### Ideas of Stewart Shapiro, by Theme

#### [American, b.1951, Professor at Ohio State University; visiting Professor at St Andrew's University.]

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###### 2. Reason / A. Nature of Reason / 6. Coherence
 10237 Coherence is a primitive, intuitive notion, not reduced to something formal
###### 2. Reason / D. Definition / 7. Contextual Definition
 10204 An 'implicit definition' gives a direct description of the relations of an entity
###### 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'
###### 4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
 13643 Aristotelian logic is complete
###### 4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
 10206 Modal operators are usually treated as quantifiers
###### 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
###### 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
 10208 Axiom of Choice: some function has a value for every set in a given set
 10252 The Axiom of Choice seems to license an infinite amount of choosing
 10301 The axiom of choice is controversial, but it could be replaced
###### 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?
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
 13654 It is central to the iterative conception that membership is well-founded, with no infinite descending chains
 13666 Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
 13640 Russell's paradox shows that there are classes which are not iterative sets
###### 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
###### 4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
 10207 Anti-realists reject set theory
###### 5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
 13627 There is no 'correct' logic for natural languages
 13642 Logic is the ideal for learning new propositions on the basis of others
###### 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
 13668 Bernays (1918) formulated and proved the completeness of propositional logic
 13669 Can one develop set theory first, then derive numbers, or are numbers more basic?
###### 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
 13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
 13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable
 13673 The notion of finitude is actually built into first-order languages
 10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems
###### 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
 15944 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Lavine]
 13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics?
 13650 Henkin semantics has separate variables ranging over the relations and over the functions
 13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables
 13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics
 10299 If the aim of logic is to codify inferences, second-order logic is useless
 10298 Some say that second-order logic is mathematics, not logic
###### 5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
 10300 Logical consequence can be defined in terms of the logical terminology
###### 5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
 10259 The two standard explanations of consequence are semantic (in models) and deductive
###### 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
 13626 Semantic consequence is ineffective in second-order logic
 13637 If a logic is incomplete, its semantic consequence relation is not effective
###### 5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
 10257 Intuitionism only sanctions modus ponens if all three components are proved
###### 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
###### 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
 8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
###### 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
###### 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'
###### 5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
 10209 A function is just an arbitrary correspondence between collections
###### 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
###### 5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
 10290 Second-order variables also range over properties, sets, relations or functions
###### 5. Theory of Logic / G. Quantification / 6. Plural Quantification
 10268 Maybe plural quantifiers should be understood in terms of classes or sets
###### 5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
 13633 'Satisfaction' is a function from models, assignments, and formulas to {true,false}
 10235 A sentence is 'satisfiable' if it has a model
###### 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
 13644 Semantics for models uses set-theory
 10240 Model theory deals with relations, reference and extensions
 10239 The central notion of model theory is the relation of 'satisfaction'
###### 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
 13670 Categoricity can't be reached in a first-order language
 10214 Theory ontology is never complete, but is only determined 'up to isomorphism'
 10238 The set-theoretical hierarchy contains as many isomorphism types as possible
###### 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
 13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models
 13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes
 13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails
 10234 Any theory with an infinite model has a model of every infinite cardinality
 10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model
 10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
 10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics
 10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic
###### 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
###### 5. Theory of Logic / K. Features of Logics / 4. Completeness
 13628 We can live well without completeness in logic
###### 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
 13646 Compactness is derived from soundness and completeness
###### 5. Theory of Logic / K. Features of Logics / 9. Expressibility
 13661 A language is 'semantically effective' if its logical truths are recursively enumerable
###### 6. Mathematics / A. Nature of Mathematics / 1. Mathematics
 10201 Virtually all of mathematics can be modeled in set theory
###### 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
 8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
###### 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
###### 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
###### 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
 18243 Understanding the real-number structure is knowing usage of the axiomatic language of analysis
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
 18249 Cauchy gave a formal definition of a converging sequence.
###### 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
###### 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
###### 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
 10236 There is no grounding for mathematics that is more secure than mathematics
 8764 Categories are the best foundation for mathematics
###### 6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
 10256 For intuitionists, proof is inherently informal
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
 13657 First-order arithmetic can't even represent basic number theory
 10202 Natural numbers just need an initial object, successors, and an induction principle
###### 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
###### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
 10205 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic)
###### 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
###### 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
###### 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
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
 10218 Baseball positions and chess pieces depend entirely on context
 10224 The even numbers have the natural-number structure, with 6 playing the role of 3
 10228 Could infinite structures be apprehended by pattern recognition?
 10230 The 4-pattern is the structure common to all collections of four objects
 10249 The main mathematical structures are algebraic, ordered, and topological
 10273 Some structures are exemplified by both abstract and concrete
 10276 Mathematical structures are defined by axioms, or in set theory
 8760 Numbers do not exist independently; the essence of a number is its relations to other numbers
 8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
 10270 The main versions of structuralism are all definitionally equivalent
###### 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?
 10248 Number statements are generalizations about number sequences, and are bound variables
###### 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
 10223 There is no 'structure of all structures', just as there is no set of all sets
 8703 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [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?
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
 10200 We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
 10210 If mathematical objects are accepted, then a number of standard principles will follow
 10215 Platonists claim we can state the essence of a number without reference to the others
 10233 Platonism must accept that the Peano Axioms could all be false
###### 6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
 10244 Intuition is an outright hindrance to five-dimensional geometry
###### 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
###### 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
###### 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
 8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
###### 6. Mathematics / C. Sources of Mathematics / 7. Formalism
 8749 Term Formalism says mathematics is just about symbols - but real numbers have no names
 8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
 8752 Deductivism says mathematics is logical consequences of uninterpreted axioms
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
 10254 Can the ideal constructor also destroy objects?
 10255 Presumably nothing can block a possible dynamic operation?
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
 8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
###### 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
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
 13663 Some reject formal properties if they are not defined, or defined impredicatively
 8730 'Impredicative' definitions refer to the thing being described
###### 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?
###### 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
 10226 Mathematicians regard arithmetic as concrete, and group theory as abstract
###### 7. Existence / D. Theories of Reality / 6. Fictionalism
 10262 Fictionalism eschews the abstract, but it still needs the possible (without model theory)
 10277 Structuralism blurs the distinction between mathematical and ordinary objects
###### 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
###### 8. Modes of Existence / B. Properties / 11. Properties as Sets
 10591 Logicians use 'property' and 'set' interchangeably, with little hanging on it
###### 9. Objects / A. Existence of Objects / 1. Physical Objects
 10272 The notion of 'object' is at least partially structural and mathematical
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
 10275 A blurry border is still a border
###### 10. Modality / A. Necessity / 6. Logical Necessity
 10258 Logical modalities may be acceptable, because they are reducible to satisfaction in models
###### 10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
 10266 Why does the 'myth' of possible worlds produce correct modal logic?
###### 12. Knowledge Sources / C. Rationalism / 1. Rationalism
 8725 Rationalism tries to apply mathematical methodology to all of knowledge
###### 15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
 10203 We apprehend small, finite mathematical structures by abstraction from patterns
###### 18. Thought / E. Abstraction / 2. Abstracta by Selection
 10229 Simple types can be apprehended through their tokens, via abstraction
###### 18. Thought / E. Abstraction / 3. Abstracta by Ignoring
 9626 A structure is an abstraction, focussing on relationships, and ignoring other features
 10217 We can apprehend structures by focusing on or ignoring features of patterns
 9554 We can focus on relations between objects (like baseballers), ignoring their other features
###### 18. Thought / E. Abstraction / 7. Abstracta by Equivalence
 10231 Abstract objects might come by abstraction over an equivalence class of base entities