Introduction to Logic

Conditionals and Biconditionals


A statement such as:"If I am rich, I will go to Hawai for a trip" is very common in plain english. Infact, this statement can not be obtained from the previous page. Indeed, this statement may be written as: "If P then Q", where

P is "I am rich"

and Q is "I will go to Hawai for a trip".

This is not a conjunction or a disjunction of propositions. Therefore we have a new logical object.


Definition. Let P and Q be two propositions. The conditional sentence "If P, then Q", which will be written by (read P implies Q), is a proposition which is true whenever P is false or Q is true.
P is called the antecedent and Q the consequent.


The truth table of is




Example. Check that the propositional forms , , and are equivalent.


Answer. Let us answer this through the following truth table




It is clear from the above truth table that the last three propositions are equivalent.


For example, if you want to prove the following statement :


If is even, then n is even
where n is natural number, it is equivalent to prove:
If n is odd, then is odd
which is very easy to prove. Note that the negation of "to be even" is "to be odd".


Definition. Consider the conditional proposition . The conditional proposition is called its converse and its contrapositive the conditional proposition .


Putting the above results together we get the following theorem.


Theorem . Any conditional proposition is equivalent to its contrapositive .


This is a very important result. It is commonly known in mathematics as "proof by contraposition".


The conditional proposition makes it possible to consider the following new notion.


Definition. Let P and Q be two propositions. The biconditional sentence "P if and only if Q", which will be written by , is a proposition which is true whenever P and Q have the same truth values.


The truth table of is




Example. Show that is equivalent to .


Answer. We will use the truth table. We have




It is clear from the above truth table that the last two propositions are equivalent.


Problem. Check the following

1.

is equivalent to ;

2.

is equivalent to ;

3.

is equivalent to ;

4.

is equivalent to .

Note that the above results give the rules of distribution of logical terms over other logical terms. The rules (1) and (2) are also referred to as De Morgan's Laws.


Section
Mohamed Khamsi
Wed Aug 27 17:17:43 MDT 1997