Diff for "Model" - Dr. A's Wiki

Differences between revisions 7 and 10 (spanning 3 versions)

Propositional Logic:

In terms of a logic formula, a ["Model"] is some assignment of variables that causes the formula to be true.

First Order Logic:

A model for a set $X$ of formulas is an interpretation $M$ for $X$ such that every formula of $X$ is true in $M$ .

A domain $D$ is any nonempty set. An interpretation for a set of formulas $X$ , is a domain $D$ together with a rule that

assigns to each $n$ -place predicate symbol (that occurs in a formula) of $X$ an $n$ -place predicate in $D$ ;
assigns to each $n$ -place operation symbol of $X$ an $n$ -place operation in $D$ ;
assigns to each constant symbol of $X$ an element of $D$ ; and
assigns to $=$ the identity predicate $=$ in $D$ , defined by: $a = b$ iff $a$ and $b$ are the same.

See: First Order Mathematical Logic by Angelo Margaris p 145

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-{{{#!latex2
\noindent A {\bf model} for a set $X$ of formulas is an interpretation $M$ for $X$ such that every formula of $X$ is true in $M$.\bigskip
+A '''model''' for a set $$X$$ of formulas is an interpretation $$M$$ for $$X$$ such that every formula of $$X$$ is true in $$M$$.
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-\noindent A {\em domain} $D$ is any nonempty set. An {\em interpretation} for a set of formulas $X$, is a domain $D$ togther with a rule that 
 $Unknown environment 'enumerate'$ 
}}}
+A ''domain''  $D$  is any nonempty set. An ''interpretation'' for a set of formulas  $X$ , is a domain  $D$  together with a rule that 

 * assigns to each  $n$ -place predicate symbol (that occurs in a formula) of  $X$  an  $n$ -place predicate in  $D$ ;
 * assigns to each  $n$ -place operation symbol of  $X$  an  $n$ -place operation in  $D$ ;
 * assigns to each constant symbol of  $X$  an element of  $D$ ; and
 * assigns to  $=$  the identity predicate  $=$  in  $D$ , defined by:  $a = b$  iff  $a$  and  $b$  are the same.


See: First Order Mathematical Logic by Angelo Margaris p 145