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Sample sentences
AND operator
IF/THEN operator
NOT operator
OR operator
XOR operator


Chain Rule
Conjunctive Addition
DeMorgan's Law
Disjunctive Addition
Disjunctive Inference
Disjunctive Infer. (XOR)
Double Negation
Modus Ponens
Modus Tollens
Mutual Exclusion


5-step or more
Bad Argument

IF/THEN Operator

The logical operator IF/THEN (symbolized ->) is called the conditional.

The conditional links two statements in the following way: If the first statement is true, then the second statement is true as well. Here is an example:

If you studied for the exam, you are not going to fail.

In formal logic, we can replace the two statements with the letters p and q (we can use any other letters that we want to). So, let the statement "If you studied for the exam" be p and let the statement "You are not going to fail" be q.

Now we can rewrite our example in the formal way:

p -> q: "If you studied for the exam, you are not going to fail."

This is a logical expression. In this expression, p is called the antecedent and q is called the consequent.


Truth table for IF/THEN

A truth table lists truth values (that is, either true or false) for every possible combination of component values in a logical expression.

In order to build a truth table for IF/THEN, we need to consider the simplest example of a conjunction:

p -> q: "If you studied for the exam, you are not going to fail."

In formal logic, p and q (called the antecedent and the consequent, respectively) can take on two possible values: true or false. How many combinations of p and q can we come up with? The answer is four. Let us list all the possible combinations:

1. p is true and q is true
2. p is true and q is false
3. p is false and q is true
4. p is false and q is false

The conditional is true when the antecedent and the consequent are both true. The conditional is also true when the antecedent is false -- in this case, the consequent does not matter. However, the conditional is false when the antecedent is true and the consequent is false. Now we can build a truth table. We are using T for true and F for false.

p -> q

As you read the table, you will notice that when p is true and q is true, the entire expression is true. When p is true and q is false, the entire expression is false, and so on.


Necessary versus sufficient

The concepts "necessary" and "sufficient" will be useful in clarifying the meaning of the conditional. Consider this example.

p -> q: "If it is noon, then it is daytime."

Is this conditional asserting that noon is a necessary condition for daytime? No. We know that it is also daytime at 11:15 AM and at 3 PM. The conditional is asserting that noon is a sufficient condition for daytime. In general, the antecedent is sufficient but not necessary for the consequent.

Another example:

x -> y: "If you have $100, then you can go to the movies."

To go to the movies, $100 is sufficient, but not necessary.

However, the consequent is a necessary condition for the antecedent. Consider this example:

r -> c: "If it is raining, then there are clouds."

As we discussed before, the antecedent is sufficient but not necessary: Rain is sufficient for there to be clouds, but is not necessary -- there can be clouds without rain. On the other hand, clouds are necessary for rain: If there are no clouds, there will be no rain.


Rules of Conditional

These are links to specific rules pertinent to the IF/THEN operator.

Modus Ponens
Modus Tollens
Chain Rule