Combining Texts
Ideas for
'Logical Properties', 'A Tour through Mathematical Logic' and 'Review of Husserl's 'Phil of Arithmetic''
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17 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 / 4. Identity in Logic
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6051
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In 'x is F and x is G' we must assume the identity of x in the two statements [McGinn]
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6055
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Both non-contradiction and excluded middle need identity in their formulation [McGinn]
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6059
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Identity is unitary, indefinable, fundamental and a genuine relation [McGinn]
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5. Theory of Logic / G. Quantification / 1. Quantification
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6042
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The quantifier is overrated as an analytical tool [McGinn]
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6067
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Existential quantifiers just express the quantity of things, leaving existence to the predicate 'exists' [McGinn]
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5. Theory of Logic / G. Quantification / 3. Objectual Quantification
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6069
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'Partial quantifier' would be a better name than 'existential quantifier', as no existence would be implied [McGinn]
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5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
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6068
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We need an Intentional Quantifier ("some of the things we talk about.."), so existence goes into the proposition [McGinn]
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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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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13532
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Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
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13531
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Model theory reveals the structures of mathematics [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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