The meaning of the quantifiers

There are two main theories about the meaning of universal propositions.

To explain them, we need a precise notion of instances of such propositions.

This is easy in specific cases, as with the natural numbers and a universal proposition such as All even numbers are sums of two prime numbers. The instances are 2 is a sum of two prime numbers, 4 is a sum of two prime numbers, . . . . Formally, we do the following. First, constants and variables together are called terms and denoted by s, t, . . . . We use a special bracket notation for the substitution of a term in a formula:

Substitution. Let A be a formula. Then A[t/x] is the formula that is obtained from A when the free occurrences of x in A are replaced by the term t.

If A has no free occurrences of x, then A[t/x] is identical to A. Otherwise, in the ‘normal’ case, it is some other formula. We say that t is free for x in A if no variable in t becomes bound in the substitution.

Given a universal formula∀x A, an instance of ∀x A is obtained through the substitution of a term t for x in A, i.e., A[t/x] is an instance of ∀x A.

The first of the two theories about the meaning of universal propositions, associated with the name of Alfred Tarski in the 1930s, goes as follows:

The quantifiers 121

Truth of universals: Tarski. Given a domain D of objects a1, a2, . . . and relations P, Q, R, . . . inD, the formula∀x A is true in Dif the instance A[ai/x] is true for each ai.∀x A is logically true if it is true in any domain D^.

The idea is that, given a schematic formula A of predicate logic, it can be

‘interpreted’ as a concrete formula about some concrete objects in a given domain and with some concrete relations in place of the schematic atomic formulas. Thus, one says that formula A is logically true if it is true under any interpretation.

Tarski’s definition is problematic in a couple of respects. First of all, it refers to an arbitrary domain of objects and some theory would be needed for stating what that means. The most common account relies on axiomatic set theory. Secondly, the definition mainly replaces the for all in∀x A by each ai in A[ai/x]. From one point of view, the definition is obtained through ellipsis: If a domainDis finite, we can list its elements, as in a1, . . . , an. The different instances of∀x A are A[a1/x], . . . , A[an/x]. Therefore ∀x A is true in a finite domainDif each of the instances A[a1/x], . . . , A[an/x]

is true inD. These instances are true inDif and only if their conjunction is true inD, so we have:

A universal formula∀x A is true in a finite domainDof n objects a1, . . . , an

if A[a1/x] & . . . & A[an/x] is true inD^.

Tarski’s meaning explanation of the universal quantifier follows if the restric-tion to finiteness is lifted, as in:

∀x A is true inDif A[a1/x] & A[a2/x] & . . . is true inD.

Another, much earlier account of the meaning of universal propositions stems from Frege in 1879. Gentzen formulated it in natural deduction as:

Provability of universals: Frege–Gentzen. The formula ∀x A is provable from assumptions if A[y/x] is provable for an arbitrary y.

The Frege–Gentzen account is syntactic and does not need any theory of what domains of objects are in general. It is just assumed that there is an unbounded supply of parameters a, b, c, . . . that represent constants, as well as variables x, y, z, . . . . As to ‘arbitrary’ variables, there is nothing arbitrary

about them. We shall make the notion precise through an introduction rule for∀ in natural deduction:

Table 8.4 The introduction rule for the universal quantifier

.... A[y/x]

∀x A ^∀I

We have in the premiss a derivation of the formula A[y/x] from the open assumptions, and the conclusion is ∀x A. The following condition states in what sense y is arbitrary:

Variable condition in rule∀I . The variable y must not occur free in any of the assumptions the premiss A[y/x] of the rule depends on.

The idea is that nothing is assumed about y except that it is an object in a domain D. Thus, any object inD can take its place in the derivation of A[y/x] from , say the object a. If every free occurrence of y in the derivation is replaced by a, a derivation of A[a/x] from  is obtained. The arbitrary variable in rule∀I is called the eigenvariable of ∀I . In practice, when an inference with an eigenvariable is planned, a variable is chosen as eigenvariable that does not occur anywhere else. Such a variable is called a fresh variable.

The condition of provability according to Frege–Gentzen is stronger than the condition of truth according to Tarski: If we have a derivation of A[y/x]

from for an arbitrary y, we can substitute in turn a1, a2, . . . for x in the derivation, to get a derivation of A[a1/x] from , then A[a2/x] from , etc.

This is not an infinitistic explanation like the one of Tarski, because we have a finitary prescription for how to produce a derivation of A[ai/x] from  for any given object ai. The prescription is not unlike, say, the prescription for doing the sum of two natural numbers that we do not need to explain by actually showing the infinity of possible sums of two numbers.

Let the domain consist of the natural numbers 0, 1, 2, . . ., denoted N. Let A be a formula of arithmetic with just x as a free variable. Each instance A[n/x] of ∀x A states something specific about some number n. Let there be reasons specific to n for why A[n/x] is true, and let these reasons be such that they would not apply to any other number m. For all we know

The quant ifie r s 123

about ar ithmetic t r uth, there is nothing to exclude the p ossibilit y of a t r ue universal for mula ∀ x A such that each instance is t r ue for its ow n reasons, so to say. If this is so, there is no finitar y way of seing that ∀ x A is t r ue.

The Fre ge–Gentzen prov abilit y condition instead requires there to be a justification for A that is unifor m , the same for each instance. We can ask again, as in the last par agaph of S ec tion 7.3: Which comes first, t r uth or provability?

The meaning of the existential quantifier in a sentence such as∃x A is explained in terms of truth by requiring that some instance A[ai/x] be true:

Truth for existence. Given a domainDof objects a1, a2, . . ., an existential formula∃x A is true inD^{if A[a}i/x] is true for some ai.∃x A is logically true if it is true in any domainD^.

If a domain is finite, truth reduces to the propositional case similarly to the universal quantifier:

An existential formula∃x A is true in a finite domainDof n objects a1, . . . , an

if A[a1/x] ∨ . . . ∨ A[an/x] is true inD^.

There is a limiting case in which the number of objects is 0 and the domain is empty. In this case we set:

Definition 8.2. Truth in an empty domain. Existential formulas are false in