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# Propositional Logic

 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. Alexander Abian (1970). Completeness of the Generalized Propositional Calculus. Notre Dame Journal of Formal Logic 11 (4):449-452.

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2. V. Michele Abrusci (2014). On Hilbert's Axiomatics of Propositional Logic. Perspectives on Science 22 (1):115-132.
Hilbert's conference lectures during the year 1922, Neuebegründung der Mathematik. Erste Mitteilung and Die logischen Grundlagen der Mathematik (both are published in (Hilbert [1935] 1965) pp. 157-195), contain his first public presentation of an axiom system for propositional logic, or at least for a fragment of propositional logic, which is largely influenced by the study on logical woks of Frege and Russell during the previous years.The year 1922 is at the beginning of Hilbert's foundational program in its definitive form. The (...)

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3. Jarosław Achinger & Andrzej W. Jankowski (1986). On Decidable Consequence Operators. Studia Logica 45 (4):415 - 424.
The main theorem says that a consequence operator is an effective part of the consequence operator for the classical prepositional calculus iff it is a consequence operator for a logic satisfying the compactness theorem, and in which every finitely axiomatizable theory is decidable.

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4. 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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5. Majid Alizadeh & Mohammad Ardeshir (2012). On Löb Algebras, II. Logic Journal of the IGPL 20 (1):27-44.
We study the variety of Löb algebras, the algebraic structures associated with Formal Propositional Calculus. Among other things, we show that there exist only two maximal intermediate logics in the lattice of intermediate logics over Basic Propositional Calculus. We introduce countably many locally finite sub-varieties of the variety of Löb algebras and show that their corresponding intermediate logics have the interpolation property. Finally, we characterize all chain basic algebras with empty set of generators, and show that there are continuum many (...)

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6. Alice Ambrose (1962). Fundamentals of Symbolic Logic. New York, Holt, Rinehart and Winston.

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7. Irving H. Anellis (2011). Peirce's Truth-Functional Analysis and the Origin of the Truth Table. History and Philosophy of Logic 33 (1):87 - 97.
We explore the technical details and historical evolution of Charles Peirce's articulation of a truth table in 1893, against the background of his investigation into the truth-functional analysis of propositions involving implication. In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on ?The Philosophy of Logical Atomism? truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig (...)

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9. R. B. Angell (1962). A Propositional Logic with Subjunctive Conditionals. Journal of Symbolic Logic 27 (3):327-343.

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10. R. B. Angell (1960). The Sentential Calculus Using Rule of Inference Re. Journal of Symbolic Logic 25 (2):143 -.

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11. R. Bradshaw Angell (1960). Note on a Less Restricted Type of Rule of Inference. Mind 69 (274):253-255.

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12. Richard B. Angell (1973). A Unique Normal Form for Synonyms in the Propositional Calculus. Journal of Symbolic Logic 38:350.
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13. Rani Lill Anjum (2012). What's Wrong with Logic? Argumentos 4 (8).
The truth functional account of conditional statements ‘if A then B’ is not only inadequate; it eliminates the very conditionality expressed by ‘if’. Focusing only on the truth-values of the statements ‘A’ and ‘B’ and different combinations of these, one is bound to miss out on the conditional relation expressed between them. All approaches that treat conditionals as functions of their antecedents and consequents will end up in some sort of logical atomism where causal matters simply are reduced to the (...)
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14. Rani Lill Anjum (2008). Three Dogmas of 'If'. In A. Leirfall & T. Sandmel (eds.), Enhet i Mangfold. Unipub
In this paper I argue that a truth functional account of conditional statements ‘if A then B’ not only is inadequate, but that it eliminates the very conditionality expressed by ‘if’. Focusing only on the truth-values of the statements ‘A’ and ‘B’ and different combinations of these, one is bound to miss out on the conditional relation expressed between them. But this is not a flaw only of truth functionality and the material conditional. All approaches that try to treat conditionals (...)
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15. In this paper I present some of Robert N. McLaughlin's critique of a truth functional approach to conditionals as it appears in his book On the Logic of Ordinary Conditionals. Based on his criticism I argue that the basic principles of logic together amount to epistemological and metaphysical implications that can only be accepted from a logical atomist perspective. Attempts to account for conditional relations within this philosophical framework will necessarily fail. I thus argue that it is not truth functionality (...)

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16. O. Anshakov & S. Rychkov (1995). On Finite-Valued Propositional Logical Calculi. Notre Dame Journal of Formal Logic 36 (4):606-629.
In this paper we describe, in a purely algebraic language, truth-complete finite-valued propositional logical calculi extending the classical Boolean calculus. We also give a new proof of the Completeness Theorem for such calculi. We investigate the quasi-varieties of algebras playing an analogous role in the theory of these finite-valued logics to the role played by the variety of Boolean algebras in classical logic.

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17. Lee C. Archie (1979). A Simple Defense of Material Implication. Notre Dame Journal of Formal Logic 20 (2):412-414.

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18. Robert L. Armstrong (1976). A Question About Incompleteness. Notre Dame Journal of Formal Logic 17 (2):295-296.

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19. E. J. Ashworth (1968). Propositional Logic in the Sixteenth and Early Seventeenth Centuries. Notre Dame Journal of Formal Logic 9 (2):179-192.

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20. Axel Arturo Barceló Aspeitia (2008). Patrones inferenciales (Inferential Patterns). 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 (de reglas a tablas). 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, (...)
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21. 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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22. Arnon Avron (2005). A Non-Deterministic View on Non-Classical Negations. 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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23. John Bacon (1975). Elementary Symbolic Logic. Teaching Philosophy 1 (2):220-221.

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24. Tomás Barrero (2004). Lógica positiva : plenitude, potencialidade e problemas (do pensar sem negação). Dissertation, Universidade Estadual de Campinas
This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend this constructive (...)
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25. Tomás Barrero & Walter Carnielli (2005). Tableaux sin refutación. Matemáticas: Enseñanza Universitaria 13 (2):81-99.
Motivated by H. Curry’s well-known objection and by a proposal of L. Henkin, this article introduces the positive tableaux, a form of tableau calculus without refutation based upon the idea of implicational triviality. The completeness of the method is proven, which establishes a new decision procedure for the (classical) positive propositional logic. We also introduce the concept of paratriviality in order to contribute to the question of paradoxes and limitations imposed by the behavior of classical implication.
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26. Robert B. Barrett & Alfred J. Stenner (1971). The Myth of the Exclusive `Or'. Mind 80 (317):116-121.

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27. Alfred W. Benn (1907). Symbolic Logic (a Reply). Mind 16 (63):470-473.

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29. Stephen L. Bloom & Roman Suszko (1971). Semantics for the Sentential Calculus with Identity. Studia Logica 28 (1):77 - 82.

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30. Ivan Boh (1966). Propositional Connectives, Supposition, and Consequence in Paul of Pergola. Notre Dame Journal of Formal Logic 7 (1):109-128.

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31. Ivan Boh (1962). Symbolic Logic. Modern Schoolman 39 (3):277-281.

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32. Jean-François Bonnefon & Guy Politzer (2011). Pragmatics, Mental Models and One Paradox of the Material Conditional. Mind and Language 26 (2):141-155.
Most instantiations of the inference ‘y; so if x, y’ seem intuitively odd, a phenomenon known as one of the paradoxes of the material conditional. A common explanation of the oddity, endorsed by Mental Model theory, is based on the intuition that the conclusion of the inference throws away semantic information. We build on this explanation to identify two joint conditions under which the inference becomes acceptable: (a) the truth of x has bearings on the relevance of asserting y; and (...)

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34. Constantin C. Brîncuș & Iulian D. Toader (2013). A Carnapian Approach to Counterexamples to Modus Ponens. Romanian Journal of Analytic Philosophy 7:78-85.
This paper defends a Carnapian approach to known counterexamples to Modus Ponens (MP). More specifically, it proposes that instead of rejecting MP as invalid in certain interpretations, one should regard the interpretations themselves as non-normal, in Carnap’s sense.

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35. Karl Britton (1936). Epistemological Remarks on the Propositional Calculus. Analysis 3 (4):57 - 63.

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36. Daniel J. Bronstein (1942). A Correction to the Sentential Calculus of Tarski's Introduction to Logic. Journal of Symbolic Logic 7 (1):34.

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37. This is part two of a complete exposition of Logic, in which there is a radically new synthesis of Aristotelian-Scholastic Logic with modern Logic. Part II is the presentation of the theory of propositions. Simple, composite, atomic, compound, modal, and tensed propositions are all examined. Valid consequences and propositional logical identities are rigorously proven. Modal logic is rigorously defined and proven. This is the first work of Logic known to unite Aristotelian logic and modern logic using scholastic logic as the (...)

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38. M. W. Bunder & R. M. Rizkalla (2009). Proof-Finding Algorithms for Classical and Subclassical Propositional Logics. Notre Dame Journal of Formal Logic 50 (3):261-273.
The formulas-as-types isomorphism tells us that every proof and theorem, in the intuitionistic implicational logic $H_\rightarrow$, corresponds to a lambda term or combinator and its type. The algorithms of Bunder very efficiently find a lambda term inhabitant, if any, of any given type of $H_\rightarrow$ and of many of its subsystems. In most cases the search procedure has a simple bound based roughly on the length of the formula involved. Computer implementations of some of these procedures were done in Dekker. (...)

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39. F. K. C. (1978). Lewis Carroll's Symbolic Logic. Review of Metaphysics 31 (3):472-473.

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40. W. C. C. (1952). Book Review:Symbolic Logic C. I. Lewis, C. H. Langford. [REVIEW] Philosophy of Science 19 (2):180-.

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41. Xavier Caicedo Ferrer (1978). A Formal System for the Non-Theorems of the Propositional Calculus. Notre Dame Journal of Formal Logic 19 (1):147-151.

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42. Rudolf Carnap (1958). Introduction to Symbolic Logic and its Applications. New York, Dover Publications.
Clear, comprehensive, intermediate introduction to logical languages, applications of symbolic logic to physics, mathematics, biology.

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43. James D. Carney (1970). Introduction to Symbolic Logic. Englewood Cliffs, N.J.,Prentice-Hall.
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44. Hector-Neri Casta Neda (1976). Leibniz's Syllogistico-Propositional Calculus. Notre Dame Journal of Formal Logic 17 (4):481-500.

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45. Hector-Neri Castañeda (1990). Leibniz's Complete Propositional Logic. Topoi 9 (1):15-28.
I have shown (to my satisfaction) that Leibniz's final attempt at a generalized syllogistico-propositional calculus in the Generales Inquisitiones was pretty successful. The calculus includes the truth-table semantics for the propositional calculus. It contains an unorthodox view of conjunction. It offers a plethora of very important logical principles. These deserve to be called a set of fundamentals of logical form. Aside from some imprecisions and redundancies the system is a good systematization of propositional logic, its semantics, and a correct account (...)

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46. Anthony Pike Cavendish (1953). Introduction to Symbolic Logic. University Tutorial Press.
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47. Douglas Cenzer & Jeffrey B. Remmel (2006). Complexity, Decidability and Completeness. Journal of Symbolic Logic 71 (2):399 - 424.
We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension (...)

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48. This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4–9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.
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49. In this paper, we study the completeness property of some implication-negation fragments of propositional logics. By the phrase implication-negation fragment of a propositional logic, we understand the system consisting of all the theses which have implication and/or negation as their sole connectives in the said logic. This means, that we have to find a means to isolate, so to speak, all these theses and then axiomatize the resultant system. Our method of proof is by constructing a Gentzen type Sequenzen Kalkul (...)
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