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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1. Philosophy / C. History of Philosophy / 5. Modern Philosophy / b. Modern philosophy beginnings
Russell started a whole movement in philosophy by providing an analysis of descriptions
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 blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
Compactness makes consequence manageable, but restricts expressive power
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 / 4. 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 / 3. Axioms for Number / e. Peano arithmetic 2nd-order
Although second-order arithmetic is incomplete, it can fully model normal arithmetic
Second-order arithmetic covers all properties, ensuring categoricity
6. Mathematics / B. Foundations for Mathematics / 4. 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 / c. Vagueness as semantic
Would a language without vagueness be usable at all?
7. Existence / D. Theories of Reality / 9. Vagueness / e. Supervaluation for vagueness
A 'supervaluation' gives a proposition consistent truth-value for classical assignments
Supervaluations say there is a cut-off somewhere, but at no particular place
Identities and the Indiscernibility of Identicals don't work with supervaluations
9. Objects / A. Existence of Objects / 4. 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
Equating necessity with informal provability is the S4 conception of necessity
10. Modality / B. Possibility / 8. Conditionals / a. Conditionals
The point of conditionals is to show that one will accept modus ponens
Some people even claim that conditionals do not express propositions
The standard view of conditionals is that they are truth-functional
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 / B. 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