ALISM . logicism, the thesis that mathematics, or at least some significant portion thereof, is part of logic. Modifying Carnap’s suggestion (in ‘The Logicist Foundation for Mathematics,’ first published in Erkenntnis, 1931), this thesis is the conjunction of two theses: expressibility logicism: mathematical propositions are (or are alternative expressions of) purely logical propositions; and derivational logicism: the axioms and theorems of mathematics can be derived from pure logic.
Here is a motivating example from the arithmetic of the natural numbers. Let the cardinality-quantifiers be those expressible in the form ‘there are exactl. . . many xs such that’, which we abbreviate ¢(. . . x),Ü with ‘. . .’ replaced by an Arabic numeral. These quantifiers are expressible with the resources of first-order logic with identity; e.g. ‘(2x)Px’ is equivalent to ‘DxDy(x&y & Ez[Pz S (z%x 7 z%y)])’, the latter involving no numerals or other specifically mathematical vocabulary. Now 2 ! 3 % 5 is surely a mathematical truth. We might take it to express the following: if we take two things and then another three things we have five things, which is a validity of second-order logic involving no mathematical vocabulary:
EXEY ([(2x) Xx & (3x)Yx & ÝDx(Xx & Yx)] / (5x) (Xx 7 Yx)). Furthermore, this is provable in any formalized fragment of second-order logic that includes all of first-order logic with identity and secondorder ‘E’-introduction. But what counts as logic? As a derivation? As a derivation from pure logic? Such unclarities keep alive the issue of whether some version or modification of logicism is true. The ‘classical’ presentations of logicism were Frege’s Grundgesetze der Arithmetik and Russell and Whitehead’s Principia Mathematica. Frege took logic to be a formalized fragment of secondorder logic supplemented by an operator forming singular terms from ‘incomplete’ expressions, such a term standing for an extension of the ‘incomplete’ expression standing for a concept of level 1 (i.e. type 1). Axiom 5 of Grundgesetze served as a comprehension-axiom implying the existence of extensions for arbitrary Fregean concepts of level 1. In his famous letter of 1901 Russell showed that axiom to be inconsistent, thus derailing Frege’s original program. Russell and Whitehead took logic to be a formalized fragment of a ramified full finite-order (i.e. type w) logic, with higher-order variables ranging over appropriate propositional functions. The Principia and their other writings left the latter notion somewhat obscure. As a defense of expressibility logicism, Principia had this peculiarity: it postulated typical ambiguity where naive mathematics seemed unambiguous; e.g., each type had its own system of natural numbers two types up. As a defense of derivational logicism, Principia was flawed by virtue of its reliance on three axioms, a version of the Axiom of Choice, and the axioms of Reducibility and Infinity, whose truth was controversial. Reducibility could be avoided by eliminating the ramification of the logic (as suggested by Ramsey). But even then, even the arithmetic of the natural numbers required use of Infinity, which in effect asserted that there are infinitely many individuals (i.e., entities of type 0). Though Infinity was ‘purely logical,’ i.e., contained only logical expressions, in his Introduction to Mathematical Philosophy (p. 141) Russell admits that it ‘cannot be asserted by logic to be true.’ Russell then (pp. 194–95) forgets this: ‘If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point in the successive definitions and deductions of Principia Mathematica they consider that logic ends and mathematics begins. It will then be obvious that any answer is arbitrary.’ The answer, ‘Section 120, in which Infinity is first assumed!,’ is not arbitrary. In Principia Russell and Whitehead say of Infinity that they ‘prefer to keep it as a hypothesis’ (Vol. 2, p. 203). Perhaps then they did not really take logicism to assert the above identity, but rather a correspondence: to each sentence f of mathematics there corresponds a conditional sentence of logic whose antecedent is the Axiom of Infinity and whose consequent is a purely logical reformulation of f.
In spite of the problems with the ‘classical’ versions of logicism, if we count so-called higherorder (at least second-order) logic as logic, and if we reformulate the thesis to read ‘Each area of mathematics is, or is part of, a logic’, logicism remains alive and well.
See also FREGE, GÖDEL’S INCOMPLETENESS THEOREMS , PHILOSOPHY OF MATHEMATICS , SET THEOR. H.T.H.