114 ideas
3426 | If one theory is reduced to another, we make fewer independent assumptions about the world [Kim] |
10073 | There cannot be a set theory which is complete [Smith,P] |
10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
10075 | A 'partial function' maps only some elements to another set [Smith,P] |
10074 | A 'total function' maps every element to one element in another set [Smith,P] |
10605 | Two functions are the same if they have the same extension [Smith,P] |
10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
10613 | No nice theory can define truth for its own language [Smith,P] |
10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
3431 | Supervenience suggest dependence without reduction (e.g. beauty) [Kim] |
3437 | 'Physical facts determine all the facts' is the physicalists' slogan [Kim] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
3430 | Resemblance or similarity is the core of our concept of a property [Kim] |
3432 | Is weight a 'resultant' property of water, but transparency an 'emergent' property? [Kim] |
3434 | Emergent properties are 'brute facts' (inexplicable), but still cause things [Kim] |
3436 | Should properties be individuated by their causal powers? [Kim] |
3406 | Counterfactuals are either based on laws, or on nearby possible worlds [Kim, by PG] |
3368 | Mind is basically qualities and intentionality, but how do they connect? [Kim] |
3392 | Mind is only interesting if it has causal powers [Kim] |
3396 | Experiment requires mental causation [Kim] |
3397 | Beliefs cause other beliefs [Kim] |
3367 | Both thought and language have intentionality [Kim] |
3365 | Intentionality involves both reference and content [Kim] |
3360 | Are pains pure qualia, or do they motivate? [Kim] |
3366 | Pain has no reference or content [Kim] |
3389 | Inverted qualia and zombies suggest experience isn't just functional [Kim] |
3391 | Crosswiring would show that pain and its function are separate [Kim, by PG] |
3422 | Externalism about content makes introspection depend on external evidence [Kim] |
3412 | How do we distinguish our anger from embarrassment? [Kim] |
3363 | We often can't decide what emotion, or even sensation, we are experiencing [Kim] |
3409 | Mental substance causation makes physics incomplete [Kim] |
3399 | If epiphenomenalism were true, we couldn't report consciousness [Kim] |
3390 | Are inverted or absent qualia coherent ideas? [Kim] |
3414 | What could demonstrate that zombies and inversion are impossible? [Kim] |
3359 | Cartesian dualism fails because it can't explain mental causation [Kim] |
3369 | Logical behaviourism translates mental language to behavioural [Kim] |
3428 | Behaviourism reduces mind to behaviour via bridging principles [Kim] |
3380 | Are dispositions real, or just a type of explanation? [Kim] |
3370 | What behaviour goes with mathematical beliefs? [Kim] |
3373 | Snakes have different pain behaviour from us [Kim] |
3371 | Behaviour depends on lots of mental states together [Kim] |
3372 | Behaviour is determined by society as well as mental states [Kim] |
3388 | Machine functionalism requires a Turing machine, causal-theoretical version doesn't [Kim] |
3379 | Neurons seem to be very similar and interchangeable [Kim] |
3384 | The person couldn't run Searle's Chinese Room without understanding Chinese [Kim] |
3393 | How do functional states give rise to mental causation? [Kim] |
3439 | Reductionism gets stuck with qualia [Kim] |
3427 | Reductionism is impossible if there aren't any 'bridge laws' between mental and physical [Kim] |
3424 | Most modern physicalists are non-reductive property dualists [Kim] |
3376 | We can't assess evidence about mind without acknowledging phenomenal properties [Kim] |
3362 | Supervenience says all souls are identical, being physically indiscernible [Kim] |
3413 | Zombies and inversion suggest non-reducible supervenience [Kim] |
3374 | Token physicalism isn't reductive; it just says all mental events have some physical properties [Kim] |
3433 | The core of the puzzle is the bridge laws between mind and brain [Kim] |
3377 | Elimination can either be by translation or by causal explanation [Kim] |
3438 | Reductionists deny new causal powers at the higher level [Kim] |
3440 | Without reductionism, mental causation is baffling [Kim] |
3375 | If an orange image is a brain state, are some parts of the brain orange? [Kim] |
3411 | How do we distinguish our attitudes from one another? [Kim] |
3410 | Folk psychology has adapted to Freudianism [Kim] |
3387 | A culture without our folk psychology would be quite baffling [Kim] |
3386 | Folk psychology has been remarkably durable [Kim] |
3394 | Maybe folk psychology is a simulation, not a theory [Kim] |
3382 | A machine with a mind might still fail the Turing Test [Kim] |
3383 | The Turing Test is too specifically human in its requirements [Kim] |
3408 | Two identical brain states could have different contents in different worlds [Kim] |
3420 | Two types of water are irrelevant to accounts of behaviour [Kim] |
3416 | Content may match several things in the environment [Kim] |
3418 | 'Arthritis in my thigh' requires a social context for its content to be meaningful [Kim] |
3421 | Content is best thought of as truth conditions [Kim] |
3419 | Pain, our own existence, and negative existentials, are not external [Kim] |
3417 | Content depends on other content as well as the facts [Kim] |
3403 | We assume people believe the obvious logical consequences of their known beliefs [Kim] |
3402 | If someone says "I do and don't like x", we don't assume a contradiction [Kim] |
1477 | Being manly and brave is the result of convention, not of human nature [Plutarch] |
1478 | Animals don't value pleasure, as they cease sexual intercourse after impregnation [Plutarch] |
1479 | Animals have not been led into homosexuality, because they value pleasure very little [Plutarch] |
3401 | A common view is that causal connections must be instances of a law [Kim] |
3407 | Laws are either 'strict', or they involve a 'ceteris paribus' clause [Kim] |