Model (Logic)

A model is some specific mapping of logical variables (both propositional variable and …) to truth values..(?)#todo

#todo Need to talk about interpretations first, then a model is just an interpretation that makes a formula true.

Models are an example of meta-logic, since we can use them to talk about any set of logic, be it propositional logic or predicate logic.

Definition §

Model ^definition

A model $M$ is a model of a formula $P$ if, and only if, $P$ is true under the mapping given by $M$. This can be written as:

$$M\models P$$

Reusing symbols

The symbol $\models$ is used to for showing models, but it is also used for showing semantic consequences. Pay key attention to what the operands on the left and right actually are. If the left is also a formula, we are talking about semantic consequences, not models.

Showing Validity §

Showing Satisfiability §

We write

$$v\models X$$

Here, $\models$ is used to mean $M$ models or satisfies $X$. In other words, $X$ is true under $v$. E.g. (in propositional logic)

$$v=\{P\mapsto 0,Q\mapsto 1\};v\models (P\vee Q)$$

Thus, $X$ is satisfiable (with $M$)

Examples §

• Valuation (in Propositional Logic)