Department of Philosophy (PHI)

Conditional Statements

Many words and phrases can be used to assemble compound statements from simpler statements. In logic such terms are called connectives, and the statements they join are called components of the compounds.

A conditional is a compound statement with two main components. It is the type of compound typically formed using the connective if, as in "Felix likes milk if Felix is a cat", or the connective only if, as in "State will win only if they make this field goal". Either or both of the main components of a conditional may also be compound.

In some types of compound, the components are interchangeable parts from a logical standpoint. If it is true that Al and Bill are here, then it is true that Bill and Al are here, and vice versa. Buzzy had coffee or tea with lunch if and only if he had tea or coffee. But the two components of a conditional are not interchangeable parts: it may be true that Rovelle is a dog if she is a collie, but false that Rovelle is a collie if she is a dog. For this reason we use different names--antecedent and consequent--for the two components of a conditional statement.

When conditionals are formed using 'only if', the component right after the connective is the consequent. When 'if' is used as the connective, the antecedent comes right after it. In either case the location of the other component, the one that does not come right after the connective, is of no logical importance. Thus there are four main variations on the expression of conditionals in English:

The functional difference between antecedent and consequent is reflected in this fact about the logic of conditionals: any conditional with a true antecedent and a false consequent must be false, whereas a conditional with a false antecedent and a true consequent may very well be true.

Material Conditionals

In fact, for one logically important variety of conditional known as a material or truth-functional conditional, having a false antecedent and/or a true consequent is precisely what is required for the truth of the conditional, as indicated in the following table:

Conditionals cannot be regarded as material conditionals if there are indications that they are meant to cover possible situations other than the actual situation. Such indicators include the use of the subjunctive mood ("if this were so, that would be the case") or any of various modal terms ('necessarily', 'inevitably', 'impossibly', etc.). But for most logical purposes, most plain vanilla conditionals in the indicative mood ("if this is so, that is the case") can be regarded as material conditionals. Material conditionals have the property that figures most prominently in the logic of conditional reasoning, namely, that they cannot be true and have both a true antecedent and a false consequent.

Material conditionals are often symbolized using an arrow flanked by letters (or formulas) that stand for the antecedent (on the left) and consequent (on the right). Thus the formula C → B symbolizes any material conditional whose antecedent is the statement represented by 'C' and whose consequent is the statement represented by 'B', such as 'the car will start only if the battery is OK'. The arrow symbol should be understood to express only the asymmetry of material conditionals--the fact that the components are not interchangeable--and not any sort of connection between the antecedent and consequent. It is often, perhaps usually, a belief in some sort of connection between antecedent and consequent that induces us to make a conditional statement, but a material conditional does not express any such connection.

Simple conditional arguments

Conditional statements, like ‘If Al is a freshman, then Al is a student’, are often used to infer the truth or falsity of one component from that of the other. There are four common inferences of this sort, two valid and two invalid.  They have traditional names that are based on what the non-conditional premise does.  In type 1, the non-conditional premise affirms the antecedent of the conditional premise, so the argument is called "affirming the antecedent".  This valid form of argument is also traditionally called "modus ponens".  The other types are the valid form "denying the consequent" (2),  also known as "modus tollens", and two invalid types called "(fallacy of) affirming the consequent" (3), and "(fallacy of) denying the antecedent" (4).   Here are examples using conditionals in the "if...then..." form.

1. (valid) If Al is a freshman, then Al is a student. Al is a freshman. So Al is a student.
2. (valid) If Al is a freshman, then Al is a student. Al is not a student. So Al is not a freshman.
3. (invalid) If Al is a freshman, then Al is a student. Al is a student. So Al is a freshman.
4. (invalid) If Al is a freshman, then Al is a student. Al is not a freshman. So Al is not a student.

Forms 3 and 4 are invalid because Al might be (for example) a sophomore. In that case he would be a student but not a freshman, and each of the arguments (3 and 4) would have true premises and a false conclusion.

Since conditionals can be expressed in various ways, the four simple conditional argument forms above have variations. For example, ‘Al is a freshman only if Al is a student’ is another way of expressing the idea that if Al is a freshman, he is a student. With this variation the four forms above become:

1. (valid) Al is a freshman only if Al is a student. Al is a freshman. So Al is a student.
2. (valid) Al is a freshman only if Al is a student. Al is not a student. So Al is not a freshman.
3. (invalid) Al is a freshman only if Al is a student. Al is a student. So Al is a freshman.
4. (invalid) Al is a freshman only if Al is a student. Al is not a freshman. So Al is not a student.

Another set of variations uses ‘Al is a student if Al is a freshman’ to express the same idea that is expressed by ‘If Al is a freshman, then Al is a student’:

1. (valid) Al is a student if Al is a freshman. Al is a freshman. So Al is a student.
2. (valid) Al is a student if Al is a freshman. Al is not a student. So Al is not a freshman.
3. (invalid) Al is a student if Al is a freshman. Al is a student. So Al is a freshman.
4. (invalid) Al is a student if Al is a freshman. Al is not a freshman. So Al is not a student.

An exercise. Use each of the following conditional statements to construct four arguments as in the examples above, and describe a possible situation in which forms 3 and 4 would have true premises and false conclusions.

1. If Pete has a billion dollars, then Pete is rich.
2. Mindy is in Greensboro only if Mindy is in Guilford County.
3. Slick likes to nap if Slick is a cat.