The term conditional ultimately refers to not just one but an entire class of unidirectional relationships between two items. These items could be from any of the domains discussed in the previous chapter (the natural, textual, or logical), but we’ll use a word from the textual domain, terms, to discuss the items generically. Different kinds of conditionals will be discussed in the second part of the chapter, but all of them have the same basic form and logical characteristics.
Assuming that X and Y are two terms, and that the arrow (→) indicates the direction of the connection, the following formulation expresses the conditional relationship:
X → Y
Consider this a representation of the basic concept of the conditional. But what does this really mean? What exactly is the connection signified by the arrow? Let’s assume for a moment that the terms X and Y denote claims, which (as detailed in Chapter 5 of MCAT CARS Review) are evaluated on the basis of their truth value—the potential to be either true or false (but not both at the same time). When working with claims, we could rewrite the conditional relationship in this slightly less formal way:
If X is true, then Y is true.
Note that while it is made up of a relationship between two claims, this is itself a kind of statement with its own truth value. We’ll call this a conditional claim; that is, it is a statement that has a meaning that could be captured equivalently by using an if–then assertion, as above. Moreover, adopting the common convention, we’ll designate the if term the antecedent and the then term the consequent. Here’s a concrete example, with the antecedent and consequent in bold:
If it’s true that you are in Pennsylvania, then it’s also true that you are in the United States of America.
Or more simply:
If you’re in Pennsylvania, then you’re in the United States.
What is distinctive about such claims is that their truth, formally speaking, depends on only the truth of the antecedent and consequent. As long as it’s impossible for X to be true at the same time that Y is false (that you could be in Pennsylvania without being in the United States), then the conditional statement is true. In other words, the conditional relationship can be defined as the impossibility of having a true antecedent and a false consequent.
The conditional relationship is formally defined as the impossibility of having a true antecedent and a false consequent simultaneously. Technically speaking, if the antecedent and consequent terms are items other than claims (such as causally connected events, considered below), true and false would be replaced by analogous evaluations (so, for cause and effect, the terms present and absent might be substituted).
This can be represented formally in what is known as a truth table. In Table 6.1, all four possible combinations of the truth of X and Y are presented, as well as the resultant truth of the conditional claim in the final column:
| X | Y | X→Y |
|---|---|---|
| true | true | true |
| true | false | false |
| false | true | true |
| false | false | true |
Returning to the location example, we know that it’s simply not possible for the antecedent (you’re in Pennsylvania) to be true while the consequent (you’re in the United States) is false. This is because Pennsylvania is completely contained within the United States. Thus, we can say with confidence that the statement If you’re in Pennsylvania, then you’re in the United States of America is true.
Notice that this does not work the other way around; a conditional is unidirectional. You say something entirely different, something that is factually false, when you claim, If you are in the United States, then you are in Pennsylvania. In addition to the other 49 states, the District of Columbia, and 16 territories including Puerto Rico, Guam, and American Samoa, the United States occupies many military bases and diplomatic embassies, effectively spanning the globe. As can be seen, there are many ways to be in the United States without being in Pennsylvania.
Now here is a crucial point that cannot be stressed enough: a conditional claim can still be true even if you know the antecedent is itself false. In our geographical example, it doesn’t matter whether you’re in Maryland (in which case X is false but Y is true) or in New Zealand (for which both X and Y would be false). Neither of those is what might be thought of as the forbidden combination: a true antecedent plus a false consequent. No matter how hard you try, there’s simply no way to be in Pennsylvania without also being in the United States (that’s true even for the foreign embassies and consulates located in Philadelphia, which would count as being on neither US nor Pennsylvania soil, from the standpoint of international law).
Besides using the language of antecedent and consequent, there is another noteworthy way of understanding the relationship embodied in a conditional that divides the relationship in two by considering it from the perspective of each term. In this alternative account, each of the two terms in X → Y counts as a type of condition. Thus, X would be the sufficient condition, while Y would be the necessary condition. Or, to state it in a way that highlights the relation between the two terms, in X → Y, X is a sufficient condition of Y, while Y is a necessary condition of X. Let’s consider each side of this, in turn.
Another way of conceptualizing a conditional is to think of it as two simultaneous relationships. In the conditional claim if X, then Y, X is a sufficient condition of Y, while Y is a necessary condition of X.
Saying that X is sufficient for Y is equivalent to saying that it’s impossible to have X without also having Y. So, to return to our intuitive, geographical example, it’s enough to know that you’re in Pennsylvania—it doesn’t matter whether it’s Pittsburgh, Philadelphia, or Punxsutawney—to be able to conclude that you’re in the United States. In short, being in Pennsylvania is sufficient for being in the United States. This is really just another way of expressing the conditional relationship as the impossibility of a true antecedent and a false consequent.
In more complex cases, you’ll find that there can be more than one necessary condition or more than one sufficient condition. Suppose that both X and Y together lead to Z. We would then say that X and Y are mutually necessary and jointly sufficient for Z. For instance, to be a mother it is not simply sufficient to be female, but one also needs to be a parent. Thus, being female (X) and being a parent (Y) are mutually necessary and jointly sufficient for being a mother (Z). Though rare, this complication has appeared in past MCAT exams, including once in an extremely challenging question on linguistics that required understanding the two notions, only briefly explained in the accompanying passage.
While the relationship of sufficiency is the aspect of the conditional that we’re already familiar with, necessity is a bit new, although it’s simply the same relationship considered from the opposite perspective. Saying that Y is necessary for X is like saying that Y is a prerequisite or requirement of X. This reversal amounts to a negative claim: if Y is not true, then X is not true. If we consider the example from before from this perspective, it becomes:
If you are not in the United States, then you are not in Pennsylvania.
This statement is logically equivalent to If you are in Pennsylvania, then you are in the United States. However, even though the two versions of the claim will always have the same truth value, their meanings are not identical—if nothing else, there is a difference in point of reference. This alternative form, which emphasizes a necessary condition, is even known by a special name, the contrapositive.
Whenever an author makes any kind of conditional claim, it is always possible to translate the conditional claim into other relationships. Most notably, one can form the logically equivalent assertion known as the contrapositive. By definition, the contrapositive of if X, then Y is if not Y, then not X. Alternatively, this can be represented using a tilde (~) to stand for negation (~X thus means not X or the negation of X):
Conditional: X → Y
Contrapositive: ~Y → ~X
The contrapositive is a statement with a different connotation (emphasizing the necessary condition, rather than the sufficient) but that is logically equivalent to a conditional assertion. For instance, the contrapositive of if X is true, then Y is true can be written as either if Y is not true, then X is not true or if Y is false, then X is false.
One of the most useful reasons for forming the contrapositive is that it’s a guaranteed inference, a logical equivalent for any conditional claim made in a passage. Whenever you recognize Logic keywords or one of the English translations of a conditional claim, as listed in Table 6.2, you’ll want to keep in mind that a rendition of its contrapositive could appear as a correct answer.
| If X, then Y. | If I’m in Pennsylvania, then I’m in the United States. |
| X is sufficient for Y. | Being in Pennsylvania is sufficient for being in the United States. |
| All X are Y. | All those in Pennsylvania are in the United States. |
| X only if Y. | I’m in Pennsylvania only if I’m in the United States. |
| If not Y, then not X. [Contrapositive] | If I’m not in the United States, then I’m not in Pennsylvania. |
| Y is necessary for X. | Being in the United States is necessary for being in Pennsylvania. |
| Only Y are X. | Only those in the United States are in Pennsylvania. |
| Not X unless Y. | I’m not in Pennsylvania unless I’m in the United States. |