69 ideas
| 3600 | Slow and accurate thought makes the greatest progress [Descartes] |
| 3601 | Most things in human life seem vain and useless [Descartes] |
| 3602 | Almost every daft idea has been expressed by some philosopher [Descartes] |
| 3603 | Methodical thinking is cautious, analytical, systematic, and panoramic [Descartes, by PG] |
| 3612 | Clear and distinct conceptions are true because a perfect God exists [Descartes] |
| 3610 | Truth is clear and distinct conception - of which it is hard to be sure [Descartes] |
| 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] |
| 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] |
| 13701 | A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider] |
| 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] |
| 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] |
| 3605 | We can believe a thing without knowing we believe it [Descartes] |
| 1583 | In morals Descartes accepts the conventional, but rejects it in epistemology [Roochnik on Descartes] |
| 3607 | In thinking everything else false, my own existence remains totally certain [Descartes] |
| 3617 | I aim to find the principles and causes of everything, using the seeds within my mind [Descartes] |
| 3611 | Understanding, rather than imagination or senses, gives knowledge [Descartes] |
| 3606 | I was searching for reliable rock under the shifting sand [Descartes] |
| 3604 | When rebuilding a house, one needs alternative lodgings [Descartes] |
| 3618 | Only experiments can settle disagreements between rival explanations [Descartes] |
| 3615 | Little reason is needed to speak, so animals have no reason at all [Descartes] |
| 3609 | I am a thinking substance, which doesn't need a place or material support [Descartes] |
| 3608 | I can deny my body and the world, but not my own existence [Descartes] |
| 3613 | Reason is universal in its responses, but a physical machine is constrained by its organs [Descartes] |
| 3616 | The soul must unite with the body to have appetites and sensations [Descartes] |
| 3614 | A machine could speak in response to physical stimulus, but not hold a conversation [Descartes] |
| 1581 | Greeks elevate virtues enormously, but never explain them [Descartes] |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| 16686 | God has established laws throughout nature, and implanted ideas of them within us [Descartes] |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |