91 ideas
| 6200 | Wisdom is knowing the highest good, and conforming the will to it [Kant] |
| 6207 | What fills me with awe are the starry heavens above me and the moral law within me [Kant] |
| 6184 | Consistency is the highest obligation of a philosopher [Kant] |
| 6203 | Metaphysics is just a priori universal principles of physics [Kant] |
| 17713 | After 1903, Husserl avoids metaphysical commitments [Mares] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| 17715 | The truth of the axioms doesn't matter for pure mathematics, but it does for applied [Mares] |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 17716 | Mathematics is relations between properties we abstract from experience [Mares] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 6181 | Necessity cannot be extracted from an empirical proposition [Kant] |
| 17703 | Light in straight lines is contingent a priori; stipulated as straight, because they happen to be so [Mares] |
| 17714 | Aristotelians dislike the idea of a priori judgements from pure reason [Mares] |
| 17705 | Empiricists say rationalists mistake imaginative powers for modal insights [Mares] |
| 17700 | The most popular view is that coherent beliefs explain one another [Mares] |
| 17704 | Operationalism defines concepts by our ways of measuring them [Mares] |
| 17710 | Aristotelian justification uses concepts abstracted from experience [Mares] |
| 17706 | The essence of a concept is either its definition or its conceptual relations? [Mares] |
| 17701 | Possible worlds semantics has a nice compositional account of modal statements [Mares] |
| 17702 | Unstructured propositions are sets of possible worlds; structured ones have components [Mares] |
| 6183 | Can pure reason determine the will, or are empirical conditions relevant? [Kant] |
| 6191 | The will is the faculty of purposes, which guide desires according to principles [Kant] |
| 6190 | The sole objects of practical reason are the good and the evil [Kant] |
| 18235 | Only human reason can confer value on our choices [Kant, by Korsgaard] |
| 6196 | People cannot come to morality through feeling, because morality must not be sensuous [Kant] |
| 18675 | Kant may rate two things as finally valuable: having a good will, and deserving happiness [Orsi on Kant] |
| 22007 | An autonomous agent has dignity [Würde], which has absolute worth [Kant, by Pinkard] |
| 18234 | The good will is unconditionally good, because it is the only possible source of value [Kant, by Korsgaard] |
| 6192 | Good or evil cannot be a thing, but only a maxim of action, making the person good or evil [Kant] |
| 6197 | Morality involves duty and respect for law, not love of the outcome [Kant] |
| 6193 | Our happiness is all that matters, not as a sensation, but as satisfaction with our whole existence [Kant] |
| 1452 | Happiness is the condition of a rational being for whom everything goes as they wish [Kant] |
| 1454 | Morality is not about making ourselves happy, but about being worthy of happiness [Kant] |
| 6194 | The highest worth for human beings lies in dispositions, not just actions [Kant] |
| 6198 | Virtue is the supreme state of our pursuit of happiness, and so is supreme good [Kant] |
| 1456 | Moral law is holy, and the best we can do is achieve virtue through respect for the law [Kant] |
| 6185 | No one would lend money unless a universal law made it secure, even after death [Kant] |
| 6187 | Universality determines the will, and hence extends self-love into altruism [Kant] |
| 6201 | Everyone (even God) must treat rational beings as ends in themselves, and not just as means [Kant] |
| 6186 | A holy will is incapable of any maxims which conflict with the moral law [Kant] |
| 6195 | Reason cannot solve the problem of why a law should motivate the will [Kant] |
| 6188 | A permanent natural order could not universalise a rule permitting suicide [Kant] |
| 17708 | Maybe space has points, but processes always need regions with a size [Mares] |
| 6199 | Obligation does not rest on the existence of God, but on the autonomy of reason [Kant] |
| 1453 | We have to postulate something outside nature which makes happiness coincide with morality [Kant] |
| 1455 | Belief in justice requires belief in a place for justice (heaven), a time (eternity), and a cause (God) [Kant, by PG] |
| 6205 | To know if this world must have been created by God, we would need to know all other possible worlds [Kant] |
| 6204 | Using God to explain nature is referring to something inconceivable to explain what is in front of you [Kant] |
| 6206 | From our limited knowledge we can infer great virtues in God, but not ultimate ones [Kant] |
| 6202 | In all naturalistic concepts of God, if you remove the human qualities there is nothing left [Kant] |