96 ideas
5196 | Philosophy is a department of logic [Ayer] |
5189 | Philosophers should abandon speculation, as philosophy is wholly critical [Ayer] |
7919 | Humeans rejected the a priori synthetic, and so rejected even Kantian metaphysics [Ayer, by Macdonald,C] |
5195 | Critics say analysis can only show the parts, and not their distinctive configuration [Ayer] |
5179 | Philosophy deals with the questions that scientists do not wish to handle [Ayer] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
4749 | We cannot analyse the concept of 'truth', because it is simply a mark that a sentence is asserted [Ayer] |
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] |
5202 | Maths and logic are true universally because they are analytic or tautological [Ayer] |
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] |
6523 | Positivists regard ontology as either meaningless or stipulated [Ayer, by Robinson,H] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
5183 | Only tautologies can be certain; other propositions can only be probable [Ayer] |
6525 | Logical positivists could never give the sense-data equivalent of 'there is a table next door' [Robinson,H on Ayer] |
5170 | Material things are constructions from actual and possible occurrences of sense-contents [Ayer] |
5198 | We could verify 'a thing can't be in two places at once' by destroying one of the things [Ierubino on Ayer] |
2619 | Whether geometry can be applied to reality is an empirical question outside of geometry [Ayer] |
5197 | By changing definitions we could make 'a thing can't be in two places at once' a contradiction [Ayer] |
5204 | To say that a proposition is true a priori is to say that it is a tautology [Ayer] |
6524 | Positivists prefer sense-data to objects, because the vocabulary covers both illusions and perceptions [Ayer, by Robinson,H] |
5193 | Causal and representative theories of perception are wrong as they refer to unobservables [Ayer] |
5200 | The main claim of rationalism is that thought is an independent source of knowledge [Ayer] |
4729 | Empiricism lacked a decent account of the a priori, until Ayer said it was entirely analytic [O'Grady on Ayer] |
5180 | All propositions (especially 'metaphysics') must begin with the senses [Ayer] |
5169 | My empiricism logically distinguishes analytic and synthetic propositions, and metaphysical verbiage [Ayer] |
5185 | It is further sense-experience which informs us of the mistakes that arise out of sense-experience [Ayer] |
5199 | Empiricism, it is said, cannot account for our knowledge of necessary truths [Ayer] |
5190 | The induction problem is to prove generalisations about the future based on the past [Ayer] |
5191 | We can't use the uniformity of nature to prove induction, as that would be circular [Ayer] |
5177 | Other minds are 'metaphysical' objects, because I can never observe their experiences [Ayer] |
5178 | A conscious object is by definition one that behaves in a certain way, so behaviour proves consciousness [Ayer] |
5172 | If the self is meaningful, it must be constructed from sense-experiences [Ayer] |
5173 | Two experiences belong to one self if their contents belong with one body [Ayer] |
5176 | Empiricists can define personal identity as bodily identity, which consists of sense-contents [Ayer] |
5171 | The supposed 'gulf' between mind and matter is based on the senseless concept of 'substances' [Ayer] |
5181 | A sentence is factually significant to someone if they know how to verify its proposition [Ayer] |
5184 | Factual propositions imply (in conjunction with a few other premises) possible experiences [Ayer] |
5186 | Tautologies and empirical hypotheses form the entire class of significant propositions [Ayer] |
5205 | Moral intuition is worthless if there is no criterion to decide between intuitions [Ayer] |
23725 | Ayer defends the emotivist version of expressivism [Ayer, by Smith,M] |
5206 | To say an act is wrong makes no further statement about it, but merely expresses disapproval [Ayer] |
6613 | The natural kinds are objects, processes and properties/relations [Ellis] |
6616 | Least action is not a causal law, but a 'global law', describing a global essence [Ellis] |
6615 | A species requires a genus, and its essence includes the essence of the genus [Ellis] |
6614 | A hierarchy of natural kinds is elaborate ontology, but needed to explain natural laws [Ellis] |
6612 | Without general principles, we couldn't predict the behaviour of dispositional properties [Ellis] |
5208 | A person with non-empirical attributes is unintelligible. [Ayer] |
5187 | When we ascribe an attribute to a thing, we covertly assert that it exists [Ayer] |
5207 | If theism is non-sensical, then so is atheism. [Ayer] |
5209 | The 'truths' expressed by theists are not literally significant [Ayer] |