Model (Propositional Logic)

Definition

Model (Propositional Logic)

An interpretation is called model of a formula iff is true under , i.e.,

The set of all models of a formula is denoted as

Equivalence

Definition

Model Characterization of Equivalence (Propositional Logic)

Two propositional formulas are equivalent if and only if they have the same models:

where is the set of models of over all interpretations .

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Examples

Attempting a countermodel

Given

is there an interpretation with the premises true and false —

By the entailment–unsatisfiability duality, such a is a model of

which forces and . By equivalent replacement the set reduces to the contradiction

so no such exists: the entailment is valid and the countermodel construction fails.