76 ideas
| 3859 | We do not wish merely to predict, we also want to explain [Newton-Smith] |
| 3870 | The real problem of science is how to choose between possible explanations [Newton-Smith] |
| 3853 | For science to be rational, we must explain scientific change rationally [Newton-Smith] |
| 3855 | Critics attack positivist division between theory and observation [Newton-Smith] |
| 3854 | Positivists hold that theoretical terms change, but observation terms don't [Newton-Smith] |
| 3869 | More truthful theories have greater predictive power [Newton-Smith] |
| 3861 | Theories generate infinite truths and falsehoods, so they cannot be used to assess probability [Newton-Smith] |
| 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] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 3867 | De re necessity arises from the way the world is [Newton-Smith] |
| 3872 | We must assess the truth of beliefs in identifying them [Newton-Smith] |
| 3857 | Defeat relativism by emphasising truth and reference, not meaning [Newton-Smith] |
| 3858 | A full understanding of 'yellow' involves some theory [Newton-Smith] |
| 3862 | All theories contain anomalies, and so are falsified! [Newton-Smith] |
| 3863 | The anomaly of Uranus didn't destroy Newton's mechanics - it led to Neptune's discovery [Newton-Smith] |
| 3864 | Anomalies are judged against rival theories, and support for the current theory [Newton-Smith] |
| 3865 | Why should it matter whether or not a theory is scientific? [Newton-Smith] |
| 3866 | If theories are really incommensurable, we could believe them all [Newton-Smith] |
| 3871 | Explaining an action is showing that it is rational [Newton-Smith] |
| 5901 | Is 'productive of happiness' the definition of 'right', or the cause of it? [Ross on Bentham] |
| 5934 | Of Bentham's 'dimensions' of pleasure, only intensity and duration matter [Ross on Bentham] |
| 3777 | Pleasure and pain control all human desires and duties [Bentham] |
| 3554 | Bentham thinks happiness is feeling good, but why use morality to achieve that? [Annas on Bentham] |
| 3781 | The value of pleasures and pains is their force [Bentham] |
| 3778 | The community's interest is a sum of individual interests [Bentham] |
| 20280 | Large mature animals are more rational than babies. But all that really matters is - can they suffer? [Bentham] |
| 3779 | Unnatural, when it means anything, means infrequent [Bentham] |
| 3780 | We must judge a thing morally to know if it conforms to God's will [Bentham] |