Ideas of John P. Burgess, by Theme

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

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3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
Truth only applies to closed formulas, but we need satisfaction of open formulas to define it
4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
With four tense operators, all complex tenses reduce to fourteen basic cases
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
The temporal Barcan formulas fix what exists, which seems absurd
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Is classical logic a part of intuitionist logic, or vice versa?
It is still unsettled whether standard intuitionist logic is complete
4. Formal Logic / E. Nonclassical Logics / 5. Relevant Logic
Relevance logic's → is perhaps expressible by 'if A, then B, for that reason'
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
Technical people see logic as any formal system that can be studied, not a study of argument validity
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
The Cut Rule expresses the classical idea that entailment is transitive
Classical logic neglects the non-mathematical, such as temporality or modality
Classical logic neglects counterfactuals, temporality and modality, because maths doesn't use them
5. Theory of Logic / A. Overview of Logic / 9. Philosophical Logic
Philosophical logic is a branch of logic, and is now centred in computer science
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Formalising arguments favours lots of connectives; proving things favours having very few
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / e. or
Asserting a disjunction from one disjunct seems odd, but can be sensible, and needed in maths
5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
All occurrences of variables in atomic formulas are free
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
The denotation of a definite description is flexible, rather than rigid
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
'Induction' and 'recursion' on complexity prove by connecting a formula to its atomic components
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
The sequent calculus makes it possible to have proof without transitivity of entailment
We can build one expanding sequence, instead of a chain of deductions
5. Theory of Logic / I. Semantics of Logic / 4. Tautological Truth
'Tautologies' are valid formulas of classical sentential logic - or substitution instances in other logics
5. Theory of Logic / I. Semantics of Logic / 5. Satisfaction
Validity (for truth) and demonstrability (for proof) have correlates in satisfiability and consistency
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Models leave out meaning, and just focus on truth values
We aim to get the technical notion of truth in all models matching intuitive truth in all instances
We only need to study mathematical models, since all other models are isomorphic to these
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
The Liar seems like a truth-value 'gap', but dialethists see it as a 'glut'
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Set theory is the standard background for modern mathematics
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure
There is no one relation for the real number 2, as relations differ in different models
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
If set theory is used to define 'structure', we can't define set theory structurally
Abstract algebra concerns relations between models, not common features of all the models
How can mathematical relations be either internal, or external, or intrinsic?
10. Modality / A. Necessity / 4. De re / De dicto modality
De re modality seems to apply to objects a concept intended for sentences
10. Modality / A. Necessity / 6. Logical Necessity
Logical necessity has two sides - validity and demonstrability - which coincide in classical logic
General consensus is S5 for logical modality of validity, and S4 for proof
10. Modality / B. Possibility / 8. Conditionals / a. Conditionals
Three conditionals theories: Materialism (material conditional), Idealism (true=assertable), Nihilism (no truth)
It is doubtful whether the negation of a conditional has any clear meaning