7.6.1. Principles for restricted universals

The laws for unrestricted universals as premises and as conclusions were based on the relation between a generalization and its instances. In the case of restricted universals, we will also be interested in instantiation but an unrestricted universal (∀x: ρx) θx does not in general imply an instance θτ. But since (∀x: ρx) θx ⇔ ∀x (ρx →θx), the restricted universal will imply the sentences ρτ → θτ that are the instances of its restatement in unrestricted form; we will call these its conditioned instances. If we expand the language by the range R of a structure, an unrestricted universal ∀x θx will be true in that structure if and only if all its instances are true and an unrestricted universal (∀x: ρx) θx will be true if and only if all its conditioned instances are true. That means that a restricted universal behaves like a conjunction of its conditioned instances and its role in deductive reasoning is analogous in some respects to the role of an unrestricted universal and in other respects to the role of a conditional.

We can get laws for the restricted universal by restating such a universal in unrestricted form and and applying laws for both the unrestricted universal and the conditional. To get our law for the restricted universal as a conclusion, we can reason as follows for any term a that does not appear in Γ or (∀x: ρx) θx:

That is, we can conclude a restricted universal if and only if we can conclude its instance θa for a parametric term a, allowing ourselves to make the assumption that ρa—i.e., that the value of the term a is in the domain of the universal. The assumption here that ρa is comparable to the assumption that ABC is a triangle which is made when we wish to offer an argument about ABC as a basis for a generalization about all triangles.

We can approach the role of (∀x: ρx) θx as a premise also by way of a restatement in unrestricted form. If we apply the law for an unrestricted universal premise and restate the result again using a restricted quantifier, we get, for any term τ,

Γ, (∀x: ρx) θx ⇒ φ if and only if Γ, (∀x: ρx) θx, ρτ → θτ ⇒ φ

To go further, we need to take account of the conditional premise we have introduced. We have three ways of doing this, two detachment principles, each of which requires still another premise, and a principle for reductio arguments. Accordingly, we get three principles for a restricted universal premise each applying to a different sort of case:

Each holds for any term τ.

The first is a principle of restricted universal instantiation. It is perhaps the most widely recognized pattern of argument; an instance of it was our first example of a valid argument in 1.1.2. In the logical tradition, it and the second pattern (which stands to it as modus tollens stands to modus ponens) were often not distinguished from certain patterns of argument studied in the theory of syllogisms whose second premises and conclusions contain quantifier phrases rather than individual terms, and the names used here are adapted from a medieval system of nomenclature for such syllogistic arguments. (Notice that vowels in the names are the first vowels appearing in the English verbs affirm and negate, which happen to be cognate to the corresponding Latin terms, and that these vowels mark the affirmative or negative character of the premises and conclusion taken in order. Many of the consonants are also significant, but their significance lies in connections among various patterns of argument in the theory of syllogisms proper.)

The third principle for restricted universal conclusions says that we can reduce (∀x: ρx) θx to absurdity given Γ if and only if the two together enable us to do two things in the case of any term τ: (i) show that the value of τ is in the domain of the generalization and (ii) reduce to absurdity the claim that the attribute of the generalization is true of the value of τ. More briefly, Γ entails the falsity of (∀x: ρx) θx (and thus the existence of a counterexample to it) if and only if the two together entail that any term τ is a counterexample. (The only-if part of this may seem an odd claim; but its truth can be seen by reflecting that, if Γ does entail the falsity of (∀x: ρx) θx, the premises and the universal together form an inconsistent set and entail every sentence.) Because this law is sweeping enough to pull in terms that are innocent bystanders at the failure of a generalization (so we do not need a further premise as in the first two principles), we will count it as our law for the restricted universal as a premise.

To summarize, we have the following basic principles for the restricted universal:

Although these two principles suffice to capture the logical properties of restricted universals, rules implementing singular Barbara and singular Camestres will play an important role in our system of derivations.