Namespaces
Variants
Actions

Difference between revisions of "Model (in logic)"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
An interpretation of a [[Formal language|formal language]] satisfying certain axioms (cf. [[Axiom|Axiom]]). The basic formal language is the first-order language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643601.png" /> of a given signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643602.png" /> including predicate symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643603.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643604.png" />, function symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643606.png" />, and constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643608.png" />. A model of the language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m0643609.png" /> is an [[Algebraic system|algebraic system]] of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436010.png" />.
+
<!--
 +
m0643601.png
 +
$#A+1 = 55 n = 0
 +
$#C+1 = 55 : ~/encyclopedia/old_files/data/M064/M.0604360 Model (in logic)
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436011.png" /> be a set of closed formulas in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436012.png" />. A model for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436013.png" /> is a model for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436014.png" /> in which all formulas from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436015.png" /> are true. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436016.png" /> is called consistent if it has at least one model. The class of all models of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436017.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436018.png" />. Consistency of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436019.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436020.png" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436021.png" /> of models of a language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436022.png" /> is called axiomatizable if there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436023.png" /> of closed formulas of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436025.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436026.png" /> of all closed formulas of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436027.png" /> that are true in each model of a given class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436028.png" /> of models of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436029.png" /> is called the [[Elementary theory|elementary theory]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436030.png" />. Thus, a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436031.png" /> of models of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436032.png" /> is axiomatizable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436033.png" />. If a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436034.png" /> consists of models isomorphic to a given model, then its elementary theory is called the elementary theory of this model.
+
An interpretation of a [[Formal language|formal language]] satisfying certain axioms (cf. [[Axiom|Axiom]]). The basic formal language is the first-order language  $  L _  \Omega  $
 +
of a given signature  $  \Omega $
 +
including predicate symbols  $  R _ {i} $,
 +
$  i \in I $,
 +
function symbols  $  f _ {j} $,
 +
$  j \in J $,
 +
and constants  $  c _ {k} $,
 +
$  k \in K $.  
 +
A model of the language  $  L _  \Omega  $
 +
is an [[Algebraic system|algebraic system]] of signature  $  \Omega $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436035.png" /> be a model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436036.png" /> having universe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436037.png" />. One may associate to each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436038.png" /> a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436039.png" /> and consider the first-order language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436040.png" /> of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436041.png" /> which is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436042.png" /> by adding the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436044.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436045.png" /> is called the diagram language of the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436046.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436047.png" /> of all closed formulas of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436048.png" /> which are true in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436049.png" /> on replacing each constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436050.png" /> by the corresponding element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436051.png" /> is called the description (or elementary diagram) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436052.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436053.png" /> of those formulas from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436054.png" /> which are atomic or negations of atomic formulas is called the diagram of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064360/m06436055.png" />.
+
Let $  \Sigma $
 +
be a set of closed formulas in  $  L _  \Omega  $.
 +
A model for  $  \Sigma $
 +
is a model for  $  L _  \Omega  $
 +
in which all formulas from  $  \Sigma $
 +
are true. A set  $  \Sigma $
 +
is called consistent if it has at least one model. The class of all models of  $  \Sigma $
 +
is denoted by  $  \mathop{\rm Mod}  \Sigma $.  
 +
Consistency of a set  $  \Sigma $
 +
means that  $  \mathop{\rm Mod}  \Sigma \neq \emptyset $.
 +
 
 +
A class  $  {\mathcal K} $
 +
of models of a language  $  L _  \Omega  $
 +
is called axiomatizable if there is a set  $  \Sigma $
 +
of closed formulas of  $  L _  \Omega  $
 +
such that  $  {\mathcal K} =  \mathop{\rm Mod}  \Sigma $.
 +
The set  $  T ( {\mathcal K} ) $
 +
of all closed formulas of  $  L _  \Omega  $
 +
that are true in each model of a given class  $  {\mathcal K} $
 +
of models of  $  L _  \Omega  $
 +
is called the [[Elementary theory|elementary theory]] of  $  {\mathcal K} $.
 +
Thus, a class  $  {\mathcal K} $
 +
of models of  $  L _  \Omega  $
 +
is axiomatizable if and only if  $  {\mathcal K} = \mathop{\rm Mod}  T ( {\mathcal K} ) $.  
 +
If a class  $  {\mathcal K} $
 +
consists of models isomorphic to a given model, then its elementary theory is called the elementary theory of this model.
 +
 
 +
Let  $  \mathbf A $
 +
be a model of  $  L _  \Omega  $
 +
having universe $  A $.  
 +
One may associate to each element $  a \in A $
 +
a constant $  c _ {a} $
 +
and consider the first-order language $  L _ {\Omega A }  $
 +
of signature $  \Omega A $
 +
which is obtained from $  \Omega $
 +
by adding the constants $  c _ {a} $,  
 +
$  a \in A $.  
 +
$  L _ {\Omega A }  $
 +
is called the diagram language of the model $  \mathbf A $.  
 +
The set $  O ( \mathbf A ) $
 +
of all closed formulas of $  L _ {\Omega A }  $
 +
which are true in $  \mathbf A $
 +
on replacing each constant $  c _ {a} $
 +
by the corresponding element $  a \in A $
 +
is called the description (or elementary diagram) of $  \mathbf A $.  
 +
The set $  D ( \mathbf A ) $
 +
of those formulas from $  O ( \mathbf A ) $
 +
which are atomic or negations of atomic formulas is called the diagram of $  A $.
  
 
Along with models of first-order languages, models of other types (infinitary logic, [[Intuitionistic logic|intuitionistic logic]], many-sorted logic, second-order logic, [[Many-valued logic|many-valued logic]], and [[Modal logic|modal logic]]) have also been considered.
 
Along with models of first-order languages, models of other types (infinitary logic, [[Intuitionistic logic|intuitionistic logic]], many-sorted logic, second-order logic, [[Many-valued logic|many-valued logic]], and [[Modal logic|modal logic]]) have also been considered.
  
 
For references see [[Model theory|Model theory]].
 
For references see [[Model theory|Model theory]].
 
 
  
 
====Comments====
 
====Comments====
 
English usage prefers the word  "structure"  where Russian speaks of a  "model of a language"  or an  "algebraic system" ;  "model"  is reserved for structures satisfying a given theory (set of closed formulas).
 
English usage prefers the word  "structure"  where Russian speaks of a  "model of a language"  or an  "algebraic system" ;  "model"  is reserved for structures satisfying a given theory (set of closed formulas).

Latest revision as of 08:01, 6 June 2020


An interpretation of a formal language satisfying certain axioms (cf. Axiom). The basic formal language is the first-order language $ L _ \Omega $ of a given signature $ \Omega $ including predicate symbols $ R _ {i} $, $ i \in I $, function symbols $ f _ {j} $, $ j \in J $, and constants $ c _ {k} $, $ k \in K $. A model of the language $ L _ \Omega $ is an algebraic system of signature $ \Omega $.

Let $ \Sigma $ be a set of closed formulas in $ L _ \Omega $. A model for $ \Sigma $ is a model for $ L _ \Omega $ in which all formulas from $ \Sigma $ are true. A set $ \Sigma $ is called consistent if it has at least one model. The class of all models of $ \Sigma $ is denoted by $ \mathop{\rm Mod} \Sigma $. Consistency of a set $ \Sigma $ means that $ \mathop{\rm Mod} \Sigma \neq \emptyset $.

A class $ {\mathcal K} $ of models of a language $ L _ \Omega $ is called axiomatizable if there is a set $ \Sigma $ of closed formulas of $ L _ \Omega $ such that $ {\mathcal K} = \mathop{\rm Mod} \Sigma $. The set $ T ( {\mathcal K} ) $ of all closed formulas of $ L _ \Omega $ that are true in each model of a given class $ {\mathcal K} $ of models of $ L _ \Omega $ is called the elementary theory of $ {\mathcal K} $. Thus, a class $ {\mathcal K} $ of models of $ L _ \Omega $ is axiomatizable if and only if $ {\mathcal K} = \mathop{\rm Mod} T ( {\mathcal K} ) $. If a class $ {\mathcal K} $ consists of models isomorphic to a given model, then its elementary theory is called the elementary theory of this model.

Let $ \mathbf A $ be a model of $ L _ \Omega $ having universe $ A $. One may associate to each element $ a \in A $ a constant $ c _ {a} $ and consider the first-order language $ L _ {\Omega A } $ of signature $ \Omega A $ which is obtained from $ \Omega $ by adding the constants $ c _ {a} $, $ a \in A $. $ L _ {\Omega A } $ is called the diagram language of the model $ \mathbf A $. The set $ O ( \mathbf A ) $ of all closed formulas of $ L _ {\Omega A } $ which are true in $ \mathbf A $ on replacing each constant $ c _ {a} $ by the corresponding element $ a \in A $ is called the description (or elementary diagram) of $ \mathbf A $. The set $ D ( \mathbf A ) $ of those formulas from $ O ( \mathbf A ) $ which are atomic or negations of atomic formulas is called the diagram of $ A $.

Along with models of first-order languages, models of other types (infinitary logic, intuitionistic logic, many-sorted logic, second-order logic, many-valued logic, and modal logic) have also been considered.

For references see Model theory.

Comments

English usage prefers the word "structure" where Russian speaks of a "model of a language" or an "algebraic system" ; "model" is reserved for structures satisfying a given theory (set of closed formulas).

How to Cite This Entry:
Model (in logic). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Model_(in_logic)&oldid=47866
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article