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
19066 Philosophy aims to understand the world, through ordinary experience and science
1. Philosophy / F. Analytic Philosophy / 4. Conceptual Analysis
10838 To explain a concept, we need its purpose, not just its rules of usage
2. Reason / A. Nature of Reason / 5. Objectivity
17621 What matters in mathematics is its objectivity, not the existence of the objects
2. Reason / D. Definition / 7. Contextual Definition
9847 A contextual definition permits the elimination of the expression by a substitution
2. Reason / E. Argument / 6. Conclusive Proof
19067 A successful proof requires recognition of truth at every step
3. Truth / A. Truth Problems / 1. Truth
10837 It is part of the concept of truth that we aim at making true statements
3. Truth / A. Truth Problems / 2. Defining Truth
10840 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
19171 Tarski's truth is like rules for winning games, without saying what 'winning' means [Davidson]
8166 Truth is part of semantics, since valid inference preserves truth
4. Formal Logic / A. Syllogistic Logic / 3. Term Logic
19053 Logic would be more natural if negation only referred to predicates
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
19060 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
16951 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
16953 Relative possibility one way may be impossible coming back, so it isn't symmetrical
16952 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
16960 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
16958 In S4 the actual world has a special place
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
18073 Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Kitcher]
18832 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
10537 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
9193 ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality
9194 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
10542 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
11066 Deduction is justified by the semantics of its metalanguage [Hanna]
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
9820 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
19058 Syntactic consequence is positive, for validity; semantic version is negative, with counterexamples
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
8173 Language can violate bivalence because of non-referring terms or ill-defined predicates
8195 Undecidable statements result from quantifying over infinites, subjunctive conditionals, and the past tense
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
7334 Anti-realism needs an intuitionist logic with no law of excluded middle [Miller,A]
8179 The law of excluded middle is the logical reflection of the principle of bivalence
9195 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
19052 Natural language 'not' doesn't apply to sentences
18801 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
9182 Ancient names like 'Obadiah' depend on tradition, not on where the name originated
5. Theory of Logic / G. Quantification / 1. Quantification
19057 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
9186 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
19063 Beth trees show semantics for intuitionistic logic, in terms of how truth has been established
19059 In standard views you could replace 'true' and 'false' with mere 0 and 1
19062 Classical two-valued semantics implies that meaning is grasped through truth-conditions
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
9187 Logical truths and inference are characterized either syntactically or semantically
5. Theory of Logic / K. Features of Logics / 4. Completeness
19065 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')
8194 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
9896 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
18255 Addition of quantities is prior to ordering, as shown in cyclic domains like angles
9191 Ordinals seem more basic than cardinals, since we count objects in sequence
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
9895 A number is a multitude composed of units
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
9852 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
15938 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
10554 Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9829 The identity of a number may be fixed by something outside structure - by counting
9828 Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1
9192 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
9876 Set theory isn't part of logic, and why reduce to something more complex?
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
15939 For intuitionists it is constructed proofs (which take time) which make statements true
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
10552 Intuitionism says that totality of numbers is only potential, but is still determinate
8190 Intuitionists rely on the proof of mathematical statements, not their truth
7. Existence / B. Change in Existence / 1. Nature of Change
8198 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
10540 We can't say that light is concrete but radio waves abstract
10515 Ostension is possible for concreta; abstracta can only be referred to via other objects [Hale]
10544 The concrete/abstract distinction seems crude: in which category is the Mistral?
10546 We don't need a sharp concrete/abstract distinction
9884 The distinction of concrete/abstract, or actual/non-actual, is a scale, not a dichotomy
7. Existence / D. Theories of Reality / 2. Realism
22297 Dummett saw realism as acceptance of bivalence, rather than of mind-independent entities [Potter]
9869 Realism is just the application of two-valued semantics to sentences
15049 Metaphysical realists are committed to all unambiguous statements being true or not true
8184 Philosophers should not presume reality, but only invoke it when language requires it
7. Existence / D. Theories of Reality / 4. Anti-realism
3303 For anti-realists there are no natural distinctions between objects [Benardete,JA]
8185 We can't make sense of a world not apprehended by a mind
8192 I no longer think what a statement about the past says is just what can justify it
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
8163 Since 'no bird here' and 'no squirrel here' seem the same, we must talk of 'atomic' facts
7. Existence / D. Theories of Reality / 8. Facts / c. Facts and truths
8161 We know we can state facts, with true statements
7. Existence / D. Theories of Reality / 10. Vagueness / b. Vagueness of reality
21628 To say reality itself is vague is not properly intelligible
7. Existence / D. Theories of Reality / 10. Vagueness / d. Vagueness as linguistic
8180 '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 / 11. Ontological Commitment / a. Ontological commitment
10548 The context principle for names rules out a special philosophical sense for 'existence'
10281 The objects we recognise the world as containing depends on the structure of our language
8. Modes of Existence / D. Universals / 1. Universals
10532 We can understand universals by studying predication
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
10534 'Nominalism' used to mean denial of universals, but now means denial of abstract objects
9880 Nominalism assumes unmediated mental contact with objects
9. Objects / A. Existence of Objects / 1. Physical Objects
10541 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
10545 Abstract objects may not cause changes, but they can be the subject of change
9885 The existence of abstract objects is a pseudo-problem
9. Objects / A. Existence of Objects / 2. Abstract Objects / b. Need for abstracta
10555 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
10543 Abstract objects must have names that fall within the range of some functional expression
9859 It is absurd to deny the Equator, on the grounds that it lacks causal powers
9860 'We've crossed the Equator' has truth-conditions, so accept the Equator - and it's an object
9858 Abstract objects nowadays are those which are objective but not actual
9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
10320 If a genuine singular term needs a criterion of identity, we must exclude abstract nouns [Hale]
10547 Abstract objects can never be confronted, and need verbal phrases for reference
9872 Abstract objects need the context principle, since they can't be encountered directly
9. Objects / A. Existence of Objects / 3. Objects in Thought
10531 There is a modern philosophical notion of 'object', first introduced by Frege
9. Objects / F. Identity among Objects / 2. Defining Identity
9848 Content is replaceable if identical, so replaceability can't define identity [Dummett]
9842 Frege introduced criteria for identity, but thought defining identity was circular
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
16957 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
16959 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
8199 The existence of a universe without sentience or intelligence is an unintelligible fantasy
12. Knowledge Sources / D. Empiricism / 5. Empiricism Critique
8178 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
19061 An explanation is often a deduction, but that may well beg the question
18. Thought / A. Modes of Thought / 1. Thought
8174 The theories of meaning and understanding are the only routes to an account of thought
8175 A theory of thought will include propositional attitudes as well as propositions
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
19168 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
9849 Maybe a concept is 'prior' to another if it can be defined without the second concept
9850 An argument for conceptual priority is greater simplicity in explanation
18. Thought / E. Abstraction / 1. Abstract Thought
10839 You can't infer a dog's abstract concepts from its behaviour
9873 Abstract terms are acceptable as long as we know how they function linguistically
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
10549 Since abstract objects cannot be picked out, we must rely on identity statements
9993 There is no reason why abstraction by equivalence classes should be called 'logical' [Tait]
9857 We arrive at the concept 'suicide' by comparing 'Cato killed Cato' with 'Brutus killed Brutus'
18. Thought / E. Abstraction / 8. Abstractionism Critique
9833 To abstract from spoons (to get the same number as the forks), the spoons must be indistinguishable too
8165 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
19055 Stating a sentence's truth-conditions is just paraphrasing the sentence
19056 If a sentence is effectively undecidable, we can never know its truth conditions
8168 To know the truth-conditions of a sentence, you must already know the meaning
19. Language / A. Nature of Meaning / 5. Meaning as Verification
8181 A justificationist theory of meaning leads to the rejection of classical logic
8182 Verificationism could be realist, if we imagined the verification by a superhuman power
8183 If truths about the past depend on memories and current evidence, the past will change
8193 Verification is not an individual but a collective activity
19. Language / A. Nature of Meaning / 6. Meaning as Use
19054 Meaning as use puts use beyond criticism, and needs a holistic view of language
8176 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
8170 Sentences are the primary semantic units, because they can say something
19. Language / A. Nature of Meaning / 10. Denial of Meanings
19064 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
10516 A realistic view of reference is possible for concrete objects, but not for abstract objects [Hale]
9181 The causal theory of reference can't distinguish just hearing a name from knowing its use
19. Language / C. Assigning Meanings / 5. Fregean Semantics
9836 Fregean semantics assumes a domain articulated into individual objects
19. Language / C. Assigning Meanings / 6. Truth-Conditions Semantics
8189 Truth-condition theorists must argue use can only be described by appeal to conditions of truth
8191 The truth-conditions theory must get agreement on a conception of truth
19. Language / D. Propositions / 1. Propositions
8169 We can't distinguish a proposition from its content
23. Ethics / C. Virtue Theory / 1. Virtue Theory / d. Virtue theory critique
16956 To explain generosity in a person, you must understand a generous action
26. Natural Theory / B. Natural Kinds / 7. Critique of Kinds
16954 Generalised talk of 'natural kinds' is unfortunate, as they vary too much
27. Natural Reality / C. Space / 3. Points in Space
18257 Why should the limit of measurement be points, not intervals?
27. Natural Reality / D. Time / 1. Nature of Time / d. Time as measure
8186 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
8197 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
8167 If Presentism is correct, we cannot even say that the present changes
27. Natural Reality / D. Time / 2. Passage of Time / k. Temporal truths
8196 The present cannot exist alone as a mere boundary; past and future truths are rendered meaningless