(P 
Q).
To prove that this implication holds, let us first construct a truth table for the proposition P
Q.
For example consider the first implication "addition": P
(P Q).
To prove that this implication holds, let us first construct
a truth table for the proposition P Q.
| P | Q | (P Q) |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
Then by the definition of 
, we
can add a column for P
(P
Q)
to obtain the following truth table.
| P | Q | (P Q) |
P (P Q) |
|---|---|---|---|
| F | F | F | T |
| F | T | T | T |
| T | F | T | T |
| T | T | T | T |
The first row in the rightmost column results since P is false,
and the others in that column follow since (P
Q) is true.
The rightmost column shows that
P
(P
Q)
is always true.
2. Some of the implications can also be proven by using identities and implications
that have already been proven.
For example suppose that
the identity
"exportation":
[(X 
Y) 
Z]
[X (Y Z)] ,
and
the implication
"hypothetical syllogism":
[(P Q)
(Q R)]
(P R)
have been proven. Then the implication No. 7:
(P Q)
[(Q R)
(P R)]
can be proven by applying the "exportation" to the "hypothetical syllogism" as
follows:
Consider (P Q) , (Q R) ,
and
(P R) in the "hypothetical syllogism" as
X, Y and Z of
the "exportation", respectively.
Then since [ (X Y )
Z ]
[ X ( Y Z ) ]
implies [ ( X Y )
Z ]
[ X ( Y
Z ) ] ,
the implication of No. 7 follows.
Similarly the modus ponens (implication No. 3) can be proven as follows:
Noting that
( P
Q )
( 
P
Q ) ,
P
( P
Q )
P
( P
Q )
( P
P )
( P
Q ) --- by the distributive law
F
( P
Q )
( P
Q )
Q
Also the exportation (identity No. 20),
( P ( Q R ) )
( P Q )
R )
can be proven using identities as follows:
( P ( Q R ) )
P
( Q R )
P
( Q
R )
( P
Q )
R
( P
Q )
R
( P
Q )
R
3. Some of them can be proven by noting that a proposition in an implication
can be replaced by an equivalent
proposition without affecting its value.
For example by substituting
( Q
P )
for ( P Q ) ,
since they are equivalent being contrapositive to each other, modus tollens
(the implication No. 4):
[ ( P Q )
Q ]
P ,
reduces to the modus ponens:
[ X ( X Y ) ]
Y.
Hence if the modus ponens and the "contrapositive" in the "Identities" have been proven,
then the modus tollens follows from them.