Combining Texts

Ideas for 'Thinking About Mathematics', 'Towards a Universal Characteristic' and 'Prior Analytics'

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8 ideas

5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
19370 'Blind thought' is reasoning without recognition of the ingredients of the reasoning [Leibniz, by Arthur,R]
     Full Idea: Leibniz invented the concept of 'blind thought' - reasoning by a manipulation of characters without being able to recognise what each character stands for.
     From: report of Gottfried Leibniz (Towards a Universal Characteristic [1677]) by Richard T.W. Arthur - Leibniz
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
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
     Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements.
     From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
     A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle.
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 / 5. Incompleteness
19391 We can assign a characteristic number to every single object [Leibniz]
     Full Idea: The true principle is that we can assign to every object its determined characteristic number.
     From: Gottfried Leibniz (Towards a Universal Characteristic [1677], p.18)
     A reaction: I add this as a predecessor of Gödel numbering. It is part of Leibniz's huge plan for a Universal Characteristic, to map reality numerically, and then calculate the truths about it. Gödel seems to allow metaphysics to be done mathematically.