Logic (Philosophy)

Introduction

Historically there has been considerable variation in the understanding of the scope of the field of logic. Our concern is with what may be called, to distinguish it from other conceptions, formal, deductive logic. In the early 21st century, once one gets beyond the introductory level, it is customary to divide the field between mathematical and philosophical logic, each with subdivisions. But let us begin with the basics and note two points. First, (deductive) logic has always had at its core a question equally relevant to philosophical dialectic and mathematical demonstration: What follows from what? Second, (formal) logic answers by pointing to and only to argument forms. Here, whether a given conclusion is deducible from or a consequence of given premises, or whether the argument from premises to conclusions is valid, is taken to depend only on their forms. Logic proper is to be distinguished from history of logic and philosophy of logic, but one question from philosophy of logic must be mentioned at the outset: Should premises and conclusions be understood to be sentences or propositions expressed thereby? Is form a matter of composition of sentences using certain items of vocabulary, or composition of propositions? To this day, the most elementary part of classical logic goes by the rival names of “sentential logic” and “propositional logic.” For current purposes, where it makes a difference we will use the former terminology, though what we have to say could mostly be reworded in the latter; some of the cited works use the one terminology; others, the other. Also, one fact from the history of logic must be mentioned: since some time in the first half of the 20th century, when the traditional logic of syllogisms going back to Aristotle was finally subsumed and superseded, introductory textbooks have generally taught a common view, now known as classical (elementary or first-order) logic, as to which forms are valid.

Classical Logic

Classical sentential logic considers form resulting from composition of sentences out of sentences, using such connectives as “not,” “and,” and “or” (negation, conjunction, disjunction); classical predicate logic also considers composition out of predicates and other subsentential components, by using the quantifiers “all” and “some” (universal and existential). A conclusion is counted as a consequence of a set of premises if their form of composition alone guarantees that if the premises are true, so is the conclusion. The notion is made more rigorous by using symbolic formulas to represent forms; for “someone loves everyone” and “everyone loves someone,” these might be ∃x∀yFxy and ∀x∃yFxy. A model is defined to be a universe for variables x and y to range over (perhaps the set of all persons), plus a specification for each predicate letter F of what relation on the objects in the universe it stands for (perhaps that of lover to beloved). A rigorous definition of what it is for a formula to be true in a model is provided, and consequence is defined in terms thereof: ∀y∃xRxy is a consequence of ∃x∀yRxy because the former is true in every model in which the latter is true. A proof or deduction is, roughly speaking, a breakdown of the route from premises to conclusion into a sequence of short steps, each of one of a few kinds. Different textbooks provide different proof procedures, but all share two features: if there is a deduction, then the conclusion is a consequence of the premises (soundness), and, conversely, if the conclusion is a consequence of the premises, there is a deduction (completeness). The “semantic” notion of consequence coincides with the “syntactic” notion of deducibility. (Texts differ over whether they include a demonstration of these “metatheorems,” and over what supplementary material if any they include on informal logic or critical thinking, inductive logic, or elementary probability, and so on.) Kneale and Kneale 1962, a work of history of logic rather than logic proper, gives an account of the long history of syllogistic logic and the slow emergence of the classical logic that replaced it. Cohen and Nagel 1934 represents the transition. Hilbert and Ackermann 1928, Tarski 1941, Quine 1950, Church 1956, and Suppes 1957 are textbooks of classical logic, pioneering in their day and representing different kinds of proof procedures. The subject can still be learned from any number of excellent modern textbooks, among which we will not play favorites.

Church, Alonzo. Introduction to Mathematical Logic. Princeton Mathematical Series 17. Princeton, NJ: Princeton University Press, 1956.
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A classic textbook, differing from Hilbert and Ackermann 1928 by still using notations derived from Bertrand Russell, though both books use an axiomatic presentation in which certain logical laws are taken as axioms, and all others are derived from them by repeated application of a few simple rules.
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Cohen, Morris Raphael, and Ernest Nagel. An Introduction to Logic and Scientific Method. New York: Harcourt, Brace, 1934.
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A long-influential textbook, half on formal logic and half on inductive logic, interesting as representing the period when classical logic was still in the process of displacing syllogistic logic in elementary college instruction; it has gone through several editions, the most recent in 2002 (Safe Harbor, FL: Simon).
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Hilbert, David, and Wilhelm Ackermann. Grundzüge der theoretischen Logik. Berlin: Springer, 1928.
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The first textbook in classical first-order logic, noted for raising the questions of completeness and decidability that were solved in subsequent work of Kurt Gödel and Church and Alan Turing; it has gone through several editions in German and English, the latter initially as Principles of Mathematical Logic (New York: Chelsea, 1950), which was translated by Lewis M. Hammond, George G. Leckie, and Fritz Steinhardt and edited with notes by Robert E. Luce.
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Jeffrey, Richard C. Formal Logic: Its Scope and Limits. New York: McGraw Hill, 1967.
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The first textbook to use the tree method, derived from E. W. Beth’s “semantic tableaux,” as its proof procedure. Notable for how far it goes into metatheory for an introductory text; it has gone through several editions, most recently a fourth edition published in 2006 (Indianapolis, IN: Hackett).
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Kneale, William, and Martha Kneale. The Development of Logic. Oxford: Clarendon, 1962.
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A work on the history of logic that, though dated in parts owing to an explosion of scholarship more recently, remains the best available panoramic overview; it is mentioned despite not belonging to logic proper because, after an account of the long history of syllogistic logic, it illuminatingly treats the gradual emergence of what has become classical logic from the time of George Boole onward.
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Quine, W. V. O. Methods of Logic. New York: Holt, 1950.
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The first attempt at a textbook presentation of a version of Stanislaw Jaśkowski’s natural-deduction proof procedure (see Jaśkowski 1934, cited under Proof Theory) for classical logic; it has gone through several editions.
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Suppes, Patrick. Introduction to Logic. Princeton, NJ: Van Nostrand, 1957.
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An early and influential presentation of classical logic with a natural-deduction proof procedure in the style of Gerhard Gentzen (see Gentzen 1935, cited under Proof Theory); it has gone through several editions, most recently in 2013 (Mineola, NY: Dover), and has been much imitated.
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Tarski, Alfred. Introduction to Logic and to the Methodology of Deductive Sciences. Rev. ed. Translated by Olaf Helmer. Oxford: Oxford University Press, 1941.
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Another early textbook using an axiomatic proof procedure, covering not only pure logic but the axiomatization of the theories of various kinds of numbers in mathematics. Notable as a product of the figure many consider the second-greatest logician of the 20th century (after Gödel); it has gone through several editions, most recently in 2013 (Mineola, NY: Dover).
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Philosophical Logic

Philosophical logic as understood in the early 21st century (not to be confused with philosophy of logic or with the branch of philosophy of language formerly called “philosophical logic”) is conventionally divided into extraclassical and anticlassical, concerned respectively with extensions of and alternatives to classical logic. But be warned: it is not always easy to say whether a given logic represents a different theory on the same range of questions as classical logic, or a separate theory on a different range of questions. Of the logics given separate consideration here, tense and modal logic are generally taken to be extraclassical and intuitionistic, relevance/relevant and paraconsistent logic are generally considered to be anticlassical, and conditional logic is harder to classify. Extraclassical logics differ greatly not only in what classical principles they reject, but also in their motivations for rejecting them. Free logic rejects the claim that “something Fs” follows from “everything Fs” on the grounds that logical purity requires abstention from any existence assumptions, even the nontriviality assumption that the universe is nonempty. Quantum logic rejects the claim that “either both p and q or both p and r” follows from “both p and either q or r” on the grounds that it has been discovered empirically that a particle may have both a certain position and one or the other of two momenta, though it cannot have both a definite position and a definite momentum. Non-monotonic logic rejects the inference from “C follows to A” to “C follows from A and B,” on the grounds that “X flies” may be presumed given that “X is a bird” but not given also that “X is a penguin.” Free and quantum logics are generally regarded as anticlassical, but non-monotonic logic is generally considered as extraclassical, concerned not with formal deductive validity but with a different relationship; warranted presumption in the absence of further evidence. But such judgments are debatable. Gabbay and Guenthner 1983–1989 is a massive compendium covering far more varieties of philosophical logic than we have space even to mention here. The Stanford Encyclopedia of Philosophy has similarly wide coverage. Beall and van Fraassen 2003, Priest 2008, and Burgess 2009 are textbooks differing in scale and the range of logics covered, each giving an idea of the diversity of the field. Shapiro 2007 and Haack 1996 belong more to philosophy of logic than philosophical logic but give attention to several varieties of the latter.

Beall, Jc, and Bas C. van Fraassen. Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic. New York: Oxford University Press, 2003.
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A friendly introduction to standard philosophical logics at the propositional level, intended for those with no prior exposure to nonclassical logics. The book uses simple tagged-tableau procedures, while focusing heavily on the semantic or model-theoretic motivations for such logics.
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Burgess, John P. Philosophical Logic. Princeton, NJ: Princeton University Press, 2009.
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An introductory textbook, concisely treating (owing to the strict word limits of the series in which it appears) tense, modal, conditional, relevance/relevant, and intuitionistic logics; additional exercises are available from the author’s personal home page.
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Gabbay, Dov, and Franz Guenthner, eds. Handbook of Philosophical Logic. 4 vols. Synthese Library 164–167. Dordrecht, The Netherlands: Reidel, 1983–1989.
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An important collection of surveys, with the first, second, and third volumes covering, respectively, classical logic, extensions thereof, and alternatives thereto, the last category including discussion of logics we have had no space to address separately. The work is in the process of expansion to a second edition and is already up to over a dozen volumes and counting.
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Haack, Susan. Deviant Logic, Fuzzy Logic: Beyond the Formalism. Chicago: University of Chicago Press, 1996.
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A work belonging to philosophy of logic that nonetheless—because its thesis is that classical logic is revisable, but none of the anticlassical logics on the market would be a good revision—includes substantial material in philosophical logic in the form of a survey of many nonclassical logics, with special attention to the kind of many-valued logic called fuzzy.
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Priest, Graham. An Introduction to Non-classical Logic: From If to Is. 2d ed. Cambridge Introductions to Philosophy. Cambridge, UK: Cambridge University Press, 2008.
DOI: 10.1017/CBO9780511801174Save Citation »Export Citation »E-mail Citation »
In its second edition, a weighty tome treating at length modal, tense, conditional, intuitionistic, many-valued, paraconsistent, “relevant,” and fuzzy logics.
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Shapiro, Stewart, ed. The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford Handbooks in Philosophy. Oxford: Oxford University Press, 2007.
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A work on the philosophy of logic, in the form of a collection of surveys mostly by prominent figures, of the pros and cons of various positions in philosophy of mathematics and of logic; it is mentioned despite not belonging to logic proper because it overlaps with philosophical logic by touching on a variety of anticlassical logics, especially intuitionistic and relevance/relevant.
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Zalta, Edward N., ed. Stanford Encyclopedia of Philosophy. Stanford, CA: Stanford University.
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Provides introductory surveys, with ample references to further literature, for all logics specifically mentioned here and others we have had no space for, including deontic, epistemic, fuzzy, linear, many-valued, provability, and dynamic logics.
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Modal Logic

The paradigmatic case of an extraclassical logic is the logic of the modalities—necessity and possibility—the prototype of a logic that adds to classical logic the extraclassical connectives of philosophical interest, in this case “necessarily” and “possibly” (sometimes given variant readings such as the deontic “obligatorily” and “permissibly” or the epistemic). It was first treated at book length in the modern era in Lewis and Langford 1932. The type of models first developed for work with this logic have found applications to many other kinds as well. Kripke 1963 is an accessible primary source for this type of model, which Goldblatt 2003 sets in its historical context. The subject is covered in all the general references for Philosophical Logic, and at much-greater length and in much-greater detail in Blackburn, et al. 2002, which illustrates how the subject has become of interest outside philosophy, especially in theoretical computer science. Cresswell 1990 and Boolos 1993 deal with specialized issues of philosophical interest.

Blackburn, Patrick, Maarten de Rijke, and Yde Venema. Modal Logic. Cambridge Tracts in Theoretical Computer Science 53. Cambridge, UK: Cambridge University Press, 2002.
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A comprehensive textbook, directed toward mathematically sophisticated readers.
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Boolos, George. The Logic of Provability. Cambridge, UK: Cambridge University Press, 1993.
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The definitive work on the specialized branch of modal logic introduced by Kurt Gödel, in which the necessity involved is interpreted as formal provability.
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Cresswell, M. J. Entities and Indices. Studies in Linguistics and Philosophy 41. Dordrecht, The Netherlands: Kluwer Academic, 1990.
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A study of the comparative expressive power of modal logic as conventionally formulated and classical first-order logic, with explicit quantification over possible worlds.
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Goldblatt, Robert. “Mathematical Modal Logic: A View of Its Evolution.” Journal of Applied Logic 1.5–6 (2003): 309–392.
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A magisterial account of the history of the technical side of modal logic, among other things setting the work of Saul Kripke in its historical context.
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Kripke, Saul A. “Semantical Considerations on Modal Logic.” Acta Philosophica Fennica 16 (1963): 83–94.
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The locus classicus for “possible-worlds semantics,” part of a large body of related work by its author that opened the door to a great deal of philosophical activity in and around modal logic, appearing in a journal issue with other papers from a memorable conference on modal logic in Helsinki in 1962.
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Lewis, Clarence I., and Cooper H. Langford. Symbolic Logic. Century Philosophy. New York: Century, 1932.
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The first book-length exposition of the subject, by the founder of modern modal logic (Lewis), in which modal logic is presented as if in opposition to classical logic, especially in the discussion of the so-called paradoxes of “material implication”; available in an undated reproduction from Dover.
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Tense Logic

Temporal or tense logic adds to classical logic such connectives of time reference as “it (once) was the case that” and “it (sometime) will be the case that.” Philosophers who have strong views about what notions are properly viewed as logical may find these logics to be “not really logic,” regarding temporal logic as a kind of physics, as they may even regard modal logic as a kind of metaphysics (at least when the modalities are understood as “metaphysical” necessity and possibility). By contrast, setting philosophy aside, mathematicians and computer scientists generally call anything “logic” that can be fruitfully studied by adaptations of techniques traditionally used in connection with assessment of the (logical) validity of arguments, just as they will call anything “geometry” that can be studied by adaptations of techniques traditionally used in connection with the analysis of the structure of space, in either case regardless of originally intended applications, if any. Prior 1967 is the pioneering work that put the subject on the map, and van Benthem 1991 is a substantial textbook, providing more detail than in the general references listed on Philosophical Logic. Burgess 1980 and Goldblatt 1980 are studies of special topics in the field of philosophical and mathematical interest, while Pnueli 1977 inaugurated the application of the subject within computer science.

Burgess, John P. “Decidability for Branching Time.” Studia Logica 39.2 (1980): 203–218.
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A more formal treatment of the philosophical issue of future contingents than is discussed in Arthur Prior’s classic work (Prior 1967); shows how modal and tense operators interact when combined.
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Goldblatt, Robert. “Diodorean Modality in Minkowski Spacetime.” Studia Logica 39.2 (1980): 219–236.
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A sophisticated case study of how tense logic reflects assumptions of theoretical physics.
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Pnueli, Amir. “The Temporal Logic of Programs.” In 18th Annual Symposium on Foundations of Computer Science, Oct. 31–Nov. 2, 1977, Providence, Rhode Island. Edited by IEEE Computer Society, 46–57. New York: Institute of Electrical and Electronics Engineers, 1977.
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The paper, quite accessible to nonspecialists, that loosed a flood of work by theoretical computer scientists in the area of tense logic.
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Prior, Arthur N. Past, Present and Future. Oxford: Clarendon, 1967.
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A classic text, by the founder of tense logic as a distinct branch of logic, on the philosophical motivations of tense logic and the early technical development of the subject; the first and perhaps still the most important book-length work in the area.
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van Benthem, Johan. The Logic of Time: A Model-Theoretic Investigation into the Varieties of Temporal Ontology and Temporal Discourse. 2d ed. Synthese Library 156. Dordrecht, The Netherlands: Kluwer Academic, 1991.
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A textbook covering not only tense logic, but the treatment of time reference within classical logic as well.
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Conditional Logic

Logics of subjunctive conditionals, such as “if p had been the case, then q would have been the case,” are usually considered extraclassical, while logics of indicative conditionals, such as “if p is the case, then q is the case,” are usually considered anticlassical if they do not equate the conditional with the disjunction “either p is not the case or q is the case.” This classification reflects the standard thought that classical logic is committed to the given disjunctive analysis of indicatives but is silent on subjunctives. But the formalisms proposed in the two cases turn out to be similar in many ways, so that neither can be adequately studied without consideration of the other. Bennett 2003 covers both from a philosophical point of view. Adams 1975 and Edgington 2001 concern mainly indicatives, while Lewis 1973 concerns subjunctives, but Adams 1975 also shows the close connections between different formalisms. Hailperin 1996 is indirectly relevant.

Adams, Ernest W. The Logic of Conditionals: An Application of Probability to Deductive Logic. Synthese Library 86. Dordrecht, The Netherlands: Reidel, 1975.
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An account of an anticlassical theory of indicative conditionals, drawing on probability theory and the notion of conditional probability but containing the remarkable result that the formalism obtained agrees with that proposed by David Lewis for subjunctive conditionals, which uses a totally different kind of model.
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Bennett, Jonathan. A Philosophical Guide to Conditionals. New York: Oxford University Press, 2003.
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A philosophically oriented account of work, intimately related with modal logic, on indicative and subjunctive conditionals alike, presenting results and examples of Adams, Lewis, and others.
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Edgington, Dorothy. “Conditionals.” In The Blackwell Guide to Philosophical Logic. Edited by Lou Goble, 385–414. Blackwell Philosophy Guides 4. Oxford: Blackwell, 2001.
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A concise account of the main positions in late-20th-century debates over the analysis of conditionals of various kinds.
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Edgington, Dorothy. “Indicative Conditionals.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford University, 2014.
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A concise account of various positions, focusing specifically on indicative as opposed to subjunctive conditionals.
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Hailperin, Theodore. Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications. Bethlehem, PA: Lehigh University Press, 1996.
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A work not directly on conditional logic, but inviting comparison with, and perhaps ultimately integration with, that of Adams.
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Lewis, David. Counterfactuals. Library of Philosophy and Logic. Oxford: Blackwell, 1973.
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An influential proposal in the logic of subjunctive conditionals, related to but distinct from proposals by Robert Stalnaker. Revised edition published in 1986 (Cambridge, MA: Harvard University Press).
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Intuitionistic Logic

Moving on to more definitely anticlassical logics, the logic associated with mathematical intuitionism—a breakaway movement dissenting from the direction in which pure mathematics was developing in the early 20th century—holds that classical logic correctly represents the forms of argument accepted by classical mathematicians, but some of these forms of argument are incorrect. Among these are the argument from “if p then q” and “if not p then q” to “q,” which relies on the law of excluded middle “p or not p,” standardly understood to deliver that the disjunction of any (declarative) sentence and its negation are true. Underlying the rejection of this law, intuitionistic logic generally rejects the explanation of logical particles in terms of the conditions under which compounds formed with them are true. Instead of holding with classical logic that “p or q” is true just in case p is true or q is true, intuitionistic logic holds a proof of “p or q” that consists either of a proof of p or a proof of q. As a result, the intuitionist will not endorse an instance of the classical law of excluded middle unless in possession either of a proof of p or a refutation of p (a proof of “not p”). As to what underlies the rejection of truth conditions in favor of proof conditions, different intuitionists have offered different accounts, ranging from early approaches resting on idealist and mystical considerations tending toward solipsism, to later approaches based on verificationist considerations tending toward behaviorism. Brouwer 1975 represents the original approach, by the founder of intuitionism, and Heyting 1956 exemplifies the somewhat more moderate views of his chief disciple, while Dummett 1975 and Dummett 2000 represent the later view, by the most prominent late-20th- to early-21st-century philosophical defender of intuitionism, and Prawitz 1977 is an evaluation thereof by a sympathetic critic. The standpoint interested in metatheoretic questions without commitment to any intuitionistic philosophy is represented by Burgess 1981, building on work of Georg Kreisel, while Martin-Löf 1984 represents the main late-20th-century form of “constructivist” mathematics making use of intuitionistic logic. Gödel 1986 links intuitionistic with modal logic.

Brouwer, L. E. J. Collected Works. Vol. 1, Philosophy and Foundations of Mathematics. Edited by Arend Heyting. Amsterdam: North-Holland, 1975.
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A collection containing Brouwer’s presentation of his original ideas on the philosophical motivation of intuitionistic logic, spread over several key papers, to be compared with later accounts.
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Burgess, John P. “The Completeness of Intuitionistic Propositional Calculus for Its Intended Interpretation.” Notre Dame Journal of Formal Logic 22.1 (1981): 17–28.
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An adaptation, using Kripke models, of a method of Kreisel to show that, on certain assumptions, every law of classical sentential logic that is intuitionistically acceptable is provable by the usual intuitionistic system.
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Dummett, Michael. “The Philosophical Basis of Intuitionistic Logic.” In Logic Colloquium ’73: Proceedings of the Logic Colloquium, Bristol, July 1973. Edited by Harvey E. Rose and John C. Shepherdson, 5–40. Studies in Logic and the Foundations of Mathematics 80. Amsterdam: North-Holland, 1975.
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A classic paper, multiply anthologized, that advances arguments for applying intuitionistic logic well beyond the confines of mathematics.
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Dummett, Michael. Elements of Intuitionism. 2d ed. Oxford Logic Guides 39. New York: Oxford University Press, 2000.
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A comprehensive introduction to intuitionistic logic and mathematics, with an appendix reiterating the author’s distinctive philosophical case for intuitionism.
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Gödel, Kurt. “Eine Interpretation des intuitionistischen Aussagenkalküls.” In Collected Works. Vol. 1, Publications 1929–1936. Edited by Solomon Feferman, John W. Dawson Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, 300–303. New York: Oxford University Press, 1986.
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A miraculous paper in which, in three pages, Gödel indicates an interpretation of classical logic in intuitionistic logic and an interpretation of intuitionistic logic in modal logic, beside sketching a new and more convenient axiomatization of modal logic and launching the new subject of provability logic. Translated by John Dawson as “An Interpretation of Intuitionistic Propositional Calculus” on facing pages.
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Heyting, Arend. Intuitionism: An Introduction. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland, 1956.
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A short account of intuitionistic mathematics and its logic, by the Brouwer disciple who first presented intuitionistic logic as an axiomatic system.
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Martin-Löf, Per. Intuitionistic Type Theory. Studies in Proof Theory 1. Naples, Italy: Bibliopolis, 1984.
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The framework for the most active ongoing program of constructivistic mathematics, making use of intuitionistic logic, a formalism for which computer scientists have produced several proof-assistant programs, and a modification of which lies at the base of currently fashionable “univalent foundations.”
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Prawitz, Dag. “Meaning and Proofs: On the Conflict between Classical and Intuitionistic Logic.” Theoria 43.1 (1977): 2–40.
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An assessment of attempts to apply ideas from proof theory to motivate a preference for intuitionistic over classical logic, by an eminent proof theorist.
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Relevance/Relevant Logic

What some call “relevance” and others call “relevant” logic rejects the classical doctrine that “p and not p” logically implies an arbitrary q, on the grounds of lack of a connection of relevance between premise and conclusion, and is soon led to reject also the inference form “p or q and not p” to q. Anderson, et al. 1992 is a major reference by pioneers of the subject, representing what has come to be called the “American” approach, and Dunn and Restall 2002 is a shorter survey. The “Australian” approach is represented in Routley, et al. 1982; the “Scottish,” in Read 1989; and a synthesis of “American” and “Australian,” in Mares 2004. Restall 2000 introduces a broader category of nonclassical logics subsuming relevance/relevant logic.

Anderson, Alan Ross, Nuel D. Belnap Jr., and J. Michael Dunn. Entailment: The Logic of Relevance and Necessity. Vol. 2. Princeton, NJ: Princeton University Press, 1992.
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The classic source for relevance/relevant logic, both the formal details (up to the time of publication) and the philosophical ideas that were involved, at least from the “American” perspective. Comparison between this and Vol. 1 (1975) shows the rapid growth in sophistication in the technical side of the subject (through contributions by Kit Fine, Saul Kripke, Robert Meyer and Richard Routley, Alasdair Urquhart, and others).
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Brady, Ross, ed. Relevant Logics and Their Rivals. Vol. 2, A Continuation of the Work of Richard Sylvan, Robert Meyer, Val Plumwood and Ross Brady. Western Philosophy 59. Aldershot, UK: Ashgate, 2003.
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A resource for formal results on a variety of relevance/relevant logics. The philosophical remarks in the first volume (Routley, et al. 1982) are not representative of relevance/relevant logicians generally but give a good example of certain philosophical perspectives in the field, especially in Australia. The second volume covers a wider range of topics, including standard challenges for relevance/relevant logics such as conditionals and quantification, as well as a variety of metalogical results.
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Dunn, J. Michael, and Greg Restall. “Relevance Logic.” In Handbook of Philosophical Logic. Vol. 6. 2d ed. Edited by Dov M. Gabbay and Fritz Guenthner, 1–128. Dordrecht, The Netherlands: Reidel, 2002.
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A panoramic survey both of philosophical and formal issues in relevance/relevant logic, written by one of the chief pioneers of the subject (Dunn) and one of the leading experts in the broader field of substructural logics (see Restall 2000).
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Mares, Edwin D. Relevant Logic: A Philosophical Interpretation. Cambridge, UK: Cambridge University Press, 2004.
DOI: 10.1017/CBO9780511520006Save Citation »Export Citation »E-mail Citation »
A volume advancing a blend of early “American” and modern “Australasian” philosophical perspectives in an effort to illuminate and defend the centrality of one particular system, Anderson and Belnap’s R.
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Read, Stephen. Relevant Logic: A Philosophical Examination of Inference. Oxford: Basil Blackwell, 1989.
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A volume advancing what has come to be called the “Scottish” perspective in relevant logic, containing a useful discussion of other perspectives (so-called American, Australian, Australasian) and highlighting many of the distinctions (and difficulties) involved in relevance/relevant logic.
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Restall, Greg. An Introduction to Substructural Logics. London: Routledge, 2000.
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A general treatment of logics that give up one or more of the standard “structural” rules in a Gentzen-style formulation of classical logic, a class that includes relevance/relevant logics as well as other nonclassical logics, such as what is called linear logic, with varying philosophical and technical motivations.
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Routley, Richard, Robert K. Meyer, Valerie Plumwood, and Ross T. Brady. Relevant Logics and Its Rivals. Vol. 1, The Basic Philosophical and Semintical Theory. Western Philosophy. Atascadero, CA: Ridgeview, 1982.
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This collection highlights early research in Australia on relevance/relevant logic. The collection focuses both on mathematical results and philosophical perspectives on relevance/relevant logic.
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Paraconsistent Logic

On the paraconsistent view the two truth values, true and false, need not be exclusive, and a single example (perhaps the liar paradox, “this very statement is false”) may have both. Often this is combined with the “paracomplete” view that they need not be exhaustive, and that an example (perhaps the truth-teller pathology, “this very statement is true”) may have neither. But like classical logic and unlike intuitionistic logic, paraconsistent logic does allow that the truth values of compounds are determined by those of their components, and that validity is a matter of form guaranteeing appropriate preservation of truth values, from the more complicated set of recognized possible truth values. Paraconsistent logic is perhaps the chief example of a many-valued logic that comes with a philosophical interpretation and motivation; many others are used mainly as technical tools. Nonclassical logics may turn out to have a utility that does not require acceptance of the original philosophical motivation for the logic as anything more than heuristically suggestive. Asenjo 1966 and Asenjo and Tamburino 1975 are pioneering works, and Priest 1979 is the starting point for many subsequent efforts; Beall, et al. 2014 compares their approaches. Routley 1979 represents an extreme view, while Routley and Meyer 1976 and especially Belnap 1977 point to the need for or utility of paraconsistent logic in a way that does not depend on assuming that there are literal contradictions that are literally true.

Asenjo, Florencio González. “A Calculus for Antinomies.” Notre Dame Journal of Formal Logic 7.1 (1966): 103–105.
DOI: 10.1305/ndjfl/1093958482Save Citation »Export Citation »E-mail Citation »
A pioneering work of the glut-theoretic approach to paradoxes (i.e., treating paradoxical sentences both as true and false), introducing a propositional three-valued paraconsistent logic, which is now more standardly called “logic of paradox” or LP on the basis of wider familiarity with the work of Graham Priest (see Priest 1979).
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Asenjo, Florencio González, and Joanna Tamburino. “Logic of Antinomies.” Notre Dame Journal of Formal Logic 16.1 (1975): 17–44.
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An extension of Asenjo’s glut-theoretic approach to paradox from sentential to predicate logic.
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Beall, Jc, Michael Hughes, and Ross Vandegrift. “Glutty Theories and the Logic of Antinomies.” In The Metaphysics of Logic. Edited by Penelope Rush, 224–232. Cambridge, UK: Cambridge University Press, 2014.
DOI: 10.1017/CBO9781139626279.017Save Citation »Export Citation »E-mail Citation »
An elementary discussion of the formal and philosophical differences between the Asenjo-Tamburino logic of antinomies or LA (see Asenjo and Tamburino 1975) and the Priest first-order logic of paradox LP (see Priest 1979).
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Belnap, Nuel D. “How a Computer Should Think.” Paper presented at the Oxford international symposium held 29 September–4 October 1975 at Christ Church College, Oxford. In Contemporary Aspects of Philosophy. Edited by Gilbert Ryle, 30–56. Stockfield, UK: Oriel, 1977.
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A classic example of a many-valued subclassical logic, and a classic example of applying paraconsistent logic without any suggestion of glut theory (i.e., entertaining the possibility of true negation-inconsistent theories). Reprinted as “A Useful Four-Valued Logic” in Anderson, et al. 1992 (cited under Relevance/Relevant Logic), pp. 506–541.
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Jaśkowski, Stanisław. “Rachunek zdań dla systemów dedukcyjnych sprzecznych.” Studia Societatis Scientiarum Torunensis: Sectio A 1.5 (1948): 55–77.
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This is the first clear construction of a paraconsistent logic, which was tied closely to philosophical issues surrounding vagueness. The idea takes different points of a model to be discussants, and truth in a model is truth at least at one such point. The points can all be consistent, but we can have models in which a sentence and its negation are true without all sentences being true in the model. Jaśkowski’s work has contributed greatly to modern ideas in philosophical logic, even though few philosophers have heard of him. English translation published as “Propositional Calculus for Contradictory Deductive Systems” in Studia Logica 24 (1969): 143–157.
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Priest, Graham. “The Logic of Paradox.” Journal of Philosophical Logic 8 (1979): 219–291.
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A paper advocating a glutty approach to standard paradoxes generally, pioneering the argument from “semantic closure” or “expressive completeness” to gluts—sentences that are both true and false (i.e., have true negations) according to the given theory.
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Routley, Richard. “Dialectical Logic, Semantics, and Metamathematics.” Erkenntnis 14.3 (1979): 301–331.
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A paper resembling Priest 1979 in advocating a paraconsistent logic for purposes of a glut-theoretic treatment of paradoxes, arguing that standard “limitative theorems” are in fact arguments for gluts, a view that is not widely held even among glut theorists.
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Routley, Richard, and Robert K. Meyer. “Dialectical Logic, Classical Logic, and the Consistency of the World.” Studies in Soviet Thought 16 (1976): 1–25.
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An early discussion of the alleged need for paraconsistent logic, pushing for a paraconsistent and paracomplete logic while arguing for a philosophical view of “agnosticism” about gluts and their dual “gaps.”
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Mathematical Logic

Mathematical logic is conventionally divided into four distinct but interacting subfields, whose relations to basic logic and whose bearing on philosophy differ considerably from case to case: model theory, proof theory, set theory, and recursion theory (which many nowadays would rechristen “computability theory”). Crossley, et al. 1972 provides a concise overview, and Barwise 1977 is a compendium of detailed topic-by-topic surveys. Hilbert and Bernays 1934–1939, Kleene 1952, and Shoenfield 1967 represent three generations of large-scale texts, indicative of the evolution of the subject over its first several decades. Van Heijenoort 1967 reprints in translation many fundamental papers, with expert commentary.

Barwise, Jon, ed. Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics 90. Amsterdam: North-Holland, 1977.
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A classic collection of high-level surveys, prototype for the many handbooks in more-specialized areas that have appeared since.
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Crossley, John N., Christopher J. Ash, Chris J. Brickhill, John C. Stillwell, and Neil H. Williams. What Is Mathematical Logic? Oxford: Oxford University Press, 1972.
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A work of high-level popularization, explaining without pretending to prove important results from all branches of mathematical logic. Reprinted in 1990 (New York: Dover).
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Hilbert, David, and Paul Bernays. Grundlagen der Mathematik. 2 vols. Grundlehren der Mathematischen Wissenschaften 40, 50. Berlin: Springer, 1934–1939.
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An absolute classic, translating as “Foundations of mathematics.” A massive work presenting the results of multiple workers through the 1930s, not only on the “Hilbert program” but on other areas of mathematical logic; a project for an English translation with the original German on facing pages is underway, with various partial translations being made available in the meantime.
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Kleene, Stephen Cole. Introduction to Metamathematics. Bibliotheca Mathematica 1. Amsterdam: North-Holland, 1952.
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Another large-scale work, by an important participant (along with Alonzo Church and Alan Turing) in the establishment of recursion theory, incorporating many of the results from the first two decades after Kurt Gödel’s fundamental work; the textbook from which mathematical logicians learned their subject for a generation and more.
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Shoenfield, Joseph R. Mathematical Logic. Addison-Wesley Series in Logic. Reading, MA: Addison-Wesley, 1967.
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A textbook at the level of beginning graduate students in mathematics, widely used ever since its appearance and full of exercises.
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van Heijenoort, Jean, ed. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Source Books in the History of the Sciences. Cambridge, MA: Harvard University Press, 1967.
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A classic collection of translations of key documents by Gottlob Frege, Gödel, Luitzen Brouwer, Georg Cantor, Hilbert, Leon Löwenheim, Bertrand Russell, Thoralf Skolem, Ernst Zermelo, and others.
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Model Theory

The theory of models begins with the work of Alfred Tarski, reprinted in translation in Tarski 1983, from which emerged the definitions of model and truth in a model used in the metatheory of classical logic. Tarski’s work also inspired (though a kind of reversal of perspective is involved) Davidsonian truth-conditional accounts of meaning, among other philosophical developments taking us outside the domain of logic. Kreisel 1969 is a philosophical reflection on the relation of technical notions of model theory and intuitive notions they attempt to capture. A number of what may be considered extraclassical logics, adding additional quantifiers rather than connectives to classical logic, work with the same Tarskian notion of model as classical first-order logic; surveys of this diverse range of logics may be found in Barwise and Feferman 1985. As for the model theory specifically of first-order logic, Robinson 1965 is an account by a pioneer in its application to abstract algebra of some early, elementary examples of such applications. Such applied-model theory has since grown to enormous size and represents the main thrust of model theory in the early 21st century, albeit not the part of the subject most inviting to philosophers. Ebbinghaus and Flum 1995 covers an important subfield not thoroughly treated in our other references, with applications in a different direction. Chang and Keisler 2012 is the latest version of a once widely used textbook, covering the subject as it was before the revolution inaugurated by Tarski.

Barwise, Jon, and Solomon Feferman, eds. Model-Theoretic Logics. Perspectives in Mathematical Logic. Berlin: Springer, 1985.
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A collection of high-level surveys of logics that go beyond (but use the same notion of model as) classical logic, including logics of generalized quantifiers, infinitary logic, second-order logic, and more.
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Chang, C. C., and H. Jerome Keisler. Model Theory. 3d ed. Dover Books on Mathematics. Mineola, NY: Dover, 2012.
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A much-used textbook pitched at an advanced undergraduate or beginning graduate level.
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Ebbinghaus, Heinz-Dieter, and Jörg Flum. Finite Model Theory. Perspectives in Mathematical Logic. Berlin: Springer, 1995.
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The first comprehensive textbook on the theory of finite models, which differs greatly in flavor from the theory of arbitrary (finite or infinite) models.
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Kreisel, Georg. “Informal Rigour and Completeness Proofs.” In The Philosophy of Mathematics. Edited by Jaakko Hintikka, 78–94. Oxford Readings in Philosophy. Oxford: Oxford University Press, 1969.
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A subtle discussion of, among other things, the relationship between the technical notion of truth in all models and the intuitive notion of logical validity.
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Robinson, Abraham. Introduction to Model Theory and to the Metamathematics of Algebra. 2d ed. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland, 1965.
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An early and accessible textbook by a pioneer in the applications of model theory to abstract algebra, including a brief account of the author’s nonstandard analysis, a modern version of infinitesimal calculus.
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Shelah, Saharon. Classification Theory and the Number of Non-isomorphic Models. Studies in Logic and the Foundations of Mathematics 92. Amsterdam: North-Holland, 1978.
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Shelah raised the subject to a level of mathematical sophistication and technical virtuosity where few philosophers can hope to follow.
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Tarski, Alfred. Logic, Semantics, Metamathematics: Papers from 1923 to 1938. 2d ed. Edited and introduced by John Corcoran. Translated by J. H. Woodger. Indianapolis, IN: Hackett, 1983.
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A collection including not only the celebrated “Concept of Truth in Formalized Languages” but also other fundamental contributions to the creation of model theory.
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Set Theory

Georg Cantor created the theory of sets in the late 19th century, at first in connection with certain problems of mathematical analysis. Some of his key papers are made available in translation in Cantor 1952, while what has become of Cantor’s original concerns with mathematical analysis can be see from Kechris 1995, and heterodox non-Cantorian approaches are surveyed in Holmes 2012. The discovery of paradoxes led to a more rigorous, axiomatic treatment of the subject by Ernst Zermelo (for whose work, see Zermelo 2010) and others. By the middle of the 20th century, it had been found that in some sense all of mathematics can be codified on the basis of the Zermelo-Fraenkel axiom system (ZFC). Set-theoretic terminology and results came to pervade mathematics, including numerous areas of interest in connection with the more formal parts of philosophy (including probability theory, to name just one). Kurt Gödel had by this time shown that in any axiom system for mathematics there will be questions that can be posed but not answered, and together with Paul Cohen he showed that for ZFC specifically, the conjecture of Cantor known as the continuum hypothesis (CH) was undecidable. The question of whether and in what sense there can nonetheless be “right” answers in such cases has occupied set theorists and philosophers of mathematics ever since, beginning with Gödel himself in the work reprinted as Gödel 1990. Hrbacek and Jech 1999 and Jech 2003 are widely used textbooks at the undergraduate and graduate levels, and Kanamori 2010 is a survey of more-advanced material.

Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Numbers. Edited, translated, and introduced by Philip E. B. Jourdain. Mineola, NY: Dover, 1952.
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A conveniently available photographic reproduction of a 1915 original, providing in English two major papers by the founder of set theory, still of more than historical interest, with useful supplements by the editor.
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Gödel, Kurt. “What Is Cantor’s Continuum Problem?” In Collected Works. Vol. 2, Publications 1938–1974. Edited by Solomon Feferman, John W. Dawson Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, 176–188. New York: Oxford University Press, 1990.
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A forceful expression of the conviction that despite the result, due in part to the author himself, the currently accepted axioms can neither prove nor refute Cantor’s continuum hypothesis; it is nonetheless either true or false, and more likely the latter.
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Holmes, M. Randall. “Alternative Axiomatic Set Theories.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford University, 2012.
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A survey of heterodox axiomatic set theories, which are many and varied, though none have a large following.
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Hrbacek, Karel, and Thomas Jech. Introduction to Set Theory. 3d ed. Monographs and Textbooks in Pure and Applied Mathematics 220. New York: Marcel Dekker, 1999.
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A much-used modern introductory textbook, intended for undergraduate mathematics courses but usable for independent study, with extensive problems.
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Jech, Thomas. Set Theory. 3d ed. Springer Monographs in Mathematics. Berlin: Springer, 2003.
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A comprehensive graduate-level textbook, covering results of Paul Cohen, Gödel, Ronald Jensen, Donald Martin, Shelah, Jack Silver, Robert Solovay, John Steel, W. Hugh Woodin, and others.
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Kanamori, Akihiro. “Introduction.” In Handbook of Set Theory. Vol. 1. Edited by Matthew Foreman and Akihiro Kanamori, 1–92. Dordrecht, The Netherlands: Springer, 2010.
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An overview of the whole field as it appears to leading figures in the early 21st century, prefacing a multivolume compendium directed at future researchers.
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Kechris, Alexander S. Classical Descriptive Set Theory. Graduate Texts in Mathematics 156. Berlin: Springer, 1995.
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A textbook on the side of set theory called “descriptive,” as contrasted with “combinatorial,” closest to the subject’s original home in mathematical analysis.
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Zermelo, Ernst. Collected Works/Gesammelte Werke. Vol. 1. Edited by Heinz-Dieter Ebbinghaus, Craig G. Fraser, and Akihiro Kanamori. Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenshcaften 23. Berlin: Springer, 2010.
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German originals with facing English translations of the fundamental papers of the founder of axiomatic set theory.
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Proof Theory

David Hilbert in the 1920s introduced a metamathematics or theory of proof, in which formal counterparts of the proofs used as the method of study everywhere in mathematics became the objects of study. His original aim was to find a consistency proof for modern, abstract, set-theoretic mathematics that might silence the intuitionists and other critics. For what became of this program, see Zach 2015. In the wake of Gödel’s work (on incompleteness results, most readily available in Gödel 1986), it is recognized that this aim cannot be achieved in its original form, but rather that we everywhere face trade-offs between the power of axiom systems (their ability to answer mathematical questions) and their riskiness (the potential danger of collapsing in contradiction). The delicate interplay of power and risk has since been intensively investigated, especially in so-called reverse mathematics, as surveyed in Simpson 1985, which by “proving axioms from theorems” attempts to the determine the least risky assumptions powerful enough to yield this or that classical mathematical result. Meanwhile, the tools developed in connection with Hilbert’s program by Gerhard Gentzen and others have come to live a life of their own and have developed into the subject represented at the textbook level by Takeuti 1987, and at a more advanced level in Buss 1998, with a high-level survey in Feferman 2000. Especially important for philosophers have been so-called natural-deduction proof procedures, introduced independently in Gentzen 1935 and Jaśkowski 1934, which not only are favored in many introductory textbooks but also have inspired the philosophical idea that the meanings of logical operators (connectives and quantifiers) are to be explained in terms of rules of proof rather than conditions of truth, an idea that has in turn been made the basis for motivating arguments by proponents of intuitionistic logics.

Buss, Samuel R., ed. Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics 137. Amsterdam: Elsevier Science, 1998.
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A collection of high-level surveys by leading contributors of the many and varied parts of a sprawling field, including connections with computing.
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Feferman, Solomon. “Highlights in Proof Theory.” Paper presented at a conference held 31 October–1 November 1997 at the University of Roskilde, Denmark. In Proof Theory: History and Philosophical Significance. Edited by Vincent F. Hendricks, Stig Andur Pedersen, and Klaus Frovin Jørgensen, 11–34. Synthese Library 292. Dordrecht, The Netherlands: Kluwer Academic, 2000.
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A historical survey of the main lines of development of proof theory from Hilbert onward, emphasizing the increasing role of infinitistic methods.
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Gentzen, Gerhard. “Untersuchungen über das logsiche Schließen I.” Mathematische Zeitschrift 39.1 (1935): 176–210.
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Continued as “Untersuchungen über das logsiche Schließen II” in the same issue on pp. 405–431. Pioneering work both on the so-called sequent calculus and on natural deduction. Translated as “Investigations into Logical Deduction” in The Collected Papers of Gerhard Gentzen, edited by M. E. Szabo (Amsterdam: North-Holland, 1969), pp. 68–131.
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Gödel, Kurt. “Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I/On Formally Undecidable Propositions of Principia Mathematica and Related Systems I.” In Collected Works. Vol. 1, Publications 1929–1936. Edited by Solomon Feferman, John W. Dawson Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, 144–195. New York: Oxford University Press, 1986.
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An article widely viewed as the single most important publication ever in mathematical logic, presented in the German original with facing English translation by van Heijenoort.
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Jaśkowski, Stanisław. “On the Rules of Suppositions in Formal Logic.” Studia Logica 1 (1934): 5–32.
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An account of version of natural deduction entirely independent of Gentzen’s work, representing a case of nearly simultaneous discovery of a major idea. Reprinted in Polish Logic, 1920–1939, edited by Storrs McCall (Oxford: Oxford University Press, 1967), pp. 232–258.
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Simpson, Stephen G. “Friedman’s Research on Subsystems of Second Order Arithmetic.” In Harvey Friedman’s Research on the Foundations of Mathematics. Edited by L. A. Harrington, M. D. Morley, Andrej Sčědrov, and Stephen G. Simpson, 137–160. Studies in Logic and the Foundations of Mathematics 117. Amsterdam: North-Holland, 1985.
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Remains perhaps the best short introduction to the aims and claims of Harvey Friedman’s reverse mathematics.
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Takeuti, Gaisi. Proof Theory. 2d ed. Studies in Logic and the Foundations of Mathematics 81. Amsterdam: North-Holland, 1987.
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A classic textbook of main-line proof theory, focused on so-called sequent calculus and cut elimination, deriving from Gentzen, with coverage broadened in appendixes (by Georg Kreisel, Wolfram Pohlers, Stephen G. Simpson, and Solomon Feferman).
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Zach, Richard. “Hilbert’s Program.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. Stanford, CA: Stanford University, 2015.
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A concise historical and philosophical account of Hilbert’s program, the damage done to it by Gödel’s incompleteness theorems, and what nonetheless survives from it, all in the light of early-21st-century research.
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Recursion Theory

For classical sentential logic there are decision procedures, such as the method of truth tables taught in introductory textbooks, that in principle will always tell one in a finite amount of time whether a given argument form is valid. For classical predicate logic there are no such decision procedures: there are proof procedures that always, if a given argument form is valid, will always tell one that it is, but there are no disproof procedures that always, if a given argument form is not valid, will tell one that it is not. The rigorous statement and proof of such results require a rigorous definition of “effective decidability” and “effective computability” such as emerged in the form of the “Church-Turing thesis” from the work of Alonzo Church and Alan Turing (he of the famous machines) in the 1930s. The whole subject of theoretical computer science eventually emerged from these studies, a process described in Davis 2011. These developments, and the notion of the Turing machine in particular, have had considerable influence on philosophy of mind as well as philosophy of mathematics. Key papers are made available in Davis 2004. Textbook accounts at elementary and more-advanced levels are to be found in Boolos, et al. 2007 and Cooper 2004, and advanced surveys are in Griffor 1999. What became of the original motivating problems for the field can be seen from Börger, et al. 1997 and Matiyasevich 1993.

Boolos, George S., John P. Burgess, and Richard C. Jeffrey. Computability and Logic. 5th ed. Cambridge, UK: Cambridge University Press, 2007.
DOI: 10.1017/CBO9780511804076Save Citation »Export Citation »E-mail Citation »
An undergraduate-level textbook of intermediate-level logic, covering standard material through the Gödel theorems, with an account (mainly by Jeffrey) of the equivalence of various notions of computability, including Kleene’s in terms of recursive functions and Turing’s in terms of idealized machines.
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Börger, Egon, Erich Grädel, and Yuri Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Berlin: Springer, 1997.
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A modern account of the Entscheidungsproblem, or the problem of determining whether a given logical formula is valid, proved undecidable by Church and by Turing but having many decidable special cases, with connections to superficially different-seeming decidability questions such as the domino problem.
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Cooper, S. Barry. Computability Theory. Chapman & Hall/CRC Mathematics. Boca Raton, FL: Chapman & Hall/CRC, 2004.
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A modern undergraduate-level textbook, covering all the standard topics.
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Davis, Martin, ed. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Mineola, NY: Dover, 2004.
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A corrected reproduction of the 1965 original anthology of fundamental papers by Gödel, Church, Turing, and others.
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Davis, Martin. The Universal Computer: The Road from Leibniz to Turing. Turing Centenary ed. Boca Raton, FL: CRC, 2011.
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A semipopular history of conceptual developments in mathematics and logic leading up to the creation of modern digital computing, requiring very little background on the part of the reader. First published in 2000 (New York: W. W. Norton).
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Griffor, Edward R., ed. Handbook of Computability Theory. Studies in Logic and the Foundations of Mathematics 140. Amsterdam: Elsevier Science, 1999.
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A collection of high-level surveys of those parts of the theory of computability that remain more in the domain of mathematical logic than of computer science (not that there is a sharp division).
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Matiyasevich, Yuri V. Hilbert’s Tenth Problem. Foundations of Computing. Cambridge, MA: MIT, 1993.
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An account of one of the most famous problems that has turned out to be effectively undecidable, by the mathematician who proved it to be so, showing the interplay of recursion theory and number theory.