25 ideas
1922 | Spiritual qualities only become advantageous with the growth of wisdom [Plato] |
10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
11259 | How can you seek knowledge of something if you don't know it? [Plato] |
10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
20219 | True opinions only become really valuable when they are tied down by reasons [Plato] |
5985 | Seeking and learning are just recollection [Plato] |
5986 | The slave boy learns geometry from questioning, not teaching, so it is recollection [Plato] |
1923 | As a guide to action, true opinion is as good as knowledge [Plato] |
1919 | You don't need to learn what you know, and how do you seek for what you don't know? [Plato] |
1927 | It seems that virtue is neither natural nor taught, but is a divine gift [Plato] |
1913 | Is virtue taught, or achieved by practice, or a natural aptitude, or what? [Plato] |
1921 | If virtue is a type of knowledge then it ought to be taught [Plato] |
1916 | Even if virtues are many and various, they must have something in common to make them virtues [Plato] |
1918 | How can you know part of virtue without knowing the whole? [Plato] |
2594 | A true cause must involve a necessary connection between cause and effect [Malebranche] |