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A Language Change
Chapter 8, Part I
Previous: Chapter 7, Longer Chains
of Reason and Mathematical
Induction
Implication rules can be stated in several ways. We need to recognize them.
In the chapter Implication
Rules, we met the rule
When Aunt Jane visits her nephew Tom's home, Tom goes out to play
Rules like this can be said in different ways. This gives variety and choice
in the way in which we write rules. The form of a rule does not matter, if we
understand exactly what it says. The above one-way rule can also be rewritten
(or restated, again without changing its meaning) using the words IF and THEN as
follows.
IF Aunt Jane visits her nephew Tom's home THEN Tom goes out to play.
The word IF introduces a condition, namely Aunt Jane's visit to her nephew
Tom's home. The word THEN introduces the consequence, what should occur,
when the condition is satisfied. Here the consequence is Tom goes out to
play. Since the original rule can be rewritten in the IF condition THEN
consequence form, we say the original rule and the if-then form are conditional
statements.
Note that a statement If A then B is only false when the situation
or condition A occurs, but the anticipated consequence B does
not.
Another way of writing the above one-way Aunt Jane and nephew Tom rule (with
no change in meaning) is given by:
Aunt Jane's visit to her nephew Tom's home IMPLIES Tom goes out to play.
The words forces or makes may be used instead of the word
implies. We could also use the word suggests, but in everyday use,
a suggestion is optionally obeyed or followed while a rule (when it is correct)
should or must be obeyed or followed. In talking about rules, we use the
words implies, forces or makes for those
rules we expect will be obeyed, or more precisely will never be disobeyed in the
circumstances at hand. The explicit identification of such circumstances is
exhaustive unless the circumstances in question are understood from a context,
an obvious one, we hope.
Postscript (Not in Printed Version)
Instead of writing If A then B we may write B if A.
The latter states that the situation B will happen if the situation
A happens. That being said we cannot say that
B if and only if A
holds when there is a third situation C different from A, a situation
which may occur when A does not, such that B if C also
holds.
In the case
B if A
and also
B if C
the situation B may occur because of situation A or situation C, that
is, due to A OR C. So when situation B occurs, the occurrence may be
implied by A, C or another situation.
However, we can assert or state B if and only if A holds when B
follows from the occurrence of A and whenever B occurs, so must A. |
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Chapter Subsections: [ 8. Equivalent Conditions ] [ 8, A Language Change ]
Next: Stating and Writing Two-Way
Implications
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