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All the ideas for 'Explanation: the state of play', 'Model Theory' and 'Logic in Mathematics'

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31 ideas

2. Reason / D. Definition / 3. Types of Definition
16877 A 'constructive' (as opposed to 'analytic') definition creates a new sign [Frege]
2. Reason / D. Definition / 7. Contextual Definition
10476 The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W]
2. Reason / D. Definition / 10. Stipulative Definition
11219 Frege suggested that mathematics should only accept stipulative definitions [Frege, by Gupta]
2. Reason / E. Argument / 6. Conclusive Proof
16878 We must be clear about every premise and every law used in a proof [Frege]
5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
16867 Logic not only proves things, but also reveals logical relations between them [Frege]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
10478 Since first-order languages are complete, |= and |- have the same meaning [Hodges,W]
5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
16863 Does some mathematical reasoning (such as mathematical induction) not belong to logic? [Frege]
16862 The closest subject to logic is mathematics, which does little apart from drawing inferences [Frege]
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
10477 |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W]
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
16865 'Theorems' are both proved, and used in proofs [Frege]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
10474 |= should be read as 'is a model for' or 'satisfies' [Hodges,W]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10473 Model theory studies formal or natural language-interpretation using set-theory [Hodges,W]
10475 A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W]
10481 Models in model theory are structures, not sets of descriptions [Hodges,W]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
16866 Tracing inference backwards closes in on a small set of axioms and postulates [Frege]
16868 The essence of mathematics is the kernel of primitive truths on which it rests [Frege]
16871 A truth can be an axiom in one system and not in another [Frege]
16870 Axioms are truths which cannot be doubted, and for which no proof is needed [Frege]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
16869 To create order in mathematics we need a full system, guided by patterns of inference [Frege]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
10480 First-order logic can't discriminate between one infinite cardinal and another [Hodges,W]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
16864 If principles are provable, they are theorems; if not, they are axioms [Frege]
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
9388 Every concept must have a sharp boundary; we cannot allow an indeterminate third case [Frege]
14. Science / D. Explanation / 1. Explanation / b. Aims of explanation
12314 Audience-relative explanation, or metaphysical explanation based on information? [Stanford]
12313 Explanation is for curiosity, control, understanding, to make meaningful, or to give authority [Stanford]
14. Science / D. Explanation / 2. Types of Explanation / j. Explanations by reduction
12315 We can explain by showing constitution, as well as showing causes [Stanford]
18. Thought / B. Mechanics of Thought / 5. Mental Files
16876 We need definitions to cram retrievable sense into a signed receptacle [Frege]
16875 We use signs to mark receptacles for complex senses [Frege]
19. Language / A. Nature of Meaning / 6. Meaning as Use
16879 A sign won't gain sense just from being used in sentences with familiar components [Frege]
19. Language / D. Propositions / 2. Abstract Propositions / a. Propositions as sense
16873 Thoughts are not subjective or psychological, because some thoughts are the same for us all [Frege]
16872 A thought is the sense expressed by a sentence, and is what we prove [Frege]
19. Language / D. Propositions / 5. Unity of Propositions
16874 The parts of a thought map onto the parts of a sentence [Frege]