7.6.1. How generality can fail

The examples considered so have not placed much emphasis on the choice of the term used in a general argument. In many of them, any term could be used. And, in cases where this is not true (such as the second example of 7.5.5), the need to use care in choosing a term was accidental. The derivations happened to already contain terms that might naturally be chosen; but, if different letters had appeared (or we were less inclined to choose letters from the beginning of the alphabet), the natural first choice would always work. That will no longer be so when we consider conclusions involving multiple generality, so, before considering them, we will look more closely at the requirements for a term to be independent.

The most basic requirement is that we not rely on special assumptions about the term from which we hope to generalize. We cannot conclude Everything is turned on from The amplifier is turned on, so we cannot generalize on the amplifier in the latter sentence if our justification for it is simply having that sentence as a premise. That is, the following derivation clearly must be disallowed

		│Ta	(2)
		├─
		│ⓐ
		││●
		│├─
	2 QED	││Ta	1
		├─
ERROR	1 UG	│∀x Tx

and it is ruled out by the requirement that the term flagging a general argument appear only to the right of the its scope line.

But, of course, that requirement rules out many other derivations, too, and among them are some that involve no logical error. As was noted in 7.5.3, the appearance of a term among the assumptions does not imply a use of special information about it in drawing a given conclusion, and we have ruled any occurrence of a term outside a scope line it flags, whether this occurrence is in an assumption or elsewhere. The chief virtue of the severe restriction is simplicity in its statement, and this simiplicity comes at little cost since, in the derivations we will consider, there will never be a shortage of new terms to use. (In principle, there can never be a shortage in any sort of derivation if we allow new terms to be generated by devices such as the addition of primes or subscripts.)

Even when it is not needed, the use of a new term does make clear just what sort of argument is provided for the many instances of the generalization other than the one from which we generalize. As one example of this, consider the following argument showing that Everything is turned on really does follow if the premise is extended to say The amplifier is turned on and so is everything else.

	│Ta ∧ ∀x (¬ x = a → Tx)	2
	├─
	│ⓑ
2 Ext	││Ta	(6)
2 Ext	││∀x (¬ x = a → Tx)	b:4
	││
	│││¬ Tb	(5), (6)
	││├─
4 UI	│││¬ b = a → Tb	5
5 MTT	│││b = a	b—a
	│││●
	││├─
6 Nc=	│││⊥
	│├─
3 IP	││Tb	1
	├─
1 UG	│∀x Tx

This analysis uses a paraphrase of else as other than it that will be discussed in 8.3.1.

The requirement that the term we generalize on does not appear in any assumption is enough to rule out many unwarranted generalizations but it does not exclude them all. To see why, suppose we are arguing from the assumption Everything is like itself. One conclusion we can draw is Wabash is like Wabash and, in doing so, we have certainly used no special assumptions about Wabash. But this conclusion says that Wabash has the property of being like Wabash, and that makes it an instance of the generalization Everything is like Wabash. Nevertheless generalizing to that conclusion is surely unwarranted. Here is what this argument might look like in a derivation.

		│∀x Lxx	b:2
		├─
		│ⓑ
	2 UI	││Lbb	(3)
		││●
		│├─
	3 QED	││Lbb	1
		├─
ERROR	1 UG	│∀x Lxb

The problem with this argument is that even though the term Wabash stands in no special relation to the assumptions, it does stand in a special relation to the universal conclusion Everything is like Wabash. In particular, it plays a special role in the predicate that the conclusion claims to be universal.

These considerations lay behind the second requirement for a general argument: if we wish to generalize from an instance θτ to a universal ∀x θx, the term τ should not appear in our conclusion; that is, it should not appear in the predicate θ. Just as in the case of the first requirement, this is more than is strictly necessary: even if a term has occurrences other than those on which we generalize (i.e., has occurrence left behind in the predicate), this fact may not have been exploited in the argument for it, and the argument might have gone through with any other term. And our approach in derivations is stricter still since we require that the term we generalize on appear nowhere after its scope and not merely that it not appear in the immediately following universal. But, as in other cases, the justification for this restrictiveness is that the restriction is both easy to enforce and easy to satisfy.

The final issue affecting generalization concerns cases where the term we generalize on does not itself appear outside the general argument but contains vocabulary which does. Suppose our assumption is Everything has its bad side. We can conclude Wabash has its bad side. But we cannot go on to conclude Wabash has everything, as in the first derivation below (where d: [ _’s bad side] and typographical limitations force a boxed rather than a circled flag).

│∀x Hx(dx) b:2 ├─ │xxxdb ││ 2 UI ││Hb(db) (3) ││● │├─ 3 QED ││Hb(db) 1 ├─ ERROR  1 UG │∀y Hby

│xxxdb ││ │││∀x Hx(dx) b:3 ││├─ 3 UI │││Hb(db) (4) │││● ││├─ 4 QED │││Hb(db) 2 │├─ 2 CP ││∀x Hx(dx) → Hb(db) 1 ├─ ERROR  1 UG │∀y (∀x Hx(dx) → Hby)

Now the instance from which this conclusion would generalize is an instance for the term Wabash’s bad side and this term does not appear in either the assumption or the conclusion, so it satisfies both of the requirements we have imposed so far. And the same issue can arise when vocabulary is shared with the conclusion, as in the second derivation, which is an attempt to show ∀y (∀x Hx(dx) → Hby)—i.e., Everything is such that (Wabash has it if everything has its bad side)—to be a tautology by deriving it from no premises at all.

A requirement that the term we generalize on not share vocabulary with sentences outside the scope line would rule out derivations like thse, and it would be more than enough to insure that an argument is general. Indeed, in the case of a compound term, it would be enough to require that the main functor not appear outside the scope line (so, in the examples above, the real problem lies in the occurences of the functor [ _ ’s bad side] not the occurences of the term Wabash). However, it is easier simply to prohibit generalization on compound terms. Unanalyzed terms that satisfy the first two requirements clearly share no vocabulary with the assumptions or conclusion so, for those terms, the first two requirements are enough.