2.4.1. Premises, assumptions, and suppositions

The rules using lemmas that we will discussion in the next couple of subsections employ a device that will be employed also in rules for most of the logical forms we will consider after conjunction. This device is the use, for stretches of a derivation, of assumptions that add to the content of the premises of the ultimate argument. We will refer to such assumptions as suppositions. Suppositions are assumptions made, not because we accept them but for the sake of argument, and such assumptions are often introduced in ordinary argumentation by the verb suppose. That is, the stretch of discourse

comes to pretty much the same thing as

And notice that the speaker here is not committed to accepting the suggestion.

Suppositions can have a variety of roles in deductive reasoning, and the ones that will be important in later chapters are mostly rather different from their role in connection with lemmas, but all uses of suppositions have common features that can be captured by each of the ways of writing proofs that we have seen. In the case of conclusion trees, our approach to suppositions will be based on a more careful consideration of the role of premises. Indeed, premises, although not assumed only for the sake of argument are assumptions that are stated for its sake.

Now, consider again the first of the conclusion trees discussed in 2.2.1.

We looked at it there as a proof showing the validity of the argument A, B ∧ C / (A ∧ B) ∧ C. But it is equally well a proof for the argument A, B ∧ C, D / (A ∧ B) ∧ C. The extra premise D in the latter argument does not interfere because it remains true and all conclusions at the tips of branches are among the premises.

We can add a specification of the argument for which the conclusion tree is a proof by means of the following notation:

The premises are listed at the upper left, and the dotted line around this list and the tree indicates the scope of these assumptions. The tie between the list and the tree then lies in a requirement that the tip of every branch must be one of the assumptions within whose scope it lies. Although this notation for the scope of assumptions is specific to conclusion trees, the idea is not. In particular, one function of a scope line in derivations is to mark the scope of any assumptions at its top.

A two-premised argument like Cnj will appear in the text of a conclusion tree in the following way:

Here the triangles above the two premises represent any part of the conclusion tree growing up from that point (and there need be none since one or another premise might be the tip of a branch). This can be thought as a representation of a rule for building conclusion trees (from tip to root) by drawing further conclusions. A rule like the one shown here takes two conclusion trees and makes a larger one from them.

Now we might imagine a different sort of rule that applies in the following way:

Here the second conclusion tree comes with an assumption θ whose scope surrounds the conclusion tree with ψ as its root. Although θ may appear as the tip of any branch above ψ, it will not be available to use as the tip of a branch above φ since that part of the conclusion tree is outside its scope. So θ will count as a supposition that is added to whatever assumptions have the whole larger tree in their scope.

Although the pattern shown above is the one we will encounter with lemmas, it is not the only one possible. Rules involving suppositions can apply to differing numbers of smaller conclusion trees, and differing numbers of these may come with suppositions. But one thing must always be true: we must in some way pay a price for using a supposition in part of a conclusion tree. In the portion of the tree below the scope of a supposition, it is no longer an assumption being made, and it is said to be discharged. In particular, is no longer in effect when we reach the conclusion of the rule. This means that a rule like R above cannot have a conclusion that simply incorporates the content of its two premises φ and ψ in the way Cnj does, for we have concluded ψ only with the help of a supposition that is no longer being made. Different rules will compensate for this in different ways, perhaps by some requirement about what is shown in the part of the tree above the other premise or by weakening the content of the conclusion. The rules for lemmas we will consider do the former, but the latter approach will be used quite often in the rules of later chapters. These concern compounds that have less content than one or more or their components, so the weakening used to compensate for a supposition is very natural.

The appearance of suppositions in argument trees and derivations follow these ideas in unsurprising ways. A segment of an argument tree that plans for the use of the rule R above would take the form shown on the right. The second of the two arguments to which the validity of the conclusion χ is reduced has the supposition as a premise in addition to the premises it shares with the argument above it.

And a derivation rule for R would look like this:

The supposition appears at the top of the scope line of the second child gap, and that scope line marks its scope in the derivation. In particular, it is not an available resource after that scope line ends, for the end of its scope line is the point at which it is discharged from our service. The sentence χ counts as a conclusion from φ and ψ, but its relation to them must reflect the fact that the gap above ψ might be filled using an assumption that is no longer being made.