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<TITLE>The Definition of Implies</TITLE>
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<P><APPLET CODE="TruthTable.class" WIDTH=500 HEIGHT=550
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<PARAM name=introline value="Finally we consider the definition of
'implies'. Use the
truth table below to experimentally determine the truth value of 'p
implies q', given the
separate truth values of 'p' and 'q'. As a suggestion, ask yourself
whether the implication
should be true if 'p' is true and 'q' is true. Then consider the case
in which 'p' is true but
'q' is false. The remaining cases, those in which 'p' is false, you
will have to find by
experimentation. Give it a try!">
<PARAM name=proposition value="implies[p,q]">
<PARAM name=comment value="OK! You are now through the least
intuitive of the connectors.
Let's look at what it says. We can certainly agree that if 'p' is
true, then 'q' should also be true.
Therefore if 'q' is indeed true we will regard the implication as
accurate, i.e. as a true statement.
And by the same reasoning, if in this case we find that 'q' is false,
then the implication is not
accurate, i.e. it is false. Okay, now let's look at the less
intuitive side if the proposition,
namely, the case when 'p' is false. In this case the table above
shows that the implication is true
regardless ofthe truth value of 'q'. This might seem odd, but it
captures the fact that if 'p' is false,
then the implication can't be used to draw any conclusion, and there
is therefore no reason to regard
it as false. That is, we regard a statement as true if there is no
obvious reason to regard it as false.
Since there is no 'don't know' categary to truth values, this is the
alternative we are left with.">
You need a Java equipped browser to do view and use this interactive
problem!
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