183 ideas
13734 | Modern Quinean metaphysics is about what exists, but Aristotelian metaphysics asks about grounding [Schaffer,J] |
13751 | If you tore the metaphysics out of philosophy, the whole enterprise would collapse [Schaffer,J] |
14600 | Analysis aims at secure necessary and sufficient conditions [Schaffer,J] |
13743 | We should not multiply basic entities, but we can have as many derivative entities as we like [Schaffer,J] |
8368 | A correct definition is what can be substituted without loss of meaning [Ducasse] |
18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
14603 | 'Reification' occurs if we mistake a concept for a thing [Schaffer,J] |
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] |
14607 | T adds □p→p for reflexivity, and is ideal for modeling lawhood [Schaffer,J] |
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] |
10373 | Logical form can't dictate metaphysics, as it may propose an undesirable property [Schaffer,J] |
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] |
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] |
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] |
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] |
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] |
13741 | If 'there are red roses' implies 'there are roses', then 'there are prime numbers' implies 'there are numbers' [Schaffer,J] |
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] |
14604 | If a notion is ontologically basic, it should be needed in our best attempt at science [Schaffer,J] |
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] |
13748 | Grounding is unanalysable and primitive, and is the basic structuring concept in metaphysics [Schaffer,J] |
17304 | As causation links across time, grounding links the world across levels [Schaffer,J] |
17306 | If ground is transitive and irreflexive, it has a strict partial ordering, giving structure [Schaffer,J] |
14599 | Three types of reduction: Theoretical (of terms), Definitional (of concepts), Ontological (of reality) [Schaffer,J] |
13747 | Supervenience is just modal correlation [Schaffer,J] |
13744 | The cosmos is the only fundamental entity, from which all else exists by abstraction [Schaffer,J] |
10367 | There is only one fact - the True [Schaffer,J] |
13739 | Maybe categories are just the different ways that things depend on basic substances [Schaffer,J] |
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] |
14605 | Tropes are the same as events [Schaffer,J] |
14601 | Individuation aims to count entities, by saying when there is one [Schaffer,J] |
14082 | No sortal could ever exactly pin down which set of particles count as this 'cup' [Schaffer,J] |
13742 | There exist heaps with no integral unity, so we should accept arbitrary composites in the same way [Schaffer,J] |
13752 | The notion of 'grounding' can explain integrated wholes in a way that mere aggregates can't [Schaffer,J] |
13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
14081 | Identities can be true despite indeterminate reference, if true under all interpretations [Schaffer,J] |
13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
14606 | Only ideal conceivability could indicate what is possible [Schaffer,J] |
13749 | Belief in impossible worlds may require dialetheism [Schaffer,J] |
13740 | 'Moorean certainties' are more credible than any sceptical argument [Schaffer,J] |
17308 | Explaining 'Adam ate the apple' depends on emphasis, and thus implies a contrast [Schaffer,J] |
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] |
17305 | I take what is fundamental to be the whole spatiotemporal manifold and its fields [Schaffer,J] |
10359 | In causation there are three problems of relata, and three metaphysical problems [Schaffer,J] |
10372 | Causation may not be transitive; the last event may follow from the first, but not be caused by it [Schaffer,J] |
10374 | There are at least ten theories about causal connections [Schaffer,J] |
17307 | Nowadays causation is usually understood in terms of equations and variable ranges [Schaffer,J] |
8367 | Causation is defined in terms of a single sequence, and constant conjunction is no part of it [Ducasse] |
10366 | Causation transcends nature, because absences can cause things [Schaffer,J] |
10377 | Causation may not be a process, if a crucial part of the process is 'disconnected' [Schaffer,J] |
10378 | A causal process needs to be connected to the effect in the right way [Schaffer,J] |
10382 | Causation can't be a process, because a process needs causation as a primitive [Schaffer,J] |
10375 | At least four rivals have challenged the view that causal direction is time direction [Schaffer,J] |
10389 | Causal order must be temporal, or else causes could be blocked, and time couldn't be explained [Schaffer,J] |
10390 | Causal order is not temporal, because of time travel, and simultanous, joint or backward causes [Schaffer,J] |
10380 | Causation is primitive; it is too intractable and central to be reduced; all explanations require it [Schaffer,J] |
10385 | If causation is just observables, or part of common sense, or vacuous, it can't be primitive [Schaffer,J] |
10387 | The notion of causation allows understanding of science, without appearing in equations [Schaffer,J] |
10388 | Causation is utterly essential for numerous philosophical explanations [Schaffer,J] |
8372 | We see what is in common between causes to assign names to them, not to perceive them [Ducasse] |
10384 | If two different causes are possible in one set of circumstances, causation is primitive [Schaffer,J] |
10386 | If causation is primitive, it can be experienced in ourselves, or inferred as best explanation [Schaffer,J] |
10361 | Events are fairly course-grained (just saying 'hello'), unlike facts (like saying 'hello' loudly) [Schaffer,J] |
10360 | Causal relata are events - or facts, features, tropes, states, situations or aspects [Schaffer,J] |
10362 | One may defend three or four causal relata, as in 'c causes e rather than e*' [Schaffer,J] |
10368 | If causal relata must be in nature and fine-grained, neither facts nor events will do [Schaffer,J] |
10383 | The relata of causation (such as events) need properties as explanation, which need causation! [Schaffer,J] |
8369 | Causes are either sufficient, or necessary, or necessitated, or contingent upon [Ducasse] |
8373 | When a brick and a canary-song hit a window, we ignore the canary if we are interested in the breakage [Ducasse] |
8370 | A cause is a change which occurs close to the effect and just before it [Ducasse] |
10393 | Our selection of 'the' cause is very predictable, so must have a basis [Schaffer,J] |
10394 | Selecting 'the' cause must have a basis; there is no causation without such a selection [Schaffer,J] |
10376 | The actual cause may make an event less likely than a possible more effective cause [Schaffer,J] |
10381 | All four probability versions of causation may need causation to be primitive [Schaffer,J] |
8371 | Recurrence is only relevant to the meaning of law, not to the meaning of cause [Ducasse] |
8374 | We are interested in generalising about causes and effects purely for practical purposes [Ducasse] |