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Ideas of John P. Burgess, by Text

[American, fl. 1997, Studied at Berkeley. Teacher at Princeton University.]

2005 Review of Chihara 'Struct. Accnt of Maths'
1 p.79 If set theory is used to define 'structure', we can't define set theory structurally
1 p.79 Set theory is the standard background for modern mathematics
1 p.79 Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure
1 p.80 Abstract algebra concerns relations between models, not common features of all the models
5 p.86 How can mathematical relations be either internal, or external, or intrinsic?
5 p.86 There is no one relation for the real number 2, as relations differ in different models
2009 Philosophical Logic
Pref p.-5 Technical people see logic as any formal system that can be studied, not a study of argument validity
Pref p.-5 Philosophical logic is a branch of logic, and is now centred in computer science
1.1 p.1 Classical logic neglects the non-mathematical, such as temporality or modality
1.4 p.4 Formalising arguments favours lots of connectives; proving things favours having very few
1.4 p.4 'Induction' and 'recursion' on complexity prove by connecting a formula to its atomic components
1.5 p.6 'Tautologies' are valid formulas of classical sentential logic - or substitution instances in other logics
1.7 p.8 All occurrences of variables in atomic formulas are free
1.8 p.9 Truth only applies to closed formulas, but we need satisfaction of open formulas to define it
1.8 p.10 We only need to study mathematical models, since all other models are isomorphic to these
2.2 p.20 Models leave out meaning, and just focus on truth values
2.8 p.32 With four tense operators, all complex tenses reduce to fourteen basic cases
2.9 p.35 The denotation of a definite description is flexible, rather than rigid
2.9 p.37 The temporal Barcan formulas fix what exists, which seems absurd
3.2 p.43 We aim to get the technical notion of truth in all models matching intuitive truth in all instances
3.3 p.46 Logical necessity has two sides - validity and demonstrability - which coincide in classical logic
3.3 p.47 Validity (for truth) and demonstrability (for proof) have correlates in satisfiability and consistency
3.8 p.65 General consensus is S5 for logical modality of validity, and S4 for proof
3.9 p.68 De re modality seems to apply to objects a concept intended for sentences
4.1 p.73 Classical logic neglects counterfactuals, temporality and modality, because maths doesn't use them
4.3 p.78 Three conditionals theories: Materialism (material conditional), Idealism (true=assertable), Nihilism (no truth)
4.9 p.96 It is doubtful whether the negation of a conditional has any clear meaning
5.2 p.102 Asserting a disjunction from one disjunct seems odd, but can be sensible, and needed in maths
5.3 p.105 We can build one expanding sequence, instead of a chain of deductions
5.3 p.105 The sequent calculus makes it possible to have proof without transitivity of entailment
5.3 p.106 The Cut Rule expresses the classical idea that entailment is transitive
5.7 p.113 The Liar seems like a truth-value 'gap', but dialethists see it as a 'glut'
5.8 p.114 Relevance logic's → is perhaps expressible by 'if A, then B, for that reason'
6.4 p.129 Is classical logic a part of intuitionist logic, or vice versa?
6.9 p.141 It is still unsettled whether standard intuitionist logic is complete