99 ideas
5486 | Essentialism says metaphysics can't be done by analysing unreliable language [Ellis] |
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] |
13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
13640 | Russell's paradox shows that there are classes which are not iterative sets [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] |
13627 | There is no 'correct' logic for natural languages [Shapiro] |
13642 | Logic is the ideal for learning new propositions on the basis of others [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] |
13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [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] |
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] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [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] |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
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] |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
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] |
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] |
13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
5468 | Properties are 'dispositional', or 'categorical' (the latter as 'block' or 'intrinsic' structures) [Ellis, by PG] |
5469 | The passive view of nature says categorical properties are basic, but others say dispositions [Ellis] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
5456 | Redness is not a property as it is not mind-independent [Ellis] |
5481 | Properties have powers; they aren't just ways for logicians to classify objects [Ellis] |
5458 | Nearly all fundamental properties of physics are dispositional [Ellis] |
5443 | Kripke and others have made essentialism once again respectable [Ellis] |
5444 | 'Individual essences' fix a particular individual, and 'kind essences' fix the kind it belongs to [Ellis] |
5462 | Essential properties are usually quantitatively determinate [Ellis] |
5448 | 'Real essence' makes it what it is; 'nominal essence' makes us categorise it a certain way [Ellis] |
5477 | One thing can look like something else, without being the something else [Ellis] |
5479 | Scientific essentialists say science should define the limits of the possible [Ellis] |
5483 | Essentialists deny possible worlds, and say possibilities are what is compatible with the actual world [Ellis] |
5447 | Metaphysical necessities are true in virtue of the essences of things [Ellis] |
5476 | Essentialists say natural laws are in a new category: necessary a posteriori [Ellis] |
5478 | Imagination tests what is possible for all we know, not true possibility [Ellis] |
5482 | Possible worlds realism is only needed to give truth conditions for modals and conditionals [Ellis] |
22167 | Our images of bodies are not produced by the bodies, but by our own minds [Augustine, by Aquinas] |
5453 | Essentialists mostly accept the primary/secondary qualities distinction [Ellis] |
5466 | Primary qualities are number, figure, size, texture, motion, configuration, impenetrability and (?) mass [Ellis] |
22117 | Our minds grasp reality by direct illumination (rather than abstraction from experience) [Augustine, by Matthews] |
5485 | Emeralds are naturally green, and only an external force could turn them blue [Ellis] |
5484 | Essentialists don't infer from some to all, but from essences to necessary behaviour [Ellis] |
5457 | Predicates assert properties, values, denials, relations, conventions, existence and fabrications [Ellis, by PG] |
22118 | Augustine created the modern concept of the will [Augustine, by Matthews] |
5488 | Regularity theories of causation cannot give an account of human agency [Ellis] |
5489 | Humans have variable dispositions, and also power to change their dispositions [Ellis] |
5490 | Essentialism fits in with Darwinism, but not with extreme politics of left or right [Ellis] |
4348 | Love, and do what you will [Augustine] |
7821 | Pagans produced three hundred definitions of the highest good [Augustine, by Grayling] |
22119 | Augustine said (unusually) that 'ought' does not imply 'can' [Augustine, by Matthews] |
5472 | Natural kinds are of objects/substances, or events/processes, or intrinsic natures [Ellis] |
5471 | Essentialism says natural kinds are fundamental to nature, and determine the laws [Ellis] |
5446 | For essentialists two members of a natural kind must be identical [Ellis] |
5480 | The whole of our world is a natural kind, so all worlds like it necessarily have the same laws [Ellis] |
5445 | Essentialists regard inanimate objects as genuine causal agents [Ellis] |
5463 | Essentialists believe causation is necessary, resulting from dispositions and circumstances [Ellis] |
5491 | A general theory of causation is only possible in an area if natural kinds are involved [Ellis] |
5442 | For 'passivists' behaviour is imposed on things from outside [Ellis] |
5473 | The laws of nature imitate the hierarchy of natural kinds [Ellis] |
5474 | Laws of nature tend to describe ideal things, or ideal circumstances [Ellis] |
5475 | We must explain the necessity, idealisation, ontology and structure of natural laws [Ellis] |
5460 | Causal relations cannot be reduced to regularities, as they could occur just once [Ellis] |
5459 | Essentialists say dispositions are basic, rather than supervenient on matter and natural laws [Ellis] |
5461 | The essence of uranium is its atomic number and its electron shell [Ellis] |
5464 | For essentialists, laws of nature are metaphysically necessary, being based on essences of natural kinds [Ellis] |
5487 | Essentialism requires a clear separation of semantics, epistemology and ontology [Ellis] |
22116 | Augustine identified Donatism, Pelagianism and Manicheism as the main heresies [Augustine, by Matthews] |
19338 | Augustine said evil does not really exist, and evil is a limitation in goodness [Augustine, by Perkins] |