Combining Texts

All the ideas for 'The philosophical basis of intuitionist logic', 'On Sense and Reference' and 'Philosophy of Mathematics'

expand these ideas     |    start again     |     specify just one area for these texts


87 ideas

2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions are wrong, because they change the set that is being defined? [Bostock]
3. Truth / A. Truth Problems / 5. Truth Bearers
Frege was strongly in favour of taking truth to attach to propositions [Frege, by Dummett]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Dummett, by Kitcher]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
There is no single agreed structure for set theory [Bostock]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
A 'proper class' cannot be a member of anything [Bostock]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
We could add axioms to make sets either as small or as large as possible [Bostock]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Replacement enforces a 'limitation of size' test for the existence of sets [Bostock]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
The completeness of first-order logic implies its compactness [Bostock]
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
We can treat designation by a few words as a proper name [Frege]
5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
Proper name in modal contexts refer obliquely, to their usual sense [Frege, by Gibbard]
A Fregean proper name has a sense determining an object, instead of a concept [Frege, by Sainsbury]
People may have different senses for 'Aristotle', like 'pupil of Plato' or 'teacher of Alexander' [Frege]
5. Theory of Logic / F. Referring in Logic / 1. Naming / c. Names as referential
The meaning of a proper name is the designated object [Frege]
5. Theory of Logic / F. Referring in Logic / 1. Naming / d. Singular terms
Frege ascribes reference to incomplete expressions, as well as to singular terms [Frege, by Hale]
5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names
If sentences have a 'sense', empty name sentences can be understood that way [Frege, by Sawyer]
It is a weakness of natural languages to contain non-denoting names [Frege]
In a logically perfect language every well-formed proper name designates an object [Frege]
5. Theory of Logic / G. Quantification / 1. Quantification
Classical quantification is an infinite conjunction or disjunction - but you may not know all the instances [Dummett]
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
Substitutional quantification is just standard if all objects in the domain have a name [Bostock]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
The Deduction Theorem is what licenses a system of natural deduction [Bostock]
5. Theory of Logic / I. Semantics of Logic / 6. Intensionalism
Frege is intensionalist about reference, as it is determined by sense; identity of objects comes first [Frege, by Jacquette]
Frege moved from extensional to intensional semantics when he added the idea of 'sense' [Frege, by Sawyer]
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
The number of reals is the number of subsets of the natural numbers [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Modern axioms of geometry do not need the real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
The Peano Axioms describe a unique structure [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
There are many criteria for the identity of numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Numbers can't be positions, if nothing decides what position a given number has [Bostock]
Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Actual measurement could never require the precision of the real numbers [Bostock]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
If Hume's Principle is the whole story, that implies structuralism [Bostock]
Many crucial logicist definitions are in fact impredicative [Bostock]
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
Higher cardinalities in sets are just fairy stories [Bostock]
A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
The best version of conceptualism is predicativism [Bostock]
Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
Predicativism makes theories of huge cardinals impossible [Bostock]
If mathematics rests on science, predicativism may be the best approach [Bostock]
If we can only think of what we can describe, predicativism may be implied [Bostock]
The usual definitions of identity and of natural numbers are impredicative [Bostock]
The predicativity restriction makes a difference with the real numbers [Bostock]
8. Modes of Existence / D. Universals / 1. Universals
We can't get a semantics from nouns and predicates referring to the same thing [Frege, by Dummett]
9. Objects / F. Identity among Objects / 1. Concept of Identity
Frege was asking how identities could be informative [Frege, by Perry]
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
'The concept "horse"' denotes a concept, yet seems also to denote an object [Frege, by McGee]
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
Frege failed to show when two sets of truth-conditions are equivalent [Frege, by Potter]
The meaning (reference) of a sentence is its truth value - the circumstance of it being true or false [Frege]
Stating a sentence's truth-conditions is just paraphrasing the sentence [Dummett]
If a sentence is effectively undecidable, we can never know its truth conditions [Dummett]
19. Language / A. Nature of Meaning / 6. Meaning as Use
Meaning as use puts use beyond criticism, and needs a holistic view of language [Dummett]
19. Language / A. Nature of Meaning / 7. Meaning Holism / b. Language holism
Holism says all language use is also a change in the rules of language [Frege, by Dummett]
19. Language / B. Reference / 1. Reference theories
The reference of a word should be understood as part of the reference of the sentence [Frege]
19. Language / B. Reference / 4. Descriptive Reference / a. Sense and reference
Frege's Puzzle: from different semantics we infer different reference for two names with the same reference [Frege, by Fine,K]
Frege's 'sense' is ambiguous, between the meaning of a designator, and how it fixes reference [Kripke on Frege]
Every descriptive name has a sense, but may not have a reference [Frege]
Frege started as anti-realist, but the sense/reference distinction led him to realism [Frege, by Benardete,JA]
The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense [Frege]
In maths, there are phrases with a clear sense, but no actual reference [Frege]
We are driven from sense to reference by our desire for truth [Frege]
19. Language / B. Reference / 4. Descriptive Reference / b. Reference by description
Expressions always give ways of thinking of referents, rather than the referents themselves [Frege, by Soames]
19. Language / C. Assigning Meanings / 5. Fregean Semantics
'Sense' gives meaning to non-referring names, and to two expressions for one referent [Frege, by Margolis/Laurence]
Frege was the first to construct a plausible theory of meaning [Frege, by Dummett]
Earlier Frege focuses on content itself; later he became interested in understanding content [Frege, by Dummett]
Frege divided the meaning of a sentence into sense, force and tone [Frege, by Dummett]
Frege uses 'sense' to mean both a designator's meaning, and the way its reference is determined [Kripke on Frege]
Frege explained meaning as sense, semantic value, reference, force and tone [Frege, by Miller,A]
19. Language / F. Communication / 2. Assertion
In logic a proposition means the same when it is and when it is not asserted [Bostock]