1.1.3. Notation for arguments

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 φ, ψ, and χ to stand for the individual sentences that may appear as the premises or conclusion of an argument. 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. 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 basing the idea of an argument on that of a set is that we will have no interest in the order of the premises or the number of times a premise appear if the premises of an argument are listed. 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.

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 the capital Greek letters were understood to stand for lists (rather than sets) of sentences, it would make sense to write Γ, φ / ψ to speak of an argument whose premises consisted 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 the members of Γ and {φ} 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 understand the notation Γ, φ in the same way. That is, imagine the members of Γ are listed, followed by φ. The premises of the argument Γ, φ / ψ are the sentences that appear anywhere in this list. The sentence φ definitely appears, so Γ, φ / ψ is an argument whose premises include φ and whose conclusion is ψ. Since Γ could be any set, this argument may or may not have premises in addition to 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 these ways of referring to an argument can be traced to the equivalence among the following ways of referring to the set whose members are φ, ψ, and χ: