References

• Belnap, Nuel D., 1977, “A Useful Four-Valued Logic”, in Modern Uses of Multiple-Valued Logic, J. Michael Dunn and George Epstein (eds.), Dordrecht/Boston: D. Reidel, 5–37. doi:10.1007/978-94-010-1161-7_2 (Scholar)
• Bernays, Paul, 1926, “Axiomatische Untersuchung des Aussagen-Kalkuls der Principia Mathematica”, Mathematische Zeitschrift, 25: 305–320. doi:10.1007/bf01283841 (Scholar)
• Cook, Stephen A., 1971, “The Complexity of Theorem-Proving Procedures”, in Proceedings of the Third Annual ACM Symposium on Theory of Computing, New York: ACM Press, 151–158. doi:10.1145/800157.805047 (Scholar)
• Cobreros, Pablo, Paul Égré, David Ripley, and Robert van Rooij, 2014, “Foreword: Three-Valued Logics and Their Applications”, Journal of Applied Non-Classical Logics, 24(1–2): 1–11. doi:10.1080/11663081.2014.909631 (Scholar)
• Church, Alonzo, 1936, “An Unsolvable Problem of Elementary Number Theory”, American Journal of Mathematics, 58(2): 345–353. doi:10.2307/2371045 (Scholar)
• Edgington, Dorothy, 1995, “On Conditionals”, Mind, 104(414): 235–329. doi:10.1093/mind/104.414.235 (Scholar)
• Franks, Curtis, 2010, “Cut as Consequence”, History and Philosophy of Logic, 31(4): 349–379. doi:10.1080/01445340.2010.522365 (Scholar)
• –––, 2021, “The Deduction Theorem (Before and After Herbrand)”, History and Philosophy of Logic, 42(2): 129–159. doi:10.1080/01445340.2021.1889117 (Scholar)
• Frege, Gottlob, 1879, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle: L. Nebert. Translated as Conceptual Notation: a formula language of pure thought modeled upon the formula language of arithmetic in Frege 1972:: 101–208. (Scholar)
• –––, 1910 [1980], “Letter to Jourdain”, translated and reprinted in his Philosophical and Mathematical Correspondence, G. Gabriel, et al. (eds.), Chicago: University of Chicago Press, 1980. (Scholar)
• –––, 1972, Conceptual Notation, and Related Articles, Terrell Ward Bynum (trans.), Oxford: Clarendon Press. (Scholar)
• Gentzen, Gerhard, 1932, “Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen”, Mathematische Annalen 107: 329–50. Translated as “On the Existence of Independent Axiomsystems for Infinite Sentence Systems”, in Gentzen 1969: 29–52. doi:10.1007/bf01448897 (de) (Scholar)
• –––, 1934–35, “Untersuchungen über das logische Schließen”, Gentzen’s doctoral thesis at the University of Göttingen, translated as “Investigations into Logical Deduction”, in Gentzen 1969: 68–131. (Scholar)
• –––, 1969, The Collected Papers of Gerhard Gentzen, M. E. Szabo (ed.), (Studies in Logic and the Foundations of Mathematics 55), Amsterdam: North-Holland. (Scholar)
• Girard, Jean-Yves, 1987, “Linear Logic”, Theoretical Computer Science, 50(1): 1–101. doi:10.1016/0304-3975(87)90045-4 (Scholar)
• Glivenko, Valéry, 1929, “Sur quelques points de la logique de M. Brouwer”, Académie royale de Belgique, Bulletin de la classe des sciences, series 5, 15: 183–188. (Scholar)
• Gödel, Kurt, 1930 [1986], “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik, 37: 349–360. Translated by S. Bauer-Mengelberg as “The completeness of the axioms of the functional calculus of logic” reprinted in Gödel 1986: 102–23. doi:10.1007/BF01696781 (Scholar)
• –––, 1932 [1986], “Zum intuitionistischen Aussagenkalkül”, Anzeiger der Akademie der Wissenschaftischen in Wien, 69: 65–66. Translated by J. Dawson as “On the intuitionistic propositional calculus”, in Gödel 1986: 223–25. (Scholar)
• –––, 1933a [1986], “Eine Interpretation des intuitionistischen Aussagenkalküls”, Ergebnisse eines mathematischen Kolloquiums, 4: 39–40. Translated as “An Interpretation of the Intuitionistic Propositional Calculus”, in Gödel 1986: 301–302. (Scholar)
• –––, 1933b [1986], “Zur intuitionistischen Arithmetik und Zahlentheorie”, Ergebnisse eines mathematischen Kolloquiums, 4: 34–38. Translated as “On Intuitionistic Arithmetic and Number Theory”, in Gödel 1986: 287–295. (Scholar)
• –––, 1986, Collected Works, Volume I: Publications 1929–1936, Solomon Feferman, John W. Dawson Jr, Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort (eds.), Oxford/New York: Clarendon Press. (Scholar)
• Herbrand, Jacques, 1930, Recherches sur la théorie de la démonstration, Herbrand’s doctoral thesis at the University of Paris. Translated by W. Goldfarb, except pp. 133–88 translated by B. Dreben and J. van Heijenoort, as “Investigations in proof theory” in Herbrand 1968 [1971: 44–202]. (Scholar)
• –––, 1968 [1971], Écrits Logigues, Jean van Heijenoort (ed.), (Bibliothèque de Philosophie Contemporaine. Logique et Philosophie Des Sciences), Paris: Presses universitaires de France. Translated as Logical Writings, Warren D. Goldfarb (trans.), Dordrecht, Holland: D. Reidel, 1971. (Scholar)
• Hilbert, David, 2013, David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933 (David Hilbert’s Lectures on the Foundations of Mathematics and Physics, 1891–1933, vol. 3), William Ewald and Wilfried Sieg (eds.), Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-540-69444-1 (Scholar)
• Hilbert, David and W. Ackermann, 1928, Grundzüge der theoretischen Logik, (Die Grundlehren der mathematischen Wissenschaften 27), Berlin: J. Springer. (Scholar)
• Johansson, Ingebrigt, 1937, “Der Minimalkalkül, ein reduzierter Intuitionistischer Formalismus”, Compositio Mathematica, 4: 119–136. [Johansson 1937 available online] (Scholar)
• Kalmár, László, 1935, “Über die Axiomatisierbarkeit des Aussagenkalküls”, Acta Scientiarum Mathematicarum, 7(4): 222–43. (Scholar)
• Kolmogorov [Kolmogoroff], Andrey N., 1925 [1967], “О принцип tertium non datur”, Matematicheskii Sbornik, 32(4): 646–667. Translated as “On the Principle of Excluded Middle”, in From Frege to Gödel A Source Book in Mathematical Logic, 1879-1931, Jean van Heijenoort (ed.), Cambridge, MA: Harvard University Press, 1967, 416–437. (Scholar)
• Kreisel, Georg and Hilary Putnam, 1957, “Eine Unableitbarkeitsbeweismethode für den Intuitionistischen Aussagenkalkül”, Archiv für Mathematische Logik und Grundlagenforschung, 3(3–4): 74–78. doi:10.1007/bf01988049 (Scholar)
• Lewis, David, 1976, “Probabilities of Conditionals and Conditional Probabilities”, The Philosophical Review, 85(3): 297–315. doi:10.2307/2184045 (Scholar)
• Lincoln, Patrick, John Mitchell, Andre Scedrov, and Natarajan Shankar, 1992, “Decision Problems for Propositional Linear Logic”, Annals of Pure and Applied Logic, 56(1–3): 239–311. doi:10.1016/0168-0072(92)90075-b (Scholar)
• Nicod, J. G. P., 1917, “A Reduction in the Number of Primitive Propositions of Logic”, Proceedings of the Cambridge Philosophical Society, 19: 32–41. (Scholar)
• Post, Emil L., 1921, “Introduction to a General Theory of Elementary Propositions”, American Journal of Mathematics, 43(3): 163–185. doi:10.2307/2370324 (Scholar)
• Priest, Graham, 1992, “What is a non-normal world?”, Logique & Analyse, 139–140: 291–302. (Scholar)
• Prucnal, Tadeusz, 1976, “Structural Completeness of Medvedev’s Propositional Calculus”, Reports on Mathematical Logic, 6: 103–105. (Scholar)
• Quine, W. V. O., 1955, “A Way to Simplify Truth Functions”, The American Mathematical Monthly, 62(9): 627–631. doi:10.1080/00029890.1955.11988710 (Scholar)
• –––, 1982, Methods of Logic, fourth edition, Cambridge, MA: Harvard University Press. (Scholar)
• Samson, E. and B. Mills, 1954, “Circuit minimization: algebra and algorithms for new Boolean canonical expressions”, Technical Report 54–21, Air Force Cambridge Research Center. (Scholar)
• Schröder, Ernst, 1890, Vorlesungen über die Algebra der Logik (exakte Logik), volume 1, Leipzig: B. G. Teubner. Reprinted 1966, New York: Chelsea. (Scholar)
• Shannon, Claude Elwood, 1940, “A Symbolic Analysis of Relay and Switching Circuits”, Thesis (M.S.), Massachusetts Institute of Technology, Dept. of Electrical Engineering. [Shannon 1940 available online] (Scholar)
• Sheffer, Henry Maurice, 1913, “A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants”, Transactions of the American Mathematical Society, 14(4): 481–488. doi:10.1090/s0002-9947-1913-1500960-1 (Scholar)
• Sorbi, Andrea, 1991, “Some Quotient Lattices of the Medvedev Lattice”, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 37(9–12): 167–182. doi:10.1002/malq.19910370905 (Scholar)
• Takeuti, Gaisi, 1987 [2013], Proof Theory, second edition, (Studies in Logic and the Foundations of Mathematics 81), Amsterdam/New York: North-Holland. Reprinted New York: Dover, 2013. (Scholar)
• Tarski, Alfred, 1933, “Einige Betrachtungen über die Begriffe der ω-Widerspruchsfreiheit und der ω-Vollständigkeit”, Monatshefte für Mathematik und Physik, 40(1): 97–112. doi:10.1007/bf01708855 (Scholar)
• Troelstra, A. S. and D. van Dalen, 1988, Constructivism in Mathematics, Volume 1: An Introduction, (Studies in Logic and the Foundations of Mathematics 121), Amsterdam: North-Holland. (Scholar)
• Turing, A. M., 1936, “On Computable Numbers, with an Application to the Entscheidungsproblem”, Proceedings of the London Mathematical Society, series 2, 42(1): 230–265. doi:10.1112/plms/s2-42.1.230 (Scholar)
• Urquhart, Alasdair, 1984, “The Undecidability of Entailment and Relevant Implication”, Journal of Symbolic Logic, 49(4): 1059–1073. doi:10.2307/2274261 (Scholar)
• Whitehead, Alfred North and Bertrand Russell, 1925–27, Principia Mathematica, Volumes 1, 2 and 3, 2nd Edition, Cambridge: Cambridge University Press. (Scholar)

More Readings

Most textbooks treat propositional logic as an elementary step en route to quantification theory or another more general topic. A notable exception is:

• Schechter, Eric, 2005, Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions, N.J: Princeton University Press. (Scholar)

Another sensitive treatment of propositional logic with attention to non-classical interpretations is:

• Gamut, L. T. F., 1990, Logic, Language, and Meaning, Volume 1: Introduction to Logic, Chicago, IL: University of Chicago Press. Translation of Logica, taal en betekenis I: inleiding in de logica, Utrecht: Het Spectrum, 1982. (L. T. F. Gamut is a collective name for Johan van Benthem, J. A. G. Groenendijk, D. H. J. de Jongh, M. J. B. Stokhof, and H. J. Verkuyl.) (Scholar)

Among standard textbook treatments, standout presentations of propositional logic can be found in Quine 1982 and in:

• Buss, Samuel R., 1998, “An Introduction to Proof Theory”, in Handbook of Proof Theory, Samuel R. Buss (ed.), (Studies in Logic and the Foundations of Mathematics 137), Amsterdam: Elsevier, 1–78. doi:10.1016/s0049-237x(98)80016-5 (Scholar)
• Kleene, Stephen Cole, 1952, Introduction to Metamathematics, (The University Series in Higher Mathematics), New York: Van Nostrand. (Scholar)
• Mendelson, Elliott, 2015, Introduction to Mathematical Logic, sixth edition, (Textbooks in Mathematics), Boca Raton, FL: CRC Press. (Scholar)
• Von Plato, Jan, 2013, Elements of Logical Reasoning, Cambridge/New York: Cambridge University Press. doi:10.1017/cbo9781139567862 (Scholar)

An excellent presentation of the rapid historical development of the subject in the early twentieth century is:

• Mancosu, Paolo, Richard Zach, and Calixto Badesa, 2009, “The Development of Mathematical Logic from Russell to Tarski, 1900–1935”, in The Development of Modern Logic, Leila Haaparanta (ed.), Oxford/New York: Oxford University Press, 318–470. doi:10.1093/acprof:oso/9780195137316.003.0029 (Scholar)