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Single Idea 13637
[filed under theme 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
]
Full Idea
Second-order logic is inherently incomplete, so its semantic consequence relation is not effective.
Gist of Idea
If a logic is incomplete, its semantic consequence relation is not effective
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.12
The
50 ideas
from 'Foundations without Foundationalism'
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15944
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Second-order logic is better than set theory, since it only adds relations and operations, and nothing else
[Shapiro, by Lavine]
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13629
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Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics?
[Shapiro]
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13626
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Semantic consequence is ineffective in second-order logic
[Shapiro]
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13631
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Are sets part of logic, or part of mathematics?
[Shapiro]
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13627
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There is no 'correct' logic for natural languages
[Shapiro]
|
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13624
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The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
[Shapiro]
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13628
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We can live well without completeness in logic
[Shapiro]
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13630
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Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures
[Shapiro]
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13625
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Mathematics and logic have no border, and logic must involve mathematics and its ontology
[Shapiro]
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13634
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Satisfaction is 'truth in a model', which is a model of 'truth'
[Shapiro]
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13632
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Finding the logical form of a sentence is difficult, and there are no criteria of correctness
[Shapiro]
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13633
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'Satisfaction' is a function from models, assignments, and formulas to {true,false}
[Shapiro]
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13635
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'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence
[Shapiro]
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13636
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An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation
[Shapiro]
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13637
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If a logic is incomplete, its semantic consequence relation is not effective
[Shapiro]
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13640
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Russell's paradox shows that there are classes which are not iterative sets
[Shapiro]
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13638
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Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects
[Shapiro]
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13641
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Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
[Shapiro]
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13642
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Logic is the ideal for learning new propositions on the basis of others
[Shapiro]
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13643
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Aristotelian logic is complete
[Shapiro]
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13644
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Semantics for models uses set-theory
[Shapiro]
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13650
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Henkin semantics has separate variables ranging over the relations and over the functions
[Shapiro]
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13645
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In standard semantics for second-order logic, a single domain fixes the ranges for the variables
[Shapiro]
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13649
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Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics
[Shapiro]
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13648
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The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
[Shapiro]
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13647
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Choice is essential for proving downward Löwenheim-Skolem
[Shapiro]
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13646
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Compactness is derived from soundness and completeness
[Shapiro]
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13651
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A set is 'transitive' if contains every member of each of its members
[Shapiro]
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13657
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First-order arithmetic can't even represent basic number theory
[Shapiro]
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13652
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The 'continuum' is the cardinality of the powerset of a denumerably infinite set
[Shapiro]
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13653
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'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element
[Shapiro]
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13654
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It is central to the iterative conception that membership is well-founded, with no infinite descending chains
[Shapiro]
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13656
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Some sets of natural numbers are definable in set-theory but not in arithmetic
[Shapiro]
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13661
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A language is 'semantically effective' if its logical truths are recursively enumerable
[Shapiro]
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13660
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Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable
[Shapiro]
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13658
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Downward Löwenheim-Skolem: each satisfiable countable set always has countable models
[Shapiro]
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13659
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Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes
[Shapiro]
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13662
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First-order logic was an afterthought in the development of modern logic
[Shapiro]
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13666
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Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
[Shapiro]
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13664
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Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions
[Shapiro]
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13663
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Some reject formal properties if they are not defined, or defined impredicatively
[Shapiro]
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13667
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Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order
[Shapiro]
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13668
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Bernays (1918) formulated and proved the completeness of propositional logic
[Shapiro]
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13669
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Can one develop set theory first, then derive numbers, or are numbers more basic?
[Shapiro]
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13670
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Categoricity can't be reached in a first-order language
[Shapiro]
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13673
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The notion of finitude is actually built into first-order languages
[Shapiro]
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13674
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We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models
[Shapiro]
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13675
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Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails
[Shapiro]
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13676
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Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are
[Shapiro]
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13677
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Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals
[Shapiro]
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