An Introduction to Critical Thinking and Creativity - always yours

SOME BASIC CONCEPTS 61
conclude that all general claims have exceptions. But the claim "All general claims
have exceptions" is actually inconsistent. It is itself a general claim, and if it were
true, it should also have an exception. But this implies that not all general claims
have exceptions. In other words, the claim cannot possibly be true and is therefore
inconsistent!
If we want to speak truly, we should avoid inconsistent statements. But sometimes
ordinary speakers use sentences that seem to be inconsistent, such as,"I am
happy and I am not happy." Why do people say things that cannot be true? One
answer is that these sentences have incomplete meaning. When we fully specify
their meaning, they are no longer inconsistent. For example, perhaps the speaker
is happy that she is getting married, but she is also not happy that her ex-boyfriend
showed up at the wedding. She is happy about one thing and not happy about a
different thing, so there is no real inconsistency.
7.1.2 Entailment
A set of statements P\...P n entails (or implies) a statement Q if and only if Q
follows logically from Pi... P n . In other words, if Pi... P n are all true, then Q must
also be true. For example, consider these statements:
P: A bomb exploded in London.
Q: Something exploded somewhere.
Here, P entails Q, but not the other way round. Just because there was an explosion
does not mean that a bomb was involved. Perhaps it was it was an egg
exploding in a microwave oven. When P entails Q, we say that Q is a logical consequence
of P. In symbolic notation, it is P => Q. Here are two important points
about entailment:
• A set of true statements cannot have false consequences.
• A set of false statements can have true consequences.
If we look at the example carefully, we can see that if P entails Q, and Q turns
out to be false, then we should conclude that P must also be false. This point is
worth remembering because we often decide that a hypothesis or a theory is false
because it entails something false. However, if P entails Q, and P is false, it does
not follow that Q is also false. A false theory can have true consequences, perhaps
as a lucky accident. Suppose someone believes that the Earth is shaped like a
banana. This false belief entails that the Earth is not like a pyramid, which is true.
This example tells us we should avoid arguments of the following kind:
Your theory entails Q.
Your theory is wrong.
Therefore, Q must be wrong.
Entailment is related to the logical strength of statements. If a statement P
entails another statement Q but not the other way round, then P is stronger than