Conditional Statements in Logic

The conditional operation is denoted by the arrow symbol →.
Q → E reads as “if Q then E.

Conditional operation example
Q → E is read as if Q, then E
Q = "If I sleep." the value of Q
E = "I lose track of time." the value of E
The conditional proposition, Q → E, is only false when Q = True and E = False. In all other variations, the conditional proposition is true.

In the examples above, Q is called the hypothesis, and E is called the conclusion. The only way for a conditional statement to be false is if the hypothesis is true and the conclusion is false. Wondering why that is the case? let’s dig deeper.

Evaluating the hypothesis and conclusion example
I sleep and I lose track of time = True because the hypothesis was correct
I sleep but I do not lose track of time = False because the hypothesis was incorrect
I don’t sleep and I lost track of time = True because the hypothesis was never tested
I don’t sleep and I don’t lose track of time = True because the hypothesis was never tested

It is important to fully grasp the idea of conditional propositions because, in the next section, we will introduce three additional types of conditional statements, the converse, contrapositive, and inverse statements.

The Converse, Contrapositive, and Inverse Logic Statements 

Converse logic

According to Wikipedia, “In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P.” As an adjective, converse means to reverse in order, such as conversely.

In symbol notation, the converse of P → Q is Q → P.

Contrapositive logic

In logic, a contrapositive is defined as a proposition that interchanges the hypothesis and conclusion. If a conditional proposition is “if G then H,” its contrapositive is “if not-H then not-G.” Notice how G and H are inversed and swapped. In symbol notation, the contrapositive of P → Q is ¬Q → ¬P. 

Inverse logic

In logic, the inverse statement takes the negation of the hypothesis and the conclusion of a conditional proposition, such as “if P then Q” becoming “if not-P then not-Q.” In symbol notation, the inverse of P → Q is ¬P → ¬Q.

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The Biconditional Operation in Logic

The biconditional operation is denoted Q R and yields a true value when both Q and R are true or false. The statement becomes false when Q and R have different truth values.

In English, the above biconditional statement is read as “Q if and only if R” and is abbreviated as “Q iff R.”

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Order of Operation for Compound Propositions With Conditional and Biconditional Operations

It is best practice to use parenthesis to indicate the order of operation. However, when parenthesis is not present, conditional → and biconditional operations are applied last.

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