Ideas from 'Thinking About Logic' by Stephen Read [1995], by Theme Structure

[found in 'Thinking About Logic' by Read,Stephen [OUP 1995,0-19-289238-x]].

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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
Three traditional names of rules are 'Simplification', 'Addition' and 'Disjunctive Syllogism'
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic
Necessity is provability in S4, and true in all worlds in S5
4. Formal Logic / E. Nonclassical Logics / 4. Fuzzy Logic
There are fuzzy predicates (and sets), and fuzzy quantifiers and modifiers
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
Same say there are positive, negative and neuter free logics
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
Realisms like the full Comprehension Principle, that all good concepts determine sets
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
Not all validity is captured in first-order logic
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
The non-emptiness of the domain is characteristic of classical logic
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Semantics must precede proof in higher-order logics, since they are incomplete
5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
We should exclude second-order logic, precisely because it captures arithmetic
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
A theory of logical consequence is a conceptual analysis, and a set of validity techniques
Logical consequence isn't just a matter of form; it depends on connections like round-square
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
A theory is logically closed, which means infinite premisses
5. Theory of Logic / G. Quantification / 1. Quantification
Quantifiers are second-order predicates
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
In second-order logic the higher-order variables range over all the properties of the objects
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
A logical truth is the conclusion of a valid inference with no premisses
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
Any first-order theory of sets is inadequate
5. Theory of Logic / K. Features of Logics / 6. Compactness
Compactness does not deny that an inference can have infinitely many premisses
Compactness is when any consequence of infinite propositions is the consequence of a finite subset
Compactness makes consequence manageable, but restricts expressive power
Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
Self-reference paradoxes seem to arise only when falsity is involved
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
Infinite cuts and successors seems to suggest an actual infinity there waiting for us
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order arithmetic covers all properties, ensuring categoricity
Although second-order arithmetic is incomplete, it can fully model normal arithmetic
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / g. Von Neumann numbers
Von Neumann numbers are helpful, but don't correctly describe numbers
7. Existence / D. Theories of Reality / 9. Vagueness / d. Vagueness as linguistic
Would a language without vagueness be usable at all?
7. Existence / D. Theories of Reality / 9. Vagueness / f. Supervaluation for vagueness
A 'supervaluation' gives a proposition consistent truth-value for classical assignments
Identities and the Indiscernibility of Identicals don't work with supervaluations
Supervaluations say there is a cut-off somewhere, but at no particular place
9. Objects / A. Existence of Objects / 5. Individuation / d. Individuation by haecceity
A haecceity is a set of individual properties, essential to each thing
10. Modality / A. Necessity / 2. Nature of Necessity
Equating necessity with truth in every possible world is the S5 conception of necessity
10. Modality / B. Possibility / 8. Conditionals / a. Conditionals
Some people even claim that conditionals do not express propositions
The standard view of conditionals is that they are truth-functional
The point of conditionals is to show that one will accept modus ponens
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
Knowledge of possible worlds is not causal, but is an ontology entailed by semantics
10. Modality / E. Possible worlds / 1. Possible Worlds / c. Possible worlds realism
How can modal Platonists know the truth of a modal proposition?
10. Modality / E. Possible worlds / 1. Possible Worlds / d. Possible worlds actualism
Actualism is reductionist (to parts of actuality), or moderate realist (accepting real abstractions)
10. Modality / E. Possible worlds / 2. Nature of Possible Worlds / c. Worlds as propositions
A possible world is a determination of the truth-values of all propositions of a domain
10. Modality / E. Possible worlds / 3. Transworld Objects / c. Counterparts
If worlds are concrete, objects can't be present in more than one, and can only have counterparts
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
The mind abstracts ways things might be, which are nonetheless real
19. Language / C. Assigning Meanings / 4. Compositionality
Negative existentials with compositionality make the whole sentence meaningless
19. Language / D. Propositions / 1. Propositions
A proposition objectifies what a sentence says, as indicative, with secure references