64 ideas
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| 9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
| 9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
| 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| 9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
| 9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| 9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
| 9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
| 9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
| 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
| 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
| 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| 9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
| 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
| 9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
| 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| 9240 | Love creates a necessity concerning what to care about [Frankfurt] |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| 9264 | Persons are distinguished by a capacity for second-order desires [Frankfurt] |
| 9266 | A person essentially has second-order volitions, and not just second-order desires [Frankfurt] |
| 9267 | Free will is the capacity to choose what sort of will you have [Frankfurt] |
| 9265 | The will is the effective desire which actually leads to an action [Frankfurt] |
| 20015 | Freedom of action needs the agent to identify with their reason for acting [Frankfurt, by Wilson/Schpall] |
| 9228 | Ranking order of desires reveals nothing, because none of them may be considered important [Frankfurt] |
| 9270 | A 'wanton' is not a person, because they lack second-order volitions [Frankfurt] |
| 9269 | A person may be morally responsible without free will [Frankfurt] |
| 9238 | Morality isn't based on reason; moral indignation is quite unlike disapproval of irrationality [Frankfurt] |
| 9232 | It is by caring about things that we infuse the world with importance [Frankfurt] |
| 9234 | If you don't care about at least one thing, you can't find reasons to care about anything [Frankfurt] |
| 9229 | What is worthwhile for its own sake alone may be worth very little [Frankfurt] |
| 9233 | Our criteria for evaluating how to live offer an answer to the problem [Frankfurt] |
| 9235 | Rather than loving things because we value them, I think we value things because we love them [Frankfurt] |
| 9236 | Love can be cool, and it may not involve liking its object [Frankfurt] |
| 9237 | The paradigm case of pure love is not romantic, but that between parents and infants [Frankfurt] |
| 9239 | I value my children for their sake, but I also value my love for them for its own sake [Frankfurt] |
| 9227 | We might not choose a very moral life, if the character or constitution was deficient [Frankfurt] |
| 9230 | People want to fulfill their desires, but also for their desires to be sustained [Frankfurt] |
| 9241 | Loving oneself is not a failing, but is essential to a successful life [Frankfurt] |
| 9300 | Boredom is serious, not just uncomfortable; it threatens our psychic survival [Frankfurt] |
| 9231 | Freedom needs autonomy (rather than causal independence) - embracing our own desires and choices [Frankfurt] |
| 21236 | Instead of gravitational force, we now have a pervasive gravitational field [Farmelo] |
| 21235 | The Schrödinger waves are just the maths of transforming energy values to positions [Farmelo] |
| 21234 | Experiments show that fundamental particles of one type are identical [Farmelo] |