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Ideas for 'works', 'Ontology and Mathematical Truth' and 'Philosophy of Mathematics'

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6 ideas

5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
18108 First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
     Full Idea: First-order logic is not decidable. That is, there is no test which can be applied to any arbitrary formula of that logic and which will tell one whether the formula is or is not valid (as proved by Church in 1936).
     From: David Bostock (Philosophy of Mathematics [2009], 5.5)
18109 The completeness of first-order logic implies its compactness [Bostock]
     Full Idea: From the fact that the usual rules for first-level logic are complete (as proved by Gödel 1930), it follows that this logic is 'compact'.
     From: David Bostock (Philosophy of Mathematics [2009], 5.5)
     A reaction: The point is that the completeness requires finite proofs.
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
18123 Substitutional quantification is just standard if all objects in the domain have a name [Bostock]
     Full Idea: Substitutional quantification and quantification understood in the usual 'ontological' way will coincide when every object in the (ontological) domain has a name.
     From: David Bostock (Philosophy of Mathematics [2009], 7.3 n23)
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
18120 The Deduction Theorem is what licenses a system of natural deduction [Bostock]
     Full Idea: The Deduction Theorem is what licenses a system of 'natural deduction' in the first place.
     From: David Bostock (Philosophy of Mathematics [2009], 7.2)
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
9968 A model is 'fundamental' if it contains only concrete entities [Jubien]
     Full Idea: A first-order model can be viewed as a kind of ordered set, and if the domain of the model contains only concrete entities then it is a 'fundamental' model.
     From: Michael Jubien (Ontology and Mathematical Truth [1977], p.117)
     A reaction: An important idea. Fundamental models are where the world of logic connects with the physical world. Any account of relationship between fundamental models and more abstract ones tells us how thought links to world.
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
18125 Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock]
     Full Idea: Berry's Paradox can be put in this form, by considering the alleged name 'The least number not named by this name'.
     From: David Bostock (Philosophy of Mathematics [2009], 8.1)