Rules of Inference

Rules of Inference are Logical Arguments that are valid. Rules of Inference can be chained together to prove more complex logical arguments.

Modus Ponens

if $p$ then $q$ $p$ $\bullet q$

Modus Tollens

if $p$ then $q$ $\sim q$ $\bullet \sim p$

Generalisation

$p$ $\bullet p\vee q$

The conclusion is a more general form of the premise: Anton is a junior $\bullet$ (More generally,) Anton is a junior or Anton is a senior

Specialisation

$p\wedge q$ $\bullet p$

Discard extraneous information to concentrate on the property of interest Ana knows graph algorithms and Ana knows numerical analysis $\bullet$ (In particular,) Ana knows graph algorithms

Elimination

$p\vee q$ $\sim q$ $\bullet p$

When there are two possibilities and one can be ruled out, the other must be the case

Transitivity

$p\to q$ $q\to r$ $\bullet p\to r$

Proof by Division into Cases

$p\vee q$ $p\to r$ $q\to r$ $\bullet r$

Suppose one thing or another is true If in both cases, a certain conclusion follows, this conclusion must also be true

Contradiction

$\sim p\to false$ $\bullet p$

Conjunction Premises

$p$ $q$ $\bullet p\wedge q$