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This is the truth table for P implies Q.

enter image description here

i get it,

  1. When P is True and Q is True then the truth value of this "implication" function is True.('implies' function is performed correctly, if a true statement is implying another true statement)
  2. When P is True and Q is False then the truth value of this "implication" function is False.('implies' function is performed incorrectly, if a true statement implies another statement that is False)
  3. When P is False and Q is False then the truth value of this "implication" function is True.('implies' function is performed correctly, if a false statement implies another statement that is False)

i don't get the case when P is False and Q is True, because that would mean that a false statement can imply a true statement.

example : 'if all sides of a triangle are equal then the triangle is an equilateral triangle.'

if 'all sides of a triangle are equal' is true and 'the triangle is equilateral' is true, then the implication is correct.

if 'all sides of a triangle are equal' is true and 'the triangle is equilateral' is false, then the implication is incorrect, because if all sides of a triangle are equal then it has to be an equilateral triangle.

if 'all sides of a triangle are equal' is false and 'the triangle is equilateral' is false, then the implication is correct.

if 'all sides of a triangle are equal' is false and 'the triangle is equilateral' is true, then the implication is incorrect, because if all sides of a triangle are not equal then implying that the triangle is an equilateral triangle is WRONG! but, according to the truth table the truth value of 'P implies Q' for this case should be true.

please clear my doubt.

Rodrigo de Azevedo
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Schefflera Arboricola
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    Careful about only using as examples implications which happen to be bi-implications. Consider instead an implication which is true only in one direction. For instance "If $x$ is a typical healthy horse, then $x$ has four legs." Now... consider the fact that many tables have four legs despite the fact that they are not horses, that birds have two legs, and so on... – JMoravitz Sep 09 '21 at 15:26
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    My answer is here https://math.stackexchange.com/a/48202/442 – GEdgar Sep 09 '21 at 15:28
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    @xij To directly address your example: the problem with your example is that you also have $Q$ implies $P$, which is not a consequence of $P$ implies $Q$. As another example, take $P$ to be "it is raining", and $Q$ to be "it is cloudy". It is true that $P$ implies $Q$: if it is raining, then there must be clouds from which the rain falls. It is consistent with this implication that there are cloudy days with no rain. That is, if $P$ is false and $Q$ is true, the implication is not contradicted. – Ben Grossmann Sep 09 '21 at 15:29
  • We are not talking about about whether the implications whether [P is true or false implies Q is true or false] are true. We are talking about the one implication: If ALL SIDES OF TRIANGLE ARE EQUAL $\implies$ TRIANGLE IS EQUILATERAL. If the sides of the triangle are not equal then the implication IS true. Because our hypothesis is certainly FALSE we can't say anything about what would happen if it were true... because it ISN"T true. And since we can't say what would happen if the world were different anything can be implied and any implication must, by default be true. – fleablood Sep 09 '21 at 15:47
  • "because if all sides of a triangle are not equal then implying that the triangle is an equilateral triangle is WRONG!" But that's not the implication that the truth table is saying is true. If we have a right triangle called FRED, then the implication "The three sides of FRED are not equal$\implies$ FRED is equilateral" is certainly wrong as you point out. But that is not implication the truth table is talking about. The truth table is talking about the implication: "The three sides of FRED are equal $\implies$ FRED is equilateral". And that implication is true. – fleablood Sep 09 '21 at 15:52
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    Consider "If JANE were a frog then she'd eat flies". Is that true of false? Now imagine that JANE is a rabbit that eats grass but never eats flies. Is it true or false? Is it false because JANE doesn't eat flies? No, the implication never said JANE does it flies. It just said IF JANE were a frog she'd eat flies. But JANE isn't a frog so "If JANE were a frog then she'd eat flies" is not contradicted by her not eating flies. (And if it's not contradicted and can't be confirmed the axioms of logic assume it is a true (not false) statement). – fleablood Sep 09 '21 at 16:00

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