Conditional statement (logic)

In philosophy, logic, and mathematics, a conditional statement (also called strict conditional) 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 (also called antecedent). The proposition immediately following the word "then" is called the conclusion (also called 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 (also called 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 a logical implication in that the requirement for a logical implication that p and not q be logically inconsistent is excluded from the definition. Conditional statements are often symbolized using an arrow (→) as p → q (read "p implies 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 true. A conditional statement is false if and only if a counterexample to the conditional statement exists. A counterexample to a conditional statement exists if and only if there is a case in which the hypothesis is true, but the conclusion is false.

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 it is not the case 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 false or q is true (or both). Yet another way of describing the conditional is that it is equivalent to: "It is necessary that: not p, or q (or both)."

Examples of conditional statements include:

1. If I am running, then my legs are moving.
2. If a person makes lots of jokes, then the person is funny.
3. If the Sun is out, then it is midnight.
4. If you locked your car keys in your car, then 7 + 6 = 2.

Variations of the conditional statement

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

1. If p, then q. (called "if-then" form)
2. If p, q.
3. p implies q.
4. p only if q. (called "only-if" form)
5. p is sufficient for q.
6. A sufficient condition for q is p.
7. q if p.
8. q whenever p.
9. q when p.
10. q every time that p.
11. q is necessary for p.
12. A necessary condition for p is q.
13. q follows from p.
14. q unless ¬p.

The converse, inverse, contrapositive, and biconditional of a conditional statement

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

The converse

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

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

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

• $\lnot q\rightarrow \lnot p$

The biconditional

The biconditional of a conditional statement is the proposition produced out of the conjunction of the conditional statement and its converse. When written in its standard English form, the hypothsis and conclusion are joined by the words "if and only if." The biconditional of a conditional statement is equivalent to the conjunction of the conditional statement and its converse. The resulting proposition is as follows:

• $p\leftrightarrow q$; or equivalently,
• $(p\rightarrow q)\land (q\rightarrow p)$

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," distinguishing it from the material conditional. The following are some logical equivalencies to the conditional statement "If p, then q":

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

Distinction between conditional statements, material conditionals, and logical implications

The terms "conditional statement," "material conditional," and "logical implication" are often used interchangeably. Since, in logic, these terms have nonequivalent definitions, using them interchangeably often creates strong ambiguities. In fact, these ambiguities are so deeply rooted that they have generated a very popular misconception that the terms "conditional statement," "material conditional," and "material implication" all mean the same thing. This is far from true; the three terms are not all equivalent. Only the latter two have equivalent meanings.

The truth table

Conditional statements and material implications 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 conditional statement vs. the material implication

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. Clarence Irving Lewis (C.I. Lewis) was the first to develop modern modal logic in order to express the general conditional statement properly.

The conditional statement vs. the logical implication

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.

Example

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":

(1) p → q

The conditional statement expressed by (1) 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 (1) 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 (1) is false: Γ = {"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 considering only their abstract logical form; their logical form being p and not q. As can be seen, the same syntactic statement p → q can have different truth values, depending on whether the statement is expressing a conditional statement, a material implication, or a logical implication. Therefore, there is a fundamental difference between a conditional statement, a material implication, and a logical implication.

C.I. Lewis and his unprecedented approach to conditional statements

C.I. Lewis took a slightly different approach to expressing conditional statements. C.I. Lewis was interested in creating a symbolic version of the conditional statement. He knew there was a difference between conditional statements and material implications, and he wanted to make that difference clear. Instead of using the arrow ($\rightarrow$) for both conditional statements and material implications (like many people do today), C.I. Lewis decided to devise a series of logical systems in which use of the arrow was restricted to only material implications. He used a different symbol to represent conditional statements, a hook ($\prec$). C.I. Lewis' unambiguous approach to setting apart conditional statements from material implications makes the difference between the two clear and straightforward. His approach, however, has not found its way into mainstream popular literature very well, as many authors fail to make any syntactic difference between the two nonequivalent concepts.

As stated earlier, a conditional statement is a necessary version of its corresponding material implication. Using C.I. Lewis' notation, this gives us the following useful equivalency:

• $p\prec q\equiv \Box (p\rightarrow q)$

This equivalency clearly shows the difference between the conditional statement "If p, then q," and the material implication "p → q."

References

• Hardegree, Gary. Introduction to Modal Logic. UMass Amherst Department of Philosophy, 2009. Web. 18 December 2011 <http://people.umass.edu/gmhwww/511/text.htm>
• Larson, Boswell, et al. Geometry. McDougal Littell, 2007. Print.
• Rosen, Kenneth H. Discrete Mathematics and Its Applications, Sixth Edition. McGraw-Hill, 2007. Print.
• Hardegree, Gary. Symbolic Logic: A First Course (2nd Edition). UMass Amherst Department of Philosophy, 1994. Web. 18 December 2011 <http://courses.umass.edu/phil110-gmh/text.htm>
• Larson, Boswell, et al. Geometry. McDougal Littell, 2001. Print.
• Barwise, Jon, and John Etchemendy. Language, Proof and Logic. CSLI (Center for the Study of Language and Information) Publications, 2008. Print.

See also

• Material implication
• Logical implication
• Strict conditional
• Modal logic
• C.I. Lewis
• Counterfactual conditional
• Indicative conditional
• Propositional logic