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Humanities Libertexts

4.1: Restricted Quantifiers

For three chapters now I have been merrily transcribing '(Ǝx)' both as 'something' and 'someone', and I have been transcribing '(Vx)' both as 'everything' and 'everyone.' I justified this by saying that when we talked only about people we would restrict the variables 'x', 'y', etc. to refer only to people, and when we talked about everything, we would let the variables be unrestricted. It is actually very easy to make precise this idea of restricting the universe of discourse. If we want the universe of discourse to be restricted to people, we simply declare that all the objects in our interpretations must be people. If we want a universe of discourse consisting only of cats, we declare that all the objects in our interpretations must be cats. And so on.

   As I mentioned, this common practice is not fully satisfactory. What if we want to talk about people and things, as when we assert, 'Everyone likes sweet things.'? Restricted quantifiers will help us out here. They also have the advantage of getting what we need explicitly stated in the predicate logic sentences themselves.

   We could proceed by using '(Ǝx)' and '(Vx)' to mean 'something' and 'everything' and introduce new quantifiers for 'someone' and 'everyone'. To see how to do this, let's use the predicate 'P' to stand for 'is a person.' Then we can introduce the new quantifier '(Ǝx)p' to stand for some x chosen from among the things that are P, that is, chosen from among people. We call this a restricted quantifier. You should think of a restricted quantifier as saying exactly what an unrestricted quantifier says except that the variable is restricted to the things of which the subscripted predicate is true. With 'P' standing for 'is a person', '(Ǝx)p' has the effect of 'someone' or 'somebody'. We can play the same game with the universal quantifier. '(Vx)p' will mean all x chosen from among the things that are P. With 'P' standing for 'is a person', ('Vx)p' means, not absolutely everything, but all people, that is, everyone or everybody or anyone or anybody.

   This notion of a restricted quantifier can be useful for other things. Suppose we want to transcribe 'somewhere' and 'everywhere' or 'sometimes' and 'always'. Let's use 'N' stand for 'is a place' or 'is a location'. 'Somewhere' means 'at some place' or 'at some location'. So we can transcribe 'somewhere' as '(Ǝx)N' and'everywhere' as '(Vx)N'. For example, to transcribe 'There is water everywhere', I would introduce the predicate 'Wx' to stand for 'there is water at x'. Then '(Vx)NWx' says that there is water everywhere. Continuing the same strategy, let's use 'Q' to stand for 'is a time'. Then '(Ǝx)Q' stands for 'sometime(s)' and '(Vx)Q' stands for 'always' ('at all times').

   In fact, we can also use the same trick when English has no special word corresponding to the restricted quantifier. Suppose I want to say something about all cats, for example, that all cats are furry. Let 'Cx' stand for 'x is a cat' and 'Fx' stand for 'x is furry'. Then '(Vx)CFx' says that all things which are cats are furry; that is, all cats are furry. Suppose I want to say that some animals have tails. Using 'Ax' for 'x is an animal' and 'Txy' for 'x is a tail of y', I write '(Ǝx)A(Ǝy)Tyx': There is an animal, x, and there is a thing, y, such that y is a tail of x.

   As you will see, restricted quantifiers are very useful in figuring out transcriptions, but there is a disadvantage in introducing them as a new kind of quantifier in our system of logic. If we have many different kinds of quantifiers, we will have to specify a new rule for each of them to tell us the conditions under which a sentence formed with the quantifier is true. And when we get to checking the validity of arguments, we will have . to have a new rule of inference to deal with each new quantifier. We could state the resulting mass of new rules in a systematic way. But the whole business would still require a lot more work. Fortunately, we don't have to do any of that, for we can get the full effect of restricted quantifiers with the tools we already have.

   Let's see how to rewrite subscripted quantifiers. Consider the restricted quantifier '(Ǝx)C' which says that there is cat such that, or there are cats such that, or some cats are such that. We say 'some cats are furry' (or 'there is a furry cat' or the like) with '(Ǝx)CFx'. Now what has to be true for it to be true that some cats are furry, or that there is a furry cat? There has to be one or more things that is both a cat and is furry. If there is not something which is both a cat and is furry, it is false that there is a furry cat. So we can say that some cats are furry by writing '(Ǝx)(Cx & Fx)'. In short, we can faithfully rewrite '(Ǝx)CFx' as '(Ǝx)(Cx & Fx)'. This strategy will work generally:

   Rule for rewriting Subscripted Existential Quantifiers: For any predicate S, any sentence of the form (Ǝu)s(. . . u. . .) is shorthand for (Ǝu)[Su & (. . . u . . .)I.

Here are some examples:

Some cats are blond.       (Ǝx)CBx               (Ǝx)(Cx & Bx)
Eve loves a cat.               (Ǝx)CLex              (Ǝx)(Cx & Lex)
Eve loves a furry cat.       (Ǝx)C(Fx & Lex)    (Ǝx)[Cx & (Fx & Lex)]

   Clearly, we can proceed in the same way with 'someone' and 'somebody':

Someone loves Eve.               (Ǝx)pLxe              (Ǝx)(Px & Lxe)
Somebody loves Eve or Adam. (Ǝx)p(Lxe v Lxa)  (Ǝx)[Px &(Lxe v Lxa)]
If somebody loves Eve, then Eve loves somebody.
                                            (Ǝx)pLxe  ⊃         (Ǝx)(Px & Lxe) ⊃ (Ǝx)(Px & Lex)
                                            (Ǝx)pLex

Notice that in the last example I used the rule for rewriting the subscript on each of two sentences X and Y, inside a compound sentence, X ⊃ Y.

   How should we proceed with restricted universal quantifiers? This is a little tricky. Let's work on '(Vx)CFx'-that is, 'All cats are furry'. Under what conditions is this sentence true? To answer the question, imagine that everything in the world is lined up in front of you: All the cats, dogs, people, stones, basketballs, everything. You go down the line and examine the items, one by one, to determine whether all cats are furry. If the first thing in line is a dog, you don't have to determine whether or not it is furry. If the second thing is a basketball, you don't have to worry about it either. But as soon as you come to a cat you must examine it further to find out if it is furry. When you finally come to the end of the line, you will have established that all cats are furry if you have found of each thing that, if it is a cat, then it is furry. In short, to say that all cats are furry is to say '(Vx)(Cx ⊃ Fx)'.

   At this point, many students balk. Why, they want to know, should we rewrite a restricted universal quantifier with the '⊃' when we rewrite a restricted existential quantifier with the '&'? Shouldn't '&' work also for restricted universal quantifiers? Well, I'm sorry. It doesn't. That is just not what restricted universal quantifiers mean.

   You can prove this for yourself by trying to use '&' in rewriting the subscripted 'C' in our transcription of 'All cats are furry.' You get

(1) (Vx)(Cx & Fx)

What does (1) say? It says that everything is a furry cat, and in particular that everything is a cat! That's much too strong. All cats could be furry even though there are lots of things which are not cats. Thus 'All cats are furry' could be true even when (1) is false, so that (1) cannot be the right way to rewrite '(Vx) CFx'.

   What has gone wrong? The unrestricted universal quantifier applies to everything. So we can use conjunction in expressing the restriction of cats only if we somehow disallow or except the cases of noncats. We can do this by saying that everything is either not a cat or is a cat and is furry:

(2) (Vx)[~Cx v (Cx & Fx)]

(2) does indeed say what 'All cats are furry' says. So (2) should satisfy your feeling that an '&' also comes into the restricted universal quantifier in some way. But you can easily show that (2) is logically equivalent to '(Vx)(Cx ⊃ Fx)'! As the formulation with the '⊃' is more compact, and is also traditional, it is the one we will use.

   In general, we rewrite restricted universal quantifiers according to the rule

   Rule for rewriting Subscripted Universal Quantifers: For any predicate S, any sentence of the form (Vu)s(. . . u . . .) is shorthand for (Vu)[Su ⊃ (. . . u . . .)].

   Here are some examples to make sure you see how this rule applies:

Eve loves all cats.           (Vx)C(Lex)     (Vx)(Cx ⊃ Lex)
Everybody loves Eve.      (Vx)PLxe         (Vx)(Px ⊃ Lxe)
Everyone loves either Adam or Eve.
                    (VX)P(Lxa v Lxe)           (Vx)[Px ⊃ (Lxa v Lxe)]
Not everyone loves both Adam and Eve.
                    ~(Vx)P(Lxa & Lxe)           ~(Vx)[Px ⊃ (Lxa & Lxe)]

In the last example, I used the rewriting rule on a sentence, X, inside a negated sentence of the form ~X.

   If you are still feeling doubtful about using the '⊃' to rewrite restricted universal quantifiers, I have yet another way to show you that this way of rewriting must be right. I am assuming that you agree that our way of rewriting restricted existential quantifiers is right. And I will use a new rule of logical equivalence. This rule tells us that the same equivalences that hold for negated unrestricted universal quantifiers hold for negated restricted universal quantifiers. In particular, saying that not all cats are furry is clearly the same as saying that some cat is not furry. In general

   Rule ~VS: A sentence of the form ~(Vu)S(. . . u . . .) is logically equivalent to (Ǝu)S(. . . u . . .).

You can prove this new rule along the same lines we used in proving the rule ~V.
   Now, watch the following chain logical equivalents:

1 (VU)S(. . . u . . .)
2 ~~(VU)S(. . . u . . .)                DN
3 ~(Ǝ~)S~(. . . u . . .)                ~VS
4 ~(Ǝu)[Su & ~(. . . u . . .)]        Rule for rewriting (Ǝu)S
5 ~(Ǝu)~~[Su & ~(. . . u . . .)]    DN
6 ~(Ǝu)~[~Su v (. . . u . . .)]       DM, DN
7 ~(Ǝu)~[Su ⊃ (. . . u . . .)]         C
8 ~~(Vu)[Su ⊃ (. . . u . . .)]         ~Ǝ
9 (Vu)[Su ⊃ (. . . u . . .)]             DN

Since the last line is logically equivalent to the first, it must be a correct way of rewriting the first.

   If you are having a hard time following this chain of equivalents, let me explain the strategy. Starting with a restricted universal quantifier, I turn it into a restricted existential quantifier in lines 2 and 3 by using double denial and pushing one of the two negation signs through the restricted quantifier. I then get line 4 by using the rule we have agreed on for rewriting restricted existential quantifiers. Notice that 1. am applying this rule inside a negated sentence, so that here (and below) I am really using substitution of logical equivalents. In lines 5, 6, and 7 I use rules of logical equivalence to transform a conjunction into a conditional. These steps are pure sentence logic. They involve no quantifiers. Line 8 comes from line 7 by pushing the negation sign back out through what is now an unrestricted existential quantifier, changing it into an unrestricted universal quantifier. Finally, in line 9, I drop the double negation. It's almost like magic!

Exercise

4-1. Give an argument which shows that the rule ~VS is correct. Similarly, show that

   Rule X: a sentence of the form ~(Ǝu)S(. . . u . . .) is logically equivalent to (VU)S~(. . . u . . .).

is also correct.

4-2. Use the rule ~ƎS to show that, starting from the rule for rewriting subscripted universal quantifiers, you can derive the rule for rewriting subscripted existential quantifiers. Your argument will closely follow the one given in the text for arguing the rule for rewriting subscripted universal quantifiers from the rule for rewriting subscripted existential quantifiers.

4-3. Transcribe the following English sentences into the language of predicate logic. Use this procedure: In a first step, transcribe into a sentence using one or more subscripted quantifiers. Then rewrite the resulting sentence using the rules for rewriting subscripted quantifiers. Show both your first and second steps. Here are two examples of sentences to transcribe and the two sentences to present in presenting the problem:

Someone loves Eve.           All cats love Eve.
      (Ǝ~)PLxe                       (Vx)CLxe
      (Ǝx)(Px & Lxe)                (Vx)(Cx ⊃ Lxe)


Transcription Guide
e: Eve                         Dx: x is a dog
Px: x is a person          Bx: x is blond
Cx: x is a cat                Lxy: x loves y

a) Everyone loves Eve.
b) Eve loves somebody.
c) Eve loves everyone.
d) Some cat loves some dog.
e) Somebody is neither a cat nor a dog.
f) Someone blond loves Eve.
g) Some cat is blond.
h) Somebody loves all cats.
i) No cat is a dog.
j) Someone loves someone.
k) Everybody loves everyone.
l) Someone loves everyone.
m) Someone is loved by everyone.
n) Everyone loves someone.
o) Everyone is loved by somebody.

 

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