Ideas of José L. Zalabardo, by Theme
[Spanish, b.1964, Lecturer at the University of Birmingham, then University College, London.]
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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10888
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Sets can be defined by 'enumeration', or by 'abstraction' (based on a property)
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
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10889
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The 'Cartesian Product' of two sets relates them by pairing every element with every element
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10890
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A 'partial ordering' is reflexive, antisymmetric and transitive
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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10886
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Determinacy: an object is either in a set, or it isn't
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
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10887
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Specification: Determinate totals of objects always make a set
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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10897
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A first-order 'sentence' is a formula with no free variables
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5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
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10899
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Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations
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10893
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Γ |= φ for sentences if φ is true when all of Γ is true
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5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / b. Basic connectives
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10896
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Propositional logic just needs ¬, and one of ∧, ∨ and →
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5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
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10898
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The semantics shows how truth values depend on instantiations of properties and relations
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10902
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We can do semantics by looking at given propositions, or by building new ones
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5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
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10892
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We make a truth assignment to T and F, which may be true and false, but merely differ from one another
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5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
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10900
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Logically true sentences are true in all structures
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10895
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'Logically true' (|= φ) is true for every truth-assignment
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5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
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10894
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A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true
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10901
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Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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10903
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A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
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10891
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If a set is defined by induction, then proof by induction can be applied to it
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