100 ideas
20910 | Everything happens necessarily, and for a reason [Democritus] |
13689 | 'Theorems' are formulas provable from no premises at all [Sider] |
13705 | Truth tables assume truth functionality, and are just pictures of truth functions [Sider] |
13706 | Intuitively, deontic accessibility seems not to be reflexive, but to be serial [Sider] |
13710 | In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary [Sider] |
13711 | System B introduces iterated modalities [Sider] |
13708 | S5 is the strongest system, since it has the most valid formulas, because it is easy to be S5-valid [Sider] |
13712 | Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K) [Sider] |
13714 | We can treat modal worlds as different times [Sider] |
13720 | Converse Barcan Formula: □∀αφ→∀α□φ [Sider] |
13718 | The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic [Sider] |
13723 | System B is needed to prove the Barcan Formula [Sider] |
13715 | You can employ intuitionist logic without intuitionism about mathematics [Sider] |
13678 | The most popular account of logical consequence is the semantic or model-theoretic one [Sider] |
13679 | Maybe logical consequence is more a matter of provability than of truth-preservation [Sider] |
13682 | Maybe logical consequence is impossibility of the premises being true and the consequent false [Sider] |
13680 | Maybe logical consequence is a primitive notion [Sider] |
13722 | A 'theorem' is an axiom, or the last line of a legitimate proof [Sider] |
13696 | When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity [Sider] |
13700 | A 'total' function must always produce an output for a given domain [Sider] |
13703 | λ can treat 'is cold and hungry' as a single predicate [Sider] |
13688 | Good axioms should be indisputable logical truths [Sider] |
13687 | No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider] |
13690 | Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms [Sider] |
13691 | Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider] |
13685 | Natural deduction helpfully allows reasoning with assumptions [Sider] |
13686 | We can build proofs just from conclusions, rather than from plain formulae [Sider] |
13697 | Valuations in PC assign truth values to formulas relative to variable assignments [Sider] |
13684 | The semantical notion of a logical truth is validity, being true in all interpretations [Sider] |
13704 | It is hard to say which are the logical truths in modal logic, especially for iterated modal operators [Sider] |
13724 | In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation [Sider] |
13698 | In a complete logic you can avoid axiomatic proofs, by using models to show consequences [Sider] |
13699 | Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider] |
16146 | Two can't be a self-contained unit, because it would need to be one to do that [Democritus, by Aristotle] |
13701 | A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider] |
20901 | True Being only occurs when it is completely full, with atoms and no void [Democritus, by Aristotle] |
20902 | Being does not exist more than non-being [Democritus, by Aristotle] |
20904 | The non-existent exists as much as the existent, because it has causal powers [Democritus] |
20903 | The only distinctions are Configuration (shape), Disposition (order) and Turning (position) [Democritus, by Aristotle] |
20893 | Nothing comes from non-existence, or passes into it [Democritus, by Diog. Laertius] |
13692 | A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation [Sider] |
13695 | Supervaluational logic is classical, except when it adds the 'Definitely' operator [Sider] |
13693 | A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification [Sider] |
13694 | We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings [Sider] |
20896 | It is not possible to know what sort each thing is [Democritus] |
13683 | A relation is a feature of multiple objects taken together [Sider] |
3357 | Democritus denies reality to large objects, because atomic entities can't combine to produce new ones [Benardete,JA on Democritus] |
598 | Democritus said that substances could never be mixed, so atoms are the substances [Democritus, by Aristotle] |
13702 | The identity of indiscernibles is necessarily true, if being a member of some set counts as a property [Sider] |
13721 | 'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist [Sider] |
13707 | Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible [Sider] |
13709 | Logical truths must be necessary if anything is [Sider] |
13716 | 'If B hadn't shot L someone else would have' if false; 'If B didn't shoot L, someone else did' is true [Sider] |
13717 | Transworld identity is not a problem in de dicto sentences, which needn't identify an individual [Sider] |
13719 | Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost [Sider] |
1532 | Sensible qualities can't be real if they appear different to different creatures [Democritus, by Theophrastus] |
20894 | Man is separated from reality [Democritus] |
517 | All evidence comes from senses, so they are indispensable to the mind [Democritus] |
20897 | Obscure knowledge belongs to the five senses, and genuine knowledge is the other type [Democritus] |
577 | Democritus says there is either no truth, or it is concealed from us [Democritus, by Aristotle] |
20895 | We actually know nothing, and opinions are mere flux [Democritus] |
1528 | We in fact know nothing, but we each restructure our reality with beliefs [Democritus] |
492 | It is obviously impossible to understand the reality of each thing [Democritus] |
515 | We know nothing in reality; for truth lies in an abyss [Democritus] |
20892 | Democritus was devoted to discovering causal explanations [Democritus, by Eusebius] |
5882 | Democritus says soul consists of smooth round bodies brought together in accidental collision [Democritus, by Cicero] |
6034 | Atomists say soul has a rational part in the chest, and a diffused non-rational part [Democritus, by Aetius] |
20912 | The soul is the same as the mind [Democritus, by Aristotle] |
20916 | Animals have a share of reason [Democritus, by Porphyry] |
20914 | The directive centre is located in the whole head [Democritus, by Ps-Plutarch] |
6033 | Democritus said everything happens of necessity, by natural motion of atoms [Democritus, by Cicero] |
5088 | Some say there is a determinate cause for every apparently spontaneous event [Democritus, by Aristotle] |
21670 | Democritus said atoms only move by their natural motions, which are therefore necessary [Democritus, by Cicero] |
24041 | Democritus says spherical atoms are fire, and constitute the soul [psuche] [Democritus, by Aristotle] |
20913 | Democritus says the soul is the body, and thinking is thus the mixture of the body [Democritus, by Theophrastus] |
1540 | Pleasure and pain guide our choices of good and bad [Democritus] |
20329 | A work of art is an artifact created for the artworld [Dickie] |
495 | Wisdom creates a healthy passion-free soul [Democritus] |
1537 | Happiness is identifying and separating the pleasures [Democritus, by Stobaeus] |
20917 | Contentment comes from moderation and proportion in life [Democritus, by Stobaeus] |
13551 | Democritus says wealth is a burden to the virtuous mind [Democritus, by Seneca] |
20899 | Atoms cling together, until a stronger necessity disperses them [Democritus, by Aristotle] |
20898 | Atoms are irregular, hooked, concave, convex, and many other shapes [Democritus, by Aristotle] |
20908 | There could be an atom the size of the world [Democritus, by Ps-Plutarch] |
1527 | There must be atoms, to avoid the absurdity of infinite division down to nothing [Democritus, by Aristotle] |
493 | Experiences are merely convention; only atoms and the void are real [Democritus] |
20909 | The basic atoms are without qualities - which only arise from encounters between atoms [Democritus, by Galen] |
1536 | If a cone is horizontally sliced the surfaces can't be equal, so it goes up in steps [Democritus] |
17542 | 'Full' and 'Void' secularised Parmenides's Being and Not-being [Democritus, by Heisenberg] |
1525 | Atomists say there are only three differences - in shape, arrangement and position [Democritus, by Aristotle] |
5947 | If only atoms are real and the rest is convention, we wouldn't bother to avoid pain [Democritus, by Diogenes of Oen.] |
13219 | When atoms touch, why don't they coalesce, like water drops? [Aristotle on Democritus] |
1533 | Because appearance is infinitely varied, atomists assume infinitely many shapes of atom [Democritus, by Aristotle] |
23314 | Greeks explained regularity by intellectual design, not by laws [Democritus, by Frede,M] |
24059 | Democritus is wrong: in a void we wouldn't see a distant ant in exact detail [Aristotle on Democritus] |
5101 | Movement is impossible in a void, because nothing can decide the direction of movement [Aristotle on Democritus] |
20905 | Growth and movement would not exist if there were no void to receive them [Democritus] |
20911 | There are unlimited worlds of varying sizes, some without life or water [Democritus, by Hippolytus] |
1535 | Democritus said people imagined gods as the source of what awed or frightened them [Democritus, by Sext.Empiricus] |
20915 | The soul is destroyed with the body [Democritus, by Ps-Plutarch] |