Mind your Game

The truth function of the material conditional $P \rightarrow Q$ is logically equivalent to $\lnot P \lor Q$ , and it will only give a false value when $P$ is true, and $Q$ is false. The given function provides a false $P$ and a true $Q$ , telling us that this whole conditional is true.

To put into perspective why it works this way, let us assign some statements to $P$ and $Q$ .

$P$ = I eat a lot of unhealthy chips.

$Q$ = I get fat.

So, we are trying to claim that, "If I eat a lot of unhealthy chips, I get fat."

There are now four possibilities:

• I ate a lot of chips, and I got fat. ( $P$ true, $Q$ true)

Obviously then, if this were the case, the claim was true. ( $P \rightarrow Q$ is true).

• I didn't eat a lot of chips, and I didn't get fat. ( $P$ false, $Q$ false)

Now, the claim remains to be true, as it was never promised that I will get fat if I didn't eat a lot of chips. ( $P \rightarrow Q$ is true).

• I ate a lot of chips, and I didn't get fat. ( $P$ true, $Q$ false)

Now here, since the claim above states otherwise (I should've gotten fat), then we render the claim as false. ( $P \rightarrow Q$ is false).

• I didn't ate a lot of chips, and still I got fat ( $P$ false, $Q$ true )

Although the fact that I got fat isn't from eating chips, the claim didn't promise that I wouldn't get fat if I didn't eat a lot of chips. ( $P \rightarrow Q$ is true).

From here we can see that only a false conclusion derived from a true antecedent renders a claim false.

Read Carefully.

Let's store 6 in a variable a.

Our statement becomes:

If $a<3$ then $a<7$ .

You tell me whether this is true.

By definition "If false then true" is true