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Single Idea 10888
[filed under theme 4. Formal Logic / F. Set Theory ST / 1. Set Theory
]
Full Idea
We can define a set by 'enumeration' (by listing the items, within curly brackets), or by 'abstraction' (by specifying the elements as instances of a property), pretending that they form a determinate totality. The latter is written {x | x is P}.
Gist of Idea
Sets can be defined by 'enumeration', or by 'abstraction' (based on a property)
Source
José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3)
Book Ref
Zalabardo,José L.: 'Introduction to the Theory of Logic' [Westview 2000], p.6
The
18 ideas
from José L. Zalabardo
10886
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Determinacy: an object is either in a set, or it isn't
[Zalabardo]
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10887
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Specification: Determinate totals of objects always make a set
[Zalabardo]
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10888
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Sets can be defined by 'enumeration', or by 'abstraction' (based on a property)
[Zalabardo]
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10889
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The 'Cartesian Product' of two sets relates them by pairing every element with every element
[Zalabardo]
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10890
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A 'partial ordering' is reflexive, antisymmetric and transitive
[Zalabardo]
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10891
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If a set is defined by induction, then proof by induction can be applied to it
[Zalabardo]
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10893
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Γ |= φ for sentences if φ is true when all of Γ is true
[Zalabardo]
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10892
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We make a truth assignment to T and F, which may be true and false, but merely differ from one another
[Zalabardo]
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10894
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A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true
[Zalabardo]
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10895
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'Logically true' (|= φ) is true for every truth-assignment
[Zalabardo]
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10896
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Propositional logic just needs ¬, and one of ∧, ∨ and →
[Zalabardo]
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10897
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A first-order 'sentence' is a formula with no free variables
[Zalabardo]
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10898
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The semantics shows how truth values depend on instantiations of properties and relations
[Zalabardo]
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10901
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Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true
[Zalabardo]
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10900
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Logically true sentences are true in all structures
[Zalabardo]
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10899
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Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations
[Zalabardo]
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10903
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A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model
[Zalabardo]
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10902
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We can do semantics by looking at given propositions, or by building new ones
[Zalabardo]
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