81 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] |
| 15312 | We get the idea of power by abstracting from ropes, magnets and electric shocks [Priestley] |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| 4669 | Persons are conscious, they relate, they think, they feel, and they are self-aware [Glover] |
| 4656 | A problem arises in any moral system that allows more than one absolute right [Glover] |
| 4657 | Double Effect: no bad acts with good consequences, but possibly good acts despite bad consequences [Glover] |
| 4658 | Acts and Omissions: bad consequences are morally better if they result from an omission rather than an act [Glover] |
| 4659 | It doesn't seem worse to switch off a life-support machine than to forget to switch it on [Glover] |
| 4660 | Harmful omissions are unavoidable, while most harmful acts can be avoided [Glover] |
| 4661 | What matters is not intrinsic value of life or rights, but worthwhile and desired life, and avoidance of pain [Glover] |
| 4648 | 'Death' is best seen as irreversible loss of consciousness, since this is why we care about brain function [Glover] |
| 3785 | You can't separate acts from the people performing them [Glover] |
| 3786 | Aggression in defence may be beneficial but morally corrupting [Glover] |
| 4650 | The quality of a life is not altogether independent of its length [Glover] |
| 3784 | Duty prohibits some acts, whatever their consequences [Glover] |
| 3782 | Satisfaction of desires is not at all the same as achieving happiness [Glover, by PG] |
| 3787 | Rule-utilitarianism is either act-utilitarianism, or not really utilitarian [Glover] |
| 3783 | How can utilitarianism decide the ideal population size? [Glover] |
| 4675 | The sanctity of life doctrine implies a serious increase of abnormality among the population [Glover] |
| 4654 | Autonomy favours present opinions over future ones, and says nothing about the interests of potential people [Glover] |
| 4655 | If a whole community did not mind death, respect for autonomy suggests that you could kill them all [Glover] |
| 4680 | Autonomy seems to acquire greater weight when the decision is more important to a person [Glover] |
| 4670 | Being alive is not intrinsically good, and there is no 'right to life' [Glover] |
| 4668 | You can't have a right to something you can't desire, so a foetus has no 'right' to life [Glover] |
| 4651 | Utilitarians object to killing directly (pain, and lost happiness), and to side-effects (loss to others, and precedents) [Glover] |
| 4671 | What is wrong with killing someone, if another equally worthwhile life is substituted? [Glover] |
| 4676 | The 'no trade-off' position: killing is only justified if it prevents other deaths [Glover] |
| 4685 | Societies spend a lot to save known persons, but very little to reduce fatal accidents [Glover] |
| 4649 | If someone's life is 'worth living', that gives one direct reason not to kill him [Glover] |
| 4683 | Involuntary euthanasia is wrong because it violates autonomy, and it has appalling side-effects [Glover] |
| 4682 | Euthanasia is voluntary (patient's wish), or involuntary (ignore wish), or non-voluntary (no wish possible) [Glover] |
| 4684 | Maybe extreme treatment is not saving life, but prolonging the act of dying [Glover] |
| 4681 | The Nazi mass murders seem to have originated in their euthanasia programme [Glover] |
| 4665 | Conception isn't the fixed boundary for a person's beginning, because twins are possible within two weeks [Glover] |
| 4652 | If killing is wrong because it destroys future happiness, not conceiving a happy child is also wrong [Glover] |
| 4662 | Defenders of abortion focus on early pregnancy, while opponents focus on later stages [Glover] |
| 4663 | If abortion is wrong, it is because a foetus is a human being or a person (or potentially so) [Glover] |
| 4664 | If abortion is wrong because of the 'potential' person, that makes contraception wrong too [Glover] |
| 4673 | Abortion differs morally from deliberate non-conception only in its side-effects [Glover] |
| 4666 | If viability is a test or boundary at the beginning of life, it should also be so for frail old people [Glover] |
| 4672 | Apart from side effects, it seems best to replace an inadequate foetus with one which has a better chance [Glover] |
| 4674 | It is always right for a qualified person to perform an abortion when requested by the mother [Glover] |
| 4667 | How would we judge abortion if mothers had transparent wombs? [Glover] |
| 4679 | One test for a worthwhile life is to assess the amount of life for which you would rather be unconscious [Glover] |
| 15311 | Attraction or repulsion are not imparted to matter, but actually constitute it [Priestley] |