7.5.2. Instantiation

The special features of the laws of entailment we will state for the universals can be traced to two sources. One is the analogy with conjunction we have just explored. The other is a pair of differences between what we have said about universals and what we may say about ordinary conjunctions.

The first of these differences lies in the fact that the principles of entailment for universals must hold for all structures and thus cannot depend on special assumptions about the range R of reference values. This means, in particular, that the set of components of a universal (i.e., its instances in an expansion by R) must be left indefinite while an ordinary conjunction has a definite and, indeed, finite set of components.

This would make universals difficult to deal with were it not for their second difference from conjunctions. The components of an ordinary conjunction can be any pair of sentences so they need have nothing in common, and we must consider them individually; but the instances of a universal all follow the same pattern, differing only in occurrences of a single term, so we can speak of them all together by speaking of this pattern. We will look at the effects of this second difference more closely in the next subsection when we consider the role of universals as conclusions.

For the moment, we will concentrate on the role of universals as premises. We can get laws for universals, either as premises or as conclusions, by taking certain laws for conjunctions as our model and modifying them to take account of the differences just outlined. In the case of universals as premises, the laws for conjunction we will work from are the following:

These are principles that were labeled left and right extraction in 2.2.1. Although they led us to the rule of Extraction, they are less far reaching than the law for conjunction as a premise that lies behind the use of that rule. The fact that we focus on them rather than that stronger principle is due to the first difference between universals and conjunction: the law for conjunction as a premise says we can replace a conjunction by its components, but there is no hope of doing anything like this for a universal since it has no one definite set of instances.

When taken together, the laws above say that a conjunction implies each of its components. The analogous claim about an unrestricted universal is that it implies each of its instances. This is a principle known as universal instantiation:

Or, using an alternative notation, ∀x … x … ⊨ … τ … . For example, the sentence Everything is fine and dandy implies the claim The weather is fine and dandy as well as other sentences of the form τ is fine and dandy.

The principle of universal instantiation is not quite what we will take as our account of the unrestricted universal as a premise. We combine it with the law of lemmas to get a principle that allows us develop a derivation by adding any instance of a universal premise as a further resource.

Since the only if part of this claim follows from the monotonicity of entailment, the key property of the universal lies in the if part: an argument with a universal as a premise is valid if the result of adding an instance as a further premise is valid. That is, when establishing the validity of an argument with universal premise, we are free to add any instance as a further premise. Note that the instance is added as a further premise. This is required for the only if part to be true. We cannot drop the universal because we cannot expect its content to be exhausted by a single instance: Everything is fine and dandy has implications for things other than the weather. As you might expect, our inability to drop the universal from the premises will force some complications in the way we speak about the exploitation of resources in a derivation. We will consider these when we state the actual derivation rule in 7.5.4.

Although we will not be considering derivations for restricted quantifiers in their own right, arguments involving them can be captured by way of their restatements using unrestricted quantifiers, and the principles governing these quantifiers can be derived directly from those governing the unrestricted quantifiers and the conditional. In the case of the restricted universal as a premise, we have the following

The first is the key principle. It reflects aspects of the laws for unrestricted universals and for conditionals as premises. It is from the latter that it derives its restriction to reductio arguments. Notice that the two entailments on the right combine to show that the term τ refers to a counterexample to the universal, with the first showing that reference value of τ is in the domain and the second reducing to absurdity the claim that this reference value has the attribute. The other two principles reflect aspects of the modus ponens and modus tollens: if we know that τ refers to something in the domain of a generalizaiton whose truth we are assuming, we can add the assumption that this thing has the attribute. On the other hand, if we know that this reference value does not have the attribute, we can add the assumption that it is not in the domain. In short, there are three key ways to use a restricted universal assumption: to reduce to absurdity any assumptions entailing that some term refers to a counterexample, to show that something has its attribute (when it is assumed to be in the domain), and to show that something is not in the domain (when it is assumed not to have the attribute).