Validity

A (well-formed) logical formula is valid if it is true under every model. Else it is invalid.

More causally, if a formula is always true, it is valid. If it false for even a single case, it is invalid

Definition §

Validity ^definition

A well-formed formula $P$ in a formal language $L$ is valid if it is true under every model. In that case, we write:

$$\models P$$

In Propositional Logic §

In propositional logic, a proposition is valid if:

• It is a tautology, that is, true under every valuation
• Equivalently, its corresponding truth table has a column of True’s

Theorems §

T1: In propositional logic, a formula is valid iff its negation is unsatisfiable

Under propositional logic, we have:

$$(\models F)\text{ iff }(\neg F\models \perp )$$

• The expression on the left hand side of the iff says that $F$ is valid by definition.
• The expression on the RHS of iff says that $\neg F$ logically entails $\perp$

In plain English, to show that $F$ is valid, we can equivalently show that any model that makes $F$ false is absurd.. (#tosee ???)