Semantics of Propositional Logic

The semantics are a model $\mathcal{M}$ consisting of a Universe, which in this case are the two formulas $T$ and $F$, and an Interpretation Function which describes the meaning of the connectives.

$\mathcal{M}=<\mathcal{U},\mathcal{I}>$

Both the Interpretation Function and the Variable Assignment function are used to define the Value Function which assigns values from the Universe to formulas.

Using the above model we can compute the semantics for a given formula. This however is quite a mess… (see slides for example). A simpler form to compute the semantics by hand would be something like a truth table.

Semantic properties

Some important properies of formulas are:

• true under Variable Assignment
• false under Variable Assignment
• satisfiable if there exists a Variable Assignment making the formula true
• valid if the model entails the formular for all assignments
• falsifiable if there exists a Variable Assignment making the formula false
• unsatisfiable if all assignments make the formula false