75 ideas
19199 | Some say metaphysics is a highly generalised empirical study of objects [Tarski] |
19193 | Disputes that fail to use precise scientific terminology are all meaningless [Tarski] |
19179 | For a definition we need the words or concepts used, the rules, and the structure of the language [Tarski] |
18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
19178 | Definitions of truth should not introduce a new version of the concept, but capture the old one [Tarski] |
19177 | A definition of truth should be materially adequate and formally correct [Tarski] |
19186 | A rigorous definition of truth is only possible in an exactly specified language [Tarski] |
19194 | We may eventually need to split the word 'true' into several less ambiguous terms [Tarski] |
19196 | Scheme (T) is not a definition of truth [Tarski] |
19180 | It is convenient to attach 'true' to sentences, and hence the language must be specified [Tarski] |
19181 | In the classical concept of truth, 'snow is white' is true if snow is white [Tarski] |
19182 | Use 'true' so that all T-sentences can be asserted, and the definition will then be 'adequate' [Tarski] |
19183 | Each interpreted T-sentence is a partial definition of truth; the whole definition is their conjunction [Tarski] |
19198 | We don't give conditions for asserting 'snow is white'; just that assertion implies 'snow is white' is true [Tarski] |
19184 | The best truth definition involves other semantic notions, like satisfaction (relating terms and objects) [Tarski] |
19191 | Specify satisfaction for simple sentences, then compounds; true sentences are satisfied by all objects [Tarski] |
19188 | We can't use a semantically closed language, or ditch our logic, so a meta-language is needed [Tarski] |
19189 | The metalanguage must contain the object language, logic, and defined semantics [Tarski] |
10824 | If listing equivalences is a reduction of truth, witchcraft is just a list of witch-victim pairs [Field,H on Tarski] |
19190 | We need an undefined term 'true' in the meta-language, specified by axioms [Tarski] |
19197 | Truth can't be eliminated from universal claims, or from particular unspecified claims [Tarski] |
19185 | Semantics is a very modest discipline which solves no real problems [Tarski] |
19195 | Truth tables give prior conditions for logic, but are outside the system, and not definitions [Tarski] |
18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
18114 | There is no single agreed structure for set theory [Bostock] |
18107 | A 'proper class' cannot be a member of anything [Bostock] |
18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
18109 | The completeness of first-order logic implies its compactness [Bostock] |
18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
19192 | The truth definition proves semantic contradiction and excluded middle laws (not the logic laws) [Tarski] |
18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
19187 | The Liar makes us assert a false sentence, so it must be taken seriously [Tarski] |
18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
18097 | The Peano Axioms describe a unique structure [Bostock] |
18149 | There are many criteria for the identity of numbers [Bostock] |
18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
18140 | The best version of conceptualism is predicativism [Bostock] |
18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |