2005 | Review of Chihara 'Struct. Accnt of Maths' |
§1 | p.79 | 10186 | If set theory is used to define 'structure', we can't define set theory structurally |
§1 | p.79 | 10185 | Set theory is the standard background for modern mathematics |
§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 |
2009 | Philosophical Logic |
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 non-mathematical, 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.9 | 15410 | Truth only applies to closed formulas, but we need satisfaction of open formulas to define it |
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 truth-value '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 |