1.1.3. Notation for arguments

It is useful to have some abstract notation so that we can speak of reasoning generally rather than in specific examples. We will use the lower case Greek letters φ, ψ, and χ to stand for the individual sentences that may appear as the conclusion of an argument or as its premises. And we will use upper case Greek Γ, Σ, and Δ to stand for sets of sentences, such as the set of premises of an argument (or a set of sentences that is rejected as unacceptable). The general form shared by all arguments 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 can often best be thought of as standing for such a list. The reason for speaking of sets at all is that we will have no interest in the order of the premises or the number of times a premise appears in the 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. This means that we regard two arguments that share a conclusion as the same when their premises constitute the same set. There are other features of sets, however, which are of little use to us. In particular, we have no need to distinguish between a sentence φ and the set {φ} that has φ as its only member, and we will not attempt to preserve the distinction between the two in our notation for arguments.

If we regard the capital Greek letters as standing for lists of sentences, it makes sense to write Γ, φ / ψ to speak of an argument whose premises consist of the members of Γ together with φ. The set of premises of this argument is the union Γ ∪ {φ} of the sets Γ and {φ}—i.e., it is the set whose members are their members taken together. Since this idea does not exclude the possibility that φ is already a member of Γ, it provides convenient way to refer to any argument whose premises include the sentence φ. We will use an analogous convention in the vertcial notation for arguments. So, if Γ is the set {φ, ψ, χ} (i.e., the set whose members are φ, ψ, and χ) and Σ is the set {ψ, χ}, then all of the following refer to the same argument:

The equivalence of the expressions after the first can be traced to the equivalence among the following ways of referring to the set whose members are φ, ψ, and χ: