Combining Texts

All the ideas for 'A Realistic Theory of Categories', 'A Tour through Mathematical Logic' and 'Thought and Reality'

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41 ideas

3. Truth / F. Semantic Truth / 2. Semantic Truth
Truth is part of semantics, since valid inference preserves truth [Dummett]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
A 'tautology' must include connectives [Wolf,RS]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV
Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
Language can violate bivalence because of non-referring terms or ill-defined predicates [Dummett]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
The law of excluded middle is the logical reflection of the principle of bivalence [Dummett]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
Model theory reveals the structures of mathematics [Wolf,RS]
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 4. Completeness
If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Modern mathematics has unified all of its objects within set theory [Wolf,RS]
7. Existence / D. Theories of Reality / 2. Realism
Philosophers should not presume reality, but only invoke it when language requires it [Dummett]
7. Existence / D. Theories of Reality / 4. Anti-realism
We can't make sense of a world not apprehended by a mind [Dummett]
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
Since 'no bird here' and 'no squirrel here' seem the same, we must talk of 'atomic' facts [Dummett]
7. Existence / D. Theories of Reality / 8. Facts / c. Facts and truths
We know we can state facts, with true statements [Dummett]
7. Existence / D. Theories of Reality / 10. Vagueness / d. Vagueness as linguistic
'That is red or orange' might be considered true, even though 'that is red' and 'that is orange' were not [Dummett]
7. Existence / E. Categories / 3. Proposed Categories
Chisholm divides things into contingent and necessary, and then individuals, states and non-states [Chisholm, by Westerhoff]
12. Knowledge Sources / D. Empiricism / 5. Empiricism Critique
Empirical and a priori knowledge are not distinct, but are extremes of a sliding scale [Dummett]
18. Thought / A. Modes of Thought / 1. Thought
The theories of meaning and understanding are the only routes to an account of thought [Dummett]
A theory of thought will include propositional attitudes as well as propositions [Dummett]
18. Thought / E. Abstraction / 8. Abstractionism Critique
To 'abstract from' is a logical process, as opposed to the old mental view [Dummett]
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
To know the truth-conditions of a sentence, you must already know the meaning [Dummett]
19. Language / A. Nature of Meaning / 5. Meaning as Verification
A justificationist theory of meaning leads to the rejection of classical logic [Dummett]
Verificationism could be realist, if we imagined the verification by a superhuman power [Dummett]
If truths about the past depend on memories and current evidence, the past will change [Dummett]
19. Language / A. Nature of Meaning / 6. Meaning as Use
We could only guess the meanings of 'true' and 'false' when sentences were used [Dummett]
19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning
Sentences are the primary semantic units, because they can say something [Dummett]
19. Language / D. Propositions / 1. Propositions
We can't distinguish a proposition from its content [Dummett]
27. Natural Reality / D. Time / 1. Nature of Time / d. Time as measure
Time is the measure of change, so we can't speak of time before all change [Dummett]
27. Natural Reality / D. Time / 1. Nature of Time / h. Presentism
If Presentism is correct, we cannot even say that the present changes [Dummett]