Inference Rules

Inference Rules

We have already seen an example of a logical argument 1 expressed as formulas of the sentential calculus. The proof that Q follows from P and PQ relies on the inference rule traditionally known as modus ponens. Modus ponens is one of a number of such rules that owe their names to the history of syllogisms and similar attempts to turn informal critical reasoning into a more rigorous discipline.

Like truth tables, such inference rules correspond to axioms and theorems of symbolic logic. Here are a few examples:

Modus Ponens

$$\begin{array}{l} \Phi \rightarrow \Psi \\ \Phi \\ \hline \\ \Psi \end{array}$$

Modus Tollens

$$\begin{array}{l} \Phi \rightarrow \Psi \\ \neg \Psi \\ \hline \\ \neg \Phi \end{array}$$

Modus Tollendo Ponens

$$\begin{array}{l} \Phi \vee \Psi \\ \neg \Phi \\ \hline \\ \Psi \end{array}$$

Modus Ponendo Tollens

$$\begin{array}{l} \neg (\Phi \wedge \Psi) \\ \Phi \\ \hline \\ \neg \Psi \end{array}$$