192 ideas
| 9408 | Science studies phenomena, but only metaphysics tells us what exists [Mumford] |
| 9429 | Many forms of reasoning, such as extrapolation and analogy, are useful but deductively invalid [Mumford] |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| 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] |
| 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] |
| 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] |
| 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] |
| 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] |
| 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] |
| 13361 | An expression is only a name if it succeeds in referring to a real object [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] |
| 13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [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] |
| 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] |
| 13616 | The Deduction Theorem greatly simplifies the search for proof [Bostock] |
| 18120 | The Deduction Theorem is what licenses a system of natural deduction [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] |
| 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] |
| 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] |
| 13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock] |
| 13760 | A sequent calculus is good for comparing proof systems [Bostock] |
| 13364 | Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG] |
| 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] |
| 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] |
| 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] |
| 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] |
| 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] |
| 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] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
| 9427 | For Humeans the world is a world primarily of events [Mumford] |
| 14334 | Modest realism says there is a reality; the presumptuous view says we can accurately describe it [Mumford] |
| 14306 | Anti-realists deny truth-values to all statements, and say evidence and ontology are inseparable [Mumford] |
| 13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
| 13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
| 14333 | Dispositions and categorical properties are two modes of presentation of the same thing [Mumford] |
| 14336 | Categorical predicates are those unconnected to functions [Mumford] |
| 14315 | Categorical properties and dispositions appear to explain one another [Mumford] |
| 14332 | There are four reasons for seeing categorical properties as the most fundamental [Mumford] |
| 14302 | A lead molecule is not leaden, and macroscopic properties need not be microscopically present [Mumford] |
| 14294 | Dispositions are attacked as mere regularities of events, or place-holders for unknown properties [Mumford] |
| 9446 | Properties are just natural clusters of powers [Mumford] |
| 14316 | If dispositions have several categorical realisations, that makes the two separate [Mumford] |
| 14310 | Dispositions are classifications of properties by functional role [Mumford] |
| 14317 | I say the categorical base causes the disposition manifestation [Mumford] |
| 14313 | All properties must be causal powers (since they wouldn't exist otherwise) [Mumford] |
| 14318 | Intrinsic properties are just causal powers, and identifying a property as causal is then analytic [Mumford] |
| 14298 | Dispositions can be contrasted either with occurrences, or with categorical properties [Mumford] |
| 14293 | Dispositions are ascribed to at least objects, substances and persons [Mumford] |
| 14326 | Unlike categorical bases, dispositions necessarily occupy a particular causal role [Mumford] |
| 14314 | If dispositions are powers, background conditions makes it hard to say what they do [Mumford] |
| 14325 | Maybe dispositions can replace powers in metaphysics, as what induces property change [Mumford] |
| 14312 | Orthodoxy says dispositions entail conditionals (rather than being equivalent to them) [Mumford] |
| 14291 | Dispositions are not just possibilities - they are features of actual things [Mumford] |
| 14299 | There could be dispositions that are never manifested [Mumford] |
| 14323 | If every event has a cause, it is easy to invent a power to explain each case [Mumford] |
| 14328 | Traditional powers initiate change, but are mysterious between those changes [Mumford] |
| 14331 | Categorical eliminativists say there are no dispositions, just categorical states or mechanisms [Mumford] |
| 9435 | A 'porridge' nominalist thinks we just divide reality in any way that suits us [Mumford] |
| 9447 | If properties are clusters of powers, this can explain why properties resemble in degrees [Mumford] |
| 18617 | Substances, unlike aggregates, can survive a change of parts [Mumford] |
| 12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
| 14295 | Many artefacts have dispositional essences, which make them what they are [Mumford] |
| 12248 | How can we show that a universally possessed property is an essential property? [Mumford] |
| 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] |
| 18618 | Maybe possibilities are recombinations of the existing elements of reality [Mumford] |
| 18619 | Combinatorial possibility has to allow all elements to be combinable, which seems unlikely [Mumford] |
| 18620 | Combinatorial possibility relies on what actually exists (even over time), but there could be more [Mumford] |
| 14309 | Truth-functional conditionals can't distinguish whether they are causal or accidental [Mumford] |
| 14311 | Dispositions are not equivalent to stronger-than-material conditionals [Mumford] |
| 14319 | Nomothetic explanations cite laws, and structural explanations cite mechanisms [Mumford] |
| 14342 | General laws depend upon the capacities of particulars, not the other way around [Mumford] |
| 14322 | If fragile just means 'breaks when dropped', it won't explain a breakage [Mumford] |
| 14337 | Maybe dispositions can replace the 'laws of nature' as the basis of explanation [Mumford] |
| 14343 | To avoid a regress in explanations, ungrounded dispositions will always have to be posited [Mumford] |
| 14320 | Subatomic particles may terminate explanation, if they lack structure [Mumford] |
| 14324 | Ontology is unrelated to explanation, which concerns modes of presentation and states of knowledge [Mumford] |
| 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] |
| 14344 | Natural kinds, such as electrons, all behave the same way because we divide them by dispositions [Mumford] |
| 19068 | Causation interests us because we want to explain change [Mumford] |
| 9430 | Singular causes, and identities, might be necessary without falling under a law [Mumford] |
| 9445 | We can give up the counterfactual account if we take causal language at face value [Mumford] |
| 9443 | It is only properties which are the source of necessity in the world [Mumford] |
| 14338 | In the 'laws' view events are basic, and properties are categorical, only existing when manifested [Mumford] |
| 9444 | There are four candidates for the logical form of law statements [Mumford] |
| 14339 | Without laws, how can a dispositionalist explain general behaviour within kinds? [Mumford] |
| 14341 | Dretske and Armstrong base laws on regularities between individual properties, not between events [Mumford] |
| 9441 | Regularity laws don't explain, because they have no governing role [Mumford] |
| 14340 | It is a regularity that whenever a person sneezes, someone (somewhere) promptly coughs [Mumford] |
| 9431 | Pure regularities are rare, usually only found in idealized conditions [Mumford] |
| 9416 | Regularities are more likely with few instances, and guaranteed with no instances! [Mumford] |
| 9415 | Would it count as a regularity if the only five As were also B? [Mumford] |
| 9422 | If the best system describes a nomological system, the laws are in nature, not in the description [Mumford] |
| 9421 | The best systems theory says regularities derive from laws, rather than constituting them [Mumford] |
| 9432 | Laws of nature are necessary relations between universal properties, rather than about particulars [Mumford] |
| 9433 | If laws can be uninstantiated, this favours the view of them as connecting universals [Mumford] |
| 14345 | The necessity of an electron being an electron is conceptual, and won't ground necessary laws [Mumford] |
| 9434 | Laws of nature are just the possession of essential properties by natural kinds [Mumford] |
| 14307 | Some dispositions are so far unknown, until we learn how to manifest them [Mumford] |
| 9437 | To distinguish accidental from essential properties, we must include possible members of kinds [Mumford] |
| 9439 | The Central Dilemma is how to explain an internal or external view of laws which govern [Mumford] |
| 9412 | You only need laws if you (erroneously) think the world is otherwise inert [Mumford] |
| 9411 | There are no laws of nature in Aristotle; they became standard with Descartes and Newton [Mumford] |