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Single Idea 22294
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
]
Full Idea
We can only establish that a concept is free from contradiction by first producing something that falls under it.
Gist of Idea
We can show that a concept is consistent by producing something which falls under it
Source
Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §095), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 19 'Exist'
Book Ref
Potter,Michael: 'The Rise of Anaytic Philosophy 1879-1930' [Routledge 2020], p.129
A Reaction
Potter quotes this as an example of proof by modelling. If it has one model then it must be consistent. Then we ask whether all the models are or are not consistent with one another. Circular squares fail the test.
The
35 ideas
with the same theme
[general features of logical models]:
22294
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We can show that a concept is consistent by producing something which falls under it
[Frege]
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13343
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A 'model' is a sequence of objects which satisfies a complete set of sentential functions
[Tarski]
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16323
|
The object language/ metalanguage distinction is the basis of model theory
[Tarski, by Halbach]
|
13826
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Model theory looks at valid sentences and consequence, but not how we know these things
[Prawitz]
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10473
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Model theory studies formal or natural language-interpretation using set-theory
[Hodges,W]
|
10475
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A 'structure' is an interpretation specifying objects and classes of quantification
[Hodges,W]
|
10481
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Models in model theory are structures, not sets of descriptions
[Hodges,W]
|
9968
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A model is 'fundamental' if it contains only concrete entities
[Jubien]
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10827
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Model theory is unusual in restricting the range of the quantifiers
[Field,H]
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13505
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Model theory studies how set theory can model sets of sentences
[Hart,WD]
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13511
|
Model theory is mostly confined to first-order theories
[Hart,WD]
|
13513
|
Models are ways the world might be from a first-order point of view
[Hart,WD]
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13512
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Modern model theory begins with the proof of Los's Conjecture in 1962
[Hart,WD]
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13644
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Semantics for models uses set-theory
[Shapiro]
|
10240
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Model theory deals with relations, reference and extensions
[Shapiro]
|
10239
|
The central notion of model theory is the relation of 'satisfaction'
[Shapiro]
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15411
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We only need to study mathematical models, since all other models are isomorphic to these
[Burgess]
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15412
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Models leave out meaning, and just focus on truth values
[Burgess]
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15416
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We aim to get the technical notion of truth in all models matching intuitive truth in all instances
[Burgess]
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10903
|
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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13724
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In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation
[Sider]
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10129
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A 'model' is a meaning-assignment which makes all the axioms true
[George/Velleman]
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5741
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If every model that makes premises true also makes conclusion true, the argument is valid
[Melia]
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19207
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Sentence logic maps truth values; predicate logic maps objects and sets
[Merricks]
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10158
|
A structure is a 'model' when the axioms are true. So which of the structures are models?
[Feferman/Feferman]
|
10162
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Tarski and Vaught established the equivalence relations between first-order structures
[Feferman/Feferman]
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10693
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Models are mathematical structures which interpret the non-logical primitives
[Beall/Restall]
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13519
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Model theory uses sets to show that mathematical deduction fits mathematical truth
[Wolf,RS]
|
13531
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Model theory reveals the structures of mathematics
[Wolf,RS]
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13532
|
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
[Wolf,RS]
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13533
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First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
[Wolf,RS]
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18694
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Permutation Theorem: any theory with a decent model has lots of models
[Button]
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10756
|
A model is a domain, and an interpretation assigning objects, predicates, relations etc.
[Rossberg]
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18744
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Models are sets with functions and relations, and truth built up from the components
[Horsten/Pettigrew]
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17747
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A 'model' of a theory specifies interpreting a language in a domain to make all theorems true
[Walicki]
|