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UC Davis Department of Philosophy

Intermediate Logic

Philosophy Department

1240 Social Sciences and Humanities
University of California, Davis
One Shields Avenue
Davis, CA 95616

Phone: (530) 752-0607

Fax: (530) 752-8964

PIcture of Gottlob Frege

Gottlob Frege

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This is the home page of G. J. Mattey’s Philosophy 112, Intermediate Symbolic Logic, for Winter Quarter, 2012.

The Course

Philosophy 112 covers first-order predicate logic at a basic level. Students will learn the symbolism of predicate logic, natural-language and formal interpretations of predicate logic, and derivations in predicate logic.

General Catalog Course Description

Lecture/discussion—4 hours. Prerequisite: course 12 or consent of instructor. Predicate logic syntax and semantics. Transcription between predicate logic and English. Proof techniques. Identity, functions, and definite descriptions. Introduction to concepts of metatheory.

The Topic

Predicate logic has its origins in the “syllogistic” logic of Aristotle in the fifth century BCE. Although it had been developed to a small extent, it remained severely limited until it was treated in a more general fashion by Peirce and Frege in the late nineteenth century. The resulting “predicate” or “quantifier” logic was studied thoroughly in the early twentieth century. The version of predicate logic to be studied in this course is based on that work.

Initially, predicate logic was developed as a symbolism for mathematical demonstrations. However it was soon realized that it can be applied to natural language as well as the language of mathematics. Bertrand Russell’s “On Denoting” of 1905 was a dramatic example of the use to which predicate logic could be put. At present, predicate logic is used throughout formalized mathematical presentations, as well as in many presentations of philosophical theories and analyses.

The focus of this course is threefold. Students will learn the symbolism of predicate logic and how to map that symbolism onto natural language (English, in our case). Students will then learn how to formalize arguments in English and prove the validity of valid arguments in English. Finally, students will learn how to use a formal approach to interpreting predicate logic, which will allow them, among other things, to prove the invalidity of invalid arguments in English.

The Icon

The icon used for these pages is the standard notation for the “existential quantifier,” which is one of the main objects of study in this course. The original notation was a rotated capital E, introduced in the late nineteenth century by Giuseppe Peano. The notation was taken over and popularized by Bertrand Russell. In the 1910 first volume of Principia Mathematica, Russell and Whitehead wrote:

Thus corresponding to any propositional function φx̂, there is a range, or collection, of values, consisting of all the propositions (true or false) which can be obtained by giving every possible determination to x in in φx. A value of x for which φx is true will be said to “satisfy” φx̂. Now in respect to the truth or falsehood of propositions of this range three important cases must be noted and symbolised. These cases are given by three propositions of which at least one must be true. Either (1) all propositions of the range are true, or (2) some propositions of the range are true, or (3) no proposition of the range is true. The statement (1) is symbolised by “(xx,” and (2) is symbolised by “(∃x).φx.” No definition is given of these two symbols, which accordingly embody two new primitive ideas in our system. The symbol “(x).φx” may be read as “φx always,” or “φx is always true,” or “φx is true for all possible values of x.” The symbol “(∃x).φx” may be read “there exists an x for which φx is true” or “there exists an x satisfying φx̂,” and thus conforms to the natural form of the expression of thought. (pp. 15-16)