Why does P -> Q hold true when P is false and Q is true in mathematical logic?

[Physicsman567]

When P -> Q, why is it true when P is false and Q is true, but why is it false when P is true and Q is false?

If I suppose P mean "Jon is a guy" and Q mean "Mary is a girl". When both P and Q are true it does make sense that this proposition is true because Jon is a guy and Mary is a girl, but when it come to the second and third agreement, where P is false but Q is true still make the proposition true, but the Proposition is a false when P is true and Q is false. I tried replacing P and Q with "Jon is a guy" and "Mary is a girl". And it doesn't make sense why do one of them is right and the another one is wrong?

I believe this is not a smart question, but if there's anyone forgo their time to explain this I would be really appreciate.
Thank you very much.

 

[Andrew Mason]

P -> Q means: If P is true then Q is true. So if P is true and Q is false, the statement is not correct (ie. P being true does not imply that Q is also true).

P->Q does not mean: If Q is true then P is true. It does not say what P is if Q is true.

AM

 

[Svein]

In propositional calculus, $P\Rightarrow Q$ is defined as $Q \vee (\neg P) = \neg (\neg Q \wedge P)$ (∧ = logical AND, ∨ = logical OR).

 

[Physicsman567]

Thank you very much Andrew Mason and Svein.

 

[Stephen Tashi]

It's possible to understand the why the truth table for $P \implies Q$ is defined the way it is by considering a more advanced situation where quantifiers are involved.

For example, consider the statement: For each x, if x is a boy then x has a mother. Which we can abbreviate by:

$\forall x$ For each x
$B(x)$ x is a boy
$M(x)$ x has a mother
$\forall x ( B(x) \implies M(x) )$ For each x, if x is a boy then x has a mother.

Statements such as $\forall x ( B(c) \implies M(x))$ are generalities. In mathematics, for a generality to be considered True, we demand that it is True without exceptions. (- not that it is True only 9 out of 10 times etc.)

Suppose someone wishes to disprove $\forall x ( B(x) \implies M(x))$ by presenting an exception. Suppose he says "Let $x =$ my coffee cup. My coffee cup does not have a mother." We do not consider $x =$ my coffee cup as a valid exception to the rule because a coffee cup is not a boy. The simplest way to prevent $x =$ my coffee cup from disproving the rule is to declare that $B(x) \implies M(x)$ is a true when $B(x)$ is false. This prevents irrelevant examples from being used to claim exceptions to a generality.

Let $S(x)$ denote "x wears suspenders". Consider the generality $\forall x ( B(x) \implies S(x))$. To show an exception to this generality we need to provide an example of a boy who does not wear suspenders. This is in accordance with the truth table for $B(x) \implies S(x)$ , which says the implication is False when $B(x)$ is True and $S(x)$ is False.

 

[Mark44]

When P -> Q, why is it true when P is false and Q is true, but why is it false when P is true and Q is false?

In addition to the other fine responses here, the truth value of the implication $P \Rightarrow Q$ is determined by the truth values of the statements P and Q, each of which can be either true or false.

The only pair of truth values for P and Q that makes the implication false is when P is true and Q is false. All other combinations of truth values for P and Q result in a value of true for the implication. Here is the truth table for the implication $P \Rightarrow Q$:

P    Q    P => Q
T    T        T
T    F        F
F    T        T
F    F        T

 

[Svein]

In propositional calculus, $P\Rightarrow Q$ is defined as $Q \vee (\neg P) = \neg (\neg Q \wedge P)$ (∧ = logical AND, ∨ = logical OR).

Using the fact that a double negation does not change the value of a logic variable, we can derive the following identity: $\neg Q\Rightarrow \neg P = \neg (\neg (\neg P) \wedge \neg Q) = \neg (P \wedge \neg Q)= \neg (\neg Q \wedge P)=P\Rightarrow Q$. This equivalence is called contraposition and is often used in mathematical proofs.

 

[Physicsman567]

Thank you all so much!