. A general name, applied to a subject, for the body of principles governing reasoning in the subject. One can talk of anAXIOM SYSTEM for the propositional calculus, etc. Sometimes such systems are themselves called calculi.
The propositional calculus (also called the sentential calculus, calculus of unanalysed propositions, calculus of truth values, or calculus of truth functions) concerns truth FUNCTIONS of propositions, but with the restriction that the propositions are regarded as either the same as each other or completely different. Partial similarities like that between ‘All cats are black’ and ‘Some cats are black’ are ignored. Its theorems are the relevant TAUTOLOGIES. When the restriction is lifted and the structure of propositions is taken into account, we have the functional or predicate calculus, or the calculus of relations. When the predicates are limited to MONADIC predicates we have the monadic predicate calculus. The predicate calculus is called extended or second-order when predicates are quantified over (see QUANTIFICATION). When only INDIVIDUALS are quantified over, it is called restricted or first-order. There is also an extended propositional calculus, where propositions are quantified over.
The calculus of classes concerns classes and their members. It is structurally the same as the monadic predicate calculus, (‘x is red’ is interchangeable with ‘x belongs to the class of red things,—though RUSSELL’S PARADOX raises a difficulty for the view that every predicate defines a class.) It is the elementary nucleus of set theory, which treats problems arising out of the calculus of classes and goes beyond it by treating, for example, classes whose members are ordered, and problems specific to infinite classes. The relations between set theory and logic are important in connexion with logicism (see philosophy of MATHEMATICS).
The calculus of individuals concerns the part/whole relationship, and is linked to MEREOLOGY.
The hedonic calculus, or calculus of pleasures, is the set of principles which would govern any system claiming that pleasures can be measured, added, and in general systematically compared. But whether such a calculus could be constructed is controversial.
N.Goodman, The Structure of Appearance, Harvard UP, 1951, chapter 2. (Calculus of individuals.)
D.Hilbert and W.Ackermann, Principles of Mathematical Logic, 1928; 2nd edn 1938, trans. Chelsea, NY, 1950. (A standard account of the main logical calculi. Elementary introductions to symbolic logic, covering similar ground, are legion.)
D.C.Makinson, Topics in Modern Logic, Methuen, 1973, chapter 5. (Set theory and logic. Cf. also Introduction to P.Benacerraf and H.Putnam (eds), Philosophy of Mathematics, Cambridge UP, 1964).
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