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1.) First of all if P is false and Q is true then result is false. Why this?

2.) If Juan has a smartphone, then 2+3=5.

If Juan has a smartphone then both propositions true and answer is true.

If Juan does not have a smartphone then first proposition is false and second is true and answer is false.

How false does not implies truth?

I want to explain this to person for which this is new topic. So very basic explanation can help.

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There are 4 possibilities: TT, TF, FT and FF, for P and Q, respectively. .. Perhaps a little counter-intuitively, 3 of the 4 give a true conditional (as $P\rightarrow Q$, or if P then Q, is called ). The combination that's considered false is TF.

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1) I assume that you are talking about a statement of the form -If P the Q . If P is TRUE AND Q IS FALSE THEN THE STATEMENT IS FALSE . In all other cases the statement is TRUE . So you are in error here . 2) No , If Juan does not have a smart phone then any Q makes the conditional true even 2+3=7

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