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Ideas of John P. Burgess, by Text
[American, b.1948, Studied at Berkeley. Teacher at Princeton University.]
2005

Review of Chihara 'Struct. Accnt of Maths'

§1

p.79

10185

Set theory is the standard background for modern mathematics

§1

p.79

10186

If set theory is used to define 'structure', we can't define set theory structurally

§1

p.79

10184

Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure

§1

p.80

10187

Abstract algebra concerns relations between models, not common features of all the models

§5

p.86

10188

How can mathematical relations be either internal, or external, or intrinsic?

§5

p.86

10189

There is no one relation for the real number 2, as relations differ in different models

Pref

p.5

15404

Technical people see logic as any formal system that can be studied, not a study of argument validity

Pref

p.5

15403

Philosophical logic is a branch of logic, and is now centred in computer science

1.1

p.1

15405

Classical logic neglects the nonmathematical, such as temporality or modality

1.4

p.4

15407

Formalising arguments favours lots of connectives; proving things favours having very few

1.4

p.4

15406

'Induction' and 'recursion' on complexity prove by connecting a formula to its atomic components

1.5

p.6

15408

'Tautologies' are valid formulas of classical sentential logic  or substitution instances in other logics

1.7

p.8

15409

All occurrences of variables in atomic formulas are free

1.8

p.10

15411

We only need to study mathematical models, since all other models are isomorphic to these

2.2

p.20

15412

Models leave out meaning, and just focus on truth values

2.8

p.32

15413

With four tense operators, all complex tenses reduce to fourteen basic cases

2.9

p.35

15414

The denotation of a definite description is flexible, rather than rigid

2.9

p.37

15415

The temporal Barcan formulas fix what exists, which seems absurd

3.2

p.43

15416

We aim to get the technical notion of truth in all models matching intuitive truth in all instances

3.3

p.46

15417

Logical necessity has two sides  validity and demonstrability  which coincide in classical logic

3.3

p.47

15418

Validity (for truth) and demonstrability (for proof) have correlates in satisfiability and consistency

3.8

p.65

15419

General consensus is S5 for logical modality of validity, and S4 for proof

3.9

p.68

15420

De re modality seems to apply to objects a concept intended for sentences

4.1

p.73

15421

Classical logic neglects counterfactuals, temporality and modality, because maths doesn't use them

4.3

p.78

15422

Three conditionals theories: Materialism (material conditional), Idealism (true=assertable), Nihilism (no truth)

4.9

p.96

15423

It is doubtful whether the negation of a conditional has any clear meaning

5.2

p.102

15424

Asserting a disjunction from one disjunct seems odd, but can be sensible, and needed in maths

5.3

p.105

15426

We can build one expanding sequence, instead of a chain of deductions

5.3

p.105

15425

The sequent calculus makes it possible to have proof without transitivity of entailment

5.3

p.106

15427

The Cut Rule expresses the classical idea that entailment is transitive

5.7

p.113

15428

The Liar seems like a truthvalue 'gap', but dialethists see it as a 'glut'

5.8

p.114

15429

Relevance logic's → is perhaps expressible by 'if A, then B, for that reason'

6.4

p.129

15430

Is classical logic a part of intuitionist logic, or vice versa?

6.9

p.141

15431

It is still unsettled whether standard intuitionist logic is complete
