166 ideas
| 7504 | Modern science comes from Descartes' view that knowledge doesn't need moral purity [Descartes, by Foucault] |
| 1569 | Descartes impoverished the classical idea of logos, and it no longer covered human experience [Roochnik on Descartes] |
| 2248 | Reason says don't assent to uncertain principles, just as much as totally false ones [Descartes] |
| 2857 | Since Plato all philosophers have followed the herd, except Descartes, stuck in superficial reason [Nietzsche on Descartes] |
| 3641 | It is circular to make truth depend on believing God's existence is true [Arnauld on Descartes] |
| 4524 | Descartes is right that in the Christian view only God can guarantee the reliability of senses [Nietzsche on Descartes] |
| 2290 | Once it is clear that there is a God who is no deceiver, I conclude that clear and distinct perceptions must be true [Descartes] |
| 2266 | My general rule is that everything that I perceive clearly and distinctly is true [Descartes] |
| 4301 | Someone may think a thing is 'clear and distinct', but be wrong [Leibniz on Descartes] |
| 13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
| 13643 | Aristotelian logic is complete [Shapiro] |
| 13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
| 13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
| 13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
| 13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
| 13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
| 13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
| 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
| 13627 | There is no 'correct' logic for natural languages [Shapiro] |
| 13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
| 13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
| 13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
| 13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
| 13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
| 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
| 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
| 13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
| 15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
| 13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
| 13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
| 13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
| 13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
| 13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
| 13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
| 10054 | Arithmetic and geometry achieve some certainty without worrying about existence [Descartes] |
| 13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
| 13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
| 13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
| 13644 | Semantics for models uses set-theory [Shapiro] |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| 13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
| 13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
| 13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
| 13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
| 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
| 13628 | We can live well without completeness in logic [Shapiro] |
| 13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
| 13646 | Compactness is derived from soundness and completeness [Shapiro] |
| 13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
| 2252 | Surely maths is true even if I am dreaming? [Descartes] |
| 2430 | I can learn the concepts of duration and number just from observing my own thoughts [Descartes] |
| 13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
| 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
| 13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
| 13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
| 13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
| 13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
| 13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
| 13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
| 13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
| 527 | Everything exists which anyone perceives [Metrodorus of Chios] |
| 13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
| 2297 | If I can separate two things in my understanding, then God can separate them in reality [Descartes] |
| 3628 | Substance cannot be conceived or explained to others [Gassendi on Descartes] |
| 16631 | If we remove surface qualities from wax, we have an extended, flexible, changeable thing [Descartes] |
| 17865 | Descartes gives an essence by an encapsulating formula [Descartes, by Almog] |
| 2301 | We know by thought that what is done cannot be undone [Descartes] |
| 3642 | Pythagoras' Theorem doesn't cease to be part of the essence of triangles just because we doubt it [Arnauld on Descartes] |
| 20190 | Belief is not an intellectual state or act, because propositions are affirmed or denied by the will [Descartes, by Zagzebski] |
| 1585 | Descartes tried to model reason on maths instead of 'logos' [Roochnik on Descartes] |
| 1582 | Labelling slightly doubtful things as false is irrational [Roochnik on Descartes] |
| 2256 | Maybe there is only one certain fact, which is that nothing is certain [Descartes] |
| 6914 | Descartes transformed 'God is thinkable, so he exists' into 'I think, so I exist' [Descartes, by Feuerbach] |
| 4641 | In the Meditations version of the Cogito he says "I am; I exist", which avoids presenting it as an argument [Descartes, by Baggini /Fosl] |
| 6929 | Modern philosophy set the self-conscious ego in place of God [Descartes, by Feuerbach] |
| 3849 | "I think therefore I am" is the absolute truth of consciousness [Sartre on Descartes] |
| 2258 | I must even exist if I am being deceived by something [Descartes] |
| 2259 | "I am, I exist" is necessarily true every time I utter it or conceive it in my mind [Descartes] |
| 3160 | The Cogito is a transcendental argument, not a piece of a priori knowledge [Rey on Descartes] |
| 2260 | If I don't think, there is no reason to think that I exist [Descartes] |
| 1117 | The Cogito proves subjective experience is basic, but makes false claims about the Self [Russell on Descartes] |
| 2873 | Maybe 'I' am not the thinker, but something produced by thought [Nietzsche on Descartes] |
| 3623 | The Cogito only works if you already understand what thought and existence are [Mersenne on Descartes] |
| 1369 | It is a precondition of the use of the word 'I' that I exist [Ayer on Descartes] |
| 5360 | The thing which experiences may be momentary, and change with the next experience [Russell on Descartes] |
| 2870 | 'I think' assumes I exist, that thinking is known and caused, and that I am doing it [Nietzsche on Descartes] |
| 5188 | A thought doesn't imply other thoughts, or enough thoughts to make up a self [Ayer on Descartes] |
| 3624 | That I perform an activity (thinking) doesn't prove what type of thing I am [Hobbes on Descartes] |
| 3120 | Autistic children seem to use the 'I' concept without seeing themselves as thinkers [Segal on Descartes] |
| 4526 | The Cogito assumes a priori the existence of substance, when actually it is a grammatical custom [Nietzsche on Descartes] |
| 5579 | How can we infer that all thinking involves self-consciousness, just from my own case? [Kant on Descartes] |
| 5580 | My self is not an inference from 'I think', but a presupposition of it [Kant on Descartes] |
| 5587 | We cannot give any information a priori about the nature of the 'thing that thinks' [Kant on Descartes] |
| 5588 | The fact that I am a subject is not enough evidence to show that I am a substantial object [Kant on Descartes] |
| 13923 | Descartes' claim to know his existence before his essence is misleading or absurd [Descartes, by Lowe] |
| 6930 | Modern self-consciousness is a doubtful abstraction; only senses and feelings are certain [Feuerbach on Descartes] |
| 2261 | My perceiving of things may be false, but my seeming to perceive them cannot be false [Descartes] |
| 2257 | I myself could be the author of all these self-delusions [Descartes] |
| 2279 | A triangle has a separate non-invented nature, shown by my ability to prove facts about it [Descartes] |
| 6490 | For Descartes, objects have one primary quality, which is geometrical [Descartes, by Robinson,H] |
| 2295 | Why does pain make us sad? [Descartes] |
| 3627 | Dogs can make the same judgements as us about variable things [Gassendi on Descartes] |
| 2265 | We perceive objects by intellect, not by senses or imagination [Descartes] |
| 2263 | The wax is not perceived by the senses, but by the mind alone [Descartes] |
| 2264 | We don't 'see' men in heavy clothes, we judge them to be men [Descartes] |
| 2247 | To achieve good science we must rebuild from the foundations [Descartes] |
| 2255 | Only one certainty is needed for progress (like a lever's fulcrum) [Descartes] |
| 2251 | Even if my body and objects are imaginary, there may be simpler things which are true [Descartes] |
| 6347 | Descartes can't begin again, because sceptics doubt cognitive processes as well as beliefs [Pollock/Cruz on Descartes] |
| 2296 | If pain is felt in a lost limb, I cannot be certain that a felt pain exists in my real limbs [Descartes] |
| 3620 | We correct sense errors with other senses, not intellect [Mersenne on Descartes] |
| 3619 | The senses can only report, so perception errors are in the judgment [Gassendi on Descartes] |
| 2249 | It is prudent never to trust your senses if they have deceived you even once [Descartes] |
| 2253 | God may have created nothing, but made his creation appear to me as it does now [Descartes] |
| 2254 | To achieve full scepticism, I imagine a devil who deceives me about the external world and my own body and senses [Descartes] |
| 2305 | Waking actions are joined by memory to all our other actions, unlike actions of which we dream [Descartes] |
| 2294 | I can only sense an object if it is present, and can't fail to sense it when it is [Descartes] |
| 4862 | Can the pineal gland be moved more slowly or quickly by the mind than by animal spirits? [Spinoza on Descartes] |
| 3850 | We discovers others as well as ourselves in the Cogito [Sartre on Descartes] |
| 2302 | Faculties of the mind aren't parts, as one mind uses them [Descartes] |
| 3151 | Descartes put thought at the centre of the mind problem, but we put sensation [Rey on Descartes] |
| 21800 | Descartes mentions many cognitive faculties, but reduces them to will and intellect [Descartes, by Schmid] |
| 1399 | Imagination and sensation are non-essential to mind [Descartes] |
| 1400 | Some cause must unite the separate temporal sections of a person [Descartes] |
| 1401 | Since I only observe myself to be thinking, I conclude that that is my essence [Descartes] |
| 2299 | I can exist without imagination and sensing, but they can't exist without me [Descartes] |
| 6907 | For Descartes a person's essence is the mind because objects are perceived by mind, not senses [Descartes, by Feuerbach] |
| 2283 | Our 'will' just consists of the feeling that when we are motivated to do something, there are no external pressures [Descartes] |
| 2282 | My capacity to make choices with my free will extends as far as any faculty ever could [Descartes] |
| 2276 | The mind is a non-extended thing which thinks [Descartes] |
| 2298 | Mind is not extended, unlike the body [Descartes] |
| 3423 | Descartes is a substance AND property dualist [Descartes, by Kim] |
| 2303 | The mind is utterly indivisible [Descartes] |
| 6153 | Interaction between mental and physical seems to violate the principle of conservation of energy [Rowlands on Descartes] |
| 3625 | The 'thinking thing' may be the physical basis of the mind [Hobbes on Descartes] |
| 2552 | Knowing different aspects of brain/mind doesn't make them different [Rorty on Descartes] |
| 4305 | Descartes gives no clear criterion for individuating mental substances [Cottingham on Descartes] |
| 4861 | Does Descartes have a clear conception of how mind unites with body? [Spinoza on Descartes] |
| 6540 | Even Descartes may concede that mental supervenes on neuroanatomical [Lycan on Descartes] |
| 7733 | Superman's strength is indubitable, Clark Kent's is doubtful, so they are not the same? [Maslin on Descartes] |
| 5686 | In some thoughts I grasp a subject, but also I will or fear or affirm or deny it [Descartes] |
| 4017 | Descartes created the modern view of rationality, as an internal feature instead of an external vision [Descartes, by Taylor,C] |
| 2284 | I make errors because my will extends beyond my understanding [Descartes] |
| 5685 | True ideas are images, such as of a man, a chimera, or God [Descartes] |
| 3629 | All ideas are adventitious, and come from the senses [Gassendi on Descartes] |
| 2273 | The ideas of God and of my self are innate in me [Descartes] |
| 2285 | I can think of innumerable shapes I have never experienced [Descartes] |
| 2286 | The idea of a supremely perfect being is within me, like the basic concepts of mathematics [Descartes] |
| 2280 | Many causes are quite baffling, so it is absurd to deduce causes from final purposes [Descartes] |
| 2272 | There must be at least as much in the cause as there is in the effect [Descartes] |
| 2289 | Nothing apart from God could have essential existence, and such a being must be unique and eternal [Descartes] |
| 2269 | God the creator is an intelligent, infinite, powerful substance [Descartes] |
| 2275 | It is self-evident that deception is a natural defect, so God could not be a deceiver [Descartes] |
| 2287 | Existence and God's essence are inseparable, like a valley and a mountain, or a triangle and its properties [Descartes] |
| 2288 | I cannot think of a supremely perfect being without the supreme perfection of existence [Descartes] |
| 2274 | The idea of God in my mind is like the mark a craftsman puts on his work [Descartes] |
| 2268 | One idea leads to another, but there must be an initial idea that contains the reality of all the others [Descartes] |
| 3632 | We mustn't worship God as an image because we have no idea of him [Hobbes on Descartes] |
| 3633 | We can never conceive of an infinite being [Gassendi on Descartes] |
| 5036 | Descartes cannot assume that a most perfect being exists without contradictions [Leibniz on Descartes] |
| 3638 | Existence is not a perfection; it is what makes perfection possible [Gassendi on Descartes] |
| 2278 | Error arises because my faculty for judging truth is not infinite [Descartes] |
| 2277 | Since God does not wish to deceive me, my judgement won't make errors if I use it properly [Descartes] |
| 3636 | God didn't give us good judgement even about our own lives [Gassendi on Descartes] |
| 2281 | If we ask whether God's works are perfect, we must not take a narrow viewpoint, but look at the universe as a whole [Descartes] |