79 ideas
| 4767 | Traditionally, rational beliefs are those which are justified by reasons [Psillos] |
| 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] |
| 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] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [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] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [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] |
| 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] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [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] |
| 4810 | Valid deduction is monotonic - that is, it remains valid if further premises are added [Psillos] |
| 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] |
| 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] |
| 4768 | The 'epistemic fallacy' is inferring what does exist from what can be known to exist [Psillos] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 16639 | Only individual bodies exist [Bacon] |
| 16033 | There are only individual bodies containing law-based powers, and the Forms are these laws [Bacon] |
| 4808 | If we say where Mars was two months ago, we offer an explanation without a prediction [Psillos] |
| 4807 | A good barometer will predict a storm, but not explain it [Psillos] |
| 21950 | Science must clear away the idols of the mind if they are ever going to find the truth [Bacon] |
| 4811 | Induction (unlike deduction) is non-monotonic - it can be invalidated by new premises [Psillos] |
| 4812 | Explanation is either showing predictability, or showing necessity, or showing causal relations [Psillos] |
| 4802 | Just citing a cause does not enable us to understand an event; we also need a relevant law [Psillos] |
| 4804 | The 'covering law model' says only laws can explain the occurrence of single events [Psillos] |
| 4805 | If laws explain the length of a flagpole's shadow, then the shadow also explains the length of the pole [Psillos] |
| 4806 | An explanation can just be a 'causal story', without laws, as when I knock over some ink [Psillos] |
| 4395 | There are non-causal explanations, most typically mathematical explanations [Psillos] |
| 4404 | Maybe explanation is entirely relative to the interests and presuppositions of the questioner [Psillos] |
| 4803 | An explanation is the removal of the surprise caused by the event [Psillos] |
| 4769 | It is hard to analyse causation, if it is presupposed in our theory of the functioning of the mind [Psillos] |
| 4770 | Nothing is more usual than to apply to external bodies every internal sensation which they occasion [Psillos] |
| 4399 | Causes clearly make a difference, are recipes for events, explain effects, and are evidence [Psillos] |
| 4400 | Theories of causation are based either on regularity, or on intrinsic relations of properties [Psillos] |
| 4403 | We can't base our account of causation on explanation, because it is the wrong way round [Psillos] |
| 4789 | Three divisions of causal theories: generalist/singularist, intrinsic/extrinsic, reductive/non-reductive [Psillos] |
| 4790 | If causation is 'intrinsic' it depends entirely on the properties and relations of the cause and effect [Psillos] |
| 4402 | Empiricists tried to reduce causation to explanation, which they reduced to logic-plus-a-law [Psillos] |
| 4774 | Counterfactual claims about causation imply that it is more than just regular succession [Psillos] |
| 4793 | "All gold cubes are smaller than one cubic mile" is a true universal generalisation, but not a law [Psillos] |
| 4397 | Regularity doesn't seem sufficient for causation [Psillos] |
| 4401 | It is not a law of nature that all the coins in my pocket are euros, though it is a regularity [Psillos] |
| 4792 | A Humean view of causation says it is regularities, and causal facts supervene on non-causal facts [Psillos] |
| 4801 | The regularity of a cock's crow is used to predict dawn, even though it doesn't cause it [Psillos] |
| 4796 | Laws are sets of regularities within a simple and strong coherent system of wider regularities [Psillos] |
| 4799 | Dispositional essentialism can't explain its key distinction between essential and non-essential properties [Psillos] |
| 4780 | In some counterfactuals, the counterfactual event happens later than its consequent [Psillos] |
| 4791 | Counterfactual theories say causes make a difference - if c hadn't occurred, then e wouldn't occur [Psillos] |