Combining Texts

All the ideas for 'works', 'The Laws of Thought' and 'Introduction to Zermelo's 1930 paper'

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9 ideas

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic

8083	Boole applied normal algebra to logic, aiming at an algebra of thought [Boole, by Devlin]
	Full Idea: Boole proposed to use the entire apparatus of a school algebra class, with operations such as addition and multiplication, methods to solve equations, and the like, to produce an algebra of thought.
	From: report of George Boole (The Laws of Thought [1854]) by Keith Devlin - Goodbye Descartes Ch.3
	A reaction: The Stoics didn’t use any algebraic notation for their study of propositions, so Boole's idea launched full blown propositional logic, and the rest of modern logic followed. Nice one.

7727	Boole's notation can represent syllogisms and propositional arguments, but not both at once [Boole, by Weiner]
	Full Idea: Boole introduced a new symbolic notation in which it was possible to represent both syllogisms and propositional arguments, ...but not both at once.
	From: report of George Boole (The Laws of Thought [1854], Ch.3) by Joan Weiner - Frege
	A reaction: How important is the development of symbolic notations for the advancement of civilisations? Is there a perfect notation, as used in logical heaven?

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

17833	The first-order ZF axiomatisation is highly non-categorical [Hallett,M]
	Full Idea: The first-order Sermelo-Fraenkel axiomatisation is highly non-categorical.
	From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1213)

17834	Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal [Hallett,M]
	Full Idea: The non-categoricity of the axioms which Zermelo demonstrates reveals an incompleteness of a sort, ....for this seems to show that there will always be a set (indeed, an unending sequence) that the basic axioms are incapable of revealing to be sets.
	From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1215)
	A reaction: Hallett says the incompleteness concerning Zermelo was the (transfinitely) indefinite iterability of the power set operation (which is what drives the 'iterative conception' of sets).

4. Formal Logic / F. Set Theory ST / 7. Natural Sets

17837	Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M]
	Full Idea: Unlike earlier writers (such as Fraenkel), Zermelo clearly allows that there might be ur-elements (that is, objects other than the empty set, which have no members). Indeed he sees in this the possibility of widespread application of set-theory.
	From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217)

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

8686	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.

5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof

22277	Boole's method was axiomatic, achieving economy, plus multiple interpretations [Boole, by Potter]
	Full Idea: Boole's work was an early example of the axiomatic method, whereby intellectual economy is achieved by studying a set of axioms in which the primitive terms have multiple interpretations.
	From: report of George Boole (The Laws of Thought [1854]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 02 'Boole'
	A reaction: Unclear about this. I suppose the axioms are just syntactic, and a range of semantic interpretations can be applied. Are De Morgan's Laws interpretations, or implications of the syntactic axioms? The latter, I think.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis

17836	The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M]
	Full Idea: In 1938, Gödel showed that ZF plus the General Continuum Hypothesis is consistent if ZF is. Cohen showed that ZF and not-GCH is also consistent if ZF is, which finally shows that neither GCH nor ¬GCH can be proved from ZF itself.
	From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217)

22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature

7667	There are two sides to men - the pleasantly social, and the violent and creative [Diderot, by Berlin]
	Full Idea: Diderot is among the first to preach that there are two men: the artificial man, who belongs in society and seeks to please, and the violent, bold, criminal instinct of a man who wishes to break out (and, if controlled, is responsible for works of genius.
	From: report of Denis Diderot (works [1769], Ch.3) by Isaiah Berlin - The Roots of Romanticism
	A reaction: This has an obvious ancestor in Plato's picture (esp. in 'Phaedrus') of the two conflicting sides to the psuché, which seem to be reason and emotion. In Diderot, though, the suppressed man has virtues, which Plato would deny.