Basic Concepts 3b: Linked Premises


Linked premises logically require each other and work together to support a conclusion. Very often arguments don't explicitly state one of the linked 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, in this lesson all linked premises will be explicit.

Linked Premises, the "Why Should I Care?" Test, and the Shared Content Test

Suppose I want to convince Mary not to spank her child as a form of discipline. My conclusion is "you shouldn't spank your child." She asks, "why shouldn't I?" and I reply: If you spank your child they will learn that violence is a legitimate form of conflict management. From the context of the conversation it's probably quite clear that I'm having this conversation with Mary because she spanks her child. In other words, the full argument contains a second implied but unstated premise (because context makes it obvious): "You spank your child".

The full argument will look like this:

*P1.i If you spank your child they will learn that violence is a legitimate form of conflict management.
*P1.ii You spank your child.
C. You shouldn't spank your child.

Notice that I have used an '*' to indicate linked premises to distinguish them from serial premises.

The important thing to understand about linked premises is that logically both premises are required to support the conclusion. Just one on its own is irrelevant to the conclusion. Let's see why.

If I only give the argument:
P1. If you spank your child they will learn that violence is a legitimate form of conflict management.
C. Therefore, you shouldn't spank your child.

Suppose I'm not the kind of parent that spanks their child. When you say to me "If you spank your child they will learn that violence is a legitimate form of conflict management" I'll say to you, OK but so what? I don't spank my child. It doesn't apply to me. The premise only applies to people who spank their children. So, in order for the premise to be relevant to the audience, we need the second premise: YOU (the audience of the argument) spank your child.

By far the most common structure with linked premises are those that have if-then premises, i.e., conditional statements. Anytime you see a conditional premise there is also another premise required. It may be unstated because context makes it too obvious to state, but from a purely logical point of view, it's required.

Notice also that linked premises will share content. The 'if' clause of the conditional statement is the same as the individual linked premise.


*P1.i. If you spank your child they will learn that violence is a legitimate form of conflict resolution.
*P1.ii. You spank your child.

The shared content test helps us to identify linked premises. If a shorter premise is contained within a longer premise, we have linked premises.

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

*P1.i P or Q.
*P1.ii Not P (or Not Q).
C. Q (or P).

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 it isn't clear how it supports the conclusion. However, when we put both premises together it's very clear:

*P1.i Either you go to school or you get a job.
*P1.ii You don't have a job.
C. You need to go to school.

Notice also that we can apply the shared content test. The shorter premise is contained in the longer premise. This indicates linked premises.

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:

*P1i. All Xs are Y.
*P1.ii a is an X.
C. Therefore, a is Y.

Let's look at an example:

*P1.i All students like pizza.
*P1.ii 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 my to believe that "Jeff likes pizza" I'd also have to know that Jeff is a student.


The important thing to consider with linked premises is that neither one on its own is directly relevant to the conclusion. Both premises are required to work together to support the conclusion. 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, two tools will help us discover whether a pair of premises are linked (rather than serial or convergent): The "why should I care test" and the shared content test. If a premise answers the question "why should I care?" rather than "why should I believe that?" then it is a linked to premise. Also, when a long premise contains the content of the shorter one, it is a linked premise.

Recall that if a premise answers the question "why should I belief 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.

Self Quiz

Examples of Arguments with Linked Premises

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