### All the ideas for Michael Burke, Gordon Graham and Stewart Shapiro

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164 ideas

###### 2. Reason / A. Nature of Reason / 6. Coherence
 10237 Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro]
###### 2. Reason / D. Definition / 7. Contextual Definition
 10204 An 'implicit definition' gives a direct description of the relations of an entity [Shapiro]
###### 3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
 13634 Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro]
###### 4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
 13643 Aristotelian logic is complete [Shapiro]
###### 4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
 10206 Modal operators are usually treated as quantifiers [Shapiro]
###### 4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
 13651 A set is 'transitive' if contains every member of each of its members [Shapiro]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
 13647 Choice is essential for proving downward Löwenheim-Skolem [Shapiro]
 10208 Axiom of Choice: some function has a value for every set in a given set [Shapiro]
 10252 The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
 10301 The axiom of choice is controversial, but it could be replaced [Shapiro]
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
 13631 Are sets part of logic, or part of mathematics? [Shapiro]
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
 13640 Russell's paradox shows that there are classes which are not iterative sets [Shapiro]
 13654 It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro]
 13666 Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro]
###### 4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
 13653 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro]
###### 4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
 10207 Anti-realists reject set theory [Shapiro]
###### 5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
 13627 There is no 'correct' logic for natural languages [Shapiro]
 13642 Logic is the ideal for learning new propositions on the basis of others [Shapiro]
###### 5. Theory of Logic / A. Overview of Logic / 2. History of Logic
 13667 Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
 13668 Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
 13669 Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
###### 5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
 13662 First-order logic was an afterthought in the development of modern logic [Shapiro]
 10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
 13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
 13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
 13673 The notion of finitude is actually built into first-order languages [Shapiro]
###### 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
 13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
 13650 Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
 15944 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
 13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
 13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
 10298 Some say that second-order logic is mathematics, not logic [Shapiro]
 10299 If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
###### 5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
 10300 Logical consequence can be defined in terms of the logical terminology [Shapiro]
###### 5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
 10259 The two standard explanations of consequence are semantic (in models) and deductive [Shapiro]
###### 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
 13626 Semantic consequence is ineffective in second-order logic [Shapiro]
 13637 If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
###### 5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
 10257 Intuitionism only sanctions modus ponens if all three components are proved [Shapiro]
###### 5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
 10253 Either logic determines objects, or objects determine logic, or they are separate [Shapiro]
###### 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
 10251 The law of excluded middle might be seen as a principle of omniscience [Shapiro]
 8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
###### 5. Theory of Logic / E. Structures of Logic / 1. Logical Form
 13632 Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro]
###### 5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
 10212 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
###### 5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
 10209 A function is just an arbitrary correspondence between collections [Shapiro]
###### 5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
 13674 We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]
###### 5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
 10290 Second-order variables also range over properties, sets, relations or functions [Shapiro]
###### 5. Theory of Logic / G. Quantification / 6. Plural Quantification
 10268 Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro]
###### 5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
 13633 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro]
 10235 A sentence is 'satisfiable' if it has a model [Shapiro]
###### 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
 13644 Semantics for models uses set-theory [Shapiro]
 10239 The central notion of model theory is the relation of 'satisfaction' [Shapiro]
 10240 Model theory deals with relations, reference and extensions [Shapiro]
###### 5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
 13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
 13670 Categoricity can't be reached in a first-order language [Shapiro]
 10238 The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
 10214 Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
###### 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
 13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
 13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
 13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
 13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
 10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
 10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
 10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
 10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
 10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
###### 5. Theory of Logic / K. Features of Logics / 3. Soundness
 13635 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro]
###### 5. Theory of Logic / K. Features of Logics / 4. Completeness
 13628 We can live well without completeness in logic [Shapiro]
###### 5. Theory of Logic / K. Features of Logics / 6. Compactness
 13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
 13646 Compactness is derived from soundness and completeness [Shapiro]
###### 5. Theory of Logic / K. Features of Logics / 9. Expressibility
 13661 A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]
###### 6. Mathematics / A. Nature of Mathematics / 1. Mathematics
 10201 Virtually all of mathematics can be modeled in set theory [Shapiro]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
 13641 Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
 8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
 13676 Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
 13677 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
 10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
 18243 Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
 18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
 18245 Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
 13652 The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
 8764 Categories are the best foundation for mathematics [Shapiro]
 10236 There is no grounding for mathematics that is more secure than mathematics [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
 10256 For intuitionists, proof is inherently informal [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
 13657 First-order arithmetic can't even represent basic number theory [Shapiro]
 10202 Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
 10294 Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
 10205 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
 8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
 13656 Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
 10222 Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
 10218 Baseball positions and chess pieces depend entirely on context [Shapiro]
 10224 The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
 10228 Could infinite structures be apprehended by pattern recognition? [Shapiro]
 10230 The 4-pattern is the structure common to all collections of four objects [Shapiro]
 10249 The main mathematical structures are algebraic, ordered, and topological [Shapiro]
 10273 Some structures are exemplified by both abstract and concrete [Shapiro]
 10276 Mathematical structures are defined by axioms, or in set theory [Shapiro]
 8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
 8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
 10270 The main versions of structuralism are all definitionally equivalent [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
 10221 Is there is no more to structures than the systems that exemplify them? [Shapiro]
 10248 Number statements are generalizations about number sequences, and are bound variables [Shapiro]
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
 10220 Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
 10223 There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
 8703 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
 10274 Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
 10210 If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
 10215 Platonists claim we can state the essence of a number without reference to the others [Shapiro]
 10233 Platonism must accept that the Peano Axioms could all be false [Shapiro]
 10200 We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
 10244 Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
 10280 A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
 13664 Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
 13625 Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
 8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 7. Formalism
 8752 Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
 8749 Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
 8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
 10254 Can the ideal constructor also destroy objects? [Shapiro]
 10255 Presumably nothing can block a possible dynamic operation? [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
 8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
 8731 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
 13663 Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
 8730 'Impredicative' definitions refer to the thing being described [Shapiro]
###### 7. Existence / A. Nature of Existence / 1. Nature of Existence
 10279 Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro]
###### 7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
 10227 The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro]
 10226 Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro]
###### 7. Existence / D. Theories of Reality / 6. Fictionalism
 10262 Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro]
 10277 Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro]
###### 8. Modes of Existence / B. Properties / 10. Properties as Predicates
 13638 Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro]
###### 8. Modes of Existence / B. Properties / 11. Properties as Sets
 10591 Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
###### 9. Objects / A. Existence of Objects / 1. Physical Objects
 10272 The notion of 'object' is at least partially structural and mathematical [Shapiro]
###### 9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind
 16235 Persistence conditions cannot contradict, so there must be a 'dominant sortal' [Burke,M, by Hawley]
 14753 The 'dominant' of two coinciding sortals is the one that entails the widest range of properties [Burke,M, by Sider]
###### 9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
 16072 'The rock' either refers to an object, or to a collection of parts, or to some stuff [Burke,M, by Wasserman]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / b. Cat and its tail
 14751 Tib goes out of existence when the tail is lost, because Tib was never the 'cat' [Burke,M, by Sider]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / c. Statue and clay
 16071 Sculpting a lump of clay destroys one object, and replaces it with another one [Burke,M, by Wasserman]
 16234 Burke says when two object coincide, one of them is destroyed in the process [Burke,M, by Hawley]
 13278 Maybe the clay becomes a different lump when it becomes a statue [Burke,M, by Koslicki]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / d. Coincident objects
 14750 Two entities can coincide as one, but only one of them (the dominant sortal) fixes persistence conditions [Burke,M, by Sider]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
 10275 A blurry border is still a border [Shapiro]
###### 10. Modality / A. Necessity / 6. Logical Necessity
 10258 Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro]
###### 10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
 10266 Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro]
###### 12. Knowledge Sources / C. Rationalism / 1. Rationalism
 8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
###### 13. Knowledge Criteria / E. Relativism / 3. Subjectivism
 6685 'Subjectivism' is an extension of relativism from the social group to the individual [Graham]
###### 15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
 10203 We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro]
###### 18. Thought / E. Abstraction / 2. Abstracta by Selection
 10229 Simple types can be apprehended through their tokens, via abstraction [Shapiro]
###### 18. Thought / E. Abstraction / 3. Abstracta by Ignoring
 9626 A structure is an abstraction, focussing on relationships, and ignoring other features [Shapiro]
 10217 We can apprehend structures by focusing on or ignoring features of patterns [Shapiro]
 9554 We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro]
###### 18. Thought / E. Abstraction / 7. Abstracta by Equivalence
 10231 Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro]
###### 22. Metaethics / B. The Good / 1. Goodness / g. Consequentialism
 6698 Negative consequences are very hard (and possibly impossible) to assess [Graham]
###### 22. Metaethics / B. The Good / 1. Goodness / i. Moral luck
 6700 We can't criticise people because of unforeseeable consequences [Graham]
###### 22. Metaethics / C. Ethics Foundations / 1. Nature of Ethics / g. Moral responsibility
 6699 The chain of consequences may not be the same as the chain of responsibility [Graham]
###### 23. Ethics / A. Egoism / 1. Ethical Egoism
 6704 Egoism submits to desires, but cannot help form them [Graham]
###### 23. Ethics / C. Virtue Theory / 2. Elements of Virtue Theory / h. Right feelings
 6701 Rescue operations need spontaneous benevolence, not careful thought [Graham]
###### 23. Ethics / D. Deontological Ethics / 4. Categorical Imperative
 6693 'What if everybody did that?' rather misses the point as an objection to cheating [Graham]
###### 23. Ethics / F. Existentialism / 1. Existentialism
 6691 It is more plausible to say people can choose between values, than that they can create them [Graham]
###### 23. Ethics / F. Existentialism / 2. Nihilism
 6688 Life is only absurd if you expected an explanation and none turns up [Graham]
###### 23. Ethics / F. Existentialism / 5. Existence-Essence
 6705 Existentialism may transcend our nature, unlike eudaimonism [Graham]
###### 23. Ethics / F. Existentialism / 6. Authentic Self
 6690 A standard problem for existentialism is the 'sincere Nazi' [Graham]
###### 23. Ethics / F. Existentialism / 7. Existential Action
 6689 The key to existentialism: the way you make choices is more important than what you choose [Graham]
###### 29. Religion / D. Religious Issues / 1. Religious Commitment / a. Religious Belief
 6706 The great religions are much more concerned with the religious life than with ethics [Graham]
###### 29. Religion / D. Religious Issues / 2. Immortality / a. Immortality
 6709 Western religion saves us from death; Eastern religion saves us from immortality [Graham]