Validity of a logical argument

Statement

Decide whether the following argument is valid; if so give a proof, otherwise a counterexample.

• If the train is on time, then the station is open.
• If the train is not on time, then it is raining.
• If we are late, then the station is not open.
• Therefore: it is raining or we are not late.

Solution

Let $a$ = “the train is on time”, $b$ = “the station is open”, $c$ = “it is raining”, $d$ = “we are late”. The premises are $a \to b$, $\lnot a \to c$, $d \to \lnot b$; the conclusion is $c \lor \lnot d$.

The argument is valid. Suppose, for contradiction, the conclusion is false: then $c$ is false and $d$ is true.

• From $d$ true and $d \to \lnot b$ we get $\lnot b$, i.e. $b$ false.
• From $b$ false and $a \to b$ we get $\lnot a$, i.e. $a$ false.
• From $\lnot a$ true and $\lnot a \to c$ we get $c$ true.

But $c$ was false: contradiction. So the conclusion cannot be false, and the argument is valid.