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Ideas of John P. Burgess, by Text
[American, b.1948, Studied at Berkeley. Teacher at Princeton University.]
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2005
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Review of Chihara 'Struct. Accnt of Maths'
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§1
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p.79
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10185
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Set theory is the standard background for modern mathematics
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§1
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p.79
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10184
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Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure
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§1
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p.79
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10186
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If set theory is used to define 'structure', we can't define set theory structurally
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§1
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p.80
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10187
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Abstract algebra concerns relations between models, not common features of all the models
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§5
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p.86
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10188
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How can mathematical relations be either internal, or external, or intrinsic?
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§5
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p.86
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10189
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There is no one relation for the real number 2, as relations differ in different models
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Pref
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p.-5
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15404
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Technical people see logic as any formal system that can be studied, not a study of argument validity
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Pref
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p.-5
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15403
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Philosophical logic is a branch of logic, and is now centred in computer science
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1.1
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p.1
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15405
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Classical logic neglects the non-mathematical, such as temporality or modality
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1.4
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p.4
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15406
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'Induction' and 'recursion' on complexity prove by connecting a formula to its atomic components
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1.4
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p.4
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15407
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Formalising arguments favours lots of connectives; proving things favours having very few
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1.5
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p.6
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15408
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'Tautologies' are valid formulas of classical sentential logic - or substitution instances in other logics
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1.7
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p.8
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15409
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All occurrences of variables in atomic formulas are free
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1.8
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p.10
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15411
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We only need to study mathematical models, since all other models are isomorphic to these
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2.2
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p.20
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15412
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Models leave out meaning, and just focus on truth values
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2.8
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p.32
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15413
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With four tense operators, all complex tenses reduce to fourteen basic cases
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2.9
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p.35
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15414
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The denotation of a definite description is flexible, rather than rigid
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2.9
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p.37
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15415
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The temporal Barcan formulas fix what exists, which seems absurd
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3.2
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p.43
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15416
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We aim to get the technical notion of truth in all models matching intuitive truth in all instances
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3.3
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p.46
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15417
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Logical necessity has two sides - validity and demonstrability - which coincide in classical logic
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3.3
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p.47
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15418
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Validity (for truth) and demonstrability (for proof) have correlates in satisfiability and consistency
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3.8
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p.65
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15419
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General consensus is S5 for logical modality of validity, and S4 for proof
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3.9
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p.68
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15420
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De re modality seems to apply to objects a concept intended for sentences
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4.1
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p.73
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15421
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Classical logic neglects counterfactuals, temporality and modality, because maths doesn't use them
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4.3
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p.78
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15422
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Three conditionals theories: Materialism (material conditional), Idealism (true=assertable), Nihilism (no truth)
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4.9
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p.96
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15423
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It is doubtful whether the negation of a conditional has any clear meaning
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5.2
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p.102
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15424
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Asserting a disjunction from one disjunct seems odd, but can be sensible, and needed in maths
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5.3
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p.105
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15426
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We can build one expanding sequence, instead of a chain of deductions
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5.3
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p.105
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15425
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The sequent calculus makes it possible to have proof without transitivity of entailment
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5.3
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p.106
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15427
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The Cut Rule expresses the classical idea that entailment is transitive
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5.7
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p.113
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15428
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The Liar seems like a truth-value 'gap', but dialethists see it as a 'glut'
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5.8
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p.114
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15429
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Relevance logic's → is perhaps expressible by 'if A, then B, for that reason'
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6.4
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p.129
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15430
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Is classical logic a part of intuitionist logic, or vice versa?
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6.9
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p.141
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15431
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It is still unsettled whether standard intuitionist logic is complete
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