229 ideas
| 12667 | Metaphysics aims at the simplest explanation, without regard to testability [Ellis] |
| 13567 | Ontology should give insight into or an explanation of the world revealed by science [Ellis] |
| 5486 | Essentialism says metaphysics can't be done by analysing unreliable language [Ellis] |
| 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] |
| 13604 | Real possibility and necessity has the logic of S5, which links equivalence classes of worlds of the same kind [Ellis] |
| 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] |
| 12666 | We can base logic on acceptability, and abandon the Fregean account by truth-preservation [Ellis] |
| 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] |
| 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] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [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] |
| 13606 | Humean conceptions of reality drive the adoption of extensional logic [Ellis] |
| 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] |
| 12688 | Mathematics is the formal study of the categorical dimensions of things [Ellis] |
| 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] |
| 12683 | Objects and substances are a subcategory of the natural kinds of processes [Ellis] |
| 12670 | A physical event is any change of distribution of energy [Ellis] |
| 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] |
| 13584 | The extension of a property is a contingent fact, so cannot be the essence of the property [Ellis] |
| 5468 | Properties are 'dispositional', or 'categorical' (the latter as 'block' or 'intrinsic' structures) [Ellis, by PG] |
| 13587 | There is no property of 'fragility', as things are each fragile in a distinctive way [Ellis] |
| 12673 | Physical properties are those relevant to how a physical system might act [Ellis] |
| 13577 | Typical 'categorical' properties are spatio-temporal, such as shape [Ellis] |
| 9436 | The property of 'being an electron' is not of anything, and only electrons could have it [Ellis] |
| 5469 | The passive view of nature says categorical properties are basic, but others say dispositions [Ellis] |
| 12665 | I support categorical properties, although most people only want causal powers [Ellis] |
| 12682 | Essentialism needs categorical properties (spatiotemporal and numerical relations) and dispositions [Ellis] |
| 12684 | Spatial, temporal and numerical relations have causal roles, without being causal [Ellis] |
| 13582 | 'Being a methane molecule' is not a property - it is just a predicate [Ellis] |
| 12672 | Properties and relations are discovered, so they can't be mere sets of individuals [Ellis] |
| 5456 | Redness is not a property as it is not mind-independent [Ellis] |
| 18398 | Space, time, and some other basics, are not causal powers [Ellis] |
| 13580 | Causal powers must necessarily act the way they do [Ellis] |
| 13598 | Causal powers are often directional (e.g. centripetal, centrifugal, circulatory) [Ellis] |
| 12676 | Causal powers can't rest on things which lack causal power [Ellis] |
| 13568 | Basic powers may not be explained by structure, if at the bottom level there is no structure [Ellis] |
| 13586 | Maybe dispositions can be explained by intrinsic properties or structures [Ellis] |
| 5481 | Properties have powers; they aren't just ways for logicians to classify objects [Ellis] |
| 23781 | Categoricals exist to influence powers. Such as structures, orientations and magnitudes [Ellis, by Williams,NE] |
| 13585 | The most fundamental properties of nature (mass, charge, spin ...) all seem to be dispositions [Ellis] |
| 5458 | Nearly all fundamental properties of physics are dispositional [Ellis] |
| 13596 | A causal power is a disposition to produce forces [Ellis] |
| 13599 | Powers are dispositions of the essences of kinds that involve them in causation [Ellis] |
| 12686 | Causal powers are a proper subset of the dispositional properties [Ellis] |
| 13573 | Universals are all types of natural kind [Ellis] |
| 13572 | There are 'substantive' (objects of some kind), 'dynamic' (events of some kind) and 'property' universals [Ellis] |
| 12685 | Categorical properties depend only on the structures they represent [Ellis] |
| 5443 | Kripke and others have made essentialism once again respectable [Ellis] |
| 5444 | 'Individual essences' fix a particular individual, and 'kind essences' fix the kind it belongs to [Ellis] |
| 13571 | Scientific essentialism doesn't really need Kripkean individual essences [Ellis] |
| 12679 | A real essence is a kind's distinctive properties [Ellis] |
| 5462 | Essential properties are usually quantitatively determinate [Ellis] |
| 5448 | 'Real essence' makes it what it is; 'nominal essence' makes us categorise it a certain way [Ellis] |
| 13578 | The old idea that identity depends on essence and behaviour is rejected by the empiricists [Ellis] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 5477 | One thing can look like something else, without being the something else [Ellis] |
| 13576 | Necessities are distinguished by their grounds, not their different modalities [Ellis] |
| 12668 | Metaphysical necessity holds between things in the world and things they make true [Ellis] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 5479 | Scientific essentialists say science should define the limits of the possible [Ellis] |
| 12687 | Metaphysical necessities are those depending on the essential nature of things [Ellis] |
| 5483 | Essentialists deny possible worlds, and say possibilities are what is compatible with the actual world [Ellis] |
| 13570 | Individual essences necessitate that individual; natural kind essences necessitate kind membership [Ellis] |
| 5447 | Metaphysical necessities are true in virtue of the essences of things [Ellis] |
| 5476 | Essentialists say natural laws are in a new category: necessary a posteriori [Ellis] |
| 5478 | Imagination tests what is possible for all we know, not true possibility [Ellis] |
| 5482 | Possible worlds realism is only needed to give truth conditions for modals and conditionals [Ellis] |
| 5453 | Essentialists mostly accept the primary/secondary qualities distinction [Ellis] |
| 5466 | Primary qualities are number, figure, size, texture, motion, configuration, impenetrability and (?) mass [Ellis] |
| 12669 | Science aims to explain things, not just describe them [Ellis] |
| 13607 | If events are unconnected, then induction cannot be solved [Ellis] |
| 5485 | Emeralds are naturally green, and only an external force could turn them blue [Ellis] |
| 13597 | Good explanations unify [Ellis] |
| 5484 | Essentialists don't infer from some to all, but from essences to necessary behaviour [Ellis] |
| 13601 | Explanations of particular events are not essentialist, as they don't reveal essential structures [Ellis] |
| 13569 | To give essentialist explanations there have to be natural kinds [Ellis] |
| 13600 | The point of models in theories is not to idealise, but to focus on what is essential [Ellis] |
| 5457 | Predicates assert properties, values, denials, relations, conventions, existence and fabrications [Ellis, by PG] |
| 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] |
| 5488 | Regularity theories of causation cannot give an account of human agency [Ellis] |
| 5489 | Humans have variable dispositions, and also power to change their dispositions [Ellis] |
| 5490 | Essentialism fits in with Darwinism, but not with extreme politics of left or right [Ellis] |
| 5472 | Natural kinds are of objects/substances, or events/processes, or intrinsic natures [Ellis] |
| 6613 | The natural kinds are objects, processes and properties/relations [Ellis] |
| 12681 | There are natural kinds of processes [Ellis] |
| 13583 | There might be uninstantiated natural kinds, such as transuranic elements which have never occurred [Ellis] |
| 13574 | Natural kinds are distinguished by resting on essences [Ellis] |
| 5471 | Essentialism says natural kinds are fundamental to nature, and determine the laws [Ellis] |
| 12680 | Natural kind structures go right down to the bottom level [Ellis] |
| 5446 | For essentialists two members of a natural kind must be identical [Ellis] |
| 5480 | The whole of our world is a natural kind, so all worlds like it necessarily have the same laws [Ellis] |
| 13575 | If there are borderline cases between natural kinds, that makes them superficial [Ellis] |
| 5445 | Essentialists regard inanimate objects as genuine causal agents [Ellis] |
| 5463 | Essentialists believe causation is necessary, resulting from dispositions and circumstances [Ellis] |
| 5491 | A general theory of causation is only possible in an area if natural kinds are involved [Ellis] |
| 5442 | For 'passivists' behaviour is imposed on things from outside [Ellis] |
| 5473 | The laws of nature imitate the hierarchy of natural kinds [Ellis] |
| 5474 | Laws of nature tend to describe ideal things, or ideal circumstances [Ellis] |
| 5475 | We must explain the necessity, idealisation, ontology and structure of natural laws [Ellis] |
| 13595 | Laws don't exist in the world; they are true of the world [Ellis] |
| 6616 | Least action is not a causal law, but a 'global law', describing a global essence [Ellis] |
| 12675 | Laws of nature are just descriptions of how things are disposed to behave [Ellis] |
| 5460 | Causal relations cannot be reduced to regularities, as they could occur just once [Ellis] |
| 13566 | A proton must have its causal role, because without it it wouldn't be a proton [Ellis] |
| 13579 | What is most distinctive of scientific essentialism is regarding processes as natural kinds [Ellis] |
| 13581 | Scientific essentialism is more concerned with explanation than with identity (Locke, not Kripke) [Ellis] |
| 13594 | The ontological fundamentals are dispositions, and also categorical (spatio-temporal and structural) properties [Ellis] |
| 5459 | Essentialists say dispositions are basic, rather than supervenient on matter and natural laws [Ellis] |
| 5461 | The essence of uranium is its atomic number and its electron shell [Ellis] |
| 6615 | A species requires a genus, and its essence includes the essence of the genus [Ellis] |
| 13603 | A primary aim of science is to show the limits of the possible [Ellis] |
| 5464 | For essentialists, laws of nature are metaphysically necessary, being based on essences of natural kinds [Ellis] |
| 6614 | A hierarchy of natural kinds is elaborate ontology, but needed to explain natural laws [Ellis] |
| 5487 | Essentialism requires a clear separation of semantics, epistemology and ontology [Ellis] |
| 6612 | Without general principles, we couldn't predict the behaviour of dispositional properties [Ellis] |
| 12671 | I deny forces as entities that intervene in causation, but are not themselves causal [Ellis] |
| 12674 | Energy is the key multi-valued property, vital to scientific realism [Ellis] |
| 12689 | Simultaneity can be temporal equidistance from the Big Bang [Ellis] |
| 12690 | The present is the collapse of the light wavefront from the Big Bang [Ellis] |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |