Ideas of Michael Dummett, by Theme

[British, 1925 - 2011, Professor at Oxford University. Fellow of New College and All Souls'.]

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1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / a. Philosophy as worldly
Philosophy aims to understand the world, through ordinary experience and science
1. Philosophy / F. Analytic Philosophy / 4. Conceptual Analysis
To explain a concept, we need its purpose, not just its rules of usage
2. Reason / A. Nature of Reason / 5. Objectivity
What matters in mathematics is its objectivity, not the existence of the objects
2. Reason / D. Definition / 7. Contextual Definition
A contextual definition permits the elimination of the expression by a substitution
2. Reason / E. Argument / 6. Conclusive Proof
A successful proof requires recognition of truth at every step
3. Truth / A. Truth Problems / 1. Truth
It is part of the concept of truth that we aim at making true statements
3. Truth / A. Truth Problems / 2. Defining Truth
We must be able to specify truths in a precise language, like winning moves in a game
3. Truth / F. Semantic Truth / 2. Semantic Truth
Tarski's truth is like rules for winning games, without saying what 'winning' means [Davidson]
Truth is part of semantics, since valid inference preserves truth
4. Formal Logic / A. Syllogistic Logic / 3. Term Logic
Logic would be more natural if negation only referred to predicates
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Truth-tables are dubious in some cases, and may be a bad way to explain connective meaning
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
It was realised that possible worlds covered all modal logics, if they had a structure
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic
Relative possibility one way may be impossible coming back, so it isn't symmetrical
If something is only possible relative to another possibility, the possibility relation is not transitive
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / d. System T
If possibilitiy is relative, that might make accessibility non-transitive, and T the correct system
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / g. System S4
In S4 the actual world has a special place
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Kitcher]
Mathematical statements and entities that result from an infinite process must lack a truth-value
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}}
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality
The main alternative to ZF is one which includes looser classes as well as sets
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
To associate a cardinal with each set, we need the Axiom of Choice to find a representative
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Deduction is justified by the semantics of its metalanguage [Hanna]
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
In classical logic, logical truths are valid formulas; in higher-order logics they are purely logical
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
Syntactic consequence is positive, for validity; semantic version is negative, with counterexamples
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
Language can violate bivalence because of non-referring terms or ill-defined predicates
Undecidable statements result from quantifying over infinites, subjunctive conditionals, and the past tense
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Anti-realism needs an intuitionist logic with no law of excluded middle [Miller,A]
The law of excluded middle is the logical reflection of the principle of bivalence
Intuitionists reject excluded middle, not for a third value, but for possibility of proof
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / c. not
Natural language 'not' doesn't apply to sentences
Classical negation is circular, if it relies on knowing negation-conditions from truth-conditions
5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
Ancient names like 'Obadiah' depend on tradition, not on where the name originated
5. Theory of Logic / G. Quantification / 1. Quantification
Classical quantification is an infinite conjunction or disjunction - but you may not know all the instances
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
First-order logic concerns objects; second-order adds properties, kinds, relations and functions
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Beth trees show semantics for intuitionistic logic, in terms of how truth has been established
In standard views you could replace 'true' and 'false' with mere 0 and 1
Classical two-valued semantics implies that meaning is grasped through truth-conditions
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Logical truths and inference are characterized either syntactically or semantically
5. Theory of Logic / K. Features of Logics / 4. Completeness
Soundness and completeness proofs test the theory of meaning, rather than the logic theory
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / b. The Heap paradox ('Sorites')
Surely there is no exact single grain that brings a heap into existence
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
A prime number is one which is measured by a unit alone
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
Addition of quantities is prior to ordering, as shown in cyclic domains like angles
Ordinals seem more basic than cardinals, since we count objects in sequence
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
A number is a multitude composed of units
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
We understand 'there are as many nuts as apples' as easily by pairing them as by counting them
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
Platonists ruin infinity, which is precisely a growing structure which is never completed
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
The identity of a number may be fixed by something outside structure - by counting
Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
The number 4 has different positions in the naturals and the wholes, with the same structure
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Set theory isn't part of logic, and why reduce to something more complex?
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
For intuitionists it is constructed proofs (which take time) which make statements true
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionism says that totality of numbers is only potential, but is still determinate
Intuitionists rely on the proof of mathematical statements, not their truth
7. Existence / B. Change in Existence / 1. Nature of Change
A 'Cambridge Change' is like saying 'the landscape changes as you travel east'
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
Ostension is possible for concreta; abstracta can only be referred to via other objects [Hale]
The concrete/abstract distinction seems crude: in which category is the Mistral?
We don't need a sharp concrete/abstract distinction
We can't say that light is concrete but radio waves abstract
The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy
7. Existence / D. Theories of Reality / 1. Realism
Dummett saw realism as acceptance of bivalence, rather than of mind-independent entities [Potter]
Realism is just the application of two-valued semantics to sentences
Metaphysical realists are committed to all unambiguous statements being true or not true
Philosophers should not presume reality, but only invoke it when language requires it
7. Existence / D. Theories of Reality / 3. Anti-realism
For anti-realists there are no natural distinctions between objects [Benardete,JA]
We can't make sense of a world not apprehended by a mind
I no longer think what a statement about the past says is just what can justify it
7. Existence / D. Theories of Reality / 7. Facts / b. Types of fact
Since 'no bird here' and 'no squirrel here' seem the same, we must talk of 'atomic' facts
7. Existence / D. Theories of Reality / 7. Facts / c. Facts and truths
We know we can state facts, with true statements
7. Existence / D. Theories of Reality / 9. Vagueness / b. Vagueness of reality
To say reality itself is vague is not properly intelligible
7. Existence / D. Theories of Reality / 9. Vagueness / d. Vagueness as linguistic
'That is red or orange' might be considered true, even though 'that is red' and 'that is orange' were not
7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
The context principle for names rules out a special philosophical sense for 'existence'
The objects we recognise the world as containing depends on the structure of our language
8. Modes of Existence / D. Universals / 1. Universals
We can understand universals by studying predication
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
'Nominalism' used to mean denial of universals, but now means denial of abstract objects
Nominalism assumes unmediated mental contact with objects
9. Objects / A. Existence of Objects / 1. Physical Objects
Concrete objects such as sounds and smells may not be possible objects of ostension
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Abstract objects may not cause changes, but they can be the subject of change
The existence of abstract objects is a pseudo-problem
9. Objects / A. Existence of Objects / 2. Abstract Objects / b. Need for abstracta
If we can intuitively apprehend abstract objects, this makes them observable and causally active
9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
Abstract objects must have names that fall within the range of some functional expression
It is absurd to deny the Equator, on the grounds that it lacks causal powers
'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object
Abstract objects nowadays are those which are objective but not actual
9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
If a genuine singular term needs a criterion of identity, we must exclude abstract nouns [Hale]
Abstract objects can never be confronted, and need verbal phrases for reference
Abstract objects need the context principle, since they can't be encountered directly
9. Objects / A. Existence of Objects / 3. Objects in Thought
There is a modern philosophical notion of 'object', first introduced by Frege
9. Objects / F. Identity among Objects / 2. Defining Identity
Content is replaceable if identical, so replaceability can't define identity [Dummett]
Frege introduced criteria for identity, but thought defining identity was circular
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
Possible worlds aren't how the world might be, but how a world might be, given some possibility
10. Modality / E. Possible worlds / 1. Possible Worlds / c. Possible worlds realism
If possible worlds have no structure (S5) they are equal, and it is hard to deny them reality
11. Knowledge Aims / C. Knowing Reality / 2. Phenomenalism
The existence of a universe without sentience or intelligence is an unintelligible fantasy
12. Knowledge Sources / D. Empiricism / 5. Empiricism Critique
Empirical and a priori knowledge are not distinct, but are extremes of a sliding scale
14. Science / D. Explanation / 2. Types of Explanation / a. Types of explanation
An explanation is often a deduction, but that may well beg the question
18. Thought / A. Modes of Thought / 1. Thought
A theory of thought will include propositional attitudes as well as propositions
The theories of meaning and understanding are the only routes to an account of thought
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
Concepts only have a 'functional character', because they map to truth values, not objects [Davidson]
18. Thought / D. Concepts / 4. Structure of Concepts / i. Conceptual priority
Maybe a concept is 'prior' to another if it can be defined without the second concept
An argument for conceptual priority is greater simplicity in explanation
18. Thought / E. Abstraction / 1. Abstract Thought
You can't infer a dog's abstract concepts from its behaviour
Abstract terms are acceptable as long as we know how they function linguistically
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Since abstract objects cannot be picked out, we must rely on identity statements
There is no reason why abstraction by equivalence classes should be called 'logical' [Tait]
We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus'
18. Thought / E. Abstraction / 8. Abstractionism Critique
To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too
To 'abstract from' is a logical process, as opposed to the old mental view
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
Stating a sentence's truth-conditions is just paraphrasing the sentence
If a sentence is effectively undecidable, we can never know its truth conditions
To know the truth-conditions of a sentence, you must already know the meaning
19. Language / A. Nature of Meaning / 5. Meaning as Verification
A justificationist theory of meaning leads to the rejection of classical logic
Verificationism could be realist, if we imagined the verification by a superhuman power
If truths about the past depend on memories and current evidence, the past will change
Verification is not an individual but a collective activity
19. Language / A. Nature of Meaning / 6. Meaning as Use
Meaning as use puts use beyond criticism, and needs a holistic view of language
We could only guess the meanings of 'true' and 'false' when sentences were used
19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning
Sentences are the primary semantic units, because they can say something
19. Language / A. Nature of Meaning / 10. Denial of Meanings
Holism is not a theory of meaning; it is the denial that a theory of meaning is possible
19. Language / B. Reference / 3. Direct Reference / b. Causal reference
A realistic view of reference is possible for concrete objects, but not for abstract objects [Hale]
The causal theory of reference can't distinguish just hearing a name from knowing its use
19. Language / C. Assigning Meanings / 5. Fregean Semantics
Fregean semantics assumes a domain articulated into individual objects
19. Language / C. Assigning Meanings / 6. Truth-Conditions Semantics
Truth-condition theorists must argue use can only be described by appeal to conditions of truth
The truth-conditions theory must get agreement on a conception of truth
19. Language / D. Propositions / 1. Propositions
We can't distinguish a proposition from its content
23. Ethics / C. Virtue Theory / 1. Virtue Theory / d. Virtue theory critique
To explain generosity in a person, you must understand a generous action
26. Natural Theory / B. Natural Kinds / 7. Critique of Kinds
Generalised talk of 'natural kinds' is unfortunate, as they vary too much
27. Natural Reality / C. Space / 3. Points in Space
Why should the limit of measurement be points, not intervals?
27. Natural Reality / D. Time / 1. Nature of Time / d. Time as measure
Time is the measure of change, so we can't speak of time before all change
27. Natural Reality / D. Time / 1. Nature of Time / f. Eternalism
Maybe past (which affects us) and future (which we can affect) are both real
27. Natural Reality / D. Time / 1. Nature of Time / h. Presentism
If Presentism is correct, we cannot even say that the present changes
27. Natural Reality / D. Time / 2. Passage of Time / k. Temporal truths
The present cannot exist alone as a mere boundary; past and future truths are rendered meaningless