# Logic and Modern Philosophy of Mathematics

## The Revival Of Logic From The 1820s, And Its Algebraic Flourishing

Some major philosophers gave logic a high status in the seventeenth and early eighteenth centuries; in particular, Gottfried Wilhelm von Leibniz advocated it as a lingua characteristica, with an attendant calculus ratiocinator, while John Locke took it as a case of semeiotiké, the theory of signs. The subject did not flourish, however; thus, while René Descartes stressed rules for correct thinking and deduction, and Immanuel Kant viewed logic as analytic knowledge (with mathematics as synthetically a priori), neither philosopher gave logic itself much attention. Most figures in all disciplines were content to appeal to Aristotelian syllogistic logic, while mathematicians normally saw the Elements of Euclid as the apotheosis of rigor.

The revival in logic came from an unexpected quarter: the Elements of Logic (1826) by the English theologian Richard Whately. Within a decade four more editions of this treatise had appeared, and various contemporaries commented at length. The strength of the reception is puzzling, for Whately did not advocate any radical new stances; nevertheless, interest in logic increased considerably and some notable advances occurred, especially the quantification of the predicate (1827) of George Bentham, in which Aristotle's modes were greatly extended by admitting forms such as all As are some Bs.

These advances attracted the mathematician Augustus De Morgan, who from the 1840s symbolized in an algebraic manner the forms of syllogistic modes and the relationships between them. He also studied the logical form of the proofs in Euclid, and came away rather perplexed; for the flow of argument in Euclid involved far more than logical deductions. A more sweeping change to logic was effected by George Boole, who started out from a new algebraic principle. Take a "universe of discourse" (his phrase) 1 of, say, boxes, and divide it into the class x of black ones and the complementary class (1 x) of nonblack ones. Then lay down the basic laws obeyed by x, including the novelty that x together with x is the same as x. Make deductions by formulating algebraically relationships between classes x, y, … and using the laws and attendant theorems to find as a deduction the relationships between say, y and the other classes. Boole elaborated his method especially in The Laws of Thought (1854), where syllogisms occupied only the last chapter on logic. Finally, in 1860 De Morgan enriched syllogistics with a logic of (two-place) relations: at last the failure properly to handle, for example, John is older than Jack had been recognized.

The contributions of these two English algebraists were distinct: the American logician Charles S. Peirce conjoined them from the 1870s. One of his major insights, arrived at with his student O. H. Mitchell, was to individuate the "quantifier," both the existential there is an X such that … and the universal for all X …, and to stress the importance of watching quantifier order. Continuing the algebraic style, Peirce and Mitchell regarded these quantifiers as generalizations of logical disjunctions and conjunctions respectively. The interpretation of quantifiers was to be an enduring theme in the philosophy of logic. Their logic was extended, especially as an algebra, by the German Ernst Schröder in his vast Vorlesungen über die Algebra der Logik (1890–1905).

Many others wrote upon logic, especially in Britain; I note three figures. John Stuart Mill's System of Logic (1843, and many later editions), while not tied to syllogistics, relied much on it for an analysis of reasoning and deduction; mathematicization was absent, but in Mill's account of "induction," which we would regard as philosophy of science, he touched upon probability theory. Mill was broadly aligned to British empiricism: by great contrast, an influential "neo-Hegelianism" later became popular in academic circles. Francis H. Bradley's Principles of Logic (1883), its landmark, is an idealistic meditation upon the basic laws of logic (such as that of the excluded middle), judgments, and the reconciliation of thesis and antithesis in synthesis. Finally, Lewis Carroll produced some rather dull books on logic (1886; 1896); but in his Alice books (1865; 1871) he had brilliantly anticipated several concerns of those logicians of the next century who were to launch a new tradition that came to eclipse the algebraic logicians.