91 ideas
| 4588 | There is no such thing as 'science'; there are just many different sciences [Heil] |
| 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] |
| 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] |
| 10605 | Two functions are the same if they have the same extension [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] |
| 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] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [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] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [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] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [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] |
| 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] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 4616 | A higher level is 'supervenient' if it is determined by lower levels, but has its own natural laws [Heil] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 4603 | Functionalists in Fodor's camp usually say that a genuine property is one that figures in some causal laws [Heil] |
| 4617 | A stone does not possess the property of being a stone; its other properties make it a stone [Heil] |
| 4615 | Complex properties are not new properties, they are merely new combinations of properties [Heil] |
| 4612 | Complex properties are just arrangements of simple properties; they do not "emerge" as separate [Heil] |
| 4587 | From the property predicates P and Q, we can get 'P or Q', but it doesn't have to designate another property [Heil] |
| 4611 | The supporters of 'tropes' treat objects as bundles of tropes, when I think objects 'possess' properties [Heil] |
| 4592 | If you can have the boat without its current planks, and the planks with no boat, the planks aren't the boat [Heil] |
| 4586 | You can't embrace the formal apparatus of possible worlds, but reject the ontology [Heil] |
| 4591 | Idealism explains appearances by identifying appearances with reality [Heil] |
| 4610 | Different generations focus on either the quality of mind, or its scientific standing, or the content of thought [Heil] |
| 4618 | If minds are realised materially, it looks as if the material laws will pre-empt any causal role for mind [Heil] |
| 4621 | Whatever exists has qualities, so it is no surprise that states of minds have qualities [Heil] |
| 4623 | Propositional attitudes are not the only intentional states; there is also mental imagery [Heil] |
| 4626 | The widespread externalist view says intentionality has content because of causal links of agent to world [Heil] |
| 4622 | Error must be possible in introspection, because error is possible in all judgements [Heil] |
| 4590 | If causation is just regularities in events, the interaction of mind and body is not a special problem [Heil] |
| 4614 | Disposition is a fundamental feature of reality, since basic particles are capable of endless possible interactions [Heil] |
| 4595 | No mental state entails inevitable behaviour, because other beliefs or desires may intervene [Heil] |
| 4599 | Hearts are material, but functionalism says the property of being a heart is not a material property [Heil] |
| 4624 | If you are a functionalist, there appears to be no room for qualia [Heil] |
| 4601 | Higher-level sciences cannot be reduced, because their concepts mark boundaries invisible at lower levels [Heil] |
| 4602 | Higher-level sciences designate real properties of objects, which are not reducible to lower levels [Heil] |
| 4593 | 'Property dualism' says mind and body are not substances, but distinct families of properties [Heil] |
| 4597 | Early identity theory talked of mind and brain 'processes', but now the focus is properties [Heil] |
| 4609 | It seems contradictory to be asked to believe that we can be eliminativist about beliefs [Heil] |
| 4596 | The appeal of the identity theory is its simplicity, and its solution to the mental causation problem [Heil] |
| 4598 | Functionalists emphasise that mental processes are not to be reduced to what realises them [Heil] |
| 4619 | 'Multiple realisability' needs to clearly distinguish low-level realisers from what is realised [Heil] |
| 4620 | Multiple realisability is not a relation among properties, but an application of predicates to resembling things [Heil] |
| 4594 | A scientist could know everything about the physiology of headaches, but never have had one [Heil] |
| 4625 | Is mental imagery pictorial, or is it propositional? [Heil] |
| 4607 | Folk psychology and neuroscience are no more competitors than cartography and geology are [Heil] |
| 4605 | Truth-conditions correspond to the idea of 'literal meaning' [Heil] |
| 4606 | To understand 'birds warble' and 'tigers growl', you must also understand 'tigers warble' [Heil] |
| 4604 | If propositions are abstract entities, how do human beings interact with them? [Heil] |
| 2705 | How can intuitionists distinguish universal convictions from local cultural ones? [Hare] |
| 2712 | You can't use intuitions to decide which intuitions you should cultivate [Hare] |
| 2706 | Emotivists mistakenly think all disagreements are about facts, and so there are no moral reasons [Hare] |
| 2709 | Prescriptivism sees 'ought' statements as imperatives which are universalisable [Hare] |
| 2704 | If morality is just a natural or intuitive description, that leads to relativism [Hare] |
| 2703 | Descriptivism say ethical meaning is just truth-conditions; prescriptivism adds an evaluation [Hare] |
| 2707 | If there can be contradictory prescriptions, then reasoning must be involved [Hare] |
| 2708 | An 'ought' statement implies universal application [Hare] |
| 2711 | Prescriptivism implies a commitment, but descriptivism doesn't [Hare] |
| 2710 | Moral judgements must invoke some sort of principle [Hare] |