269 ideas
7623 | For ancient Greeks being wise was an ethical value [Putnam] |
2352 | The job of the philosopher is to distinguish facts about the world from conventions [Putnam] |
5182 | Claims about 'the Absolute' are not even verifiable in principle [Ayer on Bradley] |
6864 | Metaphysics is finding bad reasons for instinctive beliefs [Bradley] |
6782 | Realism is the only philosophy of science that doesn't make the success of science a miracle [Putnam] |
6267 | A culture needs to admit that knowledge is more extensive than just 'science' [Putnam] |
6272 | 'True' and 'refers' cannot be made scientically precise, but are fundamental to science [Putnam] |
18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
6276 | 'The rug is green' might be warrantedly assertible even though the rug is not green [Putnam] |
4714 | Putnam's epistemic notion of truth replaces the realism of correspondence with ontological relativism [Putnam, by O'Grady] |
6266 | We need the correspondence theory of truth to understand language and science [Putnam] |
7617 | Before Kant, all philosophers had a correspondence theory of truth [Putnam] |
6277 | Correspondence between concepts and unconceptualised reality is impossible [Putnam] |
4716 | The correspondence theory is wrong, because there is no one correspondence between reality and fact [Putnam, by O'Grady] |
8828 | Truth is rational acceptability [Putnam] |
7616 | Truth is an idealisation of rational acceptability [Putnam] |
18951 | For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam] |
6264 | In Tarski's definition, you understand 'true' if you accept the notions of the object language [Putnam] |
6265 | Tarski has given a correct account of the formal logic of 'true', but there is more to the concept [Putnam] |
6269 | Only Tarski has found a way to define 'true' [Putnam] |
2345 | Semantic notions do not occur in Tarski's definitions, but assessing their correctness involves translation [Putnam] |
2347 | Asserting the truth of an indexical statement is not the same as uttering the statement [Putnam] |
18953 | Modern notation frees us from Aristotle's restriction of only using two class-names in premises [Putnam] |
18949 | The universal syllogism is now expressed as the transitivity of subclasses [Putnam] |
13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock] |
13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock] |
13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock] |
13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock] |
13350 | 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock] |
13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock] |
13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock] |
13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock] |
13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock] |
13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock] |
13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock] |
18952 | '⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)' [Putnam] |
18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
18114 | There is no single agreed structure for set theory [Bostock] |
18958 | In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam] |
18107 | A 'proper class' cannot be a member of anything [Bostock] |
9944 | We understand some statements about all sets [Putnam] |
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] |
13655 | The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate [Putnam, by Shapiro] |
9915 | V = L just says all sets are constructible [Putnam] |
18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
18954 | Before the late 19th century logic was trivialised by not dealing with relations [Putnam] |
18956 | Asserting first-order validity implicitly involves second-order reference to classes [Putnam] |
18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
18109 | The completeness of first-order logic implies its compactness [Bostock] |
13346 | Truth is the basic notion in classical logic [Bostock] |
13545 | Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock] |
13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock] |
13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock] |
13347 | Validity is a conclusion following for premises, even if there is no proof [Bostock] |
13348 | It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock] |
13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock] |
13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
18962 | Unfashionably, I think logic has an empirical foundation [Putnam] |
10066 | Putnam coined the term 'if-thenism' [Putnam, by Musgrave] |
13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock] |
13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock] |
13803 | If we are to express that there at least two things, we need identity [Bostock] |
13357 | Truth-functors are usually held to be defined by their truth-tables [Bostock] |
18961 | We can identify functions with certain sets - or identify sets with certain functions [Putnam] |
13812 | A 'zero-place' function just has a single value, so it is a name [Bostock] |
13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock] |
13360 | In logic, a name is just any expression which refers to a particular single object [Bostock] |
17505 | Using proper names properly doesn't involve necessary and sufficient conditions [Putnam] |
10999 | Names need a means of reidentifying their referents [Bradley, by Read] |
13361 | An expression is only a name if it succeeds in referring to a real object [Bostock] |
13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock] |
13814 | Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock] |
13816 | Because of scope problems, definite descriptions are best treated as quantifiers [Bostock] |
13817 | Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock] |
13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock] |
13815 | Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock] |
13438 | 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock] |
13818 | If we allow empty domains, we must allow empty names [Bostock] |
18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
13801 | An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock] |
13619 | Quantification adds two axiom-schemas and a new rule [Bostock] |
13622 | Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock] |
13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock] |
13616 | The Deduction Theorem greatly simplifies the search for proof [Bostock] |
13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock] |
13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock] |
13753 | Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock] |
13755 | Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock] |
13758 | In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock] |
13754 | Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock] |
18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock] |
13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
13760 | A sequent calculus is good for comparing proof systems [Bostock] |
13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock] |
13364 | Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG] |
18955 | Having a valid form doesn't ensure truth, as it may be meaningless [Putnam] |
13821 | Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock] |
13362 | If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock] |
14203 | Intension is not meaning, as 'cube' and 'square-faced polyhedron' are intensionally the same [Putnam] |
14207 | If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth [Putnam] |
9913 | The Löwenheim-Skolem Theorem is close to an antinomy in philosophy of language [Putnam] |
13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
13544 | Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock] |
13618 | Compactness means an infinity of sequents on the left will add nothing new [Bostock] |
18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [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] |
18959 | Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam] |
18200 | Very large sets should be studied in an 'if-then' spirit [Putnam] |
18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
9937 | I do not believe mathematics either has or needs 'foundations' [Putnam] |
18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
9939 | It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam] |
18097 | The Peano Axioms describe a unique structure [Bostock] |
13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock] |
13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all 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] |
18149 | There are many criteria for the identity of numbers [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] |
3663 | How can you contemplate Platonic entities without causal transactions with them? [Putnam] |
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] |
9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
9914 | It is unfashionable, but most mathematical intuitions come from nature [Putnam] |
9941 | Science requires more than consistency of mathematics [Putnam] |
18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
18199 | Indispensability strongly supports predicative sets, and somewhat supports impredicative sets [Putnam] |
8857 | We must quantify over numbers for science; but that commits us to their existence [Putnam] |
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] |
18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [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] |
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] |
18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
6280 | Realism is a theory, which explains the convergence of science and the success of language [Putnam] |
17644 | Metaphysical realism is committed to there being one ultimate true theory [Putnam] |
2349 | Realists believe truth is correspondence, independent of humans, is bivalent, and is unique [Putnam] |
9943 | You can't deny a hypothesis a truth-value simply because we may never know it! [Putnam] |
22181 | Putnam says anti-realism is a bad explanation of accurate predictions [Putnam, by Okasha] |
14214 | If we try to cure the abundance of theories with causal links, this is 'just more theory' [Putnam, by Lewis] |
17648 | It is an illusion to think there could be one good scientific theory of reality [Putnam] |
14205 | The sentence 'A cat is on a mat' remains always true when 'cat' means cherry and 'mat' means tree [Putnam] |
7610 | A fact is simply what it is rational to accept [Putnam] |
6422 | Internal relations are said to be intrinsic properties of two terms, and of the whole they compose [Bradley, by Russell] |
7966 | Relations must be linked to their qualities, but that implies an infinite regress of relations [Bradley] |
13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
7618 | Very nominalistic philosophers deny properties, though scientists accept them [Putnam] |
18957 | Nominalism only makes sense if it is materialist [Putnam] |
2351 | Aristotle says an object (e.g. a lamp) has identity if its parts stay together when it is moved [Putnam] |
18950 | Physics is full of non-physical entities, such as space-vectors [Putnam] |
17643 | Shape is essential relative to 'statue', but not essential relative to 'clay' [Putnam] |
11908 | Putnam bases essences on 'same kind', but same kinds may not share properties [Mackie,P on Putnam] |
18890 | Putnam smuggles essentialism about liquids into his proof that water must be H2O [Salmon,N on Putnam] |
13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
4718 | If necessity is always relative to a description in a language, then there is only 'de dicto' necessity [Putnam, by O'Grady] |
10269 | Mathematics eliminates possibility, as being simultaneous actuality in sets [Putnam] |
9169 | A statement can be metaphysically necessary and epistemologically contingent [Putnam] |
5819 | Conceivability is no proof of possibility [Putnam] |
6404 | British Idealists said reality is a single Mind which experiences itself [Bradley, by Grayling] |
22299 | Bradley's objective idealism accepts reality (the Absolute), but says we can't fully describe it [Bradley, by Potter] |
21343 | Qualities and relations are mere appearance; the Absolute is a single undifferentiated substance [Bradley, by Heil] |
6284 | If a tautology is immune from revision, why would that make it true? [Putnam] |
17642 | The old view that sense data are independent of mind is quite dotty [Putnam] |
6273 | Knowledge depends on believing others, which must be innate, as inferences are not strong enough [Putnam] |
6274 | Empathy may not give knowledge, but it can give plausibility or right opinion [Putnam] |
7620 | Some kind of objective 'rightness' is a presupposition of thought itself [Putnam] |
18960 | Most predictions are uninteresting, and are only sought in order to confirm a theory [Putnam] |
17508 | Science aims at truth, not at 'simplicity' [Putnam] |
14204 | Naïve operationalism would have meanings change every time the tests change [Putnam] |
17084 | You can't decide which explanations are good if you don't attend to the interest-relative aspects [Putnam] |
7705 | The Twin Earth theory suggests that intentionality is independent of qualia [Jacquette on Putnam] |
2590 | Dispositions need mental terms to define them [Putnam] |
3460 | Superactors and superspartans count against behaviourism [Putnam, by Searle] |
2591 | Total paralysis would mean that there were mental states but no behaviour at all [Putnam] |
2588 | Is pain a functional state of a complete organism? [Putnam] |
2589 | Functionalism is compatible with dualism, as pure mind could perform the functions [Putnam] |
2592 | Functional states correlate with AND explain pain behaviour [Putnam] |
5495 | Instances of pain are physical tokens, but the nature of pain is more abstract [Putnam, by Lycan] |
2331 | Functionalism says robots and people are the same at one level of abstraction [Putnam] |
2071 | If concepts have external meaning, computational states won't explain psychology [Putnam] |
2332 | Functionalism can't explain reference and truth, which are needed for logic [Putnam] |
2348 | Is there just one computational state for each specific belief? [Putnam] |
2587 | Temperature is mean molecular kinetic energy, but they are two different concepts [Putnam] |
2344 | If we are going to eliminate folk psychology, we must also eliminate folk logic [Putnam] |
6376 | Neuroscience does not support multiple realisability, and tends to support identity [Polger on Putnam] |
2330 | If humans and molluscs both feel pain, it can't be a single biological state [Putnam, by Kim] |
2074 | Can we give a scientific, computational account of folk psychology? [Putnam] |
7611 | Rationality is one part of our conception of human flourishing [Putnam] |
2605 | If everything uses mentalese, ALL concepts must be innate! [Putnam] |
2606 | No machine language can express generalisations [Putnam] |
4099 | If Twins talking about 'water' and 'XYZ' have different thoughts but identical heads, then thoughts aren't in the head [Putnam, by Crane] |
12026 | We say ice and steam are different forms of water, but not that they are different forms of H2O [Forbes,G on Putnam] |
3208 | Does 'water' mean a particular substance that was 'dubbed'? [Putnam, by Rey] |
14200 | 'Water' on Twin Earth doesn't refer to water, but no mental difference can account for this [Putnam] |
2343 | Reference may be different while mental representation is the same [Putnam] |
9168 | I can't distinguish elm trees, but I mean by 'elm' the same set of trees as everybody else [Putnam] |
5820 | 'Water' has an unnoticed indexical component, referring to stuff around here [Putnam] |
7612 | Reference is social not individual, because we defer to experts when referring to elm trees [Putnam] |
7613 | Concepts are (at least in part) abilities and not occurrences [Putnam] |
6282 | Theory of meaning presupposes theory of understanding and reference [Putnam] |
2346 | Meaning and translation (which are needed to define truth) both presuppose the notion of reference [Putnam] |
6281 | Truth conditions can't explain understanding a sentence, because that in turn needs explanation [Putnam] |
6278 | We should reject the view that truth is prior to meaning [Putnam] |
2354 | "Meaning is use" is not a definition of meaning [Putnam] |
2336 | Holism seems to make fixed definition more or less impossible [Putnam] |
2334 | Meaning holism tried to show that you can't get fixed meanings built out of observation terms [Putnam] |
2335 | Understanding a sentence involves background knowledge and can't be done in isolation [Putnam] |
6271 | How reference is specified is not what reference is [Putnam] |
2340 | We should separate how the reference of 'gold' is fixed from its conceptual content [Putnam] |
2341 | Like names, natural kind terms have their meaning fixed by extension and reference [Putnam] |
17506 | I now think reference by the tests of experts is a special case of being causally connected [Putnam] |
2338 | Reference (say to 'elms') is a social phenomenon which we can leave to experts [Putnam] |
14202 | Neither individual nor community mental states fix reference [Putnam] |
9170 | We need to recognise the contribution of society and of the world in determining reference [Putnam] |
14201 | Maybe the total mental state of a language community fixes the reference of a term [Putnam] |
2339 | Aristotle implies that we have the complete concepts of a language in our heads, but we don't [Putnam] |
3893 | Often reference determines sense, and not (as Frege thought) vice versa [Putnam, by Scruton] |
6268 | The claim that scientific terms are incommensurable can be blocked if scientific terms are not descriptions [Putnam] |
13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
5817 | Language is more like a cooperative steamship than an individual hammer [Putnam] |
6279 | A private language could work with reference and beliefs, and wouldn't need meaning [Putnam] |
6270 | The correct translation is the one that explains the speaker's behaviour [Putnam] |
6283 | Language maps the world in many ways (because it maps onto other languages in many ways) [Putnam] |
14206 | There are infinitely many interpretations of a sentence which can all seem to be 'correct' [Putnam] |
6275 | You can't say 'most speaker's beliefs are true'; in some areas this is not so, and you can't count beliefs [Putnam] |
7624 | The word 'inconsiderate' nicely shows the blurring of facts and values [Putnam] |
5655 | Happiness is not satisfaction of desires, but fulfilment of values [Bradley, by Scruton] |
6406 | Reality is one, because plurality implies relations, and they assert a superior unity [Bradley] |
11191 | The hidden structure of a natural kind determines membership in all possible worlds [Putnam] |
17507 | Natural kind stereotypes are 'strong' (obvious, like tiger) or 'weak' (obscure, like molybdenum) [Putnam] |
11904 | Express natural kinds as a posteriori predicate connections, not as singular terms [Putnam, by Mackie,P] |
2342 | "Water" is a natural kind term, but "H2O" is a description [Putnam] |
17645 | An alien might think oxygen was the main cause of a forest fire [Putnam] |
11192 | If causes are the essence of diseases, then disease is an example of a relational essence [Putnam, by Williams,NE] |
11190 | Archimedes meant by 'gold' the hidden structure or essence of the stuff [Putnam] |
5818 | If water is H2O in the actual world, there is no possible world where it isn't H2O [Putnam] |