# Interpretations and models in propositional logic

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?

• What books are you reading? There's a certain variety in definitions. – Fabio Somenzi Oct 16 '18 at 18:46
• It's possible that considering propositional logic is making this too simplified, and that, at the cost of some additional complexity, the distinctions would be clearer in predicate logic. For example, you could identify a truth function $\mathsf{Prop}\to\mathbf{2}$ with a subset of $\mathsf{Prop}$, the set of propositions. Doing this for first-order logic, though, doesn't make as much sense. (The equivalent would be making a family of relations, i.e. subsets of a cartesian product, indexed by arity, $n$, of the form $\mathsf{Pred}_n\times D^n$. The propositional case is then $n=0$.) – Derek Elkins left SE Oct 16 '18 at 21:59

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.

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 : $$(p_0 \to p_0), (p_0 \lor \lnot p_0)$$.

• So an interpretation can be a model of a formula, $v \models \varphi$, but then we are also able to say a set of wffs "models" a formula, e.g. $\Gamma \models \varphi$? Or is this wrong and it's more accurate to say $\Gamma \models \varphi$ means $\varphi$ is satisfied in all models of $\Gamma$? – user525966 Oct 16 '18 at 19:06
• @user525966 - Yes; the symbol $\Gamma \vDash \varphi$, with $\Gamma$ a set of formulas, means that $\varphi$ is a semantic consequence of $\Gamma$. See your previous post How is my understanding of ⊢,⊨,→, etc – Mauro ALLEGRANZA Oct 16 '18 at 19:15
• @user525966 The second one. You may be confused by the command \models. This symbol is used in two totally different ways. $\mathcal M\models \Gamma$ “the structure $\mathcal M$ is a model for the set of sentences $\Gamma$” is the use where “models” is a apt verb. The usage $\Gamma\models \phi$ would never be put into English as “$\Gamma$ is a model for $\phi.$.” (In fact, some model theory books, that don’t need to worry about confusion with the syntactical notion, use $\vdash$ here instead.) – spaceisdarkgreen Oct 16 '18 at 19:21
• @user525966 the second usage would be put into English as “$\phi$ is a semantic consequence of $\Gamma.$” – spaceisdarkgreen Oct 16 '18 at 19:27
• @user525966 no. A model of what? All valid propositions are true in any interpretation whatsoever. That’s what it means to be valid. – spaceisdarkgreen Oct 16 '18 at 19:34