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Ideas for 'Parmenides', 'Philosophical Logic: Intro to Advanced Topics' and 'On the Very Idea of a Third Dogma'

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6 ideas

5. Theory of Logic / A. Overview of Logic / 4. Pure Logic

13850	In modern logic all formal validity can be characterised syntactically [Engelbretsen/Sayward]
	Full Idea: One of the key ideas of modern formal logic is that all formally valid inferences can be specified in strictly syntactic terms.
	From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], Ch.2)

5. Theory of Logic / A. Overview of Logic / 6. Classical Logic

13849	Classical logic rests on truth and models, where constructivist logic rests on defence and refutation [Engelbretsen/Sayward]
	Full Idea: Classical logic rests on the concepts of truth and falsity (and usually makes use of a semantic theory based on models), whereas constructivist logic accounts for inference in terms of defense and refutation.
	From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], Intro)
	A reaction: My instincts go with the classical view, which is that inferences do not depend on the human capacity to defend them, but sit there awaiting revelation. My view isn't platonist, because I take the inferences to be rooted in the physical world.

5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic

13851	Unlike most other signs, = cannot be eliminated [Engelbretsen/Sayward]
	Full Idea: Unlike ∨, →, ↔, and ∀, the sign = is not eliminable from a logic.
	From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], Ch.3)

5. Theory of Logic / K. Features of Logics / 5. Incompleteness

13852	Axioms are ω-incomplete if the instances are all derivable, but the universal quantification isn't [Engelbretsen/Sayward]
	Full Idea: A set of axioms is said to be ω-incomplete if, for some universal quantification, each of its instances is derivable from those axioms but the quantification is not thus derivable.
	From: Engelbretsen,G/Sayward,C (Philosophical Logic: Intro to Advanced Topics [2011], 7)

5. Theory of Logic / L. Paradox / 3. Antinomies

13986	Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle]
	Full Idea: Plato (in 'Parmenides') shows that the theory that 'Eide' are substances, and Kant that space and time are substances, and Bradley that relations are substances, all lead to aninomies.
	From: report of Plato (Parmenides [c.364 BCE]) by Gilbert Ryle - Are there propositions? 'Objections'

14150	Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato]
	Full Idea: Plato's 'Parmenides' is perhaps the best collection of antinomies ever made.
	From: comment on Plato (Parmenides [c.364 BCE]) by Bertrand Russell - The Principles of Mathematics §337