Model (Logic)

A model is an interpretation that satisfies a given formula. Essentially, it is some mapping of symbols 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 (Propositional Logic) ^definition-propositional

In propositional logic, a model $M$ is a valuation (mapping of atoms to truth values).

With reference to a formula (proposition, in this case), $M$ is a model of a formula $P$ if, and only if, $P$ is true under $M$. This can be written as:

$$M\models P$$

Model (Predicate Logic) ^definition

In predicate logic (and higher-order logic), a model $M$ is defined to the combination of:

• An interpretation, $\mathcal{I}$
• A universe, $\mathcal{U}$
• (Optionally) A variable assignment, $v$

With reference to a formula, $M$ is a model of a formula $P$ if, and only if, $P$ is true under $M$. If there was a variable assignment, it should result in $P$ being true. This can be written as:

$$M,v\models P\mathcal{U},\mathcal{I},v\models P$$

Else, if there are no free variables, just the model is sufficient:

$$M\models P\mathcal{U},\mathcal{I}\models P$$

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 §

Examples §