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. |
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| 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. |
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| 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. |
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