If you've proven B
from hypothesis A
, then
A | B |
----------
f | f | may be possible
f | t | may be possible
t | f | definitely impossible; A is true, then B must be too!
t | t | may be possible
Since the third line can't happen, we really shouldn't include it in our truth tables; that line can never appear when (correctly) assigning truth values to propositions.
Therefore, the truth table becomes
A | B | A → B
------------------
f | f | t
f | t | t
t | t | t
and we see that A→B is identically true. So if we know A⊨B
, we can infer ⊨A→B
.
Similarly, the relevant lines for the premise that both A
and A→B
are
A | B | A → B
------------------
f | f | t not this; we need A true
f | t | t not this; we need A true
t | f | f not this; we need A→B true
t | t | t possible
and so the relevant lines of the truth table are
A | B | A → B
------------------
t | t | t
so we see B
is identically true given the premise. We conclude that A,A→B⊨B
.