There are many possible reasons, why a set is inconsistent.
Sometimes, the set's contradiction follows form an included proposition sense. For example, the sentence
"John's wife is not a married
woman"
is self-contradictory and cannot be true
for any interpretation. If we add this proposition to some consistent set, then
it becomes contradictory. It is inconsistent just since it would be impossible
for all formulas of the set to be true.
In another case, the set's contradiction follows from the
structure of the component propositions. Two mutually contradictory
propositions determine the easiest variant, e.g.
{'Adam is a student, 'Adam is not
a student.}
If we denote one of the propositions by the letter p, then the
other is ¬ p. Of course, only one of them can be true. This set of two
propositions is contradictory. Much less obvious is the contradiction of the
following set:
{'210 is an even
number', 210 = 1023'}.
If 210 is even number, then it cannot be equal to 1023
(this number is odd). On the other hand, if 210 is equal to 1023,
then it can not be even. Denote by p="210
is an even number", q="210=1023" as well as r="210
is an odd number". The propositions (q ⇒ r) and (r ⇒ ¬ p)
are true. If q was true, then ¬ p would also be true. In this case, the
proposition's contradiction is intermediate.
If we deal with a more than 2-element propositions set, then
finding out contradiction is not an easy task. It may happen that no pair of
propositions lead to contradiction but all together cannot be simultaneously
satisfied. As an example we may consider the following situation: for every
sets X, Y, Z the set of propositions
{X ⊂ Y, Y ⊂ Z, Z ⊂ U, U ⊂ X}.
is inconsistent, while no pair of set's elements is contradictory.
Notice, that the set's consistency property does not guarantee
that all its elements are true. It just assures that there exists such valuation
in which all formulas are simultaneously true, or all propositions are
satisfied under the same interpretation. If the set of propositions is
inconsistent, then we cannot find such a valuation.
Why is set's consistency very important? Well, using a
contradictory set we may conclude any proposition, according to the law (( α ⊥ ¬ α) → β). Thus, we know nothing. If some set is
inconsistent, then every set which includes it is also inconsistent. Assume
that we have some data base which stores contradictory data. Using this data
base we may reach false conclusions and this situation can have extremely
serious consequences.
Question 6.4.1 Let p, q and r denote
respectively: p ="Adam is going to buy a new car", q = "Adam is
going to sell his apartment" , r = "Adam is going to get
married". Assume that the following propositions are all true:
(p ⊦ q ⊦ r), ((p ⊥ ¬ r) → q),
((r ⊥ q) ⊦ ( ¬ r ⊥ ¬ q), (r → p).
What is Adam going to do? Is he going to buy a new car, sell his
apartment or get married?