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 2000,081336602x]].
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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. FirstOrder Logic
10897

A firstorder 'sentence' is a formula with no free variables

5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence =
10893

Γ = φ for sentences if φ is true when all of Γ is true

10899

Γ = φ if φ is true when all of Γ is true, for all structures and interpretations

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 truthassignment

10900

Logically true sentences are true in all structures

5. Theory of Logic / I. Semantics of Logic / 5. Satisfaction
10894

A sentenceset is 'satisfiable' if at least one truthassignment 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
