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
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10987
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Three traditional names of rules are 'Simplification', 'Addition' and 'Disjunctive Syllogism'
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic
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11004
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Necessity is provability in S4, and true in all worlds in S5
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4. Formal Logic / E. Nonclassical Logics / 4. Fuzzy Logic
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11018
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There are fuzzy predicates (and sets), and fuzzy quantifiers and modifiers
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4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
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11011
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Same say there are positive, negative and neuter free logics
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
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11020
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Realisms like the full Comprehension Principle, that all good concepts determine sets
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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10986
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Not all validity is captured in first-order logic
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5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
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10972
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The non-emptiness of the domain is characteristic of classical logic
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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11024
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Semantics must precede proof in higher-order logics, since they are incomplete
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5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
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10985
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We should exclude second-order logic, precisely because it captures arithmetic
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5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
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10970
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A theory of logical consequence is a conceptual analysis, and a set of validity techniques
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10984
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Logical consequence isn't just a matter of form; it depends on connections like round-square
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5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
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10973
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A theory is logically closed, which means infinite premisses
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5. Theory of Logic / G. Quantification / 1. Quantification
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11007
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Quantifiers are second-order predicates
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5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
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10978
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In second-order logic the higher-order variables range over all the properties of the objects
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5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
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10971
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A logical truth is the conclusion of a valid inference with no premisses
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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10988
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Any first-order theory of sets is inadequate
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5. Theory of Logic / K. Features of Logics / 6. Compactness
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10975
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Compactness does not deny that an inference can have infinitely many premisses
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10974
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Compactness is when any consequence of infinite propositions is the consequence of a finite subset
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10977
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Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
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10976
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Compactness makes consequence manageable, but restricts expressive power
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5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
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11014
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Self-reference paradoxes seem to arise only when falsity is involved
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
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11025
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Infinite cuts and successors seems to suggest an actual infinity there waiting for us
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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10979
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Although second-order arithmetic is incomplete, it can fully model normal arithmetic
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10980
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Second-order arithmetic covers all properties, ensuring categoricity
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / g. Von Neumann numbers
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10997
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Von Neumann numbers are helpful, but don't correctly describe numbers
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7. Existence / D. Theories of Reality / 10. Vagueness / d. Vagueness as linguistic
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11016
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Would a language without vagueness be usable at all?
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7. Existence / D. Theories of Reality / 10. Vagueness / f. Supervaluation for vagueness
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11013
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Identities and the Indiscernibility of Identicals don't work with supervaluations
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11019
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Supervaluations say there is a cut-off somewhere, but at no particular place
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11012
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A 'supervaluation' gives a proposition consistent truth-value for classical assignments
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9. Objects / A. Existence of Objects / 5. Individuation / d. Individuation by haecceity
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10995
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A haecceity is a set of individual properties, essential to each thing
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10. Modality / A. Necessity / 2. Nature of Necessity
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11001
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Equating necessity with truth in every possible world is the S5 conception of necessity
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10. Modality / B. Possibility / 8. Conditionals / a. Conditionals
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10992
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The point of conditionals is to show that one will accept modus ponens
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10989
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The standard view of conditionals is that they are truth-functional
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11017
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Some people even claim that conditionals do not express propositions
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10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
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10983
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Knowledge of possible worlds is not causal, but is an ontology entailed by semantics
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10. Modality / E. Possible worlds / 1. Possible Worlds / c. Possible worlds realism
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10982
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How can modal Platonists know the truth of a modal proposition?
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10. Modality / E. Possible worlds / 1. Possible Worlds / d. Possible worlds actualism
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10996
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Actualism is reductionist (to parts of actuality), or moderate realist (accepting real abstractions)
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10. Modality / E. Possible worlds / 2. Nature of Possible Worlds / c. Worlds as propositions
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10981
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A possible world is a determination of the truth-values of all propositions of a domain
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10. Modality / E. Possible worlds / 3. Transworld Objects / c. Counterparts
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11000
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If worlds are concrete, objects can't be present in more than one, and can only have counterparts
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15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
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10998
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The mind abstracts ways things might be, which are nonetheless real
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19. Language / C. Assigning Meanings / 4. Compositionality
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11005
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Negative existentials with compositionality make the whole sentence meaningless
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19. Language / D. Propositions / 1. Propositions
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10966
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A proposition objectifies what a sentence says, as indicative, with secure references
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