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]].
green numbers give full details |
back to texts
|
expand these ideas
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 / 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
|
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
|
13640
|
Russell's paradox shows that there are classes which are not 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
|
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
|
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
|
13642
|
Logic is the ideal for learning new propositions on the basis of others
|
|
13627
|
There is no 'correct' logic for natural languages
|
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
|
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
|
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 second-order logic
|
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 / 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 / I. Semantics of Logic / 4. Satisfaction
|
13633
|
'Satisfaction' is a function from models, assignments, and formulas to {true,false}
|
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
|
13644
|
Semantics for models uses set-theory
|
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
|
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
|
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
|
|
13648
|
The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
|
|
13675
|
Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails
|
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 / 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
|
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 / 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 / 4. Axioms for Number / d. Peano arithmetic
|
13657
|
First-order arithmetic can't even represent basic number theory
|
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 / 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
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
|
13663
|
Some reject formal properties if they are not defined, or defined impredicatively
|
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
|