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1.2: Quantifiers and Variables

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  • We still have not done enough to deal with arguments (1) and (2). The sentences in these arguments not only attribute properties and relations to things, but they involve a certain kind of generality. We need to be able to express this generality, and we must be careful to do it in a way which will make the relevant logical form quite clear. This involves a way of writing general sentences which seems very awkward from the point of view of English. But you will see how smoothly everything works when we begin proving the validity of arguments.

    English has two ways of expressing general statements. We can say 'Everyone loves Adam.' (Throughout, 'everybody' would do as well as 'everyone'.) This formulation puts the general word 'everyone' where ordinarily we might put a name, such as 'Eve'. Predicate logic does not work this way. The second way of expressing general statements in English uses expressions such as 'Everyone is such that they love Adam.' or 'Everything is such that it loves Adam.' Predicate logic uses a formulation of this kind.

    Read the symbol '(Vx)' as 'Every x is such that'. Then we transcribe 'Everyone loves Adam.' as '(Vx)Lxa'. In words, we read this as "Every x is such that x loves Adam." '(Vx)' is called a Universal Quantifer. In other logic books you may see it written as (x).

    We are going to need not only a way of saying that everyone loves Adam but also a way of saying that someone loves Adam. Again, English does this most smoothly by putting the general word 'someone' where we might have placed a name like 'Eve'. And again logic does not imitate this style. Instead, it imitates English expressions such as 'Someone is such that he or she loves Adam.', or 'Some person is such that he or she loves Adam.', or 'Something is such that it loves Adam.' Read the symbol '(Ǝx)' as 'Some x is such that'. Then we transcribe 'Someone loves Adam.' as '(Ǝx)Lxa'. '(Ǝx)' is called an Existent Quantifier.

    In one respect, '(Ǝx)' corresponds imperfectly to English expressions which use words such as 'some', 'there is a', and 'there are'. For example, we say 'Some cat has caught a mouse' and 'There is a cat which has caught a mouse' when we think that there is exactly one such cat. We say 'Some cats have caught a mouse' or 'There are cats which have caught a mouse' when we think that there are more than one. Predicate logic has only the one expression, '(Ǝx)', which does not distinguish between 'exactly one' and 'more than one'. '(Ǝx)' means that there is one or more x such that. (In chapter 9 we will learn about an extension of our logic which will enable us to make this distinction not made by '(Ǝx)'.)

    In English, we also make a distinction by using words such as 'Everyone' and 'everybody' as opposed to words like 'everything'. That is, English uses one word to talk about all people and another word to talk about all things which are not people. The universal quantifier, '(Vx)', does not mark this distinction. If we make no qualification, '(Vx), means all people and things. The same comments apply to the existential quantifier. English contrasts 'someone' and 'somebody' with 'something'. But in logic, if we make no qualification, '(Ǝx)' means something, which can be a person or a thing. All this is very inconvenient when we want to transcribe sentences such as 'Someone loves Adam.' and 'Everybody loves Eve.' into predicate logic.

    Many logicians try to deal with this difficulty by putting restrictions on the things to which the 'x' in '(Vx)' and '(Ǝx)' can refer. For example, in dealing with a problem which deals only with people, they say at the outset: For this problem 'x' will refer only to people. This practice is called establishing a Universe of Discourse or Restricting the Domain of Discourse. I am not going to fill in the details of this common logical practice because it really does not solve our present problem. If we resolved to talk only about people, how would we say something such as 'Everybody likes something.'? In chapter 4 I will show you how to get the effect of restricting the domain of discourse in a more general way which will also allow us to talk at the same time about people, things, places, or whatever we like.

    But until chapter 4 we will make do with the intuitive idea of restricting 'x' to refer only to people when we are transcribing sentences using expressions such as 'anybody', 'no one', and 'someone'. In other words, we will, for the time being indulge in the not quite correct practice of transcribing '(Vx)' as 'everyone', 'anybody', etc., and '(Ǝx)' as 'someone', 'somebody', or the like, when this is the intuitively right way to proceed, instead of the strictly correct 'everything', 'something', and similar expressions.

    The letter 'x' in '(Vx)' and '(Ǝx)' is called a Variable. Variables will do an amazing amount of work for us, work very similar to that done by English pronouns, such as 'he', 'she', and 'it'. For example, watch the work 'it' does for me when I say the following: "I felt something in the closed bag. It felt cold. I pulled it out." This little discourse involves existential quantification. The discourse begins by talking about something without saying just which thing this something is. But then the discourse goes on to make several comments about this thing. The important point is that all the comments are about the same thing. This is the work that 'it' does for us. It enables us to cross-reference, making clear that we are always referring to the same thing, even though we have not been told exactly what that thing is.

    A variable in logic functions in exactly the same way. For example, once we introduce the variable 'x' with the existential quantifier, '(Ǝx)' we can use 'x' repeatedly to refer to the same (unknown) thing. So I can say, 'Someone is blond and he or she loves Eve' with the sentence '(Ǝx)(Bx &Lxe)'. Note the use of parentheses here. They make clear that the quantifier '(Ǝx)' applies to all of the sentence 'Bx & Lxe'. Like negation, a quantifier applies to the shortest full sentence which follows it, where the shortest full following sentence may be marked with parentheses. And the 'x' in the quantifier applies to, or is linked to, all the occurrences of 'x' in this shortest full following sentence. We say that

    A quantifier Governs the shortest full sentence which follows it and Binds the variables in the sentence it governs. The latter means that the variable in the quantifier applies to all occurrences of the same variable in the shortest full following sentence.

    Unlike English pronouns, variables in logic do not make cross-references between sentences.

    These notions actually involve some complications in sentences which use two quantifiers, complications which we will study in chapter 3. But this rough characterization will suffice until then.

    Let us look at an example with the universal quantifier, '(Vx)'. Consider the English sentences 'Anyone blond loves Eve.', 'All blonds love Eve.','Any blond loves Eve.', and 'All who are blond love Eve.' All these sentences say the same thing, at least so far as logic is concerned. We can express what they say more painstakingly by saying, 'Any people are such that if they are blond then they love Eve.' This formulation guides us in transcribing into logic. Let us first transcribe a part of this sentence, the conditional, which talks about some unnamed people referred to with the pronoun 'they': 'If they are blond then they love Eve.' Using the variable 'x' for the English pronoun 'they', this comes out as 'Bx ⊃ Lxe'. Now all we have to do is to say that this is true whoever "they" may be. This gives us '(Vx)(Bx ⊃ Lxe)'. Note that I have enclosed 'Bx ⊃ Lxe' in parentheses before prefixing the quantifier. This is to make clear that the quantifier applies to the whole sentence.

    I have been using 'x' as a variable which appears in quantifiers and in sentences governed by quantifiers. Obviously, I would just as well have used some other letter, such as 'y' or 'z'. In fact, later on, we will need to use more than one variable at the same time with more than one quantifier. So we will take '(Vx)', '(Vy)', and '(Vz)' all to be universal quantifiers, as well as any other variable prefixed with 'V' and surrounded by parentheses if we should need still more universal quantifiers. In the same way, '(Ǝx)', '(Ǝy)', and '(Ǝz)' will all function as existential quantifiers, as will any similar symbol obtained by substituting some other variable for 'x', 'y', or 'z'.

    To make all this work smoothly, we should clearly distinguish the letters which will serve as variables from other letters. Henceforth, I will reserve lowercase 'w', 'x', 'y', and 'z' to use as variables. I will use lowercase 'a' through 'r' as names. If one ever wanted more variables or names, one could add to these lists indefinitely by using subscripts. Thus 'a1,' and 'd17,' are both names, and 'x1' and 'z34( are both variables. But in practice we will never need that many variables or names.

    What happened to 's', 't', 'u', and 'v'? I am going to reserve these letters to talk generally about names and variables. The point is this: As I have mentioned, when I want to talk generally in English about sentences in sentence logic, I use boldface capital 'X', 'Y', and 'Z'. For example, when I stated the & rule I wrote, "For any sentences X and Y. . . ." The idea is that what I wrote is true no matter what sentence you might write in for 'X' and 'Y'. I will need to do the same thing when I state the new rules for quantifiers. I will need to say something which will be true no matter what names you might use and no matter what variables you might use. I will do this by using boldface 's' and 't' when I talk about names and boldface 'u' and 'v' when I talk about variables.
    To summarize our conventions for notation:

    We will use lowercase letter 'a' through 'r' as names, and 'w', 'x', 'y' and '2' as variables. We will use boldface 's' and 't' to talk generally about names and boldface 'u' and 'v' to talk generally about variables.

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