200 ideas
| 6123 | Empirical investigation can't discover if holes exist, or if two things share a colour [Merricks] |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| 19215 | Arguers often turn the opponent's modus ponens into their own modus tollens [Merricks] |
| 14415 | A ground must be about its truth, and not just necessitate it [Merricks] |
| 14408 | Truthmaker needs truths to be 'about' something, and that is often unclear [Merricks] |
| 14395 | If a ball changes from red to white, Truthmaker says some thing must make the change true [Merricks] |
| 14398 | Truthmaker says if an entity is removed, some nonexistence truthmaker must replace it [Merricks] |
| 14403 | If Truthmaker says each truth is made by the existence of something, the theory had de re modality at is core [Merricks] |
| 14397 | Truthmaker demands not just a predication, but an existing state of affairs with essential ingredients [Merricks] |
| 14396 | If 'truth supervenes on being', worlds with the same entities, properties and relations have the same truths [Merricks] |
| 14400 | If truth supervenes on being, that won't explain why truth depends on being [Merricks] |
| 14394 | It is implausible that claims about non-existence are about existing things [Merricks] |
| 14390 | Truthmaker isn't the correspondence theory, because it offers no analysis of truth [Merricks] |
| 14412 | Speculations about non-existent things are not about existent things, so Truthmaker is false [Merricks] |
| 14414 | I am a truthmaker for 'that a human exists', but is it about me? [Merricks] |
| 14418 | Being true is not a relation, it is a primitive monadic property [Merricks] |
| 14391 | If the correspondence theory is right, then necessary truths must correspond to something [Merricks] |
| 19205 | 'Snow is white' only contingently expresses the proposition that snow is white [Merricks] |
| 14419 | Deflationism just says there is no property of being truth [Merricks] |
| 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] |
| 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] |
| 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] |
| 13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [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] |
| 19209 | Simple Quantified Modal Logc doesn't work, because the Converse Barcan is a theorem [Merricks] |
| 19208 | The Converse Barcan implies 'everything exists necessarily' is a consequence of 'necessarily, everything exists' [Merricks] |
| 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] |
| 15717 | Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan] |
| 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] |
| 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] |
| 13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [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] |
| 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
| 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [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] |
| 13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [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] |
| 13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock] |
| 13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [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] |
| 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] |
| 13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
| 13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
| 13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
| 13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
| 13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [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] |
| 19207 | Sentence logic maps truth values; predicate logic maps objects and sets [Merricks] |
| 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
| 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [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] |
| 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] |
| 15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
| 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] |
| 15711 | The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan] |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| 15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
| 15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
| 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] |
| 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] |
| 14393 | The totality state is the most plausible truthmaker for negative existential truths [Merricks] |
| 6143 | Prolonged events don't seem to endure or exist at any particular time [Merricks] |
| 6135 | A crumbling statue can't become vague, because vagueness is incoherent [Merricks] |
| 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] |
| 14413 | Some properties seem to be primitive, but others can be analysed [Merricks] |
| 6145 | Intrinsic properties are those an object still has even if only that object exists [Merricks] |
| 14416 | An object can have a disposition when the revelant conditional is false [Merricks] |
| 6124 | I say that most of the objects of folk ontology do not exist [Merricks] |
| 6134 | Is swimming pool water an object, composed of its mass or parts? [Merricks] |
| 14392 | Fregeans say 'hobbits do not exist' is just 'being a hobbit' is not exemplified [Merricks] |
| 6125 | We can eliminate objects without a commitment to simples [Merricks] |
| 14229 | Merricks agrees that there are no composite objects, but offers a different semantics [Merricks, by Liggins] |
| 6142 | The 'folk' way of carving up the world is not intrinsically better than quite arbitrary ways [Merricks] |
| 14472 | If atoms 'arranged baseballwise' break a window, that analytically entails that a baseball did it [Merricks, by Thomasson] |
| 14469 | Overdetermination: the atoms do all the causing, so the baseball causes no breakage [Merricks] |
| 6137 | Clay does not 'constitute' a statue, as they have different persistence conditions (flaking, squashing) [Merricks] |
| 6141 | There is no visible difference between statues, and atoms arranged statuewise [Merricks] |
| 6127 | 'Unrestricted composition' says any two things can make up a third thing [Merricks] |
| 6131 | Composition as identity is false, as identity is never between a single thing and many things [Merricks] |
| 6132 | Composition as identity is false, as it implies that things never change their parts [Merricks] |
| 6130 | 'Composition' says things are their parts; 'constitution' says a whole substance is an object [Merricks] |
| 6138 | It seems wrong that constitution entails that two objects are wholly co-located [Merricks] |
| 6128 | Objects decompose (it seems) into non-overlapping parts that fill its whole region [Merricks] |
| 14410 | You believe you existed last year, but your segment doesn't, so they have different beliefs [Merricks] |
| 19214 | In twinning, one person has the same origin as another person [Merricks] |
| 6136 | Eliminativism about objects gives the best understanding of the Sorites paradox [Merricks] |
| 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] |
| 14417 | Counterfactuals aren't about actuality, so they lack truthmakers or a supervenience base [Merricks] |
| 6133 | If my counterpart is happy, that is irrelevant to whether I 'could' have been happy [Merricks] |
| 14402 | If 'Fido is possibly black' depends on Fido's counterparts, then it has no actual truthmaker [Merricks] |
| 6150 | The 'warrant' for a belief is what turns a true belief into knowledge [Merricks] |
| 15713 | The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan] |
| 6144 | You hold a child in your arms, so it is not mental substance, or mental state, or software [Merricks] |
| 6140 | Maybe the word 'I' can only refer to persons [Merricks] |
| 6149 | Free will and determinism are incompatible, since determinism destroys human choice [Merricks] |
| 6148 | Human organisms can exercise downward causation [Merricks] |
| 6147 | The hypothesis of solipsism doesn't seem to be made incoherent by the nature of mental properties [Merricks] |
| 6146 | Before Creation it is assumed that God still had many many mental properties [Merricks] |
| 19217 | I don't accept that if a proposition is directly about an entity, it has a relation to the entity [Merricks] |
| 19203 | A sentence's truth conditions depend on context [Merricks] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 19200 | Propositions are standardly treated as possible worlds, or as structured [Merricks] |
| 19206 | 'Cicero is an orator' represents the same situation as 'Tully is an orator', so they are one proposition [Merricks] |
| 19202 | Propositions are necessary existents which essentially (but inexplicably) represent things [Merricks] |
| 19204 | True propositions existed prior to their being thought, and might never be thought [Merricks] |
| 19210 | The standard view of propositions says they never change their truth-value [Merricks] |
| 19201 | Propositions can be 'about' an entity, but that doesn't make the entity a constituent of it [Merricks] |
| 19211 | Early Russell says a proposition is identical with its truthmaking state of affairs [Merricks] |
| 19212 | Unity of the proposition questions: what unites them? can the same constituents make different ones? [Merricks] |
| 19213 | We want to explain not just what unites the constituents, but what unites them into a proposition [Merricks] |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| 17960 | Eternalism says all times are equally real, and future and past objects and properties are real [Merricks] |
| 17961 | Growing block has a subjective present and a growing edge - but these could come apart [Merricks, by PG] |
| 14407 | Presentist should deny there is a present time, and just say that things 'exist' [Merricks] |
| 14411 | Maybe only presentism allows change, by now having a property, and then lacking it [Merricks] |
| 14406 | Presentists say that things have existed and will exist, not that they are instantaneous [Merricks] |
| 14405 | How can a presentist explain an object's having existed? [Merricks] |