## Inconsistency and Proof

A set of axioms is inconsistent if both p and not-p can be proved from the axioms. It is said that if a set of axioms is inconsistent, then anything is provable from that set of axioms. At first blush this may not be immediately obvious to you. But a simple example should help. We repeat some of the logical implications in the figure below since these will be used in our example.

The figure below shows a proof of r (an arbitrary proposition). Here we are given the single premise (1) of p and not p and we will see if r can be proved from this premise alone. The lines numbered 3 through 6 show the steps of the proof. Note that although r is a simple proposition, we could substitute any complex proposition and the same pattern of proof could be used.

The rules of inference used in this proof are intuitively reasonable. It is hard to imagine any argument against their validity. For example, the rule of inference referred to as simplification simply takes a conjunctive expression that is true and allows one to assert either of the components as true. Addition allows one to take a true expression and add a disjunct to that expression. Given the truth table for disjunction, this clearly yields an expression that is True. The only slightly involved rule of inference is disjunctive syllogism. The intuition behind this rule is actually quite straighforward. That is, if we have a disjunctive statement of two expressions that is true, and we know that one of the expressions is false, then the reamining expression must be true. As you can see, this is the point where holding that a contradition is true gets us into trouble.