Combining Texts
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'The Establishment of Scientific Semantics', 'A Tour through Mathematical Logic' and 'Mind, Brain and the Quantum'
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16 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 / A. Overview of Logic / 6. Classical Logic
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13337
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A language: primitive terms, then definition rules, then sentences, then axioms, and finally inference rules [Tarski]
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5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
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13335
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Semantics is the concepts of connections of language to reality, such as denotation, definition and truth [Tarski]
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13336
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A language containing its own semantics is inconsistent - but we can use a second language [Tarski]
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5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
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13339
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A sentence is satisfied when we can assert the sentence when the variables are assigned [Tarski]
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13340
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Satisfaction is the easiest semantical concept to define, and the others will reduce to it [Tarski]
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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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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13519
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Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
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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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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 / 2. Consistency
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13341
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Using the definition of truth, we can prove theories consistent within sound logics [Tarski]
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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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