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 / 6. Metaphysics as Conceptual
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18835
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Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics
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3. Truth / A. Truth Problems / 1. Truth
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18819
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The idea that there are unrecognised truths is basic to our concept of truth
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3. Truth / B. Truthmakers / 7. Making Modal Truths
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18826
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'True at a possibility' means necessarily true if what is said had obtained
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4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
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18803
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Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
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18814
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'Absolute necessity' would have to rest on S5
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4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
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18798
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It is the second-order part of intuitionistic logic which actually negates some classical theorems
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18799
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Intuitionists can accept Double Negation Elimination for decidable propositions
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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18830
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Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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18843
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The iterated conception of set requires continual increase in axiom strength
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
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18836
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A set may well not consist of its members; the empty set, for example, is a problem
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18837
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A set can be determinate, because of its concept, and still have vague membership
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
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18845
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If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom
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5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
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18815
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Logic is higher-order laws which can expand the range of any sort of deduction
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5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
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18805
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Classical logic rules cannot be proved, but various lines of attack can be repelled
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18804
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The case for classical logic rests on its rules, much more than on the Principle of Bivalence
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18827
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If truth-tables specify the connectives, classical logic must rely on Bivalence
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5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
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18813
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Logical consequence is a relation that can extended into further statements
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5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
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18808
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Normal deduction presupposes the Cut Law
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5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
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18840
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When faced with vague statements, Bivalence is not a compelling principle
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5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
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18802
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In specifying a logical constant, use of that constant is quite unavoidable
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5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
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18800
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Introduction rules give deduction conditions, and Elimination says what can be deduced
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5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
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18809
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Logical truths are just the assumption-free by-products of logical rules
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5. Theory of Logic / K. Features of Logics / 10. Monotonicity
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18807
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Monotonicity means there is a guarantee, rather than mere inductive support
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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18842
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Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
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18834
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Infinitesimals do not stand in a determinate order relation to zero
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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18846
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Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
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9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
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18839
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An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases
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18838
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The extension of a colour is decided by a concept's place in a network of contraries
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10. Modality / A. Necessity / 5. Metaphysical Necessity
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18816
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Metaphysical modalities respect the actual identities of things
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10. Modality / A. Necessity / 6. Logical Necessity
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18825
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S5 is the logic of logical necessity
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10. Modality / B. Possibility / 1. Possibility
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18824
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Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right
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18828
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If two possibilities can't share a determiner, they are incompatible
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10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
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18821
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Possibilities are like possible worlds, but not fully determinate or complete
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11. Knowledge Aims / A. Knowledge / 2. Understanding
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18831
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Medieval logicians said understanding A also involved understanding not-A
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13. Knowledge Criteria / B. Internal Justification / 3. Evidentialism / a. Evidence
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18820
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In English 'evidence' is a mass term, qualified by 'little' and 'more'
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19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
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18817
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We understand conditionals, but disagree over their truth-conditions
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19. Language / F. Communication / 3. Denial
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18829
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The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A
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