### 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
 9724 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
 9703 'dom R' indicates the 'domain' of objects having a relation
 9704 'ran R' indicates the 'range' of objects being related to
 9705 'fld R' indicates the 'field' of all objects in the relation
 9707 'F(x)' is the unique value which F assumes for a value of x
 9710 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
 13201 ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
 13204 The 'ordered pair' is defined to be {{x}, {x,y}}
 13206 A 'linear or total ordering' must be transitive and satisfy trichotomy
 9699 The 'powerset' of a set is all the subsets of a given set
 9700 Two sets are 'disjoint' iff their intersection is empty
 9702 A 'domain' of a relation is the set of members of ordered pairs in the relation
 9714 A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects
 9717 A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second
 9708 A function 'maps A into B' if the relating things are set A, and the things related to are all in B
 9706 A 'function' is a relation in which each object is related to just one other object
 9701 A 'relation' is a set of ordered pairs
 9713 A relation is 'transitive' if it can be carried over from two ordered pairs to a third
 9712 A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
 9711 A relation is 'reflexive' on a set if every member bears the relation to itself
 9709 A function 'maps A onto B' if the relating things are set A, and the things related to are set B
###### 4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
 13200 Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ
 13199 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
 13203 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
 9716 We 'partition' a set into distinct subsets, according to each relation on its objects
 9715 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
 13202 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
 13205 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
 9722 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
 9718 Validity is either semantic (what preserves truth), or proof-theoretic (following procedures)
###### 5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
 9721 A logical truth or tautology is a logical consequence of the empty set
###### 5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
 9994 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
 9719 A proof theory is 'sound' if its valid inferences entail semantic validity
###### 5. Theory of Logic / K. Features of Logics / 4. Completeness
 9720 A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
###### 5. Theory of Logic / K. Features of Logics / 6. Compactness
 9995 Proof in finite subsets is sufficient for proof in an infinite set
###### 5. Theory of Logic / K. Features of Logics / 7. Decidability
 9996 Expressions are 'decidable' if inclusion in them (or not) can be proved
###### 5. Theory of Logic / K. Features of Logics / 8. Enumerability
 9997 For a reasonable language, the set of valid wff's can always be enumerated
###### 10. Modality / B. Possibility / 8. Conditionals / f. Pragmatics of conditionals
 9723 Sentences with 'if' are only conditionals if they can read as A-implies-B