82 ideas
12027 | There must be a plausible epistemological theory alongside any metaphysical theory [Forbes,G] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
13643 | Aristotelian logic is complete [Shapiro] |
12005 | The symbol 'ι' forms definite descriptions; (ιx)F(x) says 'the x which is such that F(x)' [Forbes,G] |
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] |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
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] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
12010 | Is the meaning of 'and' given by its truth table, or by its introduction and elimination rules? [Forbes,G] |
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] |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
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] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
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] |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
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] |
12023 | Vagueness problems arise from applying sharp semantics to vague languages [Forbes,G] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
12017 | In all instances of identity, there must be some facts to ensure the identity [Forbes,G] |
12024 | If we combined two clocks, it seems that two clocks may have become one clock. [Forbes,G] |
11885 | Only individual essences will ground identities across worlds in other properties [Forbes,G, by Mackie,P] |
12014 | An individual essence is a set of essential properties which only that object can have [Forbes,G] |
12015 | Non-trivial individual essence is properties other than de dicto, or universal, or relational [Forbes,G] |
12013 | Essential properties depend on a category, and perhaps also on particular facts [Forbes,G] |
12012 | Essential properties are those without which an object could not exist [Forbes,G] |
12022 | Same parts does not ensure same artefact, if those parts could constitute a different artefact [Forbes,G] |
12025 | Artefacts have fuzzy essences [Forbes,G] |
12020 | An individual might change their sex in a world, but couldn't have differed in sex at origin [Forbes,G] |
11888 | Identities must hold because of other facts, which must be instrinsic [Forbes,G, by Mackie,P] |
12003 | De re modal formulae, unlike de dicto, are sensitive to transworld identities [Forbes,G] |
12028 | De re necessity is a form of conceptual necessity, just as de dicto necessity is [Forbes,G] |
12008 | Unlike places and times, we cannot separate possible worlds from what is true at them [Forbes,G] |
12009 | The problem with possible worlds realism is epistemological; we can't know properties of possible objects [Forbes,G] |
12007 | Possible worlds are points of logical space, rather like other times than our own [Forbes,G] |
12011 | Transworld identity concerns the limits of possibility for ordinary things [Forbes,G] |
12016 | The problem of transworld identity can be solved by individual essences [Forbes,G] |
12004 | Counterpart theory is not good at handling the logic of identity [Forbes,G] |
12021 | Haecceitism attributes to each individual a primitive identity or thisness [Forbes,G] |
12029 | We believe in thisnesses, because we reject bizarre possibilities as not being about that individual [Forbes,G] |