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

9707	'F(x)' is the unique value which F assumes for a value of x

9705	'fld R' indicates the 'field' of all objects in the relation

9704	'ran R' indicates the 'range' of objects being related to

9703	'dom R' indicates the 'domain' of objects having a relation

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' <x,y> 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

9701	A 'relation' is a set of ordered pairs

9708	A function 'maps A into B' if the relating things are set A, and the things related to are all in B

9709	A function 'maps A onto B' if the relating things are set A, and the things related to are set B

9711	A relation is 'reflexive' on a set if every member bears the relation to itself

9712	A relation is 'symmetric' on a set if every ordered pair has the relation in both directions

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

9713	A relation is 'transitive' if it can be carried over from two ordered pairs to a third

9706	A 'function' is a relation in which each object is related to just one other object

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) Sets

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