Conflicting Definitions: THE TRUTH about Conditional Statements

Conflicting Definitions: The Truth about Conditional Statements: 

Conflicting Definitions: The Truth about Conditional Statements By Paul E. Mokrzecki

Contradictory Definitions: 

Contradictory Definitions Among the reputable and published literature on the topic, two nonequivalent definitions of conditional statements exist. One is right and one is wrong. So, which definition is the right definition?

Common Ground: The Basics: 

Common Ground: The Basics Let p and q be propositions. A conditional statement (also called a conditional , and an implication ) is a proposition that can be written in the form If p , then q . The proposition immediately following the word “if” is called the hypothesis . The proposition immediately following the word “then” is called the conclusion . In the above conditional, p is the hypothesis, and q is the conclusion.

Common Ground: The Basics: 

Common Ground: The Basics Conditional statements can be expressed in other ways. Among these ways include: p implies q . p only if q . p is sufficient for q . q is necessary for p . q whenever p . q unless ¬ p . q follows from p .

Common Ground: The Basics: 

Common Ground: The Basics As a proposition, a conditional statement can be either true or false. So when is a conditional statement true and when is it false? That’s where the definitions horrendously diverge.

Two Nonequivalent Definitions: 

Two Nonequivalent Definitions Definition A A conditional statement is true if and only if the conclusion is true every time the hypothesis is true. Definition B A conditional statement is true if and only if it is not the case that: the hypothesis is true and the conclusion is false.

It turns out Definition A is NOT EQUIVALENT to Definition B!: 

It turns out Definition A is NOT EQUIVALENT to Definition B! How so?

Consider the conditional statement If today is Monday, then 5 + 5 = 7.: 

Consider the conditional statement If today is Monday, then 5 + 5 = 7. Under Definition A The conditional is FALSE. It is not the case that every time today is Monday, that 5 + 5 = 7. In fact, 5 + 5 = 7 is never true. In other words, a counterexample exists: It could be Monday, but 5 + 5 = 7 isn’t true. Under Definition B The conditional is TRUE every day except Monday. On every day except Monday, both the hypothesis and the conclusion are false. Thus, the conditional is true. On Mondays, the hypothesis is true, but the conclusion is false. Thus, the conditional is false. Obviously there is a difference between Definition A and Definition B. They contradict each other about the truth of the given conditional statement.

So, which definition correctly captures the conditional statement?: 

So, which definition correctly captures the conditional statement? Definition A does. Why?

Why is Definition A correct?: 

Why is Definition A correct ? The truth of a conditional statement, as we intuitively know it, is determined by all possible cases . A conditional statement is intended to mean that the conclusion is true every time the hypothesis is true . It is a statement prescribing a property to all possible cases . It’s a proposition that is either true in all possible cases or false in all possible cases. Only Definition A allows for this. Definition B makes the conditional statement truth-functional. It makes the truth of a conditional statement determined merely by the truth values of its two component parts. It thus makes the conditional statement a proposition that is either true in a given case or false in a given case. This is not the everyday conditional statement we have come to know.

Definition B and Its Truth Table: 

Definition B and Its Truth Table Under Definition B, the truth of a conditional statement can be determined directly by a truth table. The truth table is shown at right. This is why the conditional statement under Definition B is truth-functional. Correct Definition A holds a non-truth-functional view of conditional statements. For Definition A, there is more to the truth of a conditional statement than just a truth table. p q If p, then q T T T T F F F T T F F T

Another Example: Let p be “I am reading” and let q be “I am reading a book.” Suppose both p and q are true. I am reading, and I am reading a book.: 

Another Example: Let p be “I am reading” and let q be “I am reading a book.” Suppose both p and q are true. I am reading, and I am reading a book. Under Definition A (Correct) The conditional “If p , then q ” is FALSE. A counterexample exists. I could be reading, but reading a news article on the Internet instead of a book. Under Definition B (Incorrect) The conditional “If p , then q ” is TRUE. According to the first row of the truth table for conditionals, the conditional is true. Both the hypothesis and the conclusion are true and therefore the conditional is true. The conditional is of the form “If T, then T,” which is T. Obviously there is a difference between Definitions A and B. Many sources, reputable textbooks included, fail to make this difference. BEWARE!

BEWARE!: 

BEWARE! Definition B is wrong. Definition A is right. Some textbooks, even reputable ones: improperly combine the two to create a giant ambiguity, or are just inconsistent about which definition to use throughout the book. Watch out for this flaw!