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A model $M$ is a tuple $(O, F, P)$, where $O$ is a list of objects, $F$ is a table of function values, and $P$ is a table of predicate values.

Let $O=\mathbb{Z_{\le4}}$. I use $even(x), prime(x)$ as two unary preciates and $plus\_one\_mod4(x)$ as a function;

Now Model is given by $$M = (O, F, P)$$ Where $$O = \{0,1, 2, 3, 4\} $$

$$F =\{(plus\_one\_mod4(0),1), (plus\_one\_mod4(0),2), (plus\_one\_mod4(0),3), (plus\_one\_mod4(0),0), (plus\_one\_mod4(0),1)\}$$

$$P= \{ ((0),T), ((1),F), ((2),T), ((3),F), ((4),T)\}$$

Is my understanding about model correct? Is it a valid example for model?

If yes, then consider the following:

A valuation I (under M) is a mapping of syntactic variables to model objects.

What can be an example of interpretation? Are the following valid example for interpretation?

$I_1 : x = 3;$

$I_2: x= 5 $

Mauro ALLEGRANZA
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hanugm
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1 Answers1

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Basically yes, also if "tables" is not the "standard" word.

The set $O$ is a set of objects, like e.g the set $\mathbb N$ of natural numbers and it is called domain of the model $\mathcal M$.

The interpretation of a unary predicate symbol $P(x)$ is a subset of $O$, and an $n$-ary relation on $O$ in the general case of a $n$-ary predicate symbol $P(x_1, \ldots,x_n)$, i.e. $P^{\mathcal M} ⊆ O \times O \times \ldots \times O$.

Having said that, you are right : $O = \{ 0,1,2,3,4 \}$ is the domain of the model $\mathcal M$ and $P^{\mathcal M} = \{ 0,2,4 \}$ is the subset of $O$ that interpret the predicate symbol $P(x)$ meaning "$x$ is even".

In order to interpret a formula with free variables, like e.g. $(x=3)$, we need an additional "device" : a valuation $I$ (under $\mathcal M$), i.e. a mapping of syntactic variables to objects in the domain $O$ of $\mathcal M$.

The following are examples of valuations : with $I_1(x)=3$ we have that the formula $(x=3)$ is satisfied, i.e. $\mathcal M, I_1 \vDash (x=3)$, while with $I_2(x)=5$ it is not.

Mauro ALLEGRANZA
  • 99,247