Dependent premises (sometimes called ‘linked’ premises) logically require each other and work together to support a conclusion; i.e., they depend on each other to support a conclusion. Very often arguments don’t explicitly state one of the dependent premises, however it’s implied from the context of the argument. These hidden premises are called enthymemes or “hidden premises/assumptions”. Since we’ll be taking an in-depth look at enthymemes later in the course, this lesson will make all dependent premises explicit.

Dependent Premises and the “Why Should I Care?” Test

Dependent premises are easy to identify once you learn their common forms. Below are the most commonly used argument structures that use dependent premises.

Conditional Arguments

Suppose I claim that it’s going to rain. Not believing me, you ask me for a reason to believe your claim. I reply, “because, if there are clouds and high levels of humidity then it will rain.” At this point I’ve given you a reason to believe that it’s going to rain. But notice that my reason only supports my conclusion if something else is also true: i.e., that there are clouds and high levels of humidity. In other words there are two premises working together that support the conclusion that it’s going to rain. Let’s put the argument into standard form to investigate further:

Example 1:

P1. If there are clouds and high levels of humidity then it will rain.
P2. There are clouds and high levels of humidity.

C. Therefore, it will rain.

In order to support the conclusion both premises have to be true. To see why, suppose that P1 is true but P2 is false, i.e., it’s false that there are clouds and the air is humid. Is the conclusion well supported? Nope. Even if one of the premises is true (P1), the conclusion is only supported if both premises are true.

Let’s try the other way. Suppose P2 is true but imagine some weird planet where P1 is false. That is to say, there’s an alien planet where clouds and high humidity never lead to rain. The conclusion isn’t supported in this case either. Again, the conclusion is only supported if both premises are true.

Whenever a conclusion requires two (or more) premises to both be true, we call these premises dependent premises since the premises logically depend on each other to support a conclusion. (Note: Some textbooks call these linked premises because they are logically linked together to support a conclusion). The premises are both logically necessary for the conclusion to be true.

The diagram for dependent premises looks like this:

Notice that the dependent premises are linked together to indicate that they logically depend on each other.

By far the most common argument structure with linked premises is one that contains if-then premises, i.e., conditional arguments. Anytime you see a conditional statement as a premise there is always another premise required to complete the argument. The second premise may be unstated because context makes it too obvious to state, but from a purely logical point of view, it’s required. Pro tip: Notice also that linked premises will share content: The ‘if’ clause of the conditional statement is the same as the other linked premise.

E.g.,

P1. If [there are clouds and high humidity] then it will rain.

P2. [There are clouds and high humidity.]

Let’s look at one more example.

Example 2:

Suppose you and your mom are walking down the street and your mom sees what appears to be a senior citizen at an intersection. She says “you should help them cross the street.” “But mooooooooom! I don’t wannu,” you reply. To which your mom says “if there’s a senior citizen at the intersection then you should help them cross the street.” Notice again that there’s an unstated premise, “there’s a senior citizen at the intersection.” From context it’s too obvious to state, but it’s there nonetheless.

P1. If there’s an senior citizen at an intersection then you should help her cross the street.
P2. There’s an senior citizen at an intersection.

C.  You should help them cross the street.

Suppose P1 is true but P2 is false. That is, there isn’t senior citizen at the intersection. Maybe unbeknownst to your mom, the boys from Jackass are filming another movie. What appears to be a senior citizen is actually Johnny Knoxville in disguise.  Even if P1 is still true, the fact that P2 is false now undermines your mom’s claim that you should help the person across the street.

Conversely, suppose P2 is true but P1 is false. There is a senior citizen at the intersection. However, you are a hardcore Libertarian and believe that you shouldn’t help senior citizens to cross the street. That is, you believe that you ought not interfere with anyone even if it’s for their own good. By helping the senior citizen across the street you are undermining the intrinsic good that comes from achieving our own rationally chosen ends. Also, by helping you are contributing to a culture of dependence where senior citizens will come to depend on others for help rather than solve their own problems.

If P1 is false then even if there’s a senior citizen at the intersection it doesn’t follow that you ought to help them. And so, the conclusion is unsupported despite one of the premises (P2) being true.

Again, our diagram would look like this:

To reiterate, with dependent premises both premises must be true in order for the conclusion to follow. Very often with dependent premises, one of the premises will be unstated because context makes it obvious. Nevertheless, the premise is still there floating around in logic space. To evaluate an argument you must identify that premise and evaluate whether it’s true.

Conditional arguments are easy to recognize because they always take the following form:

P1. If A then B.
P2. A.

C.  B

Chain arguments link together several conditional arguments:

P1. If A then B.
P2. If B then C.
P3. If C then D.
P4. A.

C.  D.

The diagram for this chain argument looks like this:

Disjunctive Syllogism

Let’s look at another common argument structure that uses linked premises: Disjunctive syllogism. A disjunctive syllogism takes the following form:

P1. P or Q.
P2 Not P (or Not Q).
C. Q (or P).

The diagram for disjunctive syllogism looks exactly the same as for a two premise conditional argument. Just as with the conditional argument, there are two premises that logically depend on each other to support the conclusion.

Let’s look at an example. Suppose your mom says to you “you need to go to school.” You ask, “Why?” Your mom says, “because either you go to school or you get a job.” In order for this premise on its own to be relevant to the conclusion, one other thing has to be true: you don’t have job (i.e., Not Q). By itself the ‘or’ statement support the conclusion. However, when we put both premises together it’s very clear how they work together:

P1. Either you go to school or you get a job.
P2. You don’t have a job.
C. You need to go to school.

There are many forms of linked premises but here is one more extremely common one called universal instantiation. A universal instantiation just means that if we know something about every member of a group we can say the same about a particular member of that group. It’s standard form looks like this:

P1. All Xs are Y.
P2. a is an X.
C. Therefore, a is Y.

Its diagram will also be exactly the same as the one for a two premise conditional argument and disjunctive syllogism.

Let’s look at an example:

P1. All students like pizza.
P2. Jeff is a student.
C. Therefore, Jeff likes pizza.

Suppose you’re trying to convince me that Jeff likes pizza and to do so you say, “he likes pizza because he’s a student”. On it’s own this premise doesn’t support the conclusion. I’d also need to know that “All students like pizza.” Similarly, you might begin with the first premise “All students like pizza” but in order for me to believe that Jeff likes pizza I’d also have to know that Jeff is a student. In other words, the conclusion is only supported if both premises are true.

Looking Ahead

Understanding the structure of arguments is crucial to evaluating them. If I recognize that a conclusion is supported with dependent premises then I know that by showing just one of the premises to be false I can undermine the whole argument. If a conclusion is supported by convergent (i.e., independent) premises then I need to show that each one is false before I can reject it. If a conclusion (or major premise) is supported by serial premises, then I know that I should try to undermine the sub-premise(s) before I can reject the major premise or conclusion.

Summary

With dependent premises, neither one on its own directly supports the conclusion. Both premises must work together to support the conclusion; that is, both must be true. Very often the context of an argument makes one of the premises too obvious to state and so it won’t be stated. However, this doesn’t mean it isn’t there. Later when we look at enthymemes we will learn some tricks for uncovering unstated linked premises. For now, aside from knowing the common structures, two tools will help us discover whether a pair of premises are dependent (rather than serial or convergent): (a) The “why should I care test” and (b) the related content test.

Why Should I Care? Test: If a premise answers the question “why should I care?” rather than “why should I believe that?” then it is a linked premise.

Related Content Test: Dependent premises have related content. The longer linked premise will contain the shorter one.

Recall also that if a premise answers the question “why should I believe that?” then it is a sub-premise in serial premises. If a premise is unrelated to any of the other premises, then it is an independent convergent premise.