7.2.1. The universal quantifier

A quantifier is an operator that takes predicates as input and yields sentences as output. The quantifiers we will consider all apply only to 1-place predicates, but we will consider them in two forms, one that is a 2-place operator applying to a pair of 1-place predicates and another that is a 1-place operator applying to a single 1-place predicate. When there is no need to distinguish them we will refer to both as universal quantifiers and describe the formulas they form also as universal (or, less formally, use universal as a common noun and refer to them as universals).

Our 2-place quantifier is the restricted universal quantifier for which we use the symbol ∀ (the symbol for all). It will eventually be convenient to use this symbol always along with a variable and some associated punctuation; but, for the moment, we will speak of it alone, as the sole notation for this quantifier. A sentence ∀_ρθ that results from applying the restricted universal quantifier to predicates ρ and θ will be referred to as a restricted universal. It says that θ is true of everything that ρ is true of—i.e., that the extension of θ includes the extension of ρ. This makes ∀_ρθ an affirmative direct generalization whose domain is the extension of ρ and whose attribute is expressed by θ. Since the scope of the generalization is limited to the extension of ρ we will refer to ρ as the restricting predicate, and we will refer to θ, which expresses the property said to hold generally, as the quantified predicate.

The simplest case of a restricted universal is one whose restricting and quantified predicates are unanalyzed. For example, if W is [ _ walks] and M is [ _ moves], then we might write ∀_WM to say that anything that walks also moves. More often, the restricting or quantified predicate will have internal structure, and we will use an abstract to express it. For example, if we want to say that anything that walks and talks both moves and communicates, we might do this with the form

where T is [ _ talks] and C is [ _ communicates].

Since we are able always to write the two abstracts using the same variable, we can use the following more abbreviated notation for the universal sentence:

And this notation has the advantage of a fairly natural English reading. The symbolic form (∀x: Wx ∧ Tx) (Mx ∧ Cx) can be read in something close to English as Everything, x, such that (x walks and x talks) is such that (x moves and x communicates).

Since any predicate θ can be expressed as an abstract [θx]_x, this notation can be used in all cases. That is, any universal ∀_ρθ can be expressed as

and can be rendered in English as

Here we can regard ρ and θ as the predicates to which the quantifier applies, with the apparatus of variable binding absorbed into the quantifier.

We may also write the form of a restricted universal schematically as

which amounts to

As a sort of grammatical pun, this can be read as Everything, x, such that (x dots) is such that (x dashes).

The formula …x… in (∀x: …x…) ---x--- (i.e., ρx in (∀x: ρx) θx) says what must be true of x for it to be in the domain of the generalization; we will refer to it as the restricting formula. The formula ---x--- (i.e., θx) says that x has the attribute of the generalization. The generalization says something how many values in the domain will make θx true when they are assigned to x (namely that they all will), so we will refer to θx as the quantified formula. This is a direct extension of our terminology for the component predicates of a generalization: the restricting formula is a predication of the restricting predicate and the quantified formula is a predication of the quantified predicate.

When reading the symbolic notation, we add the variable x as an appositive marked off by commas after the quantifier phrase (i.e., we say Everything, x, …) to indicate that this quantifier phrase serves as the antecedent of the symbolic pronouns x. If we put English pronouns in place of the variables, we can rely on the conventions of syntax to determine the antecedent and we can drop the appositive to get

This is a generalization whose class indicator is thing such that (…it…) and whose quantified predicate is [ _ is such that (---it---)]. Notice that the adjectival phrases such that (…it…) and such that (---it---) have two different functions in this sentence. The first appears as a modifier of the common noun thing while the second is a predicate adjective. Their roles are comparable to those of scarlet and red, respectively, in Everything scarlet is red.

The use of thing here also deserves some comment. Consider an English generalization that uses the same form of words as these readings—Everything such that it walks is such that it moves, for example. This generalization is direct and affirmative. The class indicator is the phrase thing such that it walks; and the predicate [ _ is such that it moves] is the quantified predicate. Now if this sentence is to make the same claim as (∀x: x walks) x moves, the indicated class of the English sentence should be the extension of [ _ walks] and the attribute expressed by the English quantified predicate should be the extension of [ _ moves]. There is certainly no problem in the latter case; [ _ is such that it moves] is just a more cumbersome way of expressing [ _ moves]. But does thing such that it walks, or thing that walks, really indicate the extension of [ _ walks]?

It does if we take the word thing to indicate the full range of reference values rather than being limited, say, to inanimate objects. We may say that, in such a use, thing is a dummy restriction. It does not itself restrict the domain of the generalization but provides a grammatical anchor for further restrictions. We have been using the word that way as an alternative to object, entity, and individual, but is it used that way ordinarily? This is not the sort of question we can settle here, but notice that if we really want emphasize that our generalization concerns things in some specialized sense, we are likely to use the two-word phrase every thing, with an emphasis on thing, rather than the single word everything. This is not to say that everything in English is typically used to generalize about all reference values, but more restricted uses can be traced to bounding classes provided by the context. One thing we can do here is to stipulate that, when we use it to read logical forms, everything will introduce no bounds narrower than the full referential range.

The second universal quantifier we will consider, the 1-place unrestricted universal quantifier, amounts to a special case of restricted universal quantification where the restricting predicate has the whole range of reference values as its extension. There are a number of predicates that are certain to be universal in this sense. Since identity is reflexive, the abstract [x = x]_x is one example, and the vacuous abstract [⊤]_x. Whenever ρ is a universal predicate, the sentence (∀x: ρx) θx says that the extension of the attribute predicate θ includes the whole of the referential range; that is, it says that θ is also universal. This sort of claim about a predicate θ is important enough that we add a one-place quantifier, enabling us to express it as ∀θ. The single predicate to which this quantifier applies will be called its quantified predicate. We will more often use the form

or

where θx is the quantified formula. Similarly, ∀x (…x…) can be read as Everything, x, is such that (… x…). For example, if F is [ _ is fine] and D is [ _ is dandy], the sentence ∀x (Fx ∧ Dx) can be read as Everything, x, is such that both x is fine and x is dandy.

We will not often write universals so that they apply directly to predicates; but such expressions capture the logical form of universals most clearly, so it would be worth trying, at least once, to read them. A direct symbol-by-symbol reading of the unrestricted universal ∀θ would be ∀ holds of θ. By departing from the order of the symbols we can put the content of the claim made by ∀ into words as

A symbol-by-symbol reading of the restricted universal ∀_ρθ would be something like ∀ holds of ρ and θ. Since ∀_ρθ says that the extension of θ includes the extension of ρ, we can put this universal into words also as

And this brings us full circle, back to a form that can be used in English. We could restate Everything that walks moves as The property of moving is (at least) as general as the property of walking. And we can understand the unrestricted quantifier in the same way: to say that θ holds universally is to say that θ is as general as can be.

We have already seen that we can get the effect of unrestricted universal quantification while using the restricted universal quantifier if we choose a universal predicate—e.g., [x = x]_x or [⊤]_x—as the restricting predicate. In the other direction, we can get the effect of restricted universal quantification using the unrestricted quantifier by hedging the claim made by the quantified formula. The nature of the hedge that is needed can be found by trying to restate a restricted universal claim in the form Everything is such that …. If we do this with Everything that walks moves, we might get

a sentence which says that the predicate [ x moves if x walks]_x is universal. In general, we can get the effect of restricted universal quantification by claiming universality for the result of making the quantified formula conditional on the restricting formula. That is, (∀x: ρx) θx can be expressed as ∀x (ρx → θx).

The two sorts of restatements we have been considering are licensed by the following principles of equivalence:

And we will have reason to make such restatements because the unrestricted universal quantifier is easier to use in stating laws of entailment while the restricted universal quantifier is easier to use in analyzing English sentences. In order to keep the connection between the two in mind, we will often express analyses made using the restricted universal also using the unrestricted quantifier.