7.5.5. General arguments in derivations
In order to manage general arguments in our system of derivations, we need a further sort of scope line. The portion of a derivation that constitutes a general argument will be marked by a scope line that is flagged by the independent term on which we generalize (as shown in Figure 7.5.5-1).
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Fig. 7.5.5-1. A veil of ignorance flagged by the independent term a.
This flagging declares that the term is independent. Indeed, we will require that a term flagging a scope line appear only to its right, so the scope line will mark the scope of the term’s use. In either form, the requirement is designed to insure that the independent term maintains no ties to the outside of the general argument so that, within the argument, it might refer to anything at all. For this reason, we will speak of a scope line flagged by a term as a veil of ignorance.
The limitation of the appearance of the independent term to the portion of the derivation marked by its scope line is more than is necessary to stay in accord with the laws for universals as conclusions. They require only that the term not appear in either the goal or the active resources of the gap that the vertical line spans, but we will never run short of terms and the stronger requirement is far easier to check.
Now, let us look at the planning rule for universal goals. It is known as Universal Generalization (UG) and is shown in 7.5.5-2.
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→ |
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Fig. 7.5.5-2. Developing a derivation at stage n by planning for an unrestricted universal; the independent term a may be any unanalyzed term that is new to the derivation.
We try to reach our goal by a general argument, so we choose as our independent term an unanalyzed term a that is new to the derivation. An instance of ∀x …x… for the term a is the goal of the general argument, and further development of the gap lies on the other side of a veil of ignorance concerning that independent term.
The short derivation shown below illustrates this rule. It shows that, if a relation R is universal in the sense of holding of any pair of things, then it is reflexive.
| │∀x ∀y Rxy | a:2 | |
| ├─ | ||
| │ⓐ | ||
| 2 UI | ││∀y Ray | a:3 |
| 3 UI | ││Raa | (4) |
| ││● | ||
| │├─ | ||
| 4 QED | ││Raa | 1 |
| ├─ | ||
| 1 UG | │∀x Rxx |
At the initial stage here, there is no vocabulary from which a term may be formed—and UI should be used to introduce new terms only as a last resort—so we apply the planning rule to the universal conclusion. After applying it, we have vocabulary for use with the exploitation rule, and we apply that rule twice for the term a. It would have been legitimate to exploit either universal resource for any other term τ as well, but that would not have contributed to closing the gap.
The following derivation illustrates the limitations on the scope of a term.
| │∀x Rax | c:2 | |
| │∀x ∀y (Rxy → ∀z Ryz) | a:3 | |
| ├─ | ||
| │ⓒ | ||
| 2 UI | ││Rac | (5) |
| 3 UI | ││∀y (Ray → ∀z Ryz) | c:4 |
| 4 UI | ││Rac → ∀z Rcz | 5 |
| 5 MPP | ││∀z Rcz | b:6 |
| 6 UI | ││Rcb | (7) |
| ││● | ||
| │├─ | ||
| 7 QED | ││Rcb | 1 |
| ├─ | ||
| 1 UG | │∀x Rxb |
The independent term used here could not have been either a or b since both appear beyond the scope line of the general argument, one in a premise and the other in the conclusion.
The derivation shown here minimizes the use of UI, and the particular choice of instances needed to do this might not be obvious. Once the first premise is instantiated for c, the next two instantiations are designed to set up the use of MPP at stage 5, but it is probably less obvious that c is the best choice for the initial instantiation. It is fine to experiment, and there is no need to back up if you do not make the best choice. A derivation is never damaged by extra uses of UI; and, when we go on to use derivations to show the failure of entailments involving generalizations in 7.7, we will require that, before a derivation can reach a dead end, any universal resource must be exploited for at least term from each alias set.