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5. Theory of Logic / A. Overview of Logic / 2. History of Logic

[origins of the various systems of formal logic]

15 ideas
Lull's combinatorial art would articulate all the basic concepts, then show how they combine [Lull, by Arthur,R]
     Full Idea: Lull proposed a combinatorial art. He wanted to reconcile Islam and Christianity by articulating the basic concepts that their belief systems held in common, and then inventing a device that would allow these concepts to be combined.
     From: report of Ramon (Ars Magna [1305]) by Richard T.W. Arthur - Leibniz 2 Intro
     A reaction: Leibniz's Universal Characteristic was an attempt at continuing Lull's project. Lull's plan rested on Aristotle's categories.
Boole made logic more mathematical, with algebra, quantifiers and probability [Boole, by Friend]
     Full Idea: Boole (followed by Frege) began to turn logic from a branch of philosophy into a branch of mathematics. He brought an algebraic approach to propositions, and introduced the notion of a quantifier and a type of probabilistic reasoning.
     From: report of George Boole (The Laws of Thought [1854], 3.2) by Michèle Friend - Introducing the Philosophy of Mathematics
     A reaction: The result was that logic not only became more mathematical, but also more specialised. We now have two types of philosopher, those steeped in mathematical logic and the rest. They don't always sing from the same songsheet.
In 1879 Frege developed second order logic [Frege, by Putnam]
     Full Idea: By 1879 Frege had discovered an algorithm, a mechanical proof procedure, that embraces what is today standard 'second order logic'.
     From: report of Gottlob Frege (Begriffsschrift [1879]) by Hilary Putnam - Reason, Truth and History Ch.5
     A reaction: Note that Frege did more than introduce quantifiers, and the logic of predicates.
We have no adequate logic at the moment, so mathematicians must create one [Veblen]
     Full Idea: Formal logic has to be taken over by mathematicians. The fact is that there does not exist an adequate logic at the present time, and unless the mathematicians create one, no one else is likely to do so.
     From: Oswald Veblen (Presidential Address of Am. Math. Soc [1924], 141), quoted by Stewart Shapiro - Philosophy of Mathematics
     A reaction: This remark was made well after Frege, but before the advent of Gödel and Tarski. That implies that he was really thinking of meta-logic.
Gentzen introduced a natural deduction calculus (NK) in 1934 [Gentzen, by Read]
     Full Idea: Gentzen introduced a natural deduction calculus (NK) in 1934.
     From: report of Gerhard Gentzen (works [1938]) by Stephen Read - Thinking About Logic Ch.8
Before the late 19th century logic was trivialised by not dealing with relations [Putnam]
     Full Idea: It was essentially the failure to develop a logic of relations that trivialised the logic studied before the end of the nineteenth century.
     From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
     A reaction: De Morgan, Peirce and Frege were, I believe, the people who put this right.
Nowadays logic is seen as the science of extensions, not intensions [Scruton]
     Full Idea: Logicians have come increasingly to realise that logic is the science not of the intension, but of the extension of terms.
     From: Roger Scruton (Short History of Modern Philosophy [1981], Ch.4)
     A reaction: I take this to be because the notion of a 'set' is basic, which is defined strictly in terms of its members. This move is probably because we can be clear about extensions, but not intensions. Tidiness is no substitute for complex truth.
The mainstream of modern logic sees it as a branch of mathematics [Mayberry]
     Full Idea: In the mainstream tradition of modern logic, beginning with Boole, Peirce and Schröder, descending through Löwenheim and Skolem to reach maturity with Tarski and his school ...saw logic as a branch of mathematics.
     From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-1)
     A reaction: [The lesser tradition, of Frege and Russell, says mathematics is a branch of logic]. Mayberry says the Fregean tradition 'has almost died out'.
Golden ages: 1900-1960 for pure logic, and 1950-1985 for applied logic [Devlin]
     Full Idea: The period from 1900 to about 1960 could be described as the golden age of 'pure' logic, and 1950 to 1985 the golden age of 'applied' logic (e.g. applied to everyday reasoning, and to theories of language).
     From: Keith Devlin (Goodbye Descartes [1997], Ch. 4)
     A reaction: Why do we always find that we have just missed the Golden Age? However this supports the uneasy feeling that the golden age for all advances in human knowledge is just coming to an end. Biology, including the brain, is the last frontier.
Montague's intensional logic incorporated the notion of meaning [Devlin]
     Full Idea: Montague's intensional logic was the first really successful attempt to develop a mathematical framework that incorporates the notion of meaning.
     From: Keith Devlin (Goodbye Descartes [1997], Ch. 8)
     A reaction: Previous logics, led by Tarski, had flourished by sharply dividing meaning from syntax, and concentrating on the latter.
Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
     Full Idea: Skolem and Gödel were the main proponents of first-order languages. The higher-order language 'opposition' was championed by Zermelo, Hilbert, and Bernays.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2)
Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
     Full Idea: Bernays (1918) formulated and proved the completeness of propositional logic, the first precise solution as part of the Hilbert programme.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.1)
Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
     Full Idea: In 1910 Weyl observed that set theory seemed to presuppose natural numbers, and he regarded numbers as more fundamental than sets, as did Fraenkel. Dedekind had developed set theory independently, and used it to formulate numbers.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.2)
The view of logic as knowing a body of truths looks out-of-date [Beall/Restall]
     Full Idea: Through much of the 20th century the conception of logic was inherited from Frege and Russell, as knowledge of a body of logical truths, as arithmetic or geometry was a knowledge of truths. This is odd, and a historical anomaly.
     From: JC Beall / G Restall (Logical Pluralism [2006], 2.2)
     A reaction: Interesting. I have always taken this idea to be false. I presume logic has minimal subject matter and truths, and preferably none at all.
Was logic a branch of mathematics, or mathematics a branch of logic? [Engelbretsen]
     Full Idea: Nineteenth century logicians debated whether logic should be treated simply as a branch of mathematics, and mathematics could be applied to it, or whether mathematics is a branch of logic, with no mathematics used in formulating logic.
     From: George Engelbretsen (Trees, Terms and Truth [2005], 3)
     A reaction: He cites Boole, De Morgan and Peirce for the first view, and Frege and Russell (and their 'logicism') for the second. The logic for mathematics slowly emerged from doing it, long before it was formalised. Mathematics is the boss?