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All the ideas for 'What is Logic?st1=Ian Hacking', 'In Defense of Absolute Essentialism' and 'What Required for Foundation for Maths?'

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49 ideas

2. Reason / D. Definition / 2. Aims of Definition
Definitions make our intuitions mathematically useful [Mayberry]
2. Reason / D. Definition / 3. Types of Definition
A decent modern definition should always imply a semantics [Hacking]
2. Reason / E. Argument / 6. Conclusive Proof
Proof shows that it is true, but also why it must be true [Mayberry]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking]
Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking]
Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]
There is a semi-categorical axiomatisation of set-theory [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of size is part of the very conception of a set [Mayberry]
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
The mainstream of modern logic sees it as a branch of mathematics [Mayberry]
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
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
First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking]
A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking]
First-order logic only has its main theorems because it is so weak [Mayberry]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order completeness seems to need intensional entities and possible worlds [Hacking]
Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
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
Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking]
No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry]
Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry]
5. Theory of Logic / K. Features of Logics / 6. Compactness
No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
Set theory is not just another axiomatised part of mathematics [Mayberry]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry]
9. Objects / D. Essence of Objects / 6. Essence as Unifier
A property is essential iff the object would not exist if it lacked that property [Forbes,G]
Properties are trivially essential if they are not grounded in a thing's specific nature [Forbes,G]
9. Objects / D. Essence of Objects / 7. Essence and Necessity / a. Essence as necessary properties
A relation is essential to two items if it holds in every world where they exist [Forbes,G]
9. Objects / D. Essence of Objects / 7. Essence and Necessity / c. Essentials are necessary
Trivially essential properties are existence, self-identity, and de dicto necessities [Forbes,G]
9. Objects / D. Essence of Objects / 9. Essence and Properties
A property is 'extraneously essential' if it is had only because of the properties of other objects [Forbes,G]
9. Objects / D. Essence of Objects / 11. Essence of Artefacts
One might be essentialist about the original bronze from which a statue was made [Forbes,G]
10. Modality / C. Sources of Modality / 4. Necessity from Concepts
The source of de dicto necessity is not concepts, but the actual properties of the thing [Forbes,G]