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.
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 proven 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
| 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.