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
Coherence is a primitive, intuitive notion, not reduced to something formal
2. Reason / D. Definition / 7. Contextual Definition
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
Satisfaction is 'truth in a model', which is a model of 'truth'
4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
Aristotelian logic is complete
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
Modal operators are usually treated as quantifiers
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
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
Choice is essential for proving downward L÷wenheim-Skolem
Axiom of Choice: some function has a value for every set in a given set
The Axiom of Choice seems to license an infinite amount of choosing
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
Are sets part of logic, or part of mathematics?
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
It is central to the iterative conception that membership is well-founded, with no infinite descending chains
Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
Russell's paradox shows that there are classes which are not iterative sets
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
'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
Anti-realists reject set theory
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
There is no 'correct' logic for natural languages
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
Skolem and G÷del championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order
Bernays (1918) formulated and proved the completeness of propositional logic
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
The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
Maybe compactness, semantic effectiveness, and the L÷wenheim-Skolem properties are desirable
The notion of finitude is actually built into first-order languages
First-order logic was an afterthought in the development of modern logic
First-order logic is Complete, and Compact, with the L÷wenheim-Skolem Theorems
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order logic is better than set theory, since it only adds relations and operations, and nothing else
Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics?
Henkin semantics has separate variables ranging over the relations and over the functions
In standard semantics for second-order logic, a single domain fixes the ranges for the variables
If the aim of logic is to codify inferences, second-order logic is useless
Some say that second-order logic is mathematics, not logic
Completeness, Compactness and L÷wenheim-Skolem fail in second-order standard semantics
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Logical consequence can be defined in terms of the logical terminology
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
The two standard explanations of consequence are semantic (in models) and deductive
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Semantic consequence is ineffective in second-order logic
If a logic is incomplete, its semantic consequence relation is not effective
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
Intuitionism only sanctions modus ponens if all three components are proved
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
Either logic determines objects, or objects determine logic, or they are separate
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
The law of excluded middle might be seen as a principle of omniscience
Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
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
Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and'
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A function is just an arbitrary correspondence between collections
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
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
Second-order variables also range over properties, sets, relations or functions
5. Theory of Logic / G. Quantification / 6. Plural Quantification
Maybe plural quantifiers should be understood in terms of classes or sets
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
'Satisfaction' is a function from models, assignments, and formulas to {true,false}
A sentence is 'satisfiable' if it has a model
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Semantics for models uses set-theory
The central notion of model theory is the relation of 'satisfaction'
Model theory deals with relations, reference and extensions
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation
Categoricity can't be reached in a first-order language
Theory ontology is never complete, but is only determined 'up to isomorphism'
The set-theoretical hierarchy contains as many isomorphism types as possible
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
Substitutional semantics only has countably many terms, so Upward L÷wenheim-Skolem trivially fails
Any theory with an infinite model has a model of every infinite cardinality
The L÷wenheim-Skolem Theorems fail for second-order languages with standard semantics
The L÷wenheim-Skolem theorem seems to be a defect of first-order logic
Downward L÷wenheim-Skolem: if there's an infinite model, there is a countable model
Up L÷wenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
The L÷wenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
Upward L÷wenheim-Skolem: each infinite model has infinite models of all sizes
Downward L÷wenheim-Skolem: each satisfiable countable set always has countable models
5. Theory of Logic / K. Features of Logics / 3. Soundness
'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
We can live well without completeness in logic
5. Theory of Logic / K. Features of Logics / 6. Compactness
Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures
Compactness is derived from soundness and completeness
5. Theory of Logic / K. Features of Logics / 9. Expressibility
A language is 'semantically effective' if its logical truths are recursively enumerable
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Virtually all of mathematics can be modeled in set theory
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
6. Mathematics / A. Nature of Mathematics / 3. Numbers / d. Natural numbers
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. Numbers / e. Ordinal numbers
Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Real numbers are thought of as either Cauchy sequences or Dedekind cuts
Understanding the real-number structure is knowing usage of the axiomatic language of analysis
6. Mathematics / A. Nature of Mathematics / 3. Numbers / h. Reals from Cauchy
Cauchy gave a formal definition of a converging sequence.
6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts
Cuts are made by the smallest upper or largest lower number, some of them not rational
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
The 'continuum' is the cardinality of the powerset of a denumerably infinite set
The Continuum Hypothesis says there are no sets between the natural numbers and reals
6. Mathematics / A. Nature of Mathematics / 6. Proof in Mathematics
For intuitionists, proof is inherently informal
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
There is no grounding for mathematics that is more secure than mathematics
Categories are the best foundation for mathematics
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
First-order arithmetic can't even represent basic number theory
Natural numbers just need an initial object, successors, and an induction principle
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order logic has the expressive power for mathematics, but an unworkable model theory
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / b. Greek arithmetic
Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic)
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
Property extensions outstrip objects, so shortage of objects caused the Caesar problem
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / f. Zermelo numbers
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Some sets of natural numbers are definable in set-theory but not in arithmetic
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
Mathematical foundations may not be sets; categories are a popular rival
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
Baseball positions and chess pieces depend entirely on context
The even numbers have the natural-number structure, with 6 playing the role of 3
Could infinite structures be apprehended by pattern recognition?
The 4-pattern is the structure common to all collections of four objects
The main mathematical structures are algebraic, ordered, and topological
Some structures are exemplified by both abstract and concrete
Mathematical structures are defined by axioms, or in set theory
Numbers do not exist independently; the essence of a number is its relations to other numbers
A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / b. Varieties of structuralism
The main versions of structuralism are all definitionally equivalent
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / c. Nominalist structuralism
Is there is no more to structures than the systems that exemplify them?
Number statements are generalizations about number sequences, and are bound variables
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism
Because one structure exemplifies several systems, a structure is a one-over-many
There is no 'structure of all structures', just as there is no set of all sets
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
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
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
If mathematical objects are accepted, then a number of standard principles will follow
Platonists claim we can state the essence of a number without reference to the others
Platonism must accept that the Peano Axioms could all be false
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
Intuition is an outright hindrance to five-dimensional geometry
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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
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
Mathematics and logic have no border, and logic must involve mathematics and its ontology
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
Term Formalism says mathematics is just about symbols - but real numbers have no names
Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
Deductivism says mathematics is logical consequences of uninterpreted axioms
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Can the ideal constructor also destroy objects?
Presumably nothing can block a possible dynamic operation?
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities
Some reject formal properties if they are not defined, or defined impredicatively
'Impredicative' definitions refer to the thing being described
7. Existence / A. Nature of Existence / 1. Nature of Existence
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
The abstract/concrete boundary now seems blurred, and would need a defence
Mathematicians regard arithmetic as concrete, and group theory as abstract
7. Existence / D. Theories of Reality / 6. Fictionalism
Fictionalism eschews the abstract, but it still needs the possible (without model theory)
Structuralism blurs the distinction between mathematical and ordinary objects
8. Modes of Existence / B. Properties / 10. Properties as Predicates
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
Logicians use 'property' and 'set' interchangeably, with little hanging on it
9. Objects / A. Existence of Objects / 1. Physical Objects
The notion of 'object' is at least partially structural and mathematical
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
A blurry border is still a border
10. Modality / A. Necessity / 6. Logical Necessity
Logical modalities may be acceptable, because they are reducible to satisfaction in models
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
Why does the 'myth' of possible worlds produce correct modal logic?
12. Knowledge Sources / C. Rationalism / 1. Rationalism
Rationalism tries to apply mathematical methodology to all of knowledge
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
We apprehend small, finite mathematical structures by abstraction from patterns
18. Thought / E. Abstraction / 2. Abstracta by Selection
Simple types can be apprehended through their tokens, via abstraction
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
A structure is an abstraction, focussing on relationships, and ignoring other features
We can apprehend structures by focusing on or ignoring features of patterns
We can focus on relations between objects (like baseballers), ignoring their other features
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Abstract objects might come by abstraction over an equivalence class of base entities