Combining Texts

Ideas for 'Parmenides', 'The Nature of Necessity' and 'Higher-Order Logic'

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11 ideas

5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic

10588

First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

10298

Some say that second-order logic is mathematics, not logic [Shapiro]

10299

If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]

5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence

10300

Logical consequence can be defined in terms of the logical terminology [Shapiro]

5. Theory of Logic / G. Quantification / 5. Second-Order Quantification

10290

Second-order variables also range over properties, sets, relations or functions [Shapiro]

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

10590

Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]

10296

The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]

10297

The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]

10292

Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]

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]

14150

Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato]