calculus book - Mathematics and Computer Science

• If 1 = 0, then 1 2 = 0. (Invalid)
• If 1 = 0, then 1 2 = 0. (Valid)
• If 1 = 0, then 1 2 = 0. (Valid)
1.3. LOGIC 11
If a hypothesis and conclusion are related by valid implication, then
the hypothesis is said to imply the conclusion. In this view, it is valid
(not logically erroneous) to deduce a conclusion from a false hypothesis:
If we start with truths and make valid deductions, we obtain only
truths, not falsehoods. An implication with false hypothesis is said to
be vacuous. To emphasize, validity has the possibly counterintuitive
property that if the hypothesis is false, then every conclusion follows
by valid implication. As strange as this convention seems, it does not
allow us to deduce falsehoods from truths. Logical validity is central to
the concept of “proof,” and is therefore crucial to the rest of the book
(and to mathematics in general).
The term “imply” has a very different meaning in logic than in ordinary
English. In English, to “imply” is to “hint” or “suggest” or “insinuate.”
In mathematics, if a hypothesis implies a conclusion, then the
truth of the conclusion is an ironclad certainty provided the hypothesis
is true. The term “valid” also has a precise meaning which is not exactly
the same as in ordinary English. Finally, note that every statement has
a truth value, but only an if-then statement can be valid or invalid.
Interesting logical implications usually depend on variables, and the
truth value of the implication therefore depends upon the truth values
of the hypothesis and conclusion. The concept of logical validity comes
into its own when an implication depends on variables. The following
examples illustrate various combinations of truth and falsehood in
hypothesis and conclusion:
• If n is an integer and if n is even, then n/2 is an integer. (Valid)
• If n is an integer, then n/2 is an integer. (Invalid)
• If n is an integer and n = n + 1, then 2 + 2 = 5. (Valid)
• If n is an integer and n = n + 1, then 2 + 2 = 4. (Valid)
The distinction between “truth” (which applies to statements) and
“validity” (which applies to logical implications) may at first seem a
bit fussy. However, it is important to be aware that the concepts are
different, though they are not wholly unrelated, either; when a logical