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
'fld R' indicates the 'field' of all objects in the relation
'ran R' indicates the 'range' of objects being related to
'dom R' indicates the 'domain' of objects having a 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
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
A 'linear or total ordering' must be transitive and satisfy trichotomy
The 'powerset' of a set is all the subsets of a given set
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}
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
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 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 'function' is a relation in which each object is related to just one other object
A function 'maps A into B' if the relating things are set A, and the things related to are all in B
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
An 'equivalence relation' is a reflexive, symmetric and transitive binary relation
We 'partition' a set into distinct subsets, according to each relation on its objects
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