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]].

Click on the Idea Number for the full details    |     back to texts     |     expand these ideas


4. Formal Logic / F. Set Theory ST / 1. Set Theory
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
The 'Cartesian Product' of two sets relates them by pairing every element with every element
A 'partial ordering' is reflexive, antisymmetric and transitive
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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
Specification: Determinate totals of objects always make a set
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
A first-order 'sentence' is a formula with no free variables
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Γ |= φ for sentences if φ is true when all of Γ is true
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
Propositional logic just needs ¬, and one of ∧, ∨ and →
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
The semantics shows how truth values depend on instantiations of properties and relations
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
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
Logically true sentences are true in all structures
'Logically true' (|= φ) is true for every truth-assignment
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true
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
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
If a set is defined by induction, then proof by induction can be applied to it