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All the ideas for 'What is Logic?st1=Ian Hacking', 'Sets, Aggregates and Numbers' and 'Conditionals'

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25 ideas

2. Reason / D. Definition / 3. Types of Definition
13838 A decent modern definition should always imply a semantics [Hacking]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
13833 'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking]
13834 Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking]
13835 Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking]
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
13845 The various logics are abstractions made from terms like 'if...then' in English [Hacking]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13840 First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking]
13844 A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
13842 Second-order completeness seems to need intensional entities and possible worlds [Hacking]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
14352 '¬', '&', and 'v' are truth functions: the truth of the compound is fixed by the truth of the components [Jackson]
13837 With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking]
5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
13839 Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13843 If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
17818 How many? must first partition an aggregate into sets, and then logic fixes its number [Yourgrau]
17822 Nothing is 'intrinsically' numbered [Yourgrau]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
17817 Defining 'three' as the principle of collection or property of threes explains set theory definitions [Yourgrau]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
17815 We can't use sets as foundations for mathematics if we must await results from the upper reaches [Yourgrau]
17821 You can ask all sorts of numerical questions about any one given set [Yourgrau]
10. Modality / B. Possibility / 8. Conditionals / b. Types of conditional
14360 Possible worlds for subjunctives (and dispositions), and no-truth for indicatives? [Jackson]
10. Modality / B. Possibility / 8. Conditionals / c. Truth-function conditionals
14353 Modus ponens requires that A→B is F when A is T and B is F [Jackson]
14354 When A and B have the same truth value, A→B is true, because A→A is a logical truth [Jackson]
14355 (A&B)→A is a logical truth, even if antecedent false and consequent true, so it is T if A is F and B is T [Jackson]
10. Modality / B. Possibility / 8. Conditionals / d. Non-truthfunction conditionals
14358 In the possible worlds account of conditionals, modus ponens and modus tollens are validated [Jackson]
14359 Only assertions have truth-values, and conditionals are not proper assertions [Jackson]
14357 Possible worlds account, unlike A⊃B, says nothing about when A is false [Jackson]
10. Modality / B. Possibility / 8. Conditionals / f. Pragmatics of conditionals
14356 We can't insist that A is relevant to B, as conditionals can express lack of relevance [Jackson]