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Ideas for 'Mahaprajnaparamitashastra', 'Prior Analytics' and 'Philosophy of Mathematics'

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8 ideas

5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
11148 Deduction is when we suppose one thing, and another necessarily follows [Aristotle]
     Full Idea: A deduction is a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so.
     From: Aristotle (Prior Analytics [c.328 BCE], 24b18)
     A reaction: Notice that it is modal ('suppose', rather than 'know'), that necessity is involved, which is presumably metaphysical necessity, and that there are assumptions about what would be true, and not just what follows from what.
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
9605 If a proposition is false, then its negation is true [Brown,JR]
     Full Idea: The law of excluded middle says if a proposition is false, then its negation is true
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1)
     A reaction: Surely that is the best statement of the law? How do you write that down? ¬(P)→¬P? No, because it is a semantic claim, not a syntactic claim, so a truth table captures it. Semantic claims are bigger than syntactic claims.
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
18896 Aristotle places terms at opposite ends, joined by a quantified copula [Aristotle, by Sommers]
     Full Idea: Aristotle often preferred to formulate predications by placing the terms at opposite ends of the sentence and joining them by predicating expressions like 'belongs-to-some' or 'belongs-to-every'.
     From: report of Aristotle (Prior Analytics [c.328 BCE]) by Fred Sommers - Intellectual Autobiography 'Conceptions'
     A reaction: This is Sommers's picture of Aristotle, which led Sommers to develop his modern Term Logic.
5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
3300 Aristotle's logic is based on the subject/predicate distinction, which leads him to substances and properties [Aristotle, by Benardete,JA]
     Full Idea: Basic to Aristotle's logic is the grammatical distinction between subject and predicate, which he glosses in terms of the contrast between a substance and its properties.
     From: report of Aristotle (Prior Analytics [c.328 BCE]) by José A. Benardete - Metaphysics: the logical approach Intro
     A reaction: The introduction of quantifiers and 'logical form' can't disguise the fact that we still talk about (and with) objects and predicates, because no one can think of any other way to talk.
5. Theory of Logic / G. Quantification / 1. Quantification
11149 Affirming/denying sentences are universal, particular, or indeterminate [Aristotle]
     Full Idea: Affirming/denying sentences are universal, particular, or indeterminate. Belonging 'to every/to none' is universal; belonging 'to some/not to some/not to every' is particular; belonging or not belonging (without universal/particular) is indeterminate.
     From: Aristotle (Prior Analytics [c.328 BCE], 24a16)
5. Theory of Logic / G. Quantification / 3. Objectual Quantification
8079 Aristotelian logic has two quantifiers of the subject ('all' and 'some') [Aristotle, by Devlin]
     Full Idea: Aristotelian logic has two quantifiers of the subject ('all' and 'some'), and two ways to combine the subject with the predicate ('have', and 'have not'), giving four propositions: all-s-have-p, all-s-have-not-p, some-s-have-p, and some-s-have-not-p.
     From: report of Aristotle (Prior Analytics [c.328 BCE]) by Keith Devlin - Goodbye Descartes Ch.2
     A reaction: Frege seems to have switched from 'some' to 'at-least-one'. Since then other quantifiers have been proposed. See, for example, Ideas 7806 and 6068.
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
9649 Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR]
     Full Idea: The three views one could adopt concerning axioms are that they are self-evident truths, or that they are arbitrary stipulations, or that they are fallible attempts to describe how things are.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10)
     A reaction: Presumably modern platonists like the third version, with others choosing the second, and hardly anyone now having the confidence to embrace the first.
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
9638 Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR]
     Full Idea: Berry's Paradox refers to 'the least integer not namable in fewer than nineteen syllables' - a paradox because it has just been named in eighteen syllables.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5)
     A reaction: Apparently George Boolos used this quirky idea as a basis for a new and more streamlined proof of Gödel's Theorem. Don't tell me you don't find that impressive.