Combining Texts

Ideas for 'fragments/reports', 'On the Plurality of Worlds' and 'Remarks on axiomatised set theory'

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4 ideas

5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens

8078	Modus ponens is one of five inference rules identified by the Stoics [Chrysippus, by Devlin]
	Full Idea: Modus ponens is just one of the five different inference rules identified by the Stoics.
	From: report of Chrysippus (fragments/reports [c.240 BCE]) by Keith Devlin - Goodbye Descartes Ch.2
	A reaction: Modus ponens strikes me as being more like a definition of implication than a 'rule'. Implication is what gets you from one truth to another. All the implications of a truth must also be true.

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle

6023	Every proposition is either true or false [Chrysippus, by Cicero]
	Full Idea: We hold fast to the position, defended by Chrysippus, that every proposition is either true or false.
	From: report of Chrysippus (fragments/reports [c.240 BCE]) by M. Tullius Cicero - On Fate ('De fato') 38
	A reaction: I am intrigued to know exactly how you defend this claim. It may depend what you mean by a proposition. A badly expressed proposition may have indeterminate truth, quite apart from the vague, the undecidable etc.

5. Theory of Logic / G. Quantification / 6. Plural Quantification

15731	Quantification sometimes commits to 'sets', but sometimes just to pluralities (or 'classes') [Lewis]
	Full Idea: I consider some apparent quantification over sets or classes of whatnots to carry genuine ontological commitment to 'sets' of them, but sometimes it is innocent plural quantification committed only to whatnots, for which I use 'class'.
	From: David Lewis (On the Plurality of Worlds [1986], 1.5 n37)
	A reaction: How do you tell whether you are committed to a set or not? Can I claim an innocent plurality each time, while you accuse me of a guilty set? Can I firmly commit to a set, to be told that I can never manage more than a plurality?

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

17878	If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem]
	Full Idea: Löwenheim's theorem reads as follows: If a first-order proposition is satisfied in any domain at all, it is already satisfied in a denumerably infinite domain.
	From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.293)