7.6.2. Multiply general arguments

Although we could enforce the requirements that the term we generalize on have no connection with assumptions or the conclusion simply by setting aside a special group of letters for general arguments, that would not be enough to handle cases where a conclusion is multiply general. For, to establish such a conclusion, we need more than one general argument, and the terms used in such arguments must be independent of one another.

The derivation below is a simple illustration of this.

	│∀x ∀y Rxy	b:3
	├─
	│ⓐ
	││ⓑ
3 UI	│││∀y Rby	a:4
4 UI	│││Rba	(5)
	│││●
	││├─
5 QED	│││Rba	2
	│├─
2 UG	││∀y Rya	1
	├─
1 UG	│∀x ∀y Ryx

We begin by applying the planning rule to the universal conclusion, introducing a as the term on which we will generalize. When the rule is applied a second time at stage 2, a second new term is introduced, and it must be independent of the first. That is insured by the rule because, since the term a will appear outside the scope line of the second general argument, a new term must be used to flag this new scope line.

The effects of not using independent terms is shown in the following faulty derivation, which attempts to conclude that R holds between every pair of objects from the assumption that it is reflexive.

		│∀x Rxx	a:3
		├─
		│ⓐ
		││ⓐ
	3 UI	│││Raa	(4)
		│││●
		││├─
	4 QED	│││Raa	2
		│├─
ERROR	2 UG	││∀y Ray	1
		├─
	1 UG	│∀x ∀y Rxy

Here the error lies in the use of UG planned for at stage 2, for the premise really would entail the conclusion if it entailed ∀y Ray. And it is innermost scope line that violates the requirement that the flagging term not appear outside the part of the derivation marked by the line.

The recognition of multiple generality in the Middle Ages was a real advance beyond Aristotle’s theory of syllogisms (in the narrow sense of 7.5.6). The argument shown below is the sort of pattern the medieval logicians were trying to account for. Both the premise and the conclusion assert affirmative generalizations. But the restricting and quantified predicates of the conclusion themselves involve generalization, and it is the relation that the premise establishes between these generalizations that makes the conclusion follow. The theory of syllogisms did not provide the means to analyze predicates, so it was not able to account for the impact of the premise in this sort of example.

	│∀x (Dx → Mx)	b:5
	├─
	│ⓐ
	│││∀y (My → Fay)	b:7
	││├─
	│││ⓑ
	│││││Db	(6)
	││││├─
5 UI	│││││Db → Mb	6
6 MPP	│││││Mb	(8)
7 UI	│││││Mb → Fab	8
8 MPP	│││││Fab	(9)
	│││││●
	││││├─
9 QED	│││││Fab	4
	│││├─
4 CP	││││Db → Fab	3
	││├─
3 UG	│││∀z (Dz → Faz)	2
	│├─
2 CP	││∀y (My → Fay) → ∀z (Dz → Faz)	1
	├─
1 UG	│∀x (∀y (My → Fxy) → ∀z (Dz → Fxz))

Since the general term thing does not restrict generalizations, the restriction in the conclusion comes solely from the relative clause that affects all mammals, and the whole sentence would be represented using restricted quantifiers as (∀x: x affects all mammals) x affects all dogs The derivation begins at stage 1 with planning for the unrestricted universal conclusion. At stage 2 we plan for the new conditional goal and at stages 3 and 4 for the universal and conditional that represent the claim a affects all dogs; we do this by introducing a new independent term and supplying a supposition that begins exploitation of the two resources in stages 5-8. Notice that the argument for Fab is doubly general—i.e., it falls within the scope of two independent terms.