129 ideas
| 5082 | Reason grasps generalities, while the senses grasp particulars [Aristotle] |
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [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] |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [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] |
| 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] |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [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] |
| 9706 | A 'function' is a relation in which each object is related to just one other object [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] |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| 13270 | Are a part and whole one or many? Either way, what is the cause? [Aristotle] |
| 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] |
| 9790 | Geometry studies naturally occurring lines, but not as they occur in nature [Aristotle] |
| 22962 | Two is the least number, but there is no least magnitude, because it is always divisible [Aristotle] |
| 18090 | Without infinity time has limits, magnitudes are indivisible, and numbers come to an end [Aristotle] |
| 22929 | Aristotle's infinity is a property of the counting process, that it has no natural limit [Aristotle, by Le Poidevin] |
| 22930 | Lengths do not contain infinite parts; parts are created by acts of division [Aristotle, by Le Poidevin] |
| 18833 | A continuous line cannot be composed of indivisible points [Aristotle] |
| 9974 | Ten sheep and ten dogs are the same numerically, but it is not the same ten [Aristotle] |
| 5105 | The incommensurability of the diagonal always exists, and so it is not in time [Aristotle] |
| 16115 | Change is the implied actuality of that which exists potentially [Aristotle] |
| 22960 | The sophists thought a man in the Lyceum is different from that man in the marketplace [Aristotle] |
| 17262 | Aristotle's formal and material 'becauses' [aitiai] arguably involve grounding [Aristotle, by Correia/Schnieder] |
| 16656 | The separation from here to there is not the same as the separation from there to here [Aristotle] |
| 16644 | The features of a thing (whether quality or quantity) are inseparable from their subjects [Aristotle] |
| 5117 | Heavy and light are defined by their tendency to move down or up [Aristotle] |
| 17041 | Natural objects include animals and their parts, plants, and the simple elements [Aristotle] |
| 16172 | Substance is not predicated of anything - but it still has something underlying it, that originates it [Aristotle] |
| 16623 | We only infer underlying natures by analogy, observing bronze of a statue, or wood of a bed [Aristotle] |
| 16174 | A nature is related to a substance as shapeless matter is to something which has a shape [Aristotle] |
| 17043 | Form, not matter, is a thing's nature, because it is actual, rather than potential [Aristotle] |
| 16970 | A thing's form and purpose are often the same, and form can be the initiator of change too [Aristotle] |
| 16104 | Unity of the form is just unity of the definition [Aristotle] |
| 11255 | In feature-generation the matter (such as bronze) endures, but in generation it doesn't [Aristotle, by Politis] |
| 9071 | We first sense whole entities, and then move to particular parts of it [Aristotle] |
| 16791 | There is no whole except for the parts [Aristotle] |
| 16972 | The four explanations are the main aspects of a thing's nature [Aristotle, by Moravcsik] |
| 5084 | A thing's nature is what causes its changes and stability [Aristotle] |
| 16173 | Coming to be is by shape-change, addition, subtraction, composition or alteration [Aristotle] |
| 17042 | Natural things are their own source of stability through change [Aristotle] |
| 16691 | A day, or the games, has one thing after another, actually and potentially occurring [Aristotle] |
| 16574 | Coming-to-be may be from nothing in a qualified way, as arising from an absence [Aristotle] |
| 11254 | Matter is potentiality [Aristotle, by Politis] |
| 13110 | Intrinsic cause is prior to coincidence, so nature and intelligence are primary causes, chance secondary [Aristotle] |
| 13108 | Chance is a coincidental cause among events involving purpose and choice [Aristotle] |
| 13106 | Maybe there is no pure chance; a man's choices cause his chance meetings [Aristotle] |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| 18415 | The actual world is just the world you are in [Lewis, by Cappelen/Dever] |
| 16392 | A content is a property, and believing it is self-ascribing that property [Lewis, by Recanati] |
| 8331 | To know something we need understanding, which is grasp of the primary cause [Aristotle] |
| 5080 | We know a thing if we grasp its first causes, principles and basic elements [Aristotle] |
| 16969 | Science refers the question Why? to four causes/explanations: matter, form, source, purpose [Aristotle] |
| 11250 | Four Explanations: the essence and form; the matter; the source; and the end [Aristotle, by Politis] |
| 12045 | Aristotle's four 'causes' are four items which figure in basic explanations of nature [Aristotle, by Annas] |
| 16968 | There are as many causes/explanations as there are different types of why-question [Aristotle] |
| 13109 | Chance is inexplicable, because we can only explain what happens always or usually [Aristotle] |
| 18416 | Attitudes involve properties (not propositions), and belief is self-ascribing the properties [Lewis, by Solomon] |
| 16390 | Lewis's popular centred worlds approach gives an attitude an index of world, subject and time [Lewis, by Recanati] |
| 9789 | You can't abstract natural properties to make Forms - objects and attributes are defined together [Aristotle] |
| 9788 | Mathematicians study what is conceptually separable, and doesn't lead to error [Aristotle] |
| 18418 | A theory of perspectival de se content gives truth conditions relative to an agent [Lewis, by Cappelen/Dever] |
| 5107 | Predicates are substance, quality, place, relation, quantity and action or affection [Aristotle] |
| 20042 | We assign the cause of someone's walking when we say why they are doing it [Aristotle] |
| 5110 | Goodness is when a thing (such as a circle) is complete, and conforms with its nature [Aristotle] |
| 5111 | All moral virtue is concerned with bodily pleasure and pain [Aristotle] |
| 5092 | Nature is a principle of change, so we must understand change first [Aristotle] |
| 5113 | Nothing natural is disorderly, because nature is responsible for all order [Aristotle] |
| 5085 | 'Nature' refers to two things - form and matter [Aristotle] |
| 5089 | Nature has purpose, and aims at what is better. Is it coincidence that crops grow when it rains? [Aristotle] |
| 5086 | The nature of a thing is its end and purpose [Aristotle] |
| 5087 | A thing's purpose is ambiguous, and from one point of view we ourselves are ends [Aristotle] |
| 5091 | Teeth and crops are predictable, so they cannot be mere chance, but must have a purpose [Aristotle] |
| 5108 | Is ceasing-to-be unnatural if it happens by force, and natural otherwise? [Aristotle] |
| 5093 | Continuity depends on infinity, because the continuous is infinitely divisible [Aristotle] |
| 5095 | The heavens seem to be infinite, because we cannot imagine their end [Aristotle] |
| 16762 | Matter desires form, as female desires male, and ugliness desires beauty [Aristotle] |
| 17464 | When Aristotle's elements compound they are stable, so why would they ever separate? [Weisberg/Needham/Hendry on Aristotle] |
| 11252 | The 'form' of a thing explains why the matter constitutes that particular thing [Aristotle, by Politis] |
| 11253 | A 'material' cause/explanation is the form of whatever is the source [Aristotle, by Politis] |
| 13107 | Causes produce a few things in their own right, and innumerable things coincidentally [Aristotle] |
| 8332 | The four causes are the material, the form, the source, and the end [Aristotle] |
| 9787 | Scientists must know the essential attributes of the things they study [Aristotle] |
| 20063 | Motion fulfils potentiality [Aristotle] |
| 5114 | If movement can arise within an animal, why can't it also arise in the universe? [Aristotle] |
| 5116 | When there is unnatural movement (e.g. fire going downwards) the cause is obvious [Aristotle] |
| 5099 | The universe as a whole is not anywhere [Aristotle] |
| 5097 | If everything has a place, this causes an infinite regress, because each place must have place [Aristotle] |
| 5098 | Place is not shape, or matter, or extension between limits; it is the limits of a body [Aristotle] |
| 20920 | If there were many cosmoses, each would have its own time, giving many times [Aristotle] |
| 5106 | Would there be time if there were no mind? [Aristotle] |
| 22967 | It is unclear whether time depends on the existence of soul [Aristotle] |
| 8590 | Time does not exist without change [Aristotle] |
| 22965 | Time measures rest, as well as change [Aristotle] |
| 22885 | For Aristotle time is not a process but a means for measuring processes [Aristotle, by Bardon] |
| 22959 | Time is not change, but the number we associate with change [Aristotle] |
| 22964 | Change only exists in time through its being temporally measure [Aristotle] |
| 5104 | Time is an aspect of change [Aristotle] |
| 22956 | How can time exist, when it is composed of what has ceased to be and is yet to be? [Aristotle] |
| 5102 | If all of time has either ceased to exist, or has not yet happened, maybe time does not exist [Aristotle] |
| 5103 | Time is not change, but requires change in our minds to be noticed [Aristotle] |
| 22961 | The present moment is obviously a necessary feature of time [Aristotle] |
| 22916 | Unlike time, change goes at different rates, and is usually localised [Aristotle, by Le Poidevin] |
| 16693 | Time has parts, but the now is not one of them, and time is not composed of nows [Aristotle] |
| 22958 | Nows can't be linked together, any more than points on a line [Aristotle] |
| 22968 | Circular motion is the most obvious measure of time, and especially the celestial sphere [Aristotle] |
| 22963 | We measure change by time, and time by change, as they are interdefined [Aristotle] |
| 22966 | The present moment is a link (of past to future), and also a limit (of past and of future) [Aristotle] |
| 22957 | We can't tell whether the changing present moment is one thing, or a succession of things [Aristotle] |
| 5083 | Do things come to be from what is, or from what is not? Both seem problematical. [Aristotle] |
| 5119 | The source of all movement must be indivisible and have no magnitude [Aristotle] |