Logical Strength

logic

Definition

Logical Strength

Let and be formulas in a formal language. The formula is logically stronger than if entails :

Equivalently, every interpretation that satisfies also satisfies .

If and , then is strictly stronger than .

If and , then and are logically equivalent.

A stronger formula has fewer models. The weakest formula is and the strongest formula is .

Weak and Strong Formulas

A formula is stronger than a formula when entails :

This means that every model of is also a model of :

Equivalently, is weaker than when is implied by :

and then

Square Subset Notation

In some contexts (e.g., lattice theory or abstract semantics), logical strength is denoted using “square subset” notation to emphasise the ordering of cases:

• : is stronger than (or equal to) .
• : is strictly stronger than .
• : is weaker than (or equal to) .

Examples

stronger than

is stronger than because equality implies inequality. Every assignment that satisfies also satisfies .

weaker than

is weaker than because every assignment satisfying also satisfies .

is the weakest formula

The formula (true) is the weakest formula. Every other formula implies it, so its satisfaction set is the whole state space.

is the strongest formula

The formula (false) is the strongest formula. It implies every formula, but no non-trivial formula implies it.

and are incomparable

Neither formula implies the other. There are assignments where holds but does not, and assignments where holds but does not.