Ideas from 'Introduction to the Theory of Logic' by José L. Zalabardo [2000], by Theme Structure

[found in 'Introduction to the Theory of Logic' by Zalabardo,José L. [Westview 2000,0-8133-6602-x]].

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