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I'm really confused with conditional statements.

For example, given $P(x) \to Q(x)$. Actually it's equivalent to $\lnot P(x) \lor Q(x)$, right? It's easy to understand its true value - only when $P(x)$ is true and $Q(x)$ is false, the statement is false, otherwise it's true.

However, it's hard to understand in concept. Why when $P(x)$ is false, the statement is unconditionally true? How to understand 'true' here?

Given a more detailed example: $P(x)$ is 'x is man' and $Q(x)$ is 'x is rich'. When x is not a man, the value is true, but for what? What is true here?

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  • $\begingroup$ You can see also this post. $\endgroup$ – Mauro ALLEGRANZA Feb 12 '15 at 20:51
  • $\begingroup$ The equivalence $\neg [ P(x) \land \neg Q(x)]$ is more intuitive. I don't know why they keep using the disjunctive form. $\endgroup$ – Dan Christensen Feb 13 '15 at 4:04
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The material implication is considered to hold if and only if it guarantees that the consequent is true whenever the antecedent is.

It makes no claim on the consequent's truth-hood when the antecedent is false.   So if the antecedent is false the implication is said to be true whatever the consequent is.

So if the implication holds, then we know that either the antecedent is false or else the consequent must be true.

$$A\to C \iff \neg A\vee C$$

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Let's take your example : let $P(X):X$ is a man and $Q(X):X$ is rich.

Now let's take a rich dragon, $X_0$. We have that $P(X_0)$ is false and $Q(X_0)$ is true.

What we are saying is that $P(X_0)\Rightarrow\text{ anything}$. That is because $P(X_0)$ is false, and therefore supposing that it is true leads to any conclusion you want.

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