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Predicate Logic

 Summary Predicate logic is the more complicated of the two modern classical logics.  It does not consider atomic propositions as indivisible, notwithstanding the etymology, but also considers the structure within propositions. In its treatment of the general, as opposed to the singular, propositions, it achieves the aims of Aristotelian logic in combination with the aims of propositional logic.  In the classification structure chosen by the general editors, second-order and higher-order logics are separate categories, and are therefore not classified as (ordinary) predicate calculus. This may seem a curiosity; it is exlpored in Eklund 1996. In its treatment of singular propositions, relations are permitted, too, as is the special predicate, identity. In classical predicate logic, molecular or compound propositions are built up from atomic propositions by means of the connectives, whose meaning is given by their truth tables.  Likewise, one way of understanding the meaning of the two classical quantifiers, existential and universal, is by taking them to be expanded disjunctions and conjunctions, respectively, over the universe of discourse.  The principle by which the meaning or truth conditions of compound propositions can be recovered by this "building up" process is known as compositionality.  Aside from an appropriate way to understand the meaning of the quantifiers, there is the additional issue of existential import. This leaf node is a sub-category of classical logic.  As such, non-standard predicate logics are not generally 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 predicate logic in which nothing not a theorem in ordinary predicate logic is a theorem in the restriction do fit here.  Also appropriate are modest extensions of predicate logic, excluding higher-order logics as noted above, 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 predicate logic are generally also classified as "proof theory," "model theory," "mathematical logic," etc.
 Key works See below.
 Introductions Because of the age of predicate 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 two things: (1) The correctness and clarity of the restrictions on universal generalization and existential instantiation; (2) 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. E. J. A. (1966). Symbolic Logic. [REVIEW] Review of Metaphysics 19 (4):808-808.

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2. Jonathan E. Adler (2000). First-Order Logic. Journal of Philosophy 97 (10):577-580.

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3. Henry Africk (1992). Classical Logic, Intuitionistic Logic, and the Peirce Rule. Notre Dame Journal of Formal Logic 33 (2):229-235.
A simple method is provided for translating proofs in Grentzen's LK into proofs in Gentzen's LJ with the Peirce rule adjoined. A consequence is a simpler cut elimination operator for LJ + Peirce that is primitive recursive.

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4. Samuel Alexander (2013). The First-Order Syntax of Variadic Functions. Notre Dame Journal of Formal Logic 54 (1):47-59.
We extend first-order logic to include variadic function symbols, and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.

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5. Robert A. Alps & Robert C. Neveln (1981). A Predicate Logic Based on Indefinite Description and Two Notions of Identity. Notre Dame Journal of Formal Logic 22 (3):251-263.

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

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7. Mohamed A. Amer (1989). First Order Logic with Empty Structures. Studia Logica 48 (2):169 - 177.
For first order languages with no individual constants, empty structures and truth values (for sentences) in them are defined. The first order theories of the empty structures and of all structures (the empty ones included) are axiomatized with modus ponens as the only rule of inference. Compactness is proved and decidability is discussed. Furthermore, some well known theorems of model theory are reconsidered under this new situation. Finally, a word is said on other approaches to the whole problem.

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8. Edgar Jose Andrade & Edward Samuel Becerra (2008). Establishing Connections Between Aristotle's Natural Deduction and First-Order Logic. History and Philosophy of Logic 29 (4):309-325.
This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the same properties (...)

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10. Mohammad Ardeshir (1999). A Translation of Intuitionistic Predicate Logic Into Basic Predicate Logic. Studia Logica 62 (3):341-352.
Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.

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11. Axel Arturo Barceló Aspeitia (2007). What Does '&' Mean? The Proceedings of the Twenty-First World Congress of Philosophy 5:45-50.
Using conjunction as an example, I show a technical and philosophical problem when trying to conciliate the currently prevailing views on the meaning of logical connectives: the inferientialist (also called 'syntactic') one based on introduction and elimination rules, and the representationalist (also called 'semantic') one given through truth tables. Mostly I show that the widespread strategy of using the truth theoretical definition of logical consequence to collapse both definitions must be rejected by inferentialists. An important consequence of my argument is (...)

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12. From proofs in any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions in polynomial time. From proofs in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions in polynomial time.

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13. Jeremy Avigad & Richard Zach (2008). The Epsilon Calculus. Stanford Encyclopedia of Philosophy.
The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term..

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14. John Bacon (1982). First-Order Logic Based on Inclusion and Abstraction. Journal of Symbolic Logic 47 (4):793-808.

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15. John Bacon (1975). Elementary Symbolic Logic. Teaching Philosophy 1 (2):220-221.

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16. A. J. Baker (1977). Classical Logical Relations. Notre Dame Journal of Formal Logic 18 (1):164-168.

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17. Brian Ball (2014). Forms of Thought, by E. J. Lowe. Mind 123 (492):1205-1208.

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18. John A. Barker (1975). Relevance Logic, Classical Logic, and Disjunctive Syllogism. Philosophical Studies 27 (6):361 - 376.

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19. Stephen Barker (1997). Material Implication and General Indicative Conditionals. Philosophical Quarterly 47 (187):195-211.
This paper falls into two parts. In the first part, I argue that consideration of general indicative conditionals, e.g., sentences like If a donkey brays it is beaten, provides a powerful argument that a pure material implication analysis of indicative if p, q is correct. In the second part I argue, opposing writers like Jackson, that a Gricean style theory of pragmatics can explain the manifest assertability conditions of if p, q in terms of its conventional content – assumed to (...)

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20. Michael Baumgartner & Timm Lampert (2008). Adequate Formalization. Synthese 164 (1):93-115.
This article identifies problems with regard to providing criteria that regulate the matching of logical formulae and natural language. We then take on to solve these problems by defining a necessary and sufficient criterion of adequate formalization. On the basis of this criterion we argue that logic should not be seen as an ars iudicandi capable of evaluating the validity or invalidity of informal arguments, but as an ars explicandi that renders transparent the formal structure of informal reasoning.

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21. Richard Beatty (1969). Peirce's Development of Quantifiers and of Predicate Logic. Notre Dame Journal of Formal Logic 10 (1):64-76.

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22. David Bell (1971). Fallacies in Predicate Logic? Mind 80 (317):145-147.

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

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24. Katalin Bimbó (2010). Schönfinkel-Type Operators for Classical Logic. Studia Logica 95 (3):355-378.
We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator ( Nallor ) that is the dual of Schönfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic ( Fol ). Having scrutinized the proof, we pinpoint the theorems of (...)

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25. Katalin Bimbó (2009). Combinatory Logic. Stanford Encyclopedia of Philosophy.

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26. V. A. Bocharov (1983). Subject-Predicate Calculus Free From Existential Import. Studia Logica 42 (2-3):209 - 221.
Two subject-predicate calculi with equality,SP = and its extensionUSP =, are presented as systems of natural deduction. Both the calculi are systems of free logic. Their presentation is preceded by an intuitive motivation.It is shown that Aristotle's syllogistics without the laws of identitySaP andSiP is definable withinSP =, and that the first-order predicate logic is definable withinUSP =.

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

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30. 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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31. Kai Brünnler (2006). Cut Elimination Inside a Deep Inference System for Classical Predicate Logic. Studia Logica 82 (1):51-71.
Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it (...)

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32. C. Butz & I. Moerdijk (1999). An Elementary Definability Theorem for First Order Logic. Journal of Symbolic Logic 64 (3):1028-1036.

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33. John Byrnes (1998). Peirce's First-Order Logic of 1885. Transactions of the Charles S. Peirce Society 34 (4):949 - 976.

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

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35. 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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36. There exist valuable methods for theorem proving in non classical logics based on translation from these logics into first-order classical logic (abbreviated henceforth FOL). The key notion in these approaches istranslation from aSource Logic (henceforth abbreviated SL) to aTarget Logic (henceforth abbreviated TL). These methods are concerned with the problem offinding a proof in TL by translating a formula in SL, but they do not address the very important problem ofpresenting proofs in SL via a backward translation. We propose a (...)

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37. Howard G. Callaway (1979). Reference variables and the empty universe. Logique Et Analyse 22 (85):85.
In this early paper I set out an argument in favor of the standard semantics of first-order logic, to the effect that (Vx)(Ey)x=y. Though my arguments from the paper have since been revised in details, The conclusion of the paper seems still viable and acceptable.
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38. James D. Carney (1970). Introduction to Symbolic Logic. Englewood Cliffs, N.J., Prentice-Hall.
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39. Introduction This is an elementary logic book designed for people who have no technical familiarity with modern logic but who have been reasoning, ...

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40. Anthony Pike Cavendish (1953). Introduction to Symbolic Logic. University Tutorial Press.
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41. 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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42. 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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43. Saloua Chatti & Fabien Schang (2013). The Cube, the Square and the Problem of Existential Import. History and Philosophy of Logic 34 (2):101 - 132.
(2013). The Cube, the Square and the Problem of Existential Import. History and Philosophy of Logic: Vol. 34, No. 2, pp. 101-132. doi: 10.1080/01445340.2013.764962.

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44. D. S. Clarke (1973). Deductive Logic. Carbondale: Southern Illinois University Press.
This introduction to the basic forms of deductive inference as evaluated by methods of modern symbolic logic is de­signed for sophomore-junior-level stu­dents ready to specialize in the study of deductive logic. It can be used also for an introductory logic course. The inde­pendence of many sections allows the instructor utmost flexibility. The text consists of eight chapters, the first six of which are designed to intro­duce the student to basic topics of sen­tence and predicate logic. The last two chapters extend (...)
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45. Brian Coffey (1948). Elements of Symbolic Logic. Modern Schoolman 25 (3):198-202.

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46. Marcelo E. Coniglio & Luiz H. Da Cruz Silvestrini (2013). An Alternative Approach for Quasi-Truth. Logic Journal of the IGPL 22 (2):387-410.
In 1986, Mikenberg et al. introduced the semantic notion of quasi-truth defined by means of partial structures. In such structures, the predicates are seen as triples of pairwise disjoint sets: the set of tuples which satisfies, does not satisfy and can satisfy or not the predicate, respectively. The syntactical counterpart of the logic of partial truth is a rather complicated first-order modal logic. In the present article, the notion of predicates as triples is recursively extended, in a natural way, to (...)

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47. Irving M. Copi (1973). Symbolic Logic. New York: Macmillan.

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48. J. Corcoran (2008). Aristotle's Many-Sorted Logic. Bulletin of Symbolic Logic 14:155-156.
As noted in 1962 by Timothy Smiley, if Aristotle’s logic is faithfully translated into modern symbolic logic, the fit is exact. If categorical sentences are translated into many-sorted logic MSL according to Smiley’s method or the two other methods presented here, an argument with arbitrarily many premises is valid according to Aristotle’s system if and only if its translation is valid according to modern standard many-sorted logic. As William Parry observed in 1973, this result can be proved using my 1972 (...)

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49. John Corcoran (2014). “Truth-Preserving and Consequence-Preserving Deduction Rules”,. Bulletin of Symbolic Logic 20:130-1.
A truth-preservation fallacy is using the concept of truth-preservation where some other concept is needed. For example, in certain contexts saying that consequences can be deduced from premises using truth-preserving deduction rules is a fallacy if it suggests that all truth-preserving rules are consequence-preserving. The arithmetic additive-associativity rule that yields 6 = (3 + (2 + 1)) from 6 = ((3 + 2) + 1) is truth-preserving but not consequence-preserving. As noted in James Gasser’s dissertation, Leibniz has been criticized for (...)