Discrete Mathematics

-------------------------SYMBOLIC

LOGIC

71

Columns 2 and 3 are the premises. Rows 1 and 3 are where the premises are all true. But in column

1,

in row

3,

we see the conclusion is false. This means the argument is invalid.

Intuitively, here is the reason. Recall that

(p

~

q) essentially means that

pis

the cause. Now,

if

q, the

effect, occurs, this does not mean that

p, the cause, caused it. For example, in the statement

"if

it rains, the

grass gets

wet", we can not conclude that the grass getting wet (the effect) is due to the rain. Someone could

have sprayed water from a hose (another cause). However, we do know that

if

it is not wet, then ...