187 ideas
| 16440 | I don't think Lewis's cost-benefit reflective equilibrium approach offers enough guidance [Stalnaker] |
| 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] |
| 16468 | Non-S5 can talk of contingent or necessary necessities [Stalnaker] |
| 18823 | To say there could have been people who don't exist, but deny those possible things, rejects Barcan [Stalnaker, by Rumfitt] |
| 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] |
| 16449 | In modal set theory, sets only exist in a possible world if that world contains all of its members [Stalnaker] |
| 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] |
| 12766 | Logical space is abstracted from the actual world [Stalnaker] |
| 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] |
| 16464 | We regiment to get semantic structure, for evaluating arguments, and understanding complexities [Stalnaker] |
| 13357 | Truth-functors are usually held to be defined by their truth-tables [Bostock] |
| 16465 | In 'S was F or some other than S was F', the disjuncts need S, but the whole disjunction doesn't [Stalnaker] |
| 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] |
| 16405 | To understand a name (unlike a description) picking the thing out is sufficient? [Stalnaker] |
| 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] |
| 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] |
| 16434 | Some say what exists must do so, and nothing else could possible exist [Stalnaker] |
| 16439 | A nominalist view says existence is having spatio-temporal location [Stalnaker] |
| 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] |
| 16443 | Properties are modal, involving possible situations where they are exemplified [Stalnaker] |
| 16471 | I accept a hierarchy of properties of properties of properties [Stalnaker] |
| 16452 | Dispositions have modal properties, of which properties things would have counterfactually [Stalnaker] |
| 14617 | Predicates can't apply to what doesn't exist [Stalnaker] |
| 12764 | For the bare particular view, properties must be features, not just groups of objects [Stalnaker] |
| 16407 | Possible worlds allow separating all the properties, without hitting a bare particular [Stalnaker] |
| 12761 | An essential property is one had in all the possible worlds where a thing exists [Stalnaker] |
| 16467 | 'Socrates is essentially human' seems to say nothing could be Socrates if it was not human [Stalnaker] |
| 12763 | Necessarily self-identical, or being what it is, or its world-indexed properties, aren't essential [Stalnaker] |
| 12762 | Bare particular anti-essentialism makes no sense within modal logic semantics [Stalnaker] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 16453 | The bundle theory makes the identity of indiscernibles a necessity, since the thing is the properties [Stalnaker] |
| 16466 | Strong necessity is always true; weak necessity is cannot be false [Stalnaker] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 14286 | In nearby worlds where A is true, 'if A,B' is true or false if B is true or false [Stalnaker] |
| 10994 | Conditionals are true if minimal revision of the antecedent verifies the consequent [Stalnaker, by Read] |
| 16438 | Necessity and possibility are fundamental, and there can be no reductive analysis of them [Stalnaker] |
| 16422 | The necessity of a proposition concerns reality, not our words or concepts [Stalnaker] |
| 16423 | Conceptual possibilities are metaphysical possibilities we can conceive of [Stalnaker] |
| 16436 | Modal concepts are central to the actual world, and shouldn't need extravagant metaphysics [Stalnaker] |
| 16421 | Critics say there are just an a priori necessary part, and an a posteriori contingent part [Stalnaker] |
| 16429 | A 'centred' world is an ordered triple of world, individual and time [Stalnaker] |
| 16397 | If it might be true, it might be true in particular ways, and possible worlds describe such ways [Stalnaker] |
| 16399 | Possible worlds are ontologically neutral, but a commitment to possibilities remains [Stalnaker] |
| 16398 | Possible worlds allow discussion of modality without controversial modal auxiliaries [Stalnaker] |
| 16433 | Given actualism, how can there be possible individuals, other than the actual ones? [Stalnaker] |
| 14285 | A possible world is the ontological analogue of hypothetical beliefs [Stalnaker] |
| 15793 | We can take 'ways things might have been' as irreducible elements in our ontology [Stalnaker, by Lycan] |
| 16396 | Kripke's possible worlds are methodological, not metaphysical [Stalnaker] |
| 16437 | Possible worlds are properties [Stalnaker] |
| 16444 | Possible worlds don't reduce modality, they regiment it to reveal its structure [Stalnaker] |
| 16445 | I think of worlds as cells (rather than points) in logical space [Stalnaker] |
| 12765 | Why imagine that Babe Ruth might be a billiard ball; nothing useful could be said about the ball [Stalnaker] |
| 16408 | Rigid designation seems to presuppose that differing worlds contain the same individuals [Stalnaker] |
| 16409 | Unlike Lewis, I defend an actualist version of counterpart theory [Stalnaker] |
| 16411 | If possible worlds really differ, I can't be in more than one at a time [Stalnaker] |
| 16412 | If counterparts exist strictly in one world only, this seems to be extreme invariant essentialism [Stalnaker] |
| 16454 | Modal properties depend on the choice of a counterpart, which is unconstrained by metaphysics [Stalnaker] |
| 16450 | Anti-haecceitism says there is no more to an individual than meeting some qualitative conditions [Stalnaker] |
| 16428 | Meanings aren't in the head, but that is because they are abstract [Stalnaker] |
| 16474 | How can we know what we are thinking, if content depends on something we don't know? [Stalnaker] |
| 16406 | If you don't know what you say you can't mean it; what people say usually fits what they mean [Stalnaker] |
| 16404 | In the use of a name, many individuals are causally involved, but they aren't all the referent [Stalnaker] |
| 16432 | One view says the causal story is built into the description that is the name's content [Stalnaker] |
| 16403 | 'Descriptive' semantics gives a system for a language; 'foundational' semantics give underlying facts [Stalnaker] |
| 16461 | We still lack an agreed semantics for quantifiers in natural language [Stalnaker] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 16401 | To understand an utterance, you must understand what the world would be like if it is true [Stalnaker] |
| 16410 | Extensional semantics has individuals and sets; modal semantics has intensions, functions of world to extension [Stalnaker] |
| 16448 | Possible world semantics may not reduce modality, but it can explain it [Stalnaker] |
| 16430 | Two-D says that a posteriori is primary and contingent, and the necessity is the secondary intension [Stalnaker] |
| 16431 | In one view, the secondary intension is metasemantic, about how the thinker relates to the content [Stalnaker] |
| 16442 | I take propositions to be truth conditions [Stalnaker] |
| 16447 | A theory of propositions at least needs primitive properties of consistency and of truth [Stalnaker] |
| 14616 | A 'Russellian proposition' is an ordered sequence of individual, properties and relations [Stalnaker] |
| 16446 | Propositions presumably don't exist if the things they refer to don't exist [Stalnaker] |
| 18052 | An assertion aims to add to the content of a context [Stalnaker, by Magidor] |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| 14718 | An assertion is an attempt to rule out certain possibilities, narrowing things down for good planning [Stalnaker, by Schroeter] |
| 7517 | I could take a healthy infant and train it up to be any type of specialist I choose [Watson,JB] |