Lesson PROOF OF THE LAW OF CONTRA-POSITION

This Lesson (PROOF OF THE LAW OF CONTRA-POSITION) was created by by Theo(13342)

This proof uses the law of equivalence.

The law of equivalence states:

If two statements have the same truth values for all possible conditions in the truth table, then the statements are equivalent and can be used inter-changeably.

The law of contra-position involves the use of the implies statement.

The implies statement is symbolically shown as:

p -> q

This means that:

If p is true, then q must be true in order for the implies statement to be true.

This logic only applies if p is true.
If p is false, then q can be true or false without making the implies statement false because the necessary condition for proving the implies statement is false is that p is true.

The truth table for the implies statement is shown below:

          p    q    (p -> q)
          -    -    --------

          T    T       T
          T    F       F
          F    T       T
          F    F       T

If p is true and q is true, then the implies statement is true.
If p is true and q is false, then the implies statement is false.
If p is false, then the implies statement is true regardless of whether q is true or false.

The contra-positive of the implies statement is symbolically shown as:

~q -> ~p

The law of contra-position states:

(p -> q) IS EQUIVALENT (~q -> ~p)

What this means is:

If p implies q is true, then ~q implies ~p must also be true.
If p implies q is false, then ~q implies ~p must also be false.

The truth table for the contra-positive to the implies statement is shown below:

          p    q    (p -> q)   ~q   ~p   (~q -> ~p)
          -    -    --------   --   --   ----------

          T    T       T        F    F        T    
          T    F       F        T    F        F
          F    T       T        F    T        T
          F    F       T        T    T        T

If (p -> q) is to be equivalent to (~q -> ~p), then they both must have the same truth values for all possible conditions in the truth table.

Note that equivalence means that their truth values agree.
If both are true or if both are false, then the equivalence statement is true.
If one is true and the other is false, then the equivalence statement is false.

You can immediately see that the truth table values for both are identical under all possible conditions, but we will show that through the use of the equivalency statement.

The equivalence statement is symbolically shown as:

(p -> q) <-> (~q -> ~p)

The equivalence truth table for (p -> q) and (~q -> ~p) is shown below:

            p   q   ~q   ~p   (p -> q)   (~q -> ~p)   (p -> q) <-> (~q -> ~p)
            -   -   --   --   --------   ----------   -----------------------
            T   T    F    F       T           T                 T
            T   F    T    F       F           F                 T
            F   T    F    T       T           T                 T
            F   F    T    T       T           T                 T

Each row is a condition of the truth table.
There are four possible conditions to be tested.
This is because we are dealing with two variables with two possible truth values each.
The variables are p and q.
The possible conditions are:
p is true and q is true.
q is true and q is false.
p is false and q is true.
p is false and q is false.
There are no other possible conditions involving these two variables.

The truth table for the equivalency statement is true for all possible conditions.
This proves that the two statements of (p -> q) and (~q -> ~p) are equivalent.

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