Propositional logic does not allow for an ambiguous third value.

You may be used to math speaking the language of numbers, but when it comes to logic, many of math’s ideas get written out in words and statements. The most common type of statement you will see in logic is an if-then statement. If-then statements look like this: “if p, then q.” If-then statements are conditional statements that are made up of two parts. The first part is the “if” clause, which asserts an idea, making it the statement’s hypothesis. The second part, the “then” clause, acts like an answer to that hypothesis, forming a conclusion.

Here is an example of an if-then statement: "If it snows, then we won’t be able to drive to school." In this if-then statement, “it snows” is the hypothesis and “we won’t be able to drive to school” is the conclusion.

Forming the Inverse Statement

To create an inverse statement from the original conditional statement, you have to negate both sides. You can put the phrases in the negative often by using the word “not.” However, even though this is math, be careful to make sure that the sentence remains grammatically correct.

The original conditional statement: "If it snows, then we won’t be able to drive to school."

The inverse statement: "If it does not snow, then we will be able to drive to school."

The conclusion of the original statement was already in the negative. To negate that part of the statement, it is necessary to make it positive again. So, “won’t” for “will” was switched out in the inverse statement.

Forming the Converse in Math

To use the converse in math, switch the positions of the hypothesis and conclusion so that that conclusion comes first and the hypothesis comes after the comma.

The original conditional statement: "If it snows, then we won’t be able to drive to school."

The converse (math) statement: "If we won’t be able to drive to school, then it snows."

Though it seems like this statement does not make sense, rest assured that you have done the operation correctly. The use of words in place of numbers and functions in logic is just a demonstration, helping you understand the concepts before you move on to using the functions in a broader sense.

Forming the Contrapositive

To create the contrapositive conditional statement, start with the inverse statement, where you have negated both the hypothesis and the conclusion. Then, switch the two sides so that the conclusion comes first and the hypothesis comes after the comma.

The original conditional statement: "If it snows, then we won’t be able to drive to school."

The contrapositive: "If we will be able to drive to school, then it does not snow."