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LOGIC MAIN PAGE

LOGICAL OPERATORS

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

RULES OF LOGIC

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

VALIDITY PROOFS

2-step
3-step
4-step
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
p -> q
T
T
T
T
F
F
F
T
T
F
F
T

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
Contrapositive