30 ideas
17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
17765 | Propositional language can only relate statements as the same or as different [Walicki] |
17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
18739 | Three stages of philosophical logic: syntactic (1905-55), possible worlds (1963-85), widening (1990-) [Horsten/Pettigrew] |
18741 | Logical formalization makes concepts precise, and also shows their interrelation [Horsten/Pettigrew] |
18744 | Models are sets with functions and relations, and truth built up from the components [Horsten/Pettigrew] |
17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
17754 | Inductive proof depends on the choice of the ordering [Walicki] |
18740 | If 'exist' doesn't express a property, we can hardly ask for its essence [Horsten/Pettigrew] |
17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
18745 | A Tarskian model can be seen as a possible state of affairs [Horsten/Pettigrew] |
18747 | The 'spheres model' was added to possible worlds, to cope with counterfactuals [Horsten/Pettigrew] |
18748 | Epistemic logic introduced impossible worlds [Horsten/Pettigrew] |
18746 | Possible worlds models contain sets of possible worlds; this is a large metaphysical commitment [Horsten/Pettigrew] |
18750 | Using possible worlds for knowledge and morality may be a step too far [Horsten/Pettigrew] |
7458 | The reliability of witnesses depends on whether they benefit from their observations [Laplace, by Hacking] |
3441 | If a supreme intellect knew all atoms and movements, it could know all of the past and the future [Laplace] |