171 ideas
5082 | Reason grasps generalities, while the senses grasp particulars [Aristotle] |
9847 | A contextual definition permits the elimination of the expression by a substitution [Dummett] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
13643 | Aristotelian logic is complete [Shapiro] |
13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
13270 | Are a part and whole one or many? Either way, what is the cause? [Aristotle] |
13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
13627 | There is no 'correct' logic for natural languages [Shapiro] |
13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
9820 | In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical [Dummett] |
15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
13644 | Semantics for models uses set-theory [Shapiro] |
13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
13670 | Categoricity can't be reached in a first-order language [Shapiro] |
13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
13628 | We can live well without completeness in logic [Shapiro] |
13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
13646 | Compactness is derived from soundness and completeness [Shapiro] |
13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
9790 | Geometry studies naturally occurring lines, but not as they occur in nature [Aristotle] |
9896 | A prime number is one which is measured by a unit alone [Dummett] |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
18255 | Addition of quantities is prior to ordering, as shown in cyclic domains like angles [Dummett] |
13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
22962 | Two is the least number, but there is no least magnitude, because it is always divisible [Aristotle] |
9895 | A number is a multitude composed of units [Dummett] |
9852 | We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett] |
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] |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
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] |
13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
9829 | The identity of a number may be fixed by something outside structure - by counting [Dummett] |
9828 | Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett] |
9974 | Ten sheep and ten dogs are the same numerically, but it is not the same ten [Aristotle] |
13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
9876 | Set theory isn't part of logic, and why reduce to something more complex? [Dummett] |
13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
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] |
9884 | The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy [Dummett] |
9869 | Realism is just the application of two-valued semantics to sentences [Dummett] |
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] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
5117 | Heavy and light are defined by their tendency to move down or up [Aristotle] |
9880 | Nominalism assumes unmediated mental contact with objects [Dummett] |
9885 | The existence of abstract objects is a pseudo-problem [Dummett] |
9858 | Abstract objects nowadays are those which are objective but not actual [Dummett] |
9859 | It is absurd to deny the Equator, on the grounds that it lacks causal powers [Dummett] |
9860 | 'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object [Dummett] |
9872 | Abstract objects need the context principle, since they can't be encountered directly [Dummett] |
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] |
9848 | Content is replaceable if identical, so replaceability can't define identity [Dummett, by Dummett] |
9842 | Frege introduced criteria for identity, but thought defining identity was circular [Dummett] |
11254 | Matter is potentiality [Aristotle, by Politis] |
13110 | Intrinsic cause is prior to coincidence, so nature and intelligence are primary causes, chance secondary [Aristotle] |
13106 | Maybe there is no pure chance; a man's choices cause his chance meetings [Aristotle] |
13108 | Chance is a coincidental cause among events involving purpose and choice [Aristotle] |
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] |
9849 | Maybe a concept is 'prior' to another if it can be defined without the second concept [Dummett] |
9850 | An argument for conceptual priority is greater simplicity in explanation [Dummett] |
9873 | Abstract terms are acceptable as long as we know how they function linguistically [Dummett] |
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] |
9993 | There is no reason why abstraction by equivalence classes should be called 'logical' [Dummett, by Tait] |
9857 | We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus' [Dummett] |
9833 | To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too [Dummett] |
5107 | Predicates are substance, quality, place, relation, quantity and action or affection [Aristotle] |
9836 | Fregean semantics assumes a domain articulated into individual objects [Dummett] |
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] |
18257 | Why should the limit of measurement be points, not intervals? [Dummett] |
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] |