78 ideas
1502 | Parmenides was much more cautious about accepting ideas than his predecessors [Simplicius on Parmenides] |
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] |
448 | No necessity could produce Being either later or earlier, so it must exist absolutely or not at all [Parmenides] |
447 | Being must be eternal and uncreated, and hence it is timeless [Parmenides] |
449 | Being is not divisible, since it is all alike [Parmenides] |
1503 | There is no such thing as nothing [Parmenides] |
445 | The realm of necessary non-existence cannot be explored, because it is unknowable [Parmenides] |
21820 | Parmenides at least saw Being as the same as Nous, and separate from the sensed realm [Parmenides, by Plotinus] |
452 | All our concepts of change and permanence are just names, not the truth [Parmenides] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
1504 | Something must be unchanging to make recognition and knowledge possible [Aristotle on Parmenides] |
444 | The first way of enquiry involves necessary existence [Parmenides] |
450 | Necessity sets limits on being, in order to give it identity [Parmenides] |
12582 | The function of beliefs is to produce beliefs-that-p when p [Millikan] |
451 | Thinking implies existence, because thinking depends on it [Parmenides] |
1506 | Parmenides treats perception and intellectual activity as the same [Theophrastus on Parmenides] |
3058 | Only reason can prove the truth of facts [Parmenides] |
555 | People who say that the cosmos is one forget that they must explain movement [Aristotle on Parmenides] |
5081 | There could be movement within one thing, as there is within water [Aristotle on Parmenides] |
1509 | The one can't be divisible, because if it was it could be infinitely divided down to nothing [Parmenides, by Simplicius] |
20900 | Defenders of the One say motion needs the void - but that is not part of Being [Parmenides, by Aristotle] |
226 | The one is without any kind of motion [Parmenides] |
1505 | Reason sees reality as one, the senses see it as many [Aristotle on Parmenides] |
453 | Reality is symmetrical and balanced, like a sphere, with no reason to be greater one way rather than another [Parmenides] |
1792 | He taught that there are two elements, fire the maker, and earth the matter [Parmenides, by Diog. Laertius] |
5115 | It is feeble-minded to look for explanations of everything being at rest [Aristotle on Parmenides] |
13217 | The void can't exist, and without the void there can't be movement or separation [Parmenides, by Aristotle] |
22918 | What could have triggered the beginning [of time and being]? [Parmenides] |
1794 | He was the first to discover the identity of the Morning and Evening Stars [Parmenides, by Diog. Laertius] |
1791 | He was the first person to say the earth is spherical [Parmenides, by Diog. Laertius] |