Ideas from 'The Boundary Stones of Thought' by Ian Rumfitt [2015], by Theme Structure

[found in 'The Boundary Stones of Thought' by Rumfitt,Ian [OUP 2015,978-0-19-873363-8]].

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1. Philosophy / E. Nature of Metaphysics / 5. Metaphysics as Conceptual
Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics
3. Truth / A. Truth Problems / 1. Truth
The idea that there are unrecognised truths is basic to our concept of truth
3. Truth / B. Truthmakers / 7. Making Modal Truths
'True at a possibility' means necessarily true if what is said had obtained
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
Frege thought traditional categories had psychological and linguistic impurities
4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
'Absolute necessity' would have to rest on S5
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
To say there could have been people who don't exist, but deny those possible things, rejects Barcan
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
It is the second-order part of intuitionistic logic which actually negates some classical theorems
Intuitionists can accept Double Negation Elimination for decidable propositions
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
The iterated conception of set requires continual increase in axiom strength
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
A set may well not consist of its members; the empty set, for example, is a problem
A set can be determinate, because of its concept, and still have vague membership
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Logic is higher-order laws which can expand the range of any sort of deduction
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
The case for classical logic rests on its rules, much more than on the Principle of Bivalence
Classical logic rules cannot be proved, but various lines of attack can be repelled
If truth-tables specify the connectives, classical logic must rely on Bivalence
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Logical consequence is a relation that can extended into further statements
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
Normal deduction presupposes the Cut Law
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
When faced with vague statements, Bivalence is not a compelling principle
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
In specifying a logical constant, use of that constant is quite unavoidable
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Introduction rules give deduction conditions, and Elimination says what can be deduced
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Logical truths are just the assumption-free by-products of logical rules
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
Monotonicity means there is a guarantee, rather than mere inductive support
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / l. Infinitesimals
Infinitesimals do not stand in a determinate order relation to zero
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2nd-order
Categoricity implies that Dedekind has characterised the numbers, because it has one domain
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases
The extension of a colour is decided by a concept's place in a network of contraries
10. Modality / A. Necessity / 5. Metaphysical Necessity
Metaphysical modalities respect the actual identities of things
10. Modality / A. Necessity / 6. Logical Necessity
S5 is the logic of logical necessity
10. Modality / B. Possibility / 1. Possibility
If two possibilities can't share a determiner, they are incompatible
Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
Possibilities are like possible worlds, but not fully determinate or complete
11. Knowledge Aims / A. Knowledge / 2. Understanding
Medieval logicians said understanding A also involved understanding not-A
13. Knowledge Criteria / B. Internal Justification / 3. Evidentialism / a. Evidence
In English 'evidence' is a mass term, qualified by 'little' and 'more'
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
We understand conditionals, but disagree over their truth-conditions
19. Language / F. Communication / 3. Denial
The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A