194 ideas
3798 | An overexamined life is as bad as an unexamined one [Dennett] |
3801 | Rationality requires the assumption that things are either for better or worse [Dennett] |
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] |
21554 | Sets always exceed terms, so all the sets must exceed all the sets [Lackey] |
21553 | It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey] |
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] |
18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
18146 | If Hume's Principle is the whole story, that implies structuralism [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] |
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] |
14308 | We can bring dispositions into existence, as in creating an identifier [Dennett, by Mumford] |
7384 | Words are fixed by being attached to similarity clusters, without mention of 'essences' [Dennett] |
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] |
2526 | Philosophers regularly confuse failures of imagination with insights into necessity [Dennett] |
3802 | Why pronounce impossible what you cannot imagine? [Dennett] |
7374 | Light wavelengths entering the eye are only indirectly related to object colours [Dennett] |
2523 | That every mammal has a mother is a secure reality, but without foundations [Dennett] |
3795 | Causal theories require the "right" sort of link (usually unspecified) [Dennett] |
7369 | Brains are essentially anticipation machines [Dennett] |
4608 | Minds are hard-wired, or trial-and-error, or experimental, or full self-aware [Dennett, by Heil] |
7393 | We can't draw a clear line between conscious and unconscious [Dennett] |
7367 | Perhaps the brain doesn't 'fill in' gaps in consciousness if no one is looking. [Dennett] |
4880 | Sentience comes in grades from robotic to super-human; we only draw a line for moral reasons [Dennett] |
2528 | Does consciousness need the concept of consciousness? [Dennett] |
2525 | Maybe language is crucial to consciousness [Dennett] |
7394 | Conscious events can only be explained in terms of unconscious events [Dennett] |
7391 | We can know a lot of what it is like to be a bat, and nothing important is unknown [Dennett] |
3158 | Theories of intentionality presuppose rationality, so can't explain it [Dennett] |
2527 | Unconscious intentionality is the foundation of the mind [Dennett] |
6624 | Dennett denies the existence of qualia [Dennett, by Lowe] |
4873 | What is it like to notice an uncomfortable position when you are asleep? [Dennett] |
7387 | "Qualia" can be replaced by complex dispositional brain states [Dennett] |
7658 | Obviously there can't be a functional anaylsis of qualia if they are defined by intrinsic properties [Dennett] |
7376 | We can't assume that dispositions will remain normal when qualia have been inverted [Dennett] |
7372 | In peripheral vision we see objects without their details, so blindsight is not that special [Dennett] |
7373 | Blindsight subjects glean very paltry information [Dennett] |
3797 | I am the sum total of what I directly control [Dennett] |
7385 | People accept blurred boundaries in many things, but insist self is All or Nothing [Dennett] |
4881 | Being a person must involve having second-order beliefs and desires (about beliefs and desires) [Dennett] |
7383 | The psychological self is an abstraction, not a thing in the brain [Dennett] |
7386 | Selves are not soul-pearls, but artefacts of social processes [Dennett] |
7381 | We tell stories about ourselves, to protect, control and define who we are [Dennett] |
7382 | We spin narratives about ourselves, and the audience posits a centre of gravity for them [Dennett] |
7370 | The brain is controlled by shifting coalitions, guided by good purposeful habits [Dennett] |
7655 | The work done by the 'homunculus in the theatre' must be spread amongst non-conscious agencies [Dennett] |
3800 | You can be free even though force would have prevented you doing otherwise [Dennett, by PG] |
3803 | Can we conceive of a being with a will freer than our own? [Dennett] |
3791 | Awareness of thought is a step beyond awareness of the world [Dennett] |
3794 | Foreknowledge permits control [Dennett] |
7379 | If an epiphenomenon has no physical effects, it has to be undetectable [Dennett] |
7365 | Dualism wallows in mystery, and to accept it is to give up [Dennett] |
3159 | Beliefs and desires aren't real; they are prediction techniques [Dennett] |
3796 | The active self is a fiction created because we are ignorant of our motivations [Dennett] |
3161 | If mind is just an explanation, the explainer must have beliefs [Rey on Dennett] |
3986 | The 'intentional stance' is a way of interpreting an entity by assuming it is rational and self-aware [Dennett] |
2530 | Could a robot be made conscious just by software? [Dennett] |
7371 | All functionalism is 'homuncular', of one grain size or another [Dennett] |
4875 | We descend from robots, and our intentionality is composed of billions of crude intentional systems [Dennett] |
4879 | There is no more anger in adrenaline than silliness in a bottle of whiskey [Dennett] |
7657 | Intelligent agents are composed of nested homunculi, of decreasing intelligence, ending in machines [Dennett] |
7656 | I don't deny consciousness; it just isn't what people think it is [Dennett] |
7366 | It is arbitrary to say which moment of brain processing is conscious [Dennett] |
4876 | Maybe there is a minimum brain speed for supporting a mind [Dennett] |
7380 | Visual experience is composed of neural activity, which we find pleasing [Dennett] |
4878 | The materials for a mind only matter because of speed, and a need for transducers and effectors [Dennett] |
3177 | You couldn't drive a car without folk psychology [Dennett] |
3987 | Like the 'centre of gravity', desires and beliefs are abstract concepts with no actual existence [Dennett] |
4874 | The predecessor and rival of the language of thought hypothesis is the picture theory of ideas [Dennett] |
2524 | A language of thought doesn't explain content [Dennett] |
7654 | What matters about neuro-science is the discovery of the functional role of the chemistry [Dennett] |
23803 | States have content if we can predict them well by assuming intentionality [Dennett, by Schulte] |
3984 | The nature of content is entirely based on its functional role [Dennett] |
4882 | Concepts are things we (unlike dogs) can think about, because we have language [Dennett] |
2529 | Maybe there can be non-conscious concepts (e.g. in bees) [Dennett] |
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] |
3983 | Learning is evolution in the brain [Dennett] |
4872 | Most people see an abortion differently if the foetus lacks a brain [Dennett] |
7368 | Originally there were no reasons, purposes or functions; since there were no interests, there were only causes [Dennett] |
3985 | Biology is a type of engineering, not a search for laws of nature [Dennett] |
4877 | Maybe plants are very slow (and sentient) animals, overlooked because we are faster? [Dennett] |
3804 | Darwin's idea was the best idea ever [Dennett] |