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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| History of logic | 9/13 | https://en.wikipedia.org/wiki/History_of_logic | reference | science, encyclopedia | 2026-05-05T03:59:53.898187+00:00 | kb-cron |
The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from Quantity in 1864, where he suggested a symbol to signify exclusive or, which allowed Boole's system to be greatly simplified. This was usefully exploited by Schröder when he set out theorems in parallel columns in his Vorlesungen (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive binary operation, "neither ... nor ..." and equally well "not both ... and ...", however, like many of Peirce's innovations, this remained unknown or unnoticed until Sheffer rediscovered it in 1913. Boole's early work also lacks the idea of the logical sum which originates in Peirce (1867), Schröder (1877) and Jevons (1890), and the concept of inclusion, first suggested by Gergonne (1816) and clearly articulated by Peirce (1870).
The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most ambitious was Schröder's monumental Vorlesungen über die Algebra der Logik ("Lectures on the Algebra of Logic", vol iii 1895), although the original idea was again anticipated by Peirce. Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought. Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought. According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat—from assessing validity to solving equations—and 3) expanding the range of applications it could handle—e.g. from propositions having only two terms to those having arbitrarily many. More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced the four propositional forms of Aristotelian logic to formulas in the form of equations—by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle".
=== Logicist period ===
After Boole, the next great advances were made by the German mathematician Gottlob Frege. Frege's objective was the program of Logicism, i.e. demonstrating that arithmetic is identical with logic. Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or Begriffsschrift is important. Frege also tried to show that the concept of number can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first writer to suggest this. In his pioneering work Die Grundlagen der Arithmetik (The Foundations of Arithmetic), sections 15–17, he acknowledges the efforts of Leibniz, J. S. Mill as well as Jevons, citing the latter's claim that "algebra is a highly developed logic, and number but logical discrimination." Frege's first work, the Begriffsschrift ("concept script") is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of inference (modus ponens and substitution), and six axioms. Frege referred to the "completeness" of this system, but was unable to prove this. The most significant innovation, however, was his explanation of the quantifier in terms of mathematical functions. Traditional logic regards the sentence "Caesar is a man" as of fundamentally the same form as "all men are mortal." Sentences with a proper name subject were regarded as universal in character, interpretable as "every Caesar is a man". At the outset Frege abandons the traditional "concepts subject and predicate", replacing them with argument and function respectively, which he believes "will stand the test of time". He goes on to say that it is "easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention". Frege argued that the quantifier expression "all men" does not have the same logical or semantic form as "all men", and that the universal proposition "every A is B" is a complex proposition involving two functions, namely ' – is A' and ' – is B' such that whatever satisfies the first, also satisfies the second. In modern notation, this would be expressed as
∀
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{\displaystyle \forall \;x{\big (}A(x)\rightarrow B(x){\big )}}