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 LöwenheimSkolem

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
13666

Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets

13654

It is central to the iterative conception that membership is wellfounded, with no infinite descending chains

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

'Wellordering' 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

Antirealists 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 firstorder, and Zermelo, Hilbert, and Bernays championed higherorder

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. FirstOrder Logic
13624

The 'triumph' of firstorder logic may be related to logicism and the Hilbert programme, which failed

13662

Firstorder logic was an afterthought in the development of modern logic

13660

Maybe compactness, semantic effectiveness, and the LöwenheimSkolem properties are desirable

13673

The notion of finitude is actually built into firstorder languages

10588

Firstorder logic is Complete, and Compact, with the LöwenheimSkolem Theorems

5. Theory of Logic / A. Overview of Logic / 7. SecondOrder Logic
15944

Secondorder 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 manysorted firstorder semantics?

13645

In standard semantics for secondorder logic, a single domain fixes the ranges for the variables

13650

Henkin semantics has separate variables ranging over the relations and over the functions

13649

Completeness, Compactness and LöwenheimSkolem fail in secondorder standard semantics

10298

Some say that secondorder logic is mathematics, not logic

10299

If the aim of logic is to codify inferences, secondorder logic is useless

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 =
13637

If a logic is incomplete, its semantic consequence relation is not effective

13626

Semantic consequence is ineffective in secondorder logic

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. SecondOrder Quantification
10290

Secondorder 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 settheory

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 firstorder language

10238

The settheoretical hierarchy contains as many isomorphism types as possible

10214

Theory ontology is never complete, but is only determined 'up to isomorphism'

5. Theory of Logic / J. Model Theory in Logic / 3. LöwenheimSkolem Theorems
13658

Downward LöwenheimSkolem: each satisfiable countable set always has countable models

13648

The LöwenheimSkolem theorems show an explosion of infinite models, so 1storder is useless for infinity

13675

Substitutional semantics only has countably many terms, so Upward LöwenheimSkolem trivially fails

13659

Upward LöwenheimSkolem: each infinite model has infinite models of all sizes

10234

Any theory with an infinite model has a model of every infinite cardinality

10292

Downward LöwenheimSkolem: if there's an infinite model, there is a countable model

10590

Up LöwenheimSkolem: if natural numbers satisfy wffs, then an infinite domain satisfies them

10297

The LöwenheimSkolem theorem seems to be a defect of firstorder logic

10296

The LöwenheimSkolem Theorems fail for secondorder languages with standard semantics

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

Noncompactness is a strength of secondorder 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 higherorder 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 realnumber 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

Firstorder 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 2ndorder
10294

Secondorder 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 settheory 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 naturalnumber structure, with 6 playing the role of 3

10228

Could infinite structures be apprehended by pattern recognition?

10230

The 4pattern 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

8761

A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them

8760

Numbers do not exist independently; the essence of a number is its relations to other numbers

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 oneovermany

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 truthvalue' (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 fivedimensional 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. Neologicism
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 nonstarter 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
10255

Presumably nothing can block a possible dynamic operation?

10254

Can the ideal constructor also destroy objects?

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 fiftytwo cards, or a person is timeslices 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
