Propositional Logic : Why is p $\Rightarrow$ q $\equiv$ $\neg$p $\lor$ q

Question

Why is :

p $\Rightarrow$ q $\equiv$ $\neg$p $\lor$ q

Please Provide an intuitive explanation and not one using a Truth Table

Answer 1

In logic $p \Rightarrow q$ means that if $p$ is true, $q$ is also true. If $p$ is not true, $q$ can be either true or false. Therefore, if $p$ is false, $p \Rightarrow q$ is true, and hence the $\neg p$ part of the formula. Now, in the case that $p$ is true, $q$ must be, and hence the $q$ part of the formula. The formula is in effect saying, if $p \Rightarrow q$, then either $q$ is true, or $p$ had better be false.

Answer 2

Because the only way the first statement can be contradicted is for $p$ to be true, but(and) $q$ be false. The second statement is complement of that situation.

Answer 3

$p\Rightarrow q$ means that if $p$ is true, $q$ must also be true.

$\neg p \mbox{ or } q$ means that at least one of $\neg p$ and $q$ must be true, which means that if $\neg p$ is false (i.e., $p$ is true), then $q$ must be true.

Answer 4

Sorry, but a truth table really is probably the most intuitive approach.

Suppose I say to you:

If it is Tuesday, then I will go buy milk.

Consider 4 possibilities:

• It is Tuesday, and I'm going to buy milk
• It is Tuesday, and I'm not going to buy milk
• It is not Tuesday, and I'm going to buy milk
• It is not Tuesday and I'm not going to buy milk

In which of those 4 cases would I be a liar? There is only 1. The other 3 cases combined (using or, one of them must be true) result in the "or" expression in question.