MATH 3265 Class Notes 02-19-2025

notepropositional-logic

Back to the First Day of the Semester

If it rains ($p$), either Reed remembers his umbrella ($q$) or he gets wet ($r$). It rains ($p$). Reed forgets his umbrella ($\neg q$). Therefore we can infer that he gets wet ($r$).

This gives us that,

$$\begin{array}{c}p\to (q\vee \neg r)\\ p\\ \neg q\\ \\ \therefore r\end{array}$$

Claim

Any valuation $V$ that satisfies $p\to (q\vee \neg r)$, $p$, and $\neg q$, must also satisfy $r$.

We can check this with a truth table.

Defined Semantic Consequence

Fact

Suppose $\Gamma$ is not satisfiable. Then $\Gamma \models \phi$ for every $\phi$.

This is because $\Gamma \models \phi$ means that for every valuation in which $\Gamma$ is satisfied, $\phi$ is also satisfied, which trivially holds when there are no valuations in which $\Gamma$ is satisfied.

Fact

Suppose $\Gamma \models \phi$ and $\Gamma \subseteq \Delta$. Then $\Delta \models \phi$.

Question

Do we always have $\Gamma \models \phi$ or $\Gamma \models \neg \phi$.