Combining Texts

Ideas for 'Defending the Axioms', 'Democracy in America (abr Renshaw)' and 'Goodbye Descartes'

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4 ideas

4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic

8081	'No councillors are bankers' and 'All bankers are athletes' implies 'Some athletes are not councillors' [Devlin]
	Full Idea: Most people find it hard to find any conclusion that fits the following premises: 'No councillors are bankers', and 'All bankers are athletes'. There is a valid conclusion ('Some athletes are not councillors') but it takes quite an effort to find it.
	From: Keith Devlin (Goodbye Descartes [1997], Ch. 2)
	A reaction: A nice illustration of the fact that syllogistic logic is by no means automatic and straightforward. There is a mechanical procedure, but a lot of intuition and common sense is also needed.

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

8085	Modern propositional inference replaces Aristotle's 19 syllogisms with modus ponens [Devlin]
	Full Idea: Where Aristotle had 19 different inference rules (his valid syllogisms), modern propositional logic carries out deductions using just one rule of inference: modus ponens.
	From: Keith Devlin (Goodbye Descartes [1997], Ch. 4)
	A reaction: At first glance it sounds as if Aristotle's guidelines might be more useful than the modern one, since he tells you something definite and what implies what, where modus ponens just seems to define the word 'implies'.

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL

8086	Predicate logic retains the axioms of propositional logic [Devlin]
	Full Idea: Since predicate logic merely extends propositional logic, all the axioms of propositional logic are axioms of predicate logic.
	From: Keith Devlin (Goodbye Descartes [1997], Ch. 4)
	A reaction: See Idea 7798 for the axioms.

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX

17610	The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
	Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original.
	From: Penelope Maddy (Defending the Axioms [2011], 1.3)
	A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure.