9724 | Until the 1960s the only semantics was truth-tables |

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 |

9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) |

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

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 |

9701 | A 'relation' is a set of ordered pairs |

9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation |

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

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 |

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 |

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

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 |

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 |

13203 | The singleton is defined using the pairing axiom (as {x,x}) |

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 |

13202 | Fraenkel added Replacement, to give a theory of ordinal numbers |

13205 | We can only define functions if Choice tells us which items are involved |

9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' |

9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) |

9721 | A logical truth or tautology is a logical consequence of the empty set |

9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True |

9719 | A proof theory is 'sound' if its valid inferences entail semantic validity |

9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity |

9995 | Proof in finite subsets is sufficient for proof in an infinite set |

9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved |

9997 | For a reasonable language, the set of valid wff's can always be enumerated |

9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B |