119 ideas
1887 | You cannot divide anything into many parts, because after the first division you are no longer dividing the original [Sext.Empiricus] |
22764 | Ordinary speech is not exact about what is true; we say we are digging a well before the well exists [Sext.Empiricus] |
22752 | Reasoning is impossible without a preconception [Sext.Empiricus] |
1885 | Proof moves from agreed premises to a non-evident inference [Sext.Empiricus] |
6021 | It is only when we say a proposition that we speak truly or falsely [Sext.Empiricus] |
9738 | Each line of a truth table is a model [Fitting/Mendelsohn] |
9727 | Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic [Fitting/Mendelsohn] |
9726 | We let 'R' be the accessibility relation: xRy is read 'y is accessible from x' [Fitting/Mendelsohn] |
9737 | The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ [Fitting/Mendelsohn] |
13136 | The prefix σ names a possible world, and σ.n names a world accessible from that one [Fitting/Mendelsohn] |
13727 | A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate [Fitting/Mendelsohn] |
9734 | Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x' [Fitting/Mendelsohn] |
9736 | A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- > [Fitting/Mendelsohn] |
9735 | A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R > [Fitting/Mendelsohn] |
9741 | Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual) [Fitting/Mendelsohn] |
13149 | S5: a) if n ◊X then kX b) if n ¬□X then k ¬X c) if n □X then k X d) if n ¬◊X then k ¬X [Fitting/Mendelsohn] |
13141 | Negation: if σ ¬¬X then σ X [Fitting/Mendelsohn] |
13138 | Disj: a) if σ ¬(X∨Y) then σ ¬X and σ ¬Y b) if σ X∨Y then σ X or σ Y [Fitting/Mendelsohn] |
13142 | Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new] [Fitting/Mendelsohn] |
13144 | T reflexive: a) if σ □X then σ X b) if σ ¬◊X then σ ¬X [Fitting/Mendelsohn] |
13145 | D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X [Fitting/Mendelsohn] |
13146 | B symmetric: a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [n occurs] [Fitting/Mendelsohn] |
13147 | 4 transitive: a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [n occurs] [Fitting/Mendelsohn] |
13148 | 4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [n occurs] [Fitting/Mendelsohn] |
9740 | If a proposition is possibly true in a world, it is true in some world accessible from that world [Fitting/Mendelsohn] |
9739 | If a proposition is necessarily true in a world, it is true in all worlds accessible from that world [Fitting/Mendelsohn] |
13137 | Conj: a) if σ X∧Y then σ X and σ Y b) if σ ¬(X∧Y) then σ ¬X or σ ¬Y [Fitting/Mendelsohn] |
13140 | Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails] [Fitting/Mendelsohn] |
13139 | Implic: a) if σ ¬(X→Y) then σ X and σ ¬Y b) if σ X→Y then σ ¬X or σ Y [Fitting/Mendelsohn] |
13143 | Universal: a) if σ ¬◊X then σ.m ¬X b) if σ □X then σ.m X [m exists] [Fitting/Mendelsohn] |
9742 | The system K has no accessibility conditions [Fitting/Mendelsohn] |
13114 | □P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed [Fitting/Mendelsohn] |
9743 | The system D has the 'serial' conditon imposed on its accessibility relation [Fitting/Mendelsohn] |
9744 | The system T has the 'reflexive' conditon imposed on its accessibility relation [Fitting/Mendelsohn] |
9746 | The system K4 has the 'transitive' condition on its accessibility relation [Fitting/Mendelsohn] |
9745 | The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation [Fitting/Mendelsohn] |
9747 | The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn] |
9748 | System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn] |
9404 | Modality affects content, because P→◊P is valid, but ◊P→P isn't [Fitting/Mendelsohn] |
13112 | In epistemic logic knowers are logically omniscient, so they know that they know [Fitting/Mendelsohn] |
13111 | Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P' [Fitting/Mendelsohn] |
13113 | F: will sometime, P: was sometime, G: will always, H: was always [Fitting/Mendelsohn] |
13728 | The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability [Fitting/Mendelsohn] |
13729 | The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity [Fitting/Mendelsohn] |
12196 | A valid hypothetical syllogism is 'that which does not begin with a truth and end with a falsehood' [Sext.Empiricus] |
6020 | 'Man is a rational mortal animal' is equivalent to 'if something is a man, that thing is a rational mortal animal' [Sext.Empiricus] |
9725 | 'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions [Fitting/Mendelsohn] |
1902 | Since Socrates either died when he was alive (a contradiction) or died when he was dead (meaningless), he didn't die [Sext.Empiricus] |
22744 | Parts are not parts if their whole is nothing more than the parts [Sext.Empiricus] |
22762 | Some properties are inseparable from a thing, such as the length, breadth and depth of a body [Sext.Empiricus] |
13730 | The Indiscernibility of Identicals has been a big problem for modal logic [Fitting/Mendelsohn] |
13725 | □ must be sensitive as to whether it picks out an object by essential or by contingent properties [Fitting/Mendelsohn] |
13731 | Objects retain their possible properties across worlds, so a bundle theory of them seems best [Fitting/Mendelsohn] |
13726 | Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them [Fitting/Mendelsohn] |
1889 | If an argument has an absurd conclusion, we should not assent to the absurdity, but avoid the absurd argument [Sext.Empiricus] |
1871 | Whether honey is essentially sweet may be doubted, as it is a matter of judgement rather than appearance [Sext.Empiricus] |
1883 | How can the intellect know if sensation is reliable if it doesn't directly see external objects? [Sext.Empiricus] |
22748 | Some say motion is perceived by sense, but others say it is by intellect [Sext.Empiricus] |
1890 | We distinguish ambiguities by seeing what is useful [Sext.Empiricus] |
22759 | Fools, infants and madmen may speak truly, but do not know [Sext.Empiricus] |
20795 | Some things are their own criterion, such as straightness, a set of scales, or light [Sext.Empiricus] |
22760 | Madmen are reliable reporters of what appears to them [Sext.Empiricus] |
1870 | The basis of scepticism is the claim that every proposition has an equal opposing proposition [Sext.Empiricus] |
1882 | The necks of doves appear different in colour depending on the angle of viewing [Sext.Empiricus] |
1881 | The same oar seems bent in water and straight when out of it [Sext.Empiricus] |
1872 | The same tower appears round from a distance, but square close at hand [Sext.Empiricus] |
1873 | If we press the side of an eyeball, objects appear a different shape [Sext.Empiricus] |
20794 | How can sceptics show there is no criterion? Weak without, contradiction with [Sext.Empiricus] |
1874 | How can we judge between our impressions and those of other animals, when we ourselves are involved? [Sext.Empiricus] |
1878 | Water that seems lukewarm can seem very hot on inflamed skin [Sext.Empiricus] |
1880 | Some actions seem shameful when sober but not when drunk [Sext.Empiricus] |
1877 | If we had no hearing or sight, we would assume no sound or sight exists, so there may be unsensed qualities [Sext.Empiricus] |
1879 | Sickness is perfectly natural to the sick, so their natural perceptions should carry some weight [Sext.Empiricus] |
1876 | If we enjoy different things, presumably we receive different impressions [Sext.Empiricus] |
1910 | With us it is shameful for men to wear earrings, but among Syrians it is considered noble [Sext.Empiricus] |
1911 | Even if all known nations agree on a practice, there may be unknown nations which disagree [Sext.Empiricus] |
20921 | How can we state relativism of sweet and sour, if they have no determinate nature? [Theophrastus] |
6026 | How can you investigate without some preconception of your object? [Sext.Empiricus] |
1886 | If you don't view every particular, you may miss the one which disproves your universal induction [Sext.Empiricus] |
22746 | If we try to conceive of a line with no breadth, it ceases to exist, and so has no length [Sext.Empiricus] |
22741 | The incorporeal is not in the nature of body, and so could not emerge from it [Sext.Empiricus] |
1884 | If we utter three steps of a logical argument, they never exist together [Sext.Empiricus] |
22763 | We can only dream of a winged man if we have experienced men and some winged thing [Sext.Empiricus] |
22754 | Saying the good is useful or choiceworth or happiness-creating is not the good, but a feature of it [Sext.Empiricus] |
22755 | Like a warming fire, what is good by nature should be good for everyone [Sext.Empiricus] |
22756 | If a desire is itself desirable, then we shouldn't desire it, as achieving it destroys it [Sext.Empiricus] |
6032 | Right actions, once done, are those with a reasonable justification [Sext.Empiricus] |
5990 | Theophrastus doubted whether nature could be explained teleologically [Theophrastus, by Gottschalk] |
1517 | The tektraktys (1+2+3+4=10) is the 'fount of ever-flowing nature' [Sext.Empiricus] |
1894 | Some say that causes are physical, some say not [Sext.Empiricus] |
1898 | Cause can't exist before effect, or exist at the same time, so it doesn't exist [Sext.Empiricus] |
1897 | Knowing an effect results from a cause means knowing that the cause belongs with the effect, which is circular [Sext.Empiricus] |
1896 | If there were no causes then everything would have been randomly produced by everything [Sext.Empiricus] |
1895 | Causes are either equal to the effect, or they link equally with other causes, or they contribute slightly [Sext.Empiricus] |
1900 | If time and place are infinitely divided, it becomes impossible for movement ever to begin [Sext.Empiricus] |
1899 | Does the original self-mover push itself from behind, or pull itself from in front? [Sext.Empiricus] |
1901 | If all atoms, times and places are the same, everything should move with equal velocity [Sext.Empiricus] |
22747 | A man walking backwards on a forwards-moving ship is moving in a fixed place [Sext.Empiricus] |
1903 | If motion and rest are abolished, so is time [Sext.Empiricus] |
1904 | Time must be unlimited, but past and present can't be non-existent, and can't be now, so time does not exist [Sext.Empiricus] |
22749 | Time doesn't end with the Universe, because tensed statements about destruction remain true [Sext.Empiricus] |
22750 | Time is divisible, into past, present and future [Sext.Empiricus] |
1905 | How can time be divisible if we can't compare one length of time with another? [Sext.Empiricus] |
22742 | Socrates either dies when he exists (before his death) or when he doesn't (after his death) [Sext.Empiricus] |
22751 | If the present is just the limit of the past or the future, it can't exist because they don't exist [Sext.Empiricus] |
22730 | All men agree that God is blessed, imperishable, happy and good [Sext.Empiricus] |
1891 | How can we agree on the concept of God, unless we agree on his substance or form or place? [Sext.Empiricus] |
22739 | God must suffer to understand suffering [Sext.Empiricus] |
22738 | The Divine must lack the virtues of continence and fortitude, because they are not needed [Sext.Empiricus] |
22734 | God is defended by agreement, order, absurdity of denying God, and refutations [Sext.Empiricus] |
22736 | God's sensations imply change, and hence perishing, which is absurd, so there is no such God [Sext.Empiricus] |
22740 | God without virtue is absurd, but God's virtues will be better than God [Sext.Empiricus] |
1892 | The existence of God can't be self-evident or everyone would have agreed on it, so it needs demonstration [Sext.Empiricus] |
22735 | The original substance lacked motion or shape, and was given these by a cause [Sext.Empiricus] |
22732 | The perfections of God were extrapolations from mankind [Sext.Empiricus] |
22728 | Gods were invented as watchers of people's secret actions [Sext.Empiricus] |
22737 | An incorporeal God could do nothing, and a bodily god would perish, so there is no God [Sext.Empiricus] |
22731 | It is mad to think that what is useful to us, like lakes and rivers, are gods [Sext.Empiricus] |
1893 | If God foresaw evil he would presumably prevent it, and if he only foresees some things, why those things? [Sext.Empiricus] |