Positive Monotonic Replacement (Propositional Logic)

Definition

Positive Monotonic Replacement (Propositional Logic)

Let $p$ be a pure atom in formula $F$ with positive polarity with $I\models F$. Define

$$I^{\prime }:=I\cup \{p\mapsto 1\}.$$

We can then infer that

$$I^{\prime }\models F,$$

i.e., $I$ with $p$ being true is also a model of $F$.

Proof

Positive Monotonic Replacement (Propositional Logic)

Assume that $I\models F$ and that every occurrence of $p$ in $F$ has positive polarity. The following implication is a tautology:

$$\models p\to \top .$$

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

$$\models F\to F_p^{\top }.$$

Since $I\models F$, it follows that

$$1=I(F)\le I(F_p^{\top }).$$

The right-hand side is a truth value, so $I(F_p^{\top })=1$. Evaluating $F_p^{\top }$ under $I$ is the same as evaluating $F$ under $I^{\prime }$, because $I^{\prime }$ assigns $1$ to $p$ and otherwise agrees with $I$. Hence

$$I^{\prime }(F)=I(F_p^{\top })=1,$$

and therefore $I^{\prime }\models F$.