Combining Texts

All the ideas for 'The Principles of Art', 'Introduction to Mathematical Logic' and 'What is Justified Belief?'

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26 ideas

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
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]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
17764 Boolean connectives are interpreted as functions on the set {1,0} [Walicki]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
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]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
17759 Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
17741 To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
17747 A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17748 The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
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]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [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]
17757 Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
17762 In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
17754 Inductive proof depends on the choice of the ordering [Walicki]
10. Modality / A. Necessity / 2. Nature of Necessity
17742 Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
13. Knowledge Criteria / A. Justification Problems / 3. Internal or External / a. Pro-internalism
8830 A belief can be justified when the person has forgotten the evidence for it [Goldman]
13. Knowledge Criteria / A. Justification Problems / 3. Internal or External / b. Pro-externalism
8832 If justified beliefs are well-formed beliefs, then animals and young children have them [Goldman]
13. Knowledge Criteria / C. External Justification / 3. Reliabilism / a. Reliable knowledge
8829 Justification depends on the reliability of its cause, where reliable processes tend to produce truth [Goldman]
16. Persons / C. Self-Awareness / 1. Introspection
8831 Introspection is really retrospection; my pain is justified by a brief causal history [Goldman]
21. Aesthetics / B. Nature of Art / 4. Art as Expression
20420 The emotion expressed is non-conscious, but feels oppressive until expression relieves it [Collingwood]
21. Aesthetics / B. Nature of Art / 7. Ontology of Art
20421 Art exists ideally, purely as experiences in the mind of the perceiver [Collingwood, by Kemp]
21. Aesthetics / C. Artistic Issues / 6. Value of Art
20406 Art clarifies the artist's mind and feelings, thus leading to self-knowledge [Collingwood, by Davies,S]