131 ideas
291 | Don't assume that wisdom is the automatic consequence of old age [Plato] |
2196 | The observation of human blindness and weakness is the result of all philosophy [Hume] |
2187 | If we suspect that a philosophical term is meaningless, we should ask what impression it derives from [Hume] |
2200 | All experimental conclusions assume that the future will be like the past [Hume] |
4636 | All reasoning concerning matters of fact is based on analogy (with similar results of similar causes) [Hume] |
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] |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
10852 | Robinson Arithmetic (Q) is not negation complete [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] |
2197 | Reason assists experience in discovering laws, and in measuring their application [Hume] |
7700 | We can't think about the abstract idea of triangles, but only of particular triangles [Hume] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
13602 | We cannot form an idea of a 'power', and the word is without meaning [Hume] |
2216 | We transfer the frequency of past observations to our future predictions [Hume] |
2215 | There is no such thing as chance [Hume] |
2209 | Belief is stronger, clearer and steadier than imagination [Hume] |
2207 | Belief can't be a concept plus an idea, or we could add the idea to fictions [Hume] |
2208 | Belief is just a particular feeling attached to ideas of objects [Hume] |
2213 | Beliefs are built up by resemblance, contiguity and causation [Hume] |
3661 | 'Natural beliefs' are unavoidable, whatever our judgements [Hume, by Strawson,G] |
2191 | Relations of ideas are known by thought, independently from the world [Hume] |
2239 | If secondary qualities (e.g. hardness) are in the mind, so are primary qualities like extension [Hume] |
2237 | It never occurs to people that they only experience representations, not the real objects [Hume] |
2192 | All reasoning about facts is causal; nothing else goes beyond memory and senses [Hume] |
2182 | Impressions are our livelier perceptions, Ideas the less lively ones [Hume] |
2190 | All objects of enquiry are Relations of Ideas, or Matters of Fact [Hume] |
2246 | If books don't relate ideas or explain facts, commit them to the flames [Hume] |
2184 | All ideas are copies of impressions [Hume] |
23631 | Hume is loose when he says perceptions of different strength are different species [Reid on Hume] |
2189 | All ideas are connected by Resemblance, Contiguity in time or place, and Cause and Effect [Hume] |
2186 | We cannot form the idea of something we haven't experienced [Hume] |
2702 | Only madmen dispute the authority of experience [Hume] |
2217 | When definitions are pushed to the limit, only experience can make them precise [Hume] |
2205 | You couldn't reason at all if you lacked experience [Hume] |
2194 | How could Adam predict he would drown in water or burn in fire? [Hume] |
2183 | We can only invent a golden mountain by combining experiences [Hume] |
23421 | If a person had a gap in their experience of blue shades, they could imaginatively fill it in [Hume] |
3902 | Hume mistakenly lumps sensations and perceptions together as 'impressions' [Scruton on Hume] |
2206 | Reasons for belief must eventually terminate in experience, or they are without foundation [Hume] |
2235 | There is no certain supreme principle, or infallible rule of inference [Hume] |
10328 | We think testimony matches reality because of experience, not some a priori connection [Hume] |
2230 | Good testimony needs education, integrity, motive and agreement [Hume, by PG] |
2238 | Reason can never show that experiences are connected to external objects [Hume] |
2242 | Mitigated scepticism draws attention to the limitations of human reason, and encourages modesty [Hume] |
2243 | Mitigated scepticism sensibly confines our enquiries to the narrow capacity of human understanding [Hume] |
2236 | Examples of illusion only show that sense experience needs correction by reason [Hume] |
2240 | It is a very extravagant aim of the sceptics to destroy reason and argument by means of reason and argument [Hume] |
2241 | The main objection to scepticism is that no good can come of it [Hume] |
2198 | We assume similar secret powers behind similar experiences, such as the nourishment of bread [Hume] |
2201 | Induction can't prove that the future will be like the past, since induction assumes this [Hume] |
2203 | If we infer causes from repetition, this explains why we infer from a thousand objects what we couldn't infer from one [Hume] |
2204 | All inferences from experience are effects of custom, not reasoning [Hume] |
2199 | Reason cannot show why reliable past experience should extend to future times and remote places [Hume] |
2202 | Fools, children and animals all learn from experience [Hume] |
6350 | Premises can support an argument without entailing it [Pollock/Cruz on Hume] |
3598 | Hume just shows induction isn't deduction [Williams,M on Hume] |
2210 | A picture of a friend strengthens our idea of him, by resemblance [Hume] |
17712 | General ideas are the connection by resemblance to some particular [Hume] |
8544 | Hume does not distinguish real resemblances among degrees of resemblance [Shoemaker on Hume] |
2211 | When I am close to (contiguous with) home, I feel its presence more nearly [Hume] |
2212 | An object made by a saint is the best way to produce thoughts of him [Hume] |
2214 | Our awareness of patterns of causation is too important to be left to slow and uncertain reasoning [Hume] |
2222 | The doctrine of free will arises from a false sensation we have of freedom in many actions [Hume] |
2223 | Liberty is merely acting according to the will, which anyone can do if they are not in chains [Hume] |
3655 | Hume makes determinism less rigid by removing the necessity from causation [Trusted on Hume] |
2220 | Only experience teaches us about our wills [Hume] |
2224 | Praise and blame can only be given if an action proceeds from a person's character and disposition [Hume] |
2225 | If you deny all necessity and causation, then our character is not responsible for our crime [Hume] |
2226 | Repentance gets rid of guilt, which shows that responsibility arose from the criminal principles in the mind [Hume] |
293 | Being unafraid (perhaps through ignorance) and being brave are two different things [Plato] |
2233 | No government has ever suffered by being too tolerant of philosophy [Hume] |
2195 | We can discover some laws of nature, but never its ultimate principles and causes [Hume] |
2245 | A priori it looks as if a cause could have absolutely any effect [Hume] |
4772 | If a singular effect is studied, its cause can only be inferred from the types of events involved [Hume] |
8341 | Hume never even suggests that there is no such thing as causation [Hume, by Strawson,G] |
8344 | At first Hume said qualities are the causal entities, but later he said events [Hume, by Davidson] |
3662 | Hume says we can only know constant conjunctions, not that that's what causation IS [Hume, by Strawson,G] |
4771 | In both of Hume's definitions, causation is extrinsic to the sequence of events [Psillos on Hume] |
5194 | Hume's definition of cause as constantly joined thoughts can't cover undiscovered laws [Ayer on Hume] |
2221 | A cause is either similar events following one another, or an experience always suggesting a second experience [Hume] |
2193 | No causes can be known a priori, but only from experience of constant conjunctions [Hume] |
2234 | It is only when two species of thing are constantly conjoined that we can infer one from the other [Hume] |
8422 | Cause is where if the first object had not been, the second had not existed [Hume] |
2218 | In observing causes we can never observe any necessary connections or binding qualities [Hume] |
15249 | Hume never shows how a strong habit could generate the concept of necessity [Harré/Madden on Hume] |
8339 | Hume's regularity theory of causation is epistemological; he believed in some sort of natural necessity [Hume, by Strawson,G] |
2244 | It can never be a logical contradiction to assert the non-existence of something thought to exist [Hume] |
2232 | You can't infer the cause to be any greater than its effect [Hume] |
2227 | A miracle violates laws which have been established by continuous unchanging experience, so should be ignored [Hume] |
2228 | All experience must be against a supposed miracle, or it wouldn't be called 'a miracle' [Hume] |
2229 | To establish a miracle the falseness of the evidence must be a greater miracle than the claimed miraculous event [Hume] |
2185 | The idea of an infinite, intelligent, wise and good God arises from augmenting the best qualities of our own minds [Hume] |