Positive Monotonic Replacement (Propositional Logic)

Definition

Positive Monotonic Replacement (Propositional Logic)

Let be a pure atom in formula with positive polarity with . Define

We can then infer that

i.e., with being true is also a model of .

Proof

Positive Monotonic Replacement (Propositional Logic)

Assume that and that every occurrence of in has positive polarity. The following implication is a tautology:

Positive polarity preserves this implication along every path from an occurrence of to the root. Replacing all occurrences simultaneously therefore gives

Since , it follows that

The right-hand side is a truth value, so . Evaluating under is the same as evaluating under , because assigns to and otherwise agrees with . Hence

and therefore .