• Introduction
• Classical Logic

## 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half of the 20th century, when the traditional logic of syllogisms going back to Aristotle was finally subsumed and superseded, introductory textbooks have generally taught a common view, now known as classical (elementary or first-order) logic, as to which forms are valid.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.

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.

• Cohen, Morris Raphael, and Ernest Nagel. An Introduction to Logic and Scientific Method. New York: Harcourt, Brace, 1934.

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).

• Hilbert, David, and Wilhelm Ackermann. Grundzüge der theoretischen Logik. Berlin: Springer, 1928.

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.

• Jeffrey, Richard C. Formal Logic: Its Scope and Limits. New York: McGraw Hill, 1967.

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).

• Kneale, William, and Martha Kneale. The Development of Logic. Oxford: Clarendon, 1962.

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.

• Quine, W. V. O. Methods of Logic. New York: Holt, 1950.

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.

• Suppes, Patrick. Introduction to Logic. Princeton, NJ: Van Nostrand, 1957.

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.

• Tarski, Alfred. Introduction to Logic and to the Methodology of Deductive Sciences. Rev. ed. Translated by Olaf Helmer. Oxford: Oxford University Press, 1941.

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).