User:Hanlon1755/Conditional Statements

In philosophy and logic, a conditional statement is a proposition of the form "If p, then q," where p and q are propositions. The proposition immediately following the word "if" is called the hypothesis (or antecedent). The proposition immediately following the word "then" is called the conclusion (or consequence). In the aforementioned form for conditional statements, p is the hypothesis and q is the conclusion. A conditional statement is often called simply a conditional (or an implication). A conditional statement is not the same as a material conditional in that a conditional statement is not necessarily truth-functional, while a material conditional is always truth-functional. Neither is a conditional statement the same as logical implication in that the requirement that p and not q be logically inconsistent is excluded from the definition. Conditional statements are often symbolized using an arrow (→) as p → q. The conditional statement in symbolic form is as follows:

• $p\rightarrow q$;

As a proposition, a conditional statement is either true or false. A conditional statement is true if and only if the conclusion is true in every case that the hypothesis is also true. A conditional statement is false if and only if a counterexample to the conditional statement exists.

A conditional statement p → q is logically equivalent to the modal claim "It is necessary that it is not the case that: p and not q." The conditional p → q is false if and only if it is not necessary that: both p is true and q is false. In other words, p → q is true if and only if it is necessary that: p is true and q is false (or both). Yet another way of describing the conditional is that it is equivalent to: "It is necessary that: not p or q."

Examples of conditional statements include:

• If I am running, then my legs are moving.
• If a person makes lots of jokes, then the person is funny.
• If the Sun is out, then it is midnight.
• If the Moon is out, then 7 + 6 = 2.

Variations of the Conditional Statement

The conditional statement "If p, then q" can be expressed in many other ways; among these ways include:

• If p, q.
• p implies q.
• p only if q. (also called "only if" form)
• p is sufficient for q.
• A sufficient condition for q is p.
• q if p.
• q whenever p.
• q when p.
• q every time that p.
• q is necessary for p.
• A necessary condition for p is q.
• q follows from p.
• q unless not p.

The Converse, Inverse, and Contrapositive of a Conditional Statement

The conditional statement "If p, then q" is related to several other conditional statements involving propositions p and q.

The converse of a conditional statement is the conditional statement produced when the hypothesis and conclusion are interchanged with each other. The resulting conditional is as follows:

• $q\rightarrow p$

The inverse of a conditional statement is the conditional statement produced when both the hypothesis and the conclusion are negated. The resulting conditional is as follows:

• $\lnot p\rightarrow \lnot q$

The contrapositive of a conditional statement is the conditional statement produced with the hypothesis and conclusion are interchanged with each other and then both negated. The resulting conditional is as follows:

• $\lnot q\rightarrow \lnot p$

Logical Equivalencies of the Conditional Statement

The conditional statement is a modal claim, and as such it requires the use of modal operators. Namely, it requires the use of the necessary operator □. A conditional statement is sometimes called a "strict conditional," to distinguish it from the material conditional. The following are some logical equivalencies to the conditional statement "If p, then q":

• $p\rightarrow q\equiv \Box \lnot (p\land \lnot q)$
• $p\rightarrow q\equiv \Box (\lnot p\lor q)$
• $p\rightarrow q\equiv \lnot q\rightarrow \lnot p$; The contrapositive of a conditional statement is equivalent to the conditional statement itself.
• $q\rightarrow p\equiv \lnot p\rightarrow \lnot q$; The converse of a conditional statement is equivalent to the inverse of the conditional statement.

Distinction Between Conditional Statements, Material Conditionals, and Logical Implication

The terms "conditional statement," "material conditional," and "logical implication" are often used interchangeably. Since, in logic, these terms have different definitions, using them interchangeably often creates strong ambiguities.

Conditional statements, material conditionals, and logical implications all are associated with the same truth table, given below. How exactly each is related to this truth table, however, is different.

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

The difference between a conditional statement p → q and a material implication p → q is that a conditional statement need not be truth-functional. While the truth of a material implication is determined directly by the truth table, the truth of a conditional statement is not. The truth of a conditonal statement cannot in general be determined merely through classical logic. The conditional statement is a modal claim, and as such it requires the use of the branch of logic known as modal logic. A conditional statement p → q is equivalent to "It is necessary that it is not the case that: p and not q. A material implication, on the other hand, is equivalent to "It is not the case that: p and not q. Note the lack of the clause "It is necessary that" in the latter equivalency. In general, a conditional statement is a necessary version of its corresponding material implication. C.I. Lewis was the first to develop modal logic in order to express the general conditional statement properly.

The difference between a conditional statement p → q and a logical implication p → q is that a conditonal statement need not have a valid logical form. Once again, a conditional statement is a modal claim equivalent to "It is necessary that it is not the case that: p and not q. A logical implication, on the other hand, is equivalent to "p and not q are logically inconsistent," which would be due to their abstract logical form. This requirement does not exist for conditional statements.

To show clearly the difference between the conditional statement p → q, the material implication p → q, and the logical implication p → q, consider the following ambiguous statement in which hypothesis p is "Today is Tuesday," and conclusion q is "5 + 5 = 4":

• If today is Tuesday, then 5 + 5 = 4.

The conditional statement expressed by this statement is false: a counterexample exists. It can be Tuesday, but 5 + 5 still does not equal 4. In fact, 5 + 5 never equals 4. The material implication expressed by this statement is true every day execpt Tuesday. This is because on every day except Tuesday, both the hypothesis and the conclusion are false, hence the material implication is true. This corresponds to the last row on the truth table for material implications. On Tuesday, however, the hypothesis is true, but the conclusion is false, hence the material implication is false. This corresponds to the second row on the truth table for material implications. The logical implication expressed by this statement is false: "Today is Tuesday" does not entail "5 + 5 = 4," since "Today is Tuesday" and "5 + 5 ≠ 4" are not logically inconsistent. Both of the former statements could (in theory) be true when only considering their abstract logical form. Their logical form being p and not q. As can be seen, the same syntactic statement can have different truth values, depending on whether the statement is expressing a conditional statement, a material implication, or a logical implication.

Related Articles

• Material Implication
• Logical Implication
• Strict Conditional
• Modal Logic
• C.I. Lewis
• Counterfactual Conditional
• Indicative Conditional
• Propositional Logic
• Validity