Information Technology Reference
In-Depth Information
Observe that in all these cases the thinking is the same. The pieces are put together
in exactly the same way, even though the sentences are about quite different things.
There could just as easily be a fourth example:
4.
The frumble is frimble or framble.
Nothing is frimble.
So: The frumble is framble.
Again, one does not need to know what
frimble
means to get the correct conclusion.
It does not really matter whether the subject is keys, Henry, Jill, or the frumble. What
does matter is the
form
of the sentences in terms of the other connecting words, and
the conclusion based on that form. For example, it would be wrong in the last case to
conclude that the frumble was frimble.
Logical entailment
Telling us what to conclude in such examples is the job of logic. A collection of
sentences
S
1
,
S
2
, ...,
S
n
logically entails
another sentence
S
if the truth of
S
is implicit
in the truth of the
S
i
sentences. In other words, no matter what certain terms (like
boojum
,
garden
,
framble
) in the
S
i
sentences really mean, if they are all true, then the
S
sentence is also true. So, in determining if a collection of sentences logically entails
another, it is not necessary to know what the terms in those sentences mean. (Certain
keywords in sentences, such as
and
, do have specific functions.)
So, for example, the sentence
The snark was a boojum.
logically entails
Something was a boojum.
Similarly, the sentences
My keys are in my coat pocket or on the fridge.
Nothing is in my coat pocket.
logically entail
My keys are on the fridge.
The fact that these symbols can be used in an uninterpreted way is what allows the
connection with computation.
