Ideas from 'Foundations without Foundationalism' by Stewart Shapiro [1991], by Theme Structure

[found in 'Foundations without Foundationalism' by Shapiro,Stewart [OUP 1991,0-19-825029-0]].

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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 / 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
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
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
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
Completeness, Compactness and L÷wenheim-Skolem fail in second-order standard semantics
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 / 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 / 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 / I. Semantics of Logic / 4. Satisfaction
'Satisfaction' is a function from models, assignments, and formulas to {true,false}
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Semantics for models uses set-theory
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
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
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 / 3. Numbers / b. Types of number
Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
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 / 4. The Infinite / g. Continuum Hypothesis
The 'continuum' is the cardinality of the powerset of a denumerably infinite set
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
First-order arithmetic can't even represent basic number theory
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 / 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
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Some reject formal properties if they are not defined, or defined impredicatively
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