishvili & Wesley H. Holliday - 2020 - Journal of Symbolic Logic 85 (1):109-148.
    The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...)
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  8. added 2019-07-04
    Algebraic and Topological Semantics for Inquisitive Logic Via Choice-Free Duality.Nick Bezhanishvili, Gianluca Grilletti & Wesley H. Holliday - 2019 - In Rosalie Iemhoff, Michael Moortgat & Ruy de Queiroz (eds.), Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science, Vol. 11541. Springer. pp. 35-52.
    We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev’s logic (...)
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  9. added 2019-06-06
    Patrones inferenciales.[author unknown] - 2008 - Critica 40 (120):3-35.
    El objetivo de este artículo es proponer un método de traducción de tablas de verdad a reglas de inferencia, para la lógica proposicional, que sea tan directo como el tradicional método inverso. Este método, además, permitirá resolver de manera elegante el viejo problema, formulado originalmente por Prior en 1960, de determinar qué reglas de inferencia definen un conectivo. /// This article aims at setting forth a method to translate truth tables into inference rules, in propositional logic, which is as straightforward (...)
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  10. added 2019-06-06
    In Defense of '⊃'.James F. Thomson - 1990 - Journal of Philosophy 87 (2):57-70.
  11. added 2019-06-06
    Introduction to Sentential Logic: A Workbook Approach. [REVIEW]Neil R. Luebke - 1977 - Teaching Philosophy 2 (1):73-74.
  12. added 2019-06-06
    The Propositional Logic of Avicenna.Nabil Avicenna & Shehaby - 1973 - Reidel.
    INTRODUCTION The main purpose of this work is to provide an English translation of and commentary on a recently published Arabic text dealing with conditional propositions and syllogisms. The text is that of Avicenna (Abu 'All ibn Sina, ...
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  13. added 2019-06-06
    Introduction to Symbolic Logic. [REVIEW]E. McMullin - 1956 - Philosophical Studies (Dublin) 6:238-239.
    This is a useful little book. Within a surprisingly short space it manages to cover most of the fundamentals of symbolic logic in a clear and accurate way. The development is step-by-step, and each new term is carefully explained at its first mention. The main topics covered are the logic of sentences, the axiomatic method, and an elementary form of the logic of predicates. In the final chapter and appendix a number of more difficult problems are briefly touched upon: higher-order (...)
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  14. added 2019-06-06
    Formal Logic. Prior - 1955 - Oxford University Press.
    This book was designed primarily as a textbook; though the author hopes that it will prove to be of interests to others beside logic students. Part I of this book covers the fundamentals of the subject the propositional calculus and the theory of quantification. Part II deals with the traditional formal logic and with the developments which have taken that as their starting-point. Part III deals with modal, three-valued, and extensional systems.
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  15. added 2019-06-06
    Symbolic Logic.Irving Copi - 1954 - New York: Macmillan.
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  16. added 2019-06-05
    Fundamentals of Symbolic Logic. Alice Ambrose, Morris Lazerowitz.Irving M. Copi - 1950 - Philosophy of Science 17 (2):199-200.
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  17. added 2019-06-05
    Symbolic Logic. Clarence Irving Lewis, Cooper Harold Landford.Henry Bradford Smith - 1934 - Philosophy of Science 1 (2):239-246.
  18. added 2019-04-29
    Dynamic Non-Classicality.Matthew Mandelkern - 2019 - Australasian Journal of Philosophy 98 (2):382-392.
    I show that standard dynamic approaches to the semantics of epistemic modals invalidate the classical laws of excluded middle and non-contradiction, as well as the law of ‘epistemic non-contradicti...
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  19. added 2019-03-30
    Stoic Sequent Logic and Proof Theory.Susanne Bobzien - 2019 - History and Philosophy of Logic 40 (3):234-265.
    This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic (...)
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  20. added 2019-03-24
    Two-Sided Trees for Sentential Logic, Predicate Logic, and Sentential Modal Logic.Jesse Fitts & David Beisecker - 2019 - Teaching Philosophy 42 (1):41-56.
    This paper will present two contributions to teaching introductory logic. The first contribution is an alternative tree proof method that differs from the traditional one-sided tree method. The second contribution combines this tree system with an index system to produce a user-friendly tree method for sentential modal logic.
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  21. added 2019-03-09
    An Arithmetization of Logical Oppositions.Fabien Schang - 2016 - In Jean-Yves Beziau & Gianfranco Basti (eds.), The Square of Opposition: A Cornerstone of Thought. Bâle, Suisse: pp. 215-237.
    An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers.
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  22. added 2018-11-21
    On the Concept of a Notational Variant.Alexander W. Kocurek - 2017 - In Alexandru Baltag, Jeremy Seligman & Tomoyuki Yamada (eds.), Logic, Rationality, and Interaction (LORI 2017, Sapporo, Japan). pp. 284-298.
    In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...)
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  23. added 2018-07-25
    A Note on Carnap’s Result and the Connectives.Tristan Haze - 2019 - Axiomathes 29 (3):285-288.
    Carnap’s result about classical proof-theories not ruling out non-normal valuations of propositional logic formulae has seen renewed philosophical interest in recent years. In this note I contribute some considerations which may be helpful in its philosophical assessment. I suggest a vantage point from which to see the way in which classical proof-theories do, at least to a considerable extent, encode the meanings of the connectives (not by determining a range of admissible valuations, but in their own way), and I demonstrate (...)
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  24. added 2018-07-24
    How May the Propositional Calculus Represent?Tristan Haze - 2017 - South American Journal of Logic 3 (1):173-184.
    This paper is a conceptual study in the philosophy of logic. The question considered is 'How may formulae of the propositional calculus be brought into a representational relation to the world?'. Four approaches are distinguished: (1) the denotational approach, (2) the abbreviational approach, (3) the truth-conditional approach, and (4) the modelling approach. (2) and (3) are very familiar, so I do not discuss them. (1), which is now largely obsolete, led to some interesting twists and turns in early analytic philosophy (...)
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  25. added 2018-06-18
    Wordmorph!: A Word Game to Introduce Natural Deduction.Ian Stoner - 2018 - Teaching Philosophy 41 (2):199-204.
    Some logic students falter at the transition from the mechanical method of truth tables to the less-mechanical method of natural deduction. This short paper introduces a word game intended to ease that transition.
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  26. added 2018-06-07
    Sistema Experto en Deducción Natural.Gabriel Garduño-Soto, David René Thierry García, Rafael Vidal Uribe & Hugo Padilla Chacón - manuscript
    Proceeding on the Automatic Deduction System developped at the Philosophy Faculty of the UNAM at Mexico City. (Deduktor Mexican Group of Logics work under the direction of the professor Hugo Padilla Chacón). Conference presented at the mexican City of Guadalajara at the Universidad de Guadalajara, Jalisco, by invitation of the latinoamerican association of philosophy SOPHIA. Early stage of the deductional systems at 2-valued logic. This work embodies the implementation of the first whole and standalone arithmetization of bivalent Logic, the theoretical (...)
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  27. added 2018-06-07
    Opuscula Logica. 2. The Tripropositional Bivalent Level (3L2) and its Relationship with the Aristotelic Syllogistic.Gabriel Garduño-Soto (ed.) - 2008 - Mexico, DF, MEXICO: Author's edition.
    In this fragment of Opuscula Logica it is displayed an arithmetical treatment of the aristotelic syllogisms upon the previous interpretations of Christine Ladd-Franklin and Jean Piaget. For the first time, the whole deductive corpus for each syllogism is presented in the two innovative modalities first proposed by Hugo Padilla Chacón. A. The Projection method (all the possible expressions that can be deduced through the conditional from a logical expression) and B. The Retrojection method (all the possible valid antecedents or premises (...)
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  28. added 2018-06-07
    Lista de distributividades y asociatividades posibles en lógica triproposicional.Gabriel Garduño-Soto - 2006 - Mexico DF, MEXICO: Author's edition.
    En este documento se despliega todas las distributividades y asociatividades posibles de construir en el formato tradicional de la lógica bivaluada, asimismo se presenta el valor que resulta de efectuar dichas operaciones de distributividad y asociatividad, sean o no tautológicas a través del operador de la equivalencia lógica. El valor resultante de realizar una operación de distributividad o asociatividad se expresa como el código binario de su columna veritativo-funcional, presentado en base decimal.
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  29. added 2018-05-31
    Introducción a la lógica moderna, 2a edición.Andrés Páez - 2010 - Bogotá: Ediciones Uniandes.
    The book is an undergraduate-level introduction to first-order logic and proposi­tional modal logic. The book presents symbolic logic in a way that is pedagogi­cally attractive and formally rigorous. The main concepts are formally defined, in­formally explained, and illustrated with several examples. The book is aimed at philosophy majors and it includes discussions of several problems in the philosophy of logic.
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  30. added 2018-03-01
    Bilateralism, Independence and Coordination.Gonçalo Santos - 2018 - Teorema: International Journal of Philosophy 37 (1):23-27.
    Bilateralism is a theory of meaning according to which assertion and denial are independent speech acts. Bilateralism also proposes two coordination principles for assertion and denial. I argue that if assertion and denial are independent speech acts, they cannot be coordinated by the bilateralist principles.
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  31. added 2018-02-17
    Beth Too, but Only If.Achille C. Varzi - 2005 - Analysis 65 (3):224-229.
    On the difficulty of extracting the logical form of a seemingly simple sentence such as ‘If Andy went to the movie then Beth went too, but only if she found a taxi cab’, with some morals and questions on the nature of the difficulty.
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  32. added 2018-02-17
    Conditionals.Frank Jackson (ed.) - 1991 - Oxford University Press.
    This collection introduces the reader to some of the most interesting current work on conditionals. Particular attention is paid to possible world semantics for conditionals, the role of conditional probability in helping us to understand conditionals, implicature and the material conditional, and subjunctive versus indicative conditionals. Contributors include V.H. Dudman, Dorothy Edgington, Nelson Goodman, H.P. Grice, David Lewis, and Robert Stalnaker.
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  33. added 2018-02-10
    Truthlikeness and the Convexity of Propositions.Graham Oddie - 1987 - In Kuipers T. (ed.), What is Closer-to-the-Truth. Rodopi. pp. 197-217.
  34. added 2017-09-15
    Logical Squares for Classical Logic Sentences.Urszula Wybraniec-Skardowska - 2016 - Logica Universalis 10 (2-3):293-312.
    In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus. We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.
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  35. added 2017-05-20
    A Technique for Determing Closure in Semantic Tableaux.Steven James Bartlett - 1983 - Methodology and Science: Interdisciplinary Journal for the Empirical Study of the Foundations of Science and Their Methodology 16 (1):1-16.
    The author considers the model-theoretic character of proofs and disproofs by means of attempted counterexample constructions, distinguishes this proof format from formal derivations, then contrasts two approaches to semantic tableaux proposed by Beth and Lambert-van Fraassen. It is noted that Beth's original approach has not as yet been provided with a precisely formulated rule of closure for detecting tableau sequences terminating in contradiction. To remedy this deficiency, a technique is proposed to clarify tableau operations.
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  36. added 2017-02-14
    Proof Terms for Classical Derivations.Restall Greg - manuscript
    I give an account of proof terms for derivations in a sequent calculus for classical propositional logic. The term for a derivation δ of a sequent Σ≻Δ encodes how the premises Σ and conclusions Δ are related in δ. This encoding is many–to–one in the sense that different derivations can have the same proof term, since different derivations may be different ways of representing the same underlying connection between premises and conclusions. However, not all proof terms for a sequent Σ≻Δ (...)
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  37. added 2016-12-21
    What is Wrong with Classical Negation?Nils Kurbis - 2015 - Grazer Philosophische Studien 92 (1):51-86.
    The focus of this paper are Dummett's meaning-theoretical arguments against classical logic based on consideration about the meaning of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett's outlook on the theory of meaning. In particular, I (...)
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  38. added 2016-12-12
    Valuations for Direct Propositional Logic.Franz Von Kutschera - 1983 - Erkenntnis 19 (1-3):253-260.
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  39. added 2016-12-08
    Expressive Power and Incompleteness of Propositional Logics.James W. Garson - 2010 - Journal of Philosophical Logic 39 (2):159-171.
    Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard readings, (...)
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  40. added 2016-12-08
    The Relevance of a Relevantly Assertable Disjunction for Material Implication.Liza Verhoeven - 2007 - Journal of Philosophical Logic 36 (3):339-366.
    In this paper Grice's requirements for assertability are imposed on the disjunction of Classical Logic. Defining material implication in terms of negation and disjunction supplemented by assertability conditions, results in the disappearance of the most important paradoxes of material implication. The resulting consequence relation displays a very strong resemblance to Schurz's conclusion-relevant consequence relation.
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  41. added 2016-12-08
    A Proof System for Classical Logic.Witold A. Pogorzelski & Piotr Wojtylak - 2005 - Studia Logica 80 (1):95-104.
    An operation on inferential rules, called H-operation, is used to minimize the axiom basis for classical logic.
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  42. added 2016-12-08
    Rethinking Bivalence.A. Iacona - 2005 - Synthese 146 (3):283-302.
    Classical logic rests on the assumption that there are two mutually exclusive and jointly exhaustive truth values. This assumption has always been surrounded by philosophical controversy. Doubts have been raised about its legitimacy, and hence about the legitimacy of classical logic. Usually, the assumption is stated in the form of a general principle, namely the principle that every proposition is either true or false. Then, the philosophical controversy is often framed in terms of the question whether every proposition is either (...)
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  43. added 2016-12-08
    A Non-Deterministic View on Non-Classical Negations.Arnon Avron - 2005 - Studia Logica 80 (2-3):159-194.
    We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which is (...)
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  44. added 2016-12-08
    Consequence Operations Based on Hypergraph Satisfiability.Kolany Adam - 1997 - Studia Logica 58 (2):261-272.
    Four consequence operators based on hypergraph satisfiability are defined. Their properties are explored and interconnections are displayed. Finally their relation to the case of the Classical Propositional Calculus is shown.
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  45. added 2016-12-08
    Symbolic Logic. [REVIEW]E. J. A. - 1966 - Review of Metaphysics 19 (4):808-808.
    Among the more important changes in this revised edition: the incompleteness of the first set of natural deduction rules is proved; many proofs are shortened and simplified, especially in the development of the first-order functional calculus; there is a more lucid exposition of the quantification rules; more exercises are provided, with answers given for a number of them. The changes are all improvements, but none of them are of a sufficiently radical nature to be likely to alter anyone's original opinion (...)
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  46. added 2016-12-08
    Fundamentals of Symbolic Logic. [REVIEW]D. J. - 1963 - Review of Metaphysics 16 (3):579-579.
    A revised edition of the author's 1948 introductory text, the present version differs only in minor points from the original work. Truth functions are discussed in some detail, and a propositional calculus derived from the five axioms of Principia Mathematica is presented. Quantification, however, is less extensively treated. The "traditional Aristotelian" interpretation of categorical propositions is contrasted with the non-existential interpretation of modern logic; a theory of the syllogism is offered; and various methods for evaluating the validity of syllogisms are (...)
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  47. added 2016-12-08
    Introduction to Symbolic Logic and Its Applications. [REVIEW]P. R. - 1959 - Review of Metaphysics 12 (4):662-662.
    An accelerated introductory text in symbolic logic, this work is a translation and revision of Carnap's 1954 Einführung.... The latter portion of the book is devoted to semantics and to axiom systems in areas as diverse as geometry and biology.--R. P.
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  48. added 2016-12-05
    Exercises in Introductory Symbolic Logic. [REVIEW]T. E. - 1956 - Review of Metaphysics 10 (1):180-180.
    Offers the logical tyro a varied diet, from Aristotle to Lewis Carroll, including the "neglected" forms of argument as well as examples from the logic of classes and relations. To avoid translation among systems, the examples are all in "English." Copious explanatory footnotes and references recommend it to the self-taught.--E. T.
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  49. added 2016-11-04
    A Carnapian Approach to Counterexamples to Modus Ponens.Constantin C. Brîncuș & Iulian D. Toader - 2013 - Romanian Journal of Analytic Philosophy 7:78-85.
  50. added 2016-10-28
    Bilateralist Detours: From Intuitionist to Classical Logic and Back.Nils Kürbis - 2017 - Logique Et Analyse 60 (239):301-316.
    There is widespread agreement that while on a Dummettian theory of meaning the justified logic is intuitionist, as its constants are governed by harmonious rules of inference, the situation is reversed on Huw Price's bilateralist account, where meanings are specified in terms of primitive speech acts assertion and denial. In bilateral logics, the rules for classical negation are in harmony. However, as it is possible to construct an intuitionist bilateral logic with harmonious rules, there is no formal argument against intuitionism (...)
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