# Formal logic

Formal logic is logic that deals with the form or logical structure of statements and propositions and the logical implications and relations that exist or come about because of those logical forms. In particular, formal logic is concerned with the forms that yield or guarantee valid inferences from a premise or premises to a conclusion. Formal logic is a subset of formal systems. Today formal logic is usually carried out in symbolic form, although this is not strictly necessary in order to have a formal logic. Formal logic can be distinguished from informal logic, which is logic outside of or apart from a formal logical system or theory.

## Types of Formal Logic

Formal logic encompasses predicate logic, truth-functional logic, sentential or propositional logic (the logic of sentences)—also known as the propositional calculus—quantification logic (the logic of statements containing the terms "all," "none" or "some," or surrogates for those), mathematical logic, and set theoretic logic (the logic of set theory).

## Topics and Issues

Among the topics covered in formal logic are: translation of statements from a natural language (such as English, Spanish, or Japanese) into formal logical language; logical equivalence, logical truth, contradictions and tautologies; validity and invalidity; truth-preservation of theorems; logical soundness; conditionals and their logic ("if___, then..." statements); truth tables; deductions, both natural deductions and formal deductions; well formed formulae (known as wffs); logical operators and their definitions and truth conditions (especially "and," "or," "not," and "if-then"); quantifications and quantification logic; identity and equality (the "=" sign), logical functions, and definite descriptions (a description that applies correctly to an individual person or object); axioms and axiomatic systems; axioms for mathematics; axioms for set theory; valid derivation rules, meaning principles or rules for correctly deriving statements from axioms or other assumptions in such a way that if those premises or axioms or assumptions are true, then what is derived form them is also necessarily true; existence within a logical system; variables; the theory of types (from Russell and Whitehead's Principia Mathematica); consistency and completeness of logical and other formal systems; elimination of unnecessary theorems and axioms; logical substitution and replacement of terms and statements; the laws of reflexivity (x=x), symmetry (if x=y, then y=x), and transitivity (if x=y and y=z, then x=z), the logic of relations, modal logic (use of the concepts of necessity, possibility, strict implication, and strict co-implication); tense logic ("always," "at some time," and similar operators), and logical paradoxes.

Among the most important contributors to formal logic have been Gottlob Frege, Bertrand Russell and Alfred North Whitehead, Alfred Tarski, Kurt Gödel, Alonzo Church, and Willard Van Orman Quine.

## References

All logic textbooks—and there are hundreds and possibly thousands of them today—except for those few dealing only with informal logic present formal logic at least to some extent.

• Church, Alonzo. Introduction to Mathematical Logic. Princeton, N.J.: Princeton University Press, 1996.
• Church, Alonzo. ed. from Mar. 1936 – Dec. 1939. The Journal of Symbolic Logic. Published in Menasha, Wis., Mar. 1936 – Mar. 1938; in Baltimore, June 1938 – Dec. 1939; in Providence, R.I. thereafter. Also available via the Internet Retrieved October 2, 2007.
• Frege, Gottlob. Begriffsschrift und andere Aufsätze. Hildesheim: G. Olms, 1964.
• Gödel, Kurt. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. translated by B. Meltzer, introduction by R.B. Braithwaite. New York: Dover Publications, 1992.
• Quine, Willard Van Orman. Elementary Logic, rev. ed. Cambridge: Harvard University Press, 1966.
• Quine, Willard Van Orman. Methods in Logic, rev. ed. New York: Holt, 1959.
• Quine, Willard Van Orman. Mathematical Logic, rev. ed., New York: Harper & Row, 1962.
• Quine, Willard Van Orman. Philosophy of Logic. Englewood Cliffs, N.J.: Prentice-Hall, 1970. ISBN 013663625X
• Quine, Willard Van Orman. Set Theory and Its Logic, rev. ed. Cambridge: Belknap Press of Harvard University Press, 1969.
• Quine, Willard Van Orman. The Ways of Paradox: And Other Essays. New York: Random House, 1966.
• Reese, William L. "Logic." pp. 418-423 in Dictionary of Philosophy and Religion, New and enlarged edition. Highlands, NJ: Humanities Press, 1996. ISBN 0-391-03865-6
• Tarski, Alfred. A Decision Method for Elementary Algebra and Geometry. Berkeley: University of California Press, 1951.
• Teller, Paul. A Modern Formal Logic Primer. Orig. pub. by Prentice Hall, 1989.
• Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols. Cambridge [Eng.]: The University Press, 1910-1913.