Ideas of Herbert B. Enderton, by Theme

[American, fl. 1972, At the University of California, Los Angeles.]

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4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Until the 1960s the only semantics was truth-tables
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
'dom R' indicates the 'domain' of objects having a relation
'ran R' indicates the 'range' of objects being related to
'fld R' indicates the 'field' of all objects in the relation
'F(x)' is the unique value which F assumes for a value of x
We write F:A→B to indicate that A maps into B (the output of F on A is in B)
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
A 'linear or total ordering' must be transitive and satisfy trichotomy
The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}
The 'powerset' of a set is all the subsets of a given set
A function 'maps A into B' if the relating things are set A, and the things related to are all in B
A 'function' is a relation in which each object is related to just one other object
A relation is 'transitive' if it can be carried over from two ordered pairs to a third
A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
A relation is 'reflexive' on a set if every member bears the relation to itself
A function 'maps A onto B' if the relating things are set A, and the things related to are set B
A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects
A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second
Two sets are 'disjoint' iff their intersection is empty
A 'domain' of a relation is the set of members of ordered pairs in the relation
A 'relation' is a set of ordered pairs
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ
The empty set may look pointless, but many sets can be constructed from it
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Set
The singleton is defined using the pairing axiom (as {x,x})
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
We 'partition' a set into distinct subsets, according to each relation on its objects
An 'equivalence relation' is a reflexive, symmetric and transitive binary relation
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Fraenkel added Replacement, to give a theory of ordinal numbers
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
We can only define functions if Choice tells us which items are involved
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Inference not from content, but from the fact that it was said, is 'conversational implicature'
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
Validity is either semantic (what preserves truth), or proof-theoretic (following procedures)
5. Theory of Logic / I. Semantics of Logic / 4. Tautological Truth
A logical truth or tautology is a logical consequence of the empty set
5. Theory of Logic / I. Semantics of Logic / 5. Satisfaction
A truth assignment to the components of a wff 'satisfy' it if the wff is then True
5. Theory of Logic / K. Features of Logics / 3. Soundness
A proof theory is 'sound' if its valid inferences entail semantic validity
5. Theory of Logic / K. Features of Logics / 4. Completeness
A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
5. Theory of Logic / K. Features of Logics / 6. Compactness
Proof in finite subsets is sufficient for proof in an infinite set
5. Theory of Logic / K. Features of Logics / 7. Decidability
Expressions are 'decidable' if inclusion in them (or not) can be proved
5. Theory of Logic / K. Features of Logics / 8. Enumerability
For a reasonable language, the set of valid wff's can always be enumerated
10. Modality / B. Possibility / 8. Conditionals / f. Pragmatics of conditionals
Sentences with 'if' are only conditionals if they can read as A-implies-B