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Single Idea 10902
[filed under theme 5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
]
Full Idea
We can look at semantics from the point of view of how truth values are determined by instantiations of properties and relations, or by asking how we can build, using the resources of the language, a proposition corresponding to a given semantic pattern.
Gist of Idea
We can do semantics by looking at given propositions, or by building new ones
Source
José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6)
Book Ref
Zalabardo,José L.: 'Introduction to the Theory of Logic' [Westview 2000], p.109
A Reaction
The second version of semantics is model theory.
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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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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10893
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Γ |= φ for sentences if φ is true when all of Γ is true
[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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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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10900
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Logically true sentences are true in all structures
[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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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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