73 ideas
20768 | Like spiderswebs, dialectical arguments are clever but useless [Ariston, by Diog. Laertius] |
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] |
13710 | In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary [Sider] |
13706 | Intuitively, deontic accessibility seems not to be reflexive, but to be serial [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] |
15845 | It seems absurd that seeing a person's limbs, the one is many, and yet the many are one [Plato] |
13680 | Maybe logical consequence is a primitive notion [Sider] |
13679 | Maybe logical consequence is more a matter of provability than of truth-preservation [Sider] |
13678 | The most popular account of logical consequence is the semantic or model-theoretic one [Sider] |
13682 | Maybe logical consequence is impossibility of the premises being true and the consequent false [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] |
13687 | No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider] |
13688 | Good axioms should be indisputable logical truths [Sider] |
13691 | Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider] |
13690 | Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms [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] |
9867 | It is absurd to define a circle, but not be able to recognise a real one [Plato] |
9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
13701 | A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider] |
15857 | Any mixture which lacks measure and proportion doesn't even count as a mixture at all [Plato] |
14503 | If a mixture does not contain measure and proportion, it is corrupted and destroyed [Plato] |
13692 | A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation [Sider] |
13693 | A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification [Sider] |
13695 | Supervaluational logic is classical, except when it adds the 'Definitely' operator [Sider] |
13694 | We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings [Sider] |
13683 | A relation is a feature of multiple objects taken together [Sider] |
4447 | If the good is one, is it unchanged when it is in particulars, and is it then separated from itself? [Plato] |
15856 | A thing can become one or many, depending on how we talk about it [Plato] |
374 | If one object is divided into its parts, someone can then say that one are many and many is one [Plato] |
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] |
389 | How can you be certain about aspects of the world if they aren't constant? [Plato] |
390 | If goodness involves moderation and proportion, then it seems to be found in beauty [Plato] |
391 | The good involves beauty, proportion and truth [Plato] |
392 | Neither intellect nor pleasure are the good, because they are not perfect and self-sufficient [Plato] |
393 | Good first, then beauty, then reason, then knowledge, then pleasure [Plato, by PG] |
385 | Some of the pleasures and pains we feel are false [Plato] |
387 | A small pure pleasure is much finer than a large one contaminated with pain [Plato] |
373 | Pleasure is certainly very pleasant, but it doesn't follow that all pleasures are good [Plato] |
382 | It is unlikely that the gods feel either pleasure or pain [Plato] |
371 | Reason, memory, truth and wisdom are far better than pleasure, for those who can attain them [Plato] |
376 | Would you prefer a life of pleasure without reason, or one of reason without pleasure? [Plato] |
379 | The good must be sufficient and perfect, and neither intellect nor pleasure are that [Plato] |
381 | We feel pleasure when we approach our natural state of harmony [Plato] |
386 | Intense pleasure and pain are not felt in a good body, but in a worthless one [Plato] |
388 | Hedonists must say that someone in pain is bad, even if they are virtuous [Plato] |
377 | If you lived a life of maximum pleasure, would you still be lacking anything? [Plato] |
378 | A life of pure pleasure with no intellect is the life of a jellyfish [Plato] |
3049 | The chief good is indifference to what lies midway between virtue and vice [Ariston, by Diog. Laertius] |
3549 | Ariston says rules are useless for the virtuous and the non-virtuous [Ariston, by Annas] |