Interpretations and models in propositional logic

Question

I am trying to understand model vs. interpretation of sentences vs. theories, with respect to a propositional calculus.

Is a model of a wff just a sentence that is true under some interpretation?

Or a set of wffs that are all true under some interpretation?

Or when an interpretation satisfies a wff (or set of wffs)?

Does it make sense to say that an interpretation is a model if all valid propositions are true? Or an interpretation being a model iff all theorems from the theory are still valid under the interpretation?

Answer 1

See Interpretation of a truth-functional propositional calculus :

An interpretation of a truth-functional [i.e. classic] propositional calculus $\text P$ is an assignment to each propositional symbol $\text P$ of one or the other (but not both) of the truth values truth ($\text T$) and falsity ($\text F$), and an assignment to the connective symbols of $\text P$ of their usual truth-functional meanings.

Example : Let the language of $\text P$ made of the following list of prop symbols : $\text {At} = \{p_0, p_1,\ldots \}$ and let $\{ \lor, \lnot \}$ the set of connectives.

An interpretation is an assignment $v : \text {At} \to \{ \text T, \text F \}$ such that, e.g. $v(p_0)= \text T$ and $v(p_1)= \text F$, etc.

Using $v$ and the truth tables for $\lor$ and $\lnot$ we can easily compute the truth value of a formula whatever of $\text P$, like e.g. $(p_0 \lor \lnot p_1)$.

If $\varphi$ is a formula of $\text P$ and we have $v(\varphi)= \text T$, we say that the interpretation $v$ satisfies formula $\varphi$ (and we can write : $v \vDash \varphi$).

A formula of propositional logic is true under an interpretation iff the interpretation assigns the truth value $\text T$ to that formula. If a formula is true under an interpretation, then that interpretation is called a model of that formula.

Thus, an interpetation satisfies a formula $\varphi$ iff it is a model of the formula.

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In (classic) propositional logic a formula $\varphi$ is a tautology (or valid) iff it is true in every interpretation, i.e. such that :

$v \vDash \varphi$, for every assignment $v$.

Examples of tautologies : $(p_0 \to p_0), (p_0 \lor \lnot p_0)$.