Ideas of Ian Rumfitt, by Theme
[British, fl. 2014, Pupil of Dummett. At University College, Oxford. Then at Birkbeck, then Birmingham, then All Souls Oxford]
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1. Philosophy / E. Nature of Metaphysics / 6. Metaphysics as Conceptual
18835

Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics

3. Truth / A. Truth Problems / 1. Truth
18819

The idea that there are unrecognised truths is basic to our concept of truth

3. Truth / B. Truthmakers / 7. Making Modal Truths
18826

'True at a possibility' means necessarily true if what is said had obtained

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
18803

Semantics for propositions: 1) validity preserves truth 2) noncontradition 3) bivalence 4) truth tables

4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
12204

The logic of metaphysical necessity is S5

18814

'Absolute necessity' would have to rest on S5

4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
18798

It is the secondorder part of intuitionistic logic which actually negates some classical theorems

18799

Intuitionists can accept Double Negation Elimination for decidable propositions

4. Formal Logic / F. Set Theory ST / 1. Set Theory
18830

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
18843

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
18836

A set may well not consist of its members; the empty set, for example, is a problem

18837

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
18845

If the totality of sets is not welldefined, there must be doubt about the Power Set Axiom

5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
11211

If a sound conclusion comes from two errors that cancel out, the path of the argument must matter

18815

Logic is higherorder laws which can expand the range of any sort of deduction

5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
9390

Logic guides thinking, but it isn't a substitute for it

5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
18804

The case for classical logic rests on its rules, much more than on the Principle of Bivalence

18805

Classical logic rules cannot be proved, but various lines of attack can be repelled

18827

If truthtables specify the connectives, classical logic must rely on Bivalence

5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
12195

Soundness in argument varies with context, and may be achieved very informally indeed

12199

There is a modal element in consequence, in assessing reasoning from suppositions

12201

We reject deductions by bad consequence, so logical consequence can't be deduction

18813

Logical consequence is a relation that can extended into further statements

5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence 
18808

Normal deduction presupposes the Cut Law

5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
18840

When faced with vague statements, Bivalence is not a compelling principle

5. Theory of Logic / D. Assumptions for Logic / 3. Contradiction
12194

Contradictions include 'This is red and not coloured', as well as the formal 'B and notB'

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
11210

Standardly 'and' and 'but' are held to have the same sense by having the same truth table

11212

The sense of a connective comes from primitively obvious rules of inference

18802

In specifying a logical constant, use of that constant is quite unavoidable

5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
12198

Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths)

5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
18800

Introduction rules give deduction conditions, and Elimination says what can be deduced

5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
18809

Logical truths are just the assumptionfree byproducts of logical rules

5. Theory of Logic / K. Features of Logics / 10. Monotonicity
18807

Monotonicity means there is a guarantee, rather than mere inductive support

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
18842

Maybe an ordinal is a property of isomorphic wellordered sets, and not itself a set

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
17462

A single object must not be counted twice, which needs knowledge of distinctness (negative identity)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18834

Infinitesimals do not stand in a determinate order relation to zero

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
18846

Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
17461

Some 'how many?' answers are not predications of a concept, like 'how many gallons?'

9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
9389

Vague membership of sets is possible if the set is defined by its concept, not its members

18839

An object that is not clearly red or orange can still be redororange, which sweeps up problem cases

18838

The extension of a colour is decided by a concept's place in a network of contraries

10. Modality / A. Necessity / 3. Types of Necessity
14532

A distinctive type of necessity is found in logical consequence [Hale/Hoffmann,A]

10. Modality / A. Necessity / 5. Metaphysical Necessity
18816

Metaphysical modalities respect the actual identities of things

10. Modality / A. Necessity / 6. Logical Necessity
12193

Logical necessity is when 'necessarily A' implies 'notA is contradictory'

12200

A logically necessary statement need not be a priori, as it could be unknowable

12202

Narrow nonmodal logical necessity may be metaphysical, but real logical necessity is not

18825

S5 is the logic of logical necessity

10. Modality / B. Possibility / 1. Possibility
18824

Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right

18828

If two possibilities can't share a determiner, they are incompatible

10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
12203

If a world is a fully determinate way things could have been, can anyone consider such a thing?

18821

Possibilities are like possible worlds, but not fully determinate or complete

11. Knowledge Aims / A. Knowledge / 2. Understanding
18831

Medieval logicians said understanding A also involved understanding notA

13. Knowledge Criteria / B. Internal Justification / 3. Evidentialism / a. Evidence
18820

In English 'evidence' is a mass term, qualified by 'little' and 'more'

19. Language / A. Nature of Meaning / 4. Meaning as TruthConditions
18817

We understand conditionals, but disagree over their truthconditions

19. Language / F. Communication / 3. Denial
11214

We learn 'not' along with affirmation, by learning to either affirm or deny a sentence

18829

The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A
