166 ideas
19066 | Philosophy aims to understand the world, through ordinary experience and science [Dummett] |
10838 | To explain a concept, we need its purpose, not just its rules of usage [Dummett] |
17621 | What matters in mathematics is its objectivity, not the existence of the objects [Dummett] |
9847 | A contextual definition permits the elimination of the expression by a substitution [Dummett] |
19067 | A successful proof requires recognition of truth at every step [Dummett] |
10837 | It is part of the concept of truth that we aim at making true statements [Dummett] |
10840 | We must be able to specify truths in a precise language, like winning moves in a game [Dummett] |
19171 | Tarski's truth is like rules for winning games, without saying what 'winning' means [Dummett, by Davidson] |
8166 | Truth is part of semantics, since valid inference preserves truth [Dummett] |
19053 | Logic would be more natural if negation only referred to predicates [Dummett] |
19060 | Truth-tables are dubious in some cases, and may be a bad way to explain connective meaning [Dummett] |
16951 | It was realised that possible worlds covered all modal logics, if they had a structure [Dummett] |
16953 | Relative possibility one way may be impossible coming back, so it isn't symmetrical [Dummett] |
16952 | If something is only possible relative to another possibility, the possibility relation is not transitive [Dummett] |
16960 | If possibilitiy is relative, that might make accessibility non-transitive, and T the correct system [Dummett] |
16958 | In S4 the actual world has a special place [Dummett] |
18073 | Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Dummett, by Kitcher] |
18832 | Mathematical statements and entities that result from an infinite process must lack a truth-value [Dummett] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
10537 | The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}} [Dummett] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
9193 | ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett] |
9194 | The main alternative to ZF is one which includes looser classes as well as sets [Dummett] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
10542 | To associate a cardinal with each set, we need the Axiom of Choice to find a representative [Dummett] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
11066 | Deduction is justified by the semantics of its metalanguage [Dummett, by Hanna] |
9820 | In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical [Dummett] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
19058 | Syntactic consequence is positive, for validity; semantic version is negative, with counterexamples [Dummett] |
8173 | Language can violate bivalence because of non-referring terms or ill-defined predicates [Dummett] |
8195 | Undecidable statements result from quantifying over infinites, subjunctive conditionals, and the past tense [Dummett] |
7334 | Anti-realism needs an intuitionist logic with no law of excluded middle [Dummett, by Miller,A] |
8179 | The law of excluded middle is the logical reflection of the principle of bivalence [Dummett] |
9195 | Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
19052 | Natural language 'not' doesn't apply to sentences [Dummett] |
18801 | Classical negation is circular, if it relies on knowing negation-conditions from truth-conditions [Dummett] |
9182 | Ancient names like 'Obadiah' depend on tradition, not on where the name originated [Dummett] |
19057 | Classical quantification is an infinite conjunction or disjunction - but you may not know all the instances [Dummett] |
9186 | First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett] |
19063 | Beth trees show semantics for intuitionistic logic, in terms of how truth has been established [Dummett] |
19059 | In standard views you could replace 'true' and 'false' with mere 0 and 1 [Dummett] |
19062 | Classical two-valued semantics implies that meaning is grasped through truth-conditions [Dummett] |
9187 | Logical truths and inference are characterized either syntactically or semantically [Dummett] |
19065 | Soundness and completeness proofs test the theory of meaning, rather than the logic theory [Dummett] |
8194 | Surely there is no exact single grain that brings a heap into existence [Dummett] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
9896 | A prime number is one which is measured by a unit alone [Dummett] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
18255 | Addition of quantities is prior to ordering, as shown in cyclic domains like angles [Dummett] |
9191 | Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
9895 | A number is a multitude composed of units [Dummett] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
9852 | We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15938 | Platonists ruin infinity, which is precisely a growing structure which is never completed [Dummett] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
10554 | Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
9829 | The identity of a number may be fixed by something outside structure - by counting [Dummett] |
9828 | Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett] |
9192 | The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
9876 | Set theory isn't part of logic, and why reduce to something more complex? [Dummett] |
15939 | For intuitionists it is constructed proofs (which take time) which make statements true [Dummett] |
10552 | Intuitionism says that totality of numbers is only potential, but is still determinate [Dummett] |
8190 | Intuitionists rely on the proof of mathematical statements, not their truth [Dummett] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
8198 | A 'Cambridge Change' is like saying 'the landscape changes as you travel east' [Dummett] |
10540 | We can't say that light is concrete but radio waves abstract [Dummett] |
10515 | Ostension is possible for concreta; abstracta can only be referred to via other objects [Dummett, by Hale] |
10544 | The concrete/abstract distinction seems crude: in which category is the Mistral? [Dummett] |
10546 | We don't need a sharp concrete/abstract distinction [Dummett] |
9884 | The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy [Dummett] |
22297 | Dummett saw realism as acceptance of bivalence, rather than of mind-independent entities [Dummett, by Potter] |
9869 | Realism is just the application of two-valued semantics to sentences [Dummett] |
15049 | Metaphysical realists are committed to all unambiguous statements being true or not true [Dummett] |
8184 | Philosophers should not presume reality, but only invoke it when language requires it [Dummett] |
3303 | For anti-realists there are no natural distinctions between objects [Dummett, by Benardete,JA] |
8185 | We can't make sense of a world not apprehended by a mind [Dummett] |
8192 | I no longer think what a statement about the past says is just what can justify it [Dummett] |
8163 | Since 'no bird here' and 'no squirrel here' seem the same, we must talk of 'atomic' facts [Dummett] |
8161 | We know we can state facts, with true statements [Dummett] |
21628 | To say reality itself is vague is not properly intelligible [Dummett] |
8180 | 'That is red or orange' might be considered true, even though 'that is red' and 'that is orange' were not [Dummett] |
10548 | The context principle for names rules out a special philosophical sense for 'existence' [Dummett] |
10281 | The objects we recognise the world as containing depends on the structure of our language [Dummett] |
10532 | We can understand universals by studying predication [Dummett] |
10534 | 'Nominalism' used to mean denial of universals, but now means denial of abstract objects [Dummett] |
9880 | Nominalism assumes unmediated mental contact with objects [Dummett] |
10541 | Concrete objects such as sounds and smells may not be possible objects of ostension [Dummett] |
10545 | Abstract objects may not cause changes, but they can be the subject of change [Dummett] |
9885 | The existence of abstract objects is a pseudo-problem [Dummett] |
10555 | If we can intuitively apprehend abstract objects, this makes them observable and causally active [Dummett] |
9858 | Abstract objects nowadays are those which are objective but not actual [Dummett] |
10543 | Abstract objects must have names that fall within the range of some functional expression [Dummett] |
9859 | It is absurd to deny the Equator, on the grounds that it lacks causal powers [Dummett] |
9860 | 'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object [Dummett] |
10320 | If a genuine singular term needs a criterion of identity, we must exclude abstract nouns [Dummett, by Hale] |
10547 | Abstract objects can never be confronted, and need verbal phrases for reference [Dummett] |
9872 | Abstract objects need the context principle, since they can't be encountered directly [Dummett] |
10531 | There is a modern philosophical notion of 'object', first introduced by Frege [Dummett] |
9848 | Content is replaceable if identical, so replaceability can't define identity [Dummett, by Dummett] |
9842 | Frege introduced criteria for identity, but thought defining identity was circular [Dummett] |
16957 | Possible worlds aren't how the world might be, but how a world might be, given some possibility [Dummett] |
16959 | If possible worlds have no structure (S5) they are equal, and it is hard to deny them reality [Dummett] |
8199 | The existence of a universe without sentience or intelligence is an unintelligible fantasy [Dummett] |
8178 | Empirical and a priori knowledge are not distinct, but are extremes of a sliding scale [Dummett] |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
19061 | An explanation is often a deduction, but that may well beg the question [Dummett] |
8174 | The theories of meaning and understanding are the only routes to an account of thought [Dummett] |
8175 | A theory of thought will include propositional attitudes as well as propositions [Dummett] |
19168 | Concepts only have a 'functional character', because they map to truth values, not objects [Dummett, by Davidson] |
9849 | Maybe a concept is 'prior' to another if it can be defined without the second concept [Dummett] |
9850 | An argument for conceptual priority is greater simplicity in explanation [Dummett] |
10839 | You can't infer a dog's abstract concepts from its behaviour [Dummett] |
9873 | Abstract terms are acceptable as long as we know how they function linguistically [Dummett] |
10549 | Since abstract objects cannot be picked out, we must rely on identity statements [Dummett] |
9993 | There is no reason why abstraction by equivalence classes should be called 'logical' [Dummett, by Tait] |
9857 | We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus' [Dummett] |
9833 | To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too [Dummett] |
8165 | To 'abstract from' is a logical process, as opposed to the old mental view [Dummett] |
19055 | Stating a sentence's truth-conditions is just paraphrasing the sentence [Dummett] |
19056 | If a sentence is effectively undecidable, we can never know its truth conditions [Dummett] |
8168 | To know the truth-conditions of a sentence, you must already know the meaning [Dummett] |
8193 | Verification is not an individual but a collective activity [Dummett] |
8181 | A justificationist theory of meaning leads to the rejection of classical logic [Dummett] |
8182 | Verificationism could be realist, if we imagined the verification by a superhuman power [Dummett] |
8183 | If truths about the past depend on memories and current evidence, the past will change [Dummett] |
19054 | Meaning as use puts use beyond criticism, and needs a holistic view of language [Dummett] |
8176 | We could only guess the meanings of 'true' and 'false' when sentences were used [Dummett] |
8170 | Sentences are the primary semantic units, because they can say something [Dummett] |
19064 | Holism is not a theory of meaning; it is the denial that a theory of meaning is possible [Dummett] |
10516 | A realistic view of reference is possible for concrete objects, but not for abstract objects [Dummett, by Hale] |
9181 | The causal theory of reference can't distinguish just hearing a name from knowing its use [Dummett] |
9836 | Fregean semantics assumes a domain articulated into individual objects [Dummett] |
8189 | Truth-condition theorists must argue use can only be described by appeal to conditions of truth [Dummett] |
8191 | The truth-conditions theory must get agreement on a conception of truth [Dummett] |
8169 | We can't distinguish a proposition from its content [Dummett] |
16956 | To explain generosity in a person, you must understand a generous action [Dummett] |
16954 | Generalised talk of 'natural kinds' is unfortunate, as they vary too much [Dummett] |
18257 | Why should the limit of measurement be points, not intervals? [Dummett] |
8186 | Time is the measure of change, so we can't speak of time before all change [Dummett] |
8197 | Maybe past (which affects us) and future (which we can affect) are both real [Dummett] |
8167 | If Presentism is correct, we cannot even say that the present changes [Dummett] |
8196 | The present cannot exist alone as a mere boundary; past and future truths are rendered meaningless [Dummett] |