what is the definition of an interpretation of first order theory $T$ ? what is a model for $T$?

Question

what is the definition of an interpretation of first order theory $T$ ? what is a model for $T$ ?

can you give me the definition supported with some simple examples ?

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i read the definition in first order mathematical logic by angelo margaris , 1990 ed , dover publications , in page 35

but the definition made no sense for me..

thanx

Answer 1

The entry on Wikipedia: First-order Logic helps to disambiguate the terms, and sort of gives examples.

See especially, under the heading "Semantics":

• "First order structure"

  "The most common way of specifying an interpretation (especially in mathematics) is to specify a structure (also called a model; see below). The structure [model] consists of a nonempty set D that forms the domain of discourse and an interpretation I of the non-logical terms of the signature. This interpretation is itself a function..."

• "Satisfiability",

• the sub-section "first-order theories and models",

  • "A first-order structure that satisfies all sentences in a given theory is said to be a model of the theory."

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One great resource, as you embark on your study of Enderton (and perhaps good to read before digging deeply into Enderton) is Paul Teller's Logic Primer. (It's free to download, and importantly, discusses interpretations and models extensively, as it does tautological implication, wff, and First Order Logic in general.)

Answer 2

Consider for example the theory of partial ordered groups:

In their language there is a binary operation symbol, $\cdot$ (the multiplication), a constant symbol $1$ (the unit), a unary operation symbol ${()}^{-1}$, and a binary relation symbol $\le$. (The equality symbol is usually assumed to be also present, by default.) An axiom system for it is the $6$ element set $T$ of the following formulas ($T$ itself is also called the theory):

1. $\forall x\forall y\forall z \ ((x\cdot y)\cdot z=x\cdot(y\cdot z))$
2. $\forall x\ (1\cdot x=x)$
3. $\forall x\ (x^{-1}\cdot x=1)$
4. $\forall x\ (x\le x)$
5. $\forall x\forall y\forall z\ ((x\le y\,\land\, y\le z )\implies x\le z)$
6. $\forall x\forall y\forall z\ (x\le y \implies (x\cdot z\le y\cdot z\ \land\ z\cdot x\le z\cdot y) )$

A model of $T$ is a realization of the operation and relation symbols on a base set such that they satisfy the axioms. So, first we have to consider a model $\mathfrak M$ for the given first order language: it is thus far nothing else than a set $M$, equipped with a binary ($\mu:M\times M\to M$), a unary ($\sigma:M\to M$) and a constant ($c\in M$) operation, and a binary relation $(L\subseteq M\times M)$.

This $\mathfrak M$ is said to be a model for $T$, if all elemets of $T$ is satisfied on $\mathfrak M$ when interpreting the symbols accordingly, with respect to all evaluations of the variables (whenever the variable $x$ is evaluated in $M$, $x\cdot y$ is evaluated as $\mu(x,y)$, $x^{-1}$ as $\sigma(x)$, $1$ as $c$, and $x\le y$ as $(x,y)\in L$).