## logical deductions

Sequences, counting (including probability), logic and truth tables, algorithms, number theory, set theory, etc.
lsand2525
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Joined: Sun Jun 29, 2014 2:09 am
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### logical deductions

I'm trying to understand why the following is not a sound inference rule:

Given that NOT P IMPLIES NOT Q, P implies Q.

However, this is a sound rule:

Given NOT P IMPLIES NOT Q, Q implies P.

Thanks.

nona.m.nona
Posts: 288
Joined: Sun Dec 14, 2008 11:07 pm
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### Re: logical deductions

Given that NOT P IMPLIES NOT Q, P implies Q.
What did you see when you did truth tables for each of "~P => ~Q" and "P => Q"? Did they have the same truth values, given the same inputs?

What real-life examples did you try out? For instance, one could say that P stood for "the street is wet" and Q stood for "it is raining now". Then assume that any time the street is dry (~P), it therefore also is true that it is not raining right now (~Q). Does that necessarily mean that, if the street is wet (P), it must be raining right now (Q)? Or might the neighbor's sprinkler be wetting the road, or maybe the fire department opened a hydrant for a seasonal test?

What happens if you apply this example to the other statement?