61 ideas
| 6575 | Philosophy may never find foundations, and may undermine our lives in the process [Fogelin] |
| 6585 | Rationality is threatened by fear of inconsistency, illusions of absolutes or relativism, and doubt [Fogelin] |
| 6557 | Humans may never be able to attain a world view which is both rich and consistent [Fogelin] |
| 6568 | A game can be played, despite having inconsistent rules [Fogelin] |
| 6560 | The law of noncontradiction is traditionally the most basic principle of rationality [Fogelin] |
| 6565 | The law of noncontradiction makes the distinction between asserting something and denying it [Fogelin] |
| 6574 | Legal reasoning is analogical, not deductive [Fogelin] |
| 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] |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| 6582 | Conventions can only work if they are based on something non-conventional [Fogelin] |
| 6576 | My view is 'circumspect rationalism' - that only our intellect can comprehend the world [Fogelin] |
| 6589 | Knowledge is legitimate only if all relevant defeaters have been eliminated [Fogelin] |
| 6596 | For coherentists, circularity is acceptable if the circle is large, rich and coherent [Fogelin] |
| 6597 | A rule of justification might be: don't raise the level of scrutiny without a good reason [Fogelin] |
| 6588 | Scepticism is cartesian (sceptical scenarios), or Humean (future), or Pyrrhonian (suspend belief) [Fogelin] |
| 6590 | Scepticism deals in remote possibilities that are ineliminable and set the standard very high [Fogelin] |
| 6583 | Radical perspectivism replaces Kant's necessary scheme with many different schemes [Fogelin] |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| 6555 | We are also irrational, with a unique ability to believe in bizarre self-created fictions [Fogelin] |
| 6605 | Critics must be causally entangled with their subject matter [Fogelin] |
| 6607 | The word 'beautiful', when deprived of context, is nearly contentless [Fogelin] |
| 6604 | Saying 'It's all a matter to taste' ignores the properties of the object discussed [Fogelin] |
| 6586 | Cynics are committed to morality, but disappointed or disgusted by human failings [Fogelin] |
| 6572 | Deterrence, prevention, rehabilitation and retribution can come into conflict in punishments [Fogelin] |
| 6573 | Retributivists say a crime can be 'paid for'; deterrentists still worry about potential victims [Fogelin] |