83 ideas
| 6095 | The business of metaphysics is to describe the world [Russell] |
| 6106 | Reducing entities and premisses makes error less likely [Russell] |
| 6090 | Facts make propositions true or false, and are expressed by whole sentences [Russell] |
| 18348 | Not only atomic truths, but also general and negative truths, have truth-makers [Russell, by Rami] |
| 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] |
| 6103 | Normally a class with only one member is a problem, because the class and the member are identical [Russell] |
| 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] |
| 6092 | In a logically perfect language, there will be just one word for every simple object [Russell] |
| 6101 | Romulus does not occur in the proposition 'Romulus did not exist' [Russell] |
| 6102 | You can understand 'author of Waverley', but to understand 'Scott' you must know who it applies to [Russell] |
| 10423 | There are a set of criteria for pinning down a logically proper name [Russell, by Sainsbury] |
| 7744 | Treat description using quantifiers, and treat proper names as descriptions [Russell, by McCullogh] |
| 10426 | A name has got to name something or it is not a name [Russell] |
| 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] |
| 6104 | Numbers are classes of classes, and hence fictions of fictions [Russell] |
| 13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
| 21708 | Russell's new logical atomist was of particulars, universals and facts (not platonic propositions) [Russell, by Linsky,B] |
| 19051 | Russell's atomic facts are actually compounds, and his true logical atoms are sense data [Russell, by Quine] |
| 6089 | Logical atomism aims at logical atoms as the last residue of analysis [Russell] |
| 6100 | Once you have enumerated all the atomic facts, there is a further fact that those are all the facts [Russell] |
| 6105 | Logical atoms aims to get down to ultimate simples, with their own unique reality [Russell] |
| 21709 | You can't name all the facts, so they are not real, but are what propositions assert [Russell] |
| 18376 | Russell asserts atomic, existential, negative and general facts [Russell, by Armstrong] |
| 5465 | Modern trope theory tries, like logical atomism, to reduce things to elementary states [Russell, by Ellis] |
| 6060 | 'Existence' means that a propositional function is sometimes true [Russell] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 6099 | Modal terms are properties of propositional functions, not of propositions [Russell] |
| 6098 | Perception goes straight to the fact, and not through the proposition [Russell] |
| 6097 | The theory of error seems to need the existence of the non-existent [Russell] |
| 16369 | There is a single file per object, memorised, reactivated, consolidated and expanded [Papineau, by Recanati] |
| 9022 | Russell uses 'propositional function' to refer to both predicates and to attributes [Quine on Russell] |
| 6091 | Propositions don't name facts, because each fact corresponds to a proposition and its negation [Russell] |
| 21702 | In 1918 still believes in nonlinguistic analogues of sentences, but he now calls them 'facts' [Russell, by Quine] |
| 6094 | An inventory of the world does not need to include propositions [Russell] |
| 6096 | I no longer believe in propositions, especially concerning falsehoods [Russell] |
| 21712 | I know longer believe in shadowy things like 'that today is Wednesday' when it is actually Tuesday [Russell] |
| 6093 | The names in a logically perfect language would be private, and could not be shared [Russell] |
| 6119 | You can discuss 'God exists', so 'God' is a description, not a name [Russell] |