# 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 *P* → *Q* 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

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

## Further Reading

There are many more such inference rules with traditional names. See https://en.wikipedia.org/wiki/Category:Rules_of_inference InferenceRules for more examples.

Just as the simple truth tables for individual connectives can be
combined to form truth tables for arbitrarily complex formulas,
applications of these inference rules can be combined to prove
arbitrarily complex logical arguments. Kalish & Montague refined work
done by Jaskowski and others, devising what have become standard
techniques for carrying out such proofs in their classic text,
*Logic: Techniques of Formal Reasoning*.