A Dictionary of Philosophy, Third Edition
. (This entry is confined to logic and mathematics, except for its bibliography.) Some expressions have either numerical values or TRUTH VALUES which we can calculate once we give values to the VARIABLES in the expression. The rest of the expression is then called a function of the variables. Thus 3x+7 contains a variable, x, and a function 3()+7. But ‘function’ is often applied to the whole expression, including the variables. The function is then called a function of x. Its value in either case depends on he value given to x. Again if p and q are propositions, p and q (but not p because q is a truth function of p and of q. We can know the truth value of p and q (but not that of p because q) once we know the truth values of p and of q.
A value assigned to a variable in a function is called an argument of the function, and contributes to the value of the function. x+7 has the value 19 for the argument 4.
Frege appealed to functions in analysing CONCEPTS and thereby predicates:…is red can be thought of as a function, because we can assign a truth value to x is red by assigning a value to x (i.e. replacing x by the name of something; x is red becomes true if we replace x by ‘blood’), or alternatively by quantifying over x (see QUANTIFICATION).
x is red can be called a propositional, statemental (rare) or sentential function, according as blood is red is regarded as a proposition, statement or sentence. Sentential functions are often called open sentences. The term closed sentential function is occasionally used of ordinary sentences.
J.Bigelow and R.Pargetter, ‘Functions’, Journal of Philosophy, 1987. (Discusses and defends one of four biological senses of the term.)
R.Carnap, Introduction to Semantics, Harvard UP, 1942, § 37. (Ambiguities, here ignored, of ‘function’ and ‘propositional function’.)
R.Cummins, ‘Functional analysis’, Journal of Philosophy, 1975. (Criticizes two basic assumptions of previous analyses, and then offers his own. ‘To ascribe a function to something is to ascribe a capacity to it which is singled out by its role in an analysis of some capacity of a containing system’ (p. 765).)
P.T.Geach and M.Black, Translations from the Philosophical Writings of Gottlob Frege, Blackwell, 1952. (Contains Frege’s ‘Function and Concept’ (cf. CONCEPT). Frege’s symbolism is awkward and outdated. For sentences as referring to truth-values see his ‘Sense and reference’, ibid.)
P.Kitcher, ‘Function and design’, Midwest Studies in Philosophy, vol. 18, 1993. (Claims function implies design, but not intention or designer.)
R.Sorabji, ‘Function’, Philosophical Quarterly, 1964. (Non-logical senses.)
L.Wright, ‘Functions’, Philosophical Review, 1973. (Emphasizes their explanatory role, and offers an analysis which covers both conscious and natural functions, after discussing other views. For fuller development of appeal to natural selection which he makes see R.G.Millikan, Language, Thought, and Other Biological Categories, MIT Press, esp. chapter 1.)
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