79 ideas
| 20186 | Unlike knowledge, wisdom cannot be misused [Zagzebski] |
| 19694 | Wisdom is the property of a person, not of their cognitive state [Zagzebski, by Whitcomb] |
| 20221 | Precision is only one of the virtues of a good definition [Zagzebski] |
| 20220 | Objection by counterexample is weak, because it only reveals inaccuracies in one theory [Zagzebski] |
| 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] |
| 17606 | Axioms reveal the underlying assumptions, and reveal relationships between different areas [Kline] |
| 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] |
| 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] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [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] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 20188 | Modern epistemology is too atomistic, and neglects understanding [Zagzebski] |
| 20223 | Epistemology is excessively atomic, by focusing on justification instead of understanding [Zagzebski] |
| 20217 | Truth is valuable, but someone knowing the truth is more valuable [Zagzebski] |
| 20191 | Some beliefs are fairly voluntary, and others are not at all so [Zagzebski] |
| 20222 | Knowledge either aims at a quantity of truths, or a quality of understanding of truths [Zagzebski] |
| 20225 | For internalists Gettier situations are where internally it is fine, but there is an external mishap [Zagzebski] |
| 20226 | Gettier problems are always possible if justification and truth are not closely linked [Zagzebski] |
| 20228 | We avoid the Gettier problem if the support for the belief entails its truth [Zagzebski] |
| 20227 | Gettier cases arise when good luck cancels out bad luck [Zagzebski] |
| 20194 | Intellectual virtues are forms of moral virtue [Zagzebski] |
| 20210 | A reliable process is no use without the virtues to make use of them [Zagzebski] |
| 20206 | Intellectual and moral prejudice are the same vice (and there are other examples) [Zagzebski] |
| 20208 | We can name at least thirteen intellectual vices [Zagzebski] |
| 20215 | A justified belief emulates the understanding and beliefs of an intellectually virtuous person [Zagzebski] |
| 20187 | Epistemic perfection for reliabilism is a truth-producing machine [Zagzebski] |
| 20218 | The self is known as much by its knowledge as by its action [Zagzebski] |
| 20205 | The feeling accompanying curiosity is neither pleasant nor painful [Zagzebski] |
| 20202 | Motives involve desires, but also how the desires connect to our aims [Zagzebski] |
| 20216 | Modern moral theory concerns settling conflicts, rather than human fulfilment [Zagzebski] |
| 20193 | Moral luck means our praise and blame may exceed our control or awareness [Zagzebski] |
| 20199 | Nowadays we doubt the Greek view that the flourishing of individuals and communities are linked [Zagzebski] |
| 20196 | Virtue theory is hopeless if there is no core of agreed universal virtues [Zagzebski] |
| 20200 | A virtue must always have a corresponding vice [Zagzebski] |
| 20201 | Eight marks distingush skills from virtues [Zagzebski, by PG] |
| 20203 | Virtues are deep acquired excellences of persons, which successfully attain desire ends [Zagzebski] |
| 20207 | Every moral virtue requires a degree of intelligence [Zagzebski] |
| 20214 | Virtue theory can have lots of rules, as long as they are grounded in virtues and in facts [Zagzebski] |
| 20213 | We need phronesis to coordinate our virtues [Zagzebski] |
| 20209 | For the virtue of honesty you must be careful with the truth, and not just speak truly [Zagzebski] |
| 20197 | The courage of an evil person is still a quality worth having [Zagzebski] |