Combining Philosophers
Ideas for Ian Dunt, Georg W.F.Hegel and Robert S. Wolf
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15 ideas
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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13534
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In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
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13535
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First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
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5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
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21595
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Excluded middle is the maxim of definite understanding, but just produces contradictions [Hegel]
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5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / c. not
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21777
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Negation of negation doubles back into a self-relationship [Hegel, by Houlgate]
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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13531
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Model theory reveals the structures of mathematics [Wolf,RS]
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13532
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Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
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13519
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Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
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13533
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First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
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5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
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13537
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An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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13539
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The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
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5. Theory of Logic / K. Features of Logics / 4. Completeness
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13538
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If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
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5. Theory of Logic / K. Features of Logics / 10. Monotonicity
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13525
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Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
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5. Theory of Logic / L. Paradox / 3. Antinomies
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15628
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The idea that contradiction is essential to rational understanding is a key modern idea [Hegel]
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15629
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Tenderness for the world solves the antinomies; contradiction is in our reason, not in the essence of the world [Hegel]
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15630
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Antinomies are not just in four objects, but in all objects, all representations, all objects and all ideas [Hegel]
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