1.1.3. Arguments

It is convenient to have a term for a conclusion taken together with the premises or assumptions on which it is based. We will follow tradition and label such a combination of premises and conclusion an argument. A particularly graphic way of writing an argument is to list the premises (in any order) with the conclusion following and marked off by a horizontal line (as shown in Figure 1.1.3-1). The sample argument shown here is a version of a widely used traditional example and has often served as a paradigm of the sort of reasoning studied by deductive logic.

When we need to represent an argument horizontally, we will use / (virgule or slash) to divide the premises from the conclusion, so the argument above might also be written as All humans are mortal, Socrates is human / Socrates is mortal.

Notice that the information expressed in the conclusion of this argument is the result of an interaction between the two premises. In its broadest sense, the traditional term syllogism (whose etymology might be rendered as reckoning together) applies in the first instance to inference that is based on such interaction, and the argument above is a traditional example of a syllogism. Another traditional term, immediate inference, applied to arguments with a single premise. The term immediate is not used here in a temporal sense but instead to capture the idea of a conclusion that is inferred from a premise directly and thus without the mediation of any further premises.

It is useful to have some abstract notation so that we can state generalizations about reasoning without pointing to specific examples. We will use the lower case Greek letters—most often φ, ψ, and χ—to stand for the individual sentences. And we will use an upper case Greek letter—most often Γ, Σ, and Δ—to stand for a set of sentences, such as the set of premises of an argument. The general form of an argument can then be expressed horizontally as Γ / φ, where Γ is the set of premises and φ is the conclusion.

Although we speak of the premises of an argument as forming a set, in practice what appears above a vertical line or to the left of the sign / will often be a list of sentences, and a symbol like Γ may often be thought of as standing for such a list. The reason for speaking of a set of premises rather than a list is that we will have no interest in the order of the premises or the number of times a premise appears when the premises of an argument are given by a list. We ignore just such features of a list when we move from the list to the set whose members it lists—as we do when we use the notation {a₁, a₂, …, a_n} for a set with members a₁, a₂, …, a_n. So, although premises will always be listed in concrete examples, we will regard two arguments that share a conclusion as the same when their premises constitute the same set. However, when a symbol for a set appears in a list, we will understand it to abbreviate some list of its members. One common use of this idea will be to employ the notation Γ, φ to stand for any list of sentences that includes the sentence φ. That is, we know that φ is included in the list because it is shown; in addition, the list will also include any sentences contributed by Γ, and this may be none at all since Γ might have no members.