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9.1: Identity

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    Identity, Functions, and Definite Descriptions

    9-1. IDENTITY

    Clark Kent and Superman would seem to be entirely different people. Yet it turns out they are one and the same. We say that they are Identical. Since identity plays a special role in logic, we give it a permanent relation symbol. We express 'a is identical to b' with 'a= b', and the negation with either '-(a = b)' or 'a # b'.

    '=' is not a connective, which forms longer sentences from shorter sentences. '= ' is a new logical symbol which we use to form atomic sentences out of names and variables. But as we did with the connectives, we can explain exactly how to understand '=' by giving truth conditions for closed sentences in interpretations. Just follow the intuitive meaning of identity: To say that s= t is to say that the thing named by s is identical to the thing named by t; that is, that the names s and t refer to the same object. (Logicians say that s and t have the same referent, or that they are Co-Referential.) To summarize

    '=' flanked by a name or a variable on either side is an atomic sentence. If s and t are names, t= s is true in an interpretation if s and t name the same thing. s=t is false if s and t name different things. The negation of an identity sentence can be written either as -(s=t) or as sf t.

    Identity is easy to understand, and it is extraordinarily useful in expressing things we could not say before. For example, '(3x)' means that there is one or more x such that. . . . Let's try to say that there is exactly one x such that . . . , for which we will introduce the traditional expres- . sion '(3x!)' (read "E shriek"). We could, of course, introduce '(3x!)' as a new connective, saying, for example, that '(3x!)Bx1 is true in an interpretation just in case exactly one thing in the interpretation is B. But, with the help of identity, we can get the same effect with the tools at hand, giving a rewriting rule for '(gx!)' much as we did for subscripted quantifiers in chapter 4.

    To say that there is exactly one person (or thing) who is blond is to say, first of all, that someone is blond. But it is further to say that nothing else is blond, which w'e can reexpress by saying that if anything is blond, it must be (that is, be identical to) that first blond thing. In symbols, this is '(3x)[Bx & (Vy)(By 3 y = x)r.

    Before giving a general statement, I want to introduce a small, new expository device. Previously I have used the expression '(. . . u . . .)' to stand for an arbitrary sentence with u the only free variable. From now on I am going to use expressions such as P(u) and Q(u) for the same thing:

    Boldface capital letters followed by a variable in parentheses, such as P(u) and Q(u), stand for arbitrary sentences in which u, and only u, may be free. Similarly, R(u,v) stands for an arbitrary sentence in which at most u and v are free.

    In practice P(u), Q(u), and R(u,v) stand for open sentences with the indicated variable or variables the only free variable. However, for work in Part I1 of this Volume, I have written the definition to accommodate degenerate cases in which u, or u and v, don't actually occur or don't occur free. If you are not a stickler for detail, don't worry about this complication: Just think of P(u), Q(u), and R(u,v) as arbitrary open sentences. But if you want to know why I need, to be strictly correct, to cover degenerate cases, you can get an idea from exercise 13-3. With this notation we can give the E! rewrite rule:

    Rule for reuniting 31: For any open formula P(u) with u a free variable, (3u!)P(u) is shorthand for (3u)[P(u) & (v)(P(v) 3 v=u)], where v is free for u in P(u), that is, where v is free at all the places where u is free in P(u).

    Once you understand how we have used '=' to express the idea that exactly one of something exists, you will be able to see how to use '=' to express many related ideas. Think through the following exemplars until you see why the predicate logic sentences expresses what the English expresses:

    • There are at least two x such that Fx: (3x)(3y)[x f y & Fx & Fy]. 1
    • There are exactly two x such that Fx: (3x)(3y){xf y & Fx & Fy & (Vz)[Fz 3 (z =x v z=y)]).
    • There are at most two x such that Fx: (Vx)(Vy)(Vz)[(Fx & Fy & Fz) 3 (x= y v x= z v y = z)].

    We can also use '=' to say some things more accurately which previously we could not say quite correctly in predicate logic. For example, when we say that everyone loves Adam, we usually intend to say that everyone other than Adam loves Adam, leaving it open whether Adam loves himself. But '(Vx)' means absolutely everyone (and thing), and thus won't exempt Adam. Now we can use '=' explicitly to exempt Adam:

    Everyone loves Adam (meaning, everyone except possibly Adam himself): (Vx)(x f a 3 Lxa).

    In a similar way we can solve a problem with transcribing 'Adam is the tallest one in the class'. The problem is that no one is taller than themself, so we can't just use '(Vx)', which means absolutely everyone. We have to say explicitly that Adam is taller than all class members except Adam.

    Adam is the tallest one in the class:

    (Vx)[(Cx & xf a) 3 Tax].

    To become familiar with what work '=' can do for us in transcribing, make sure you understand the following further examples:

    Everyone except Adam loves Eve: (Vx)(xf a 3 Lxe) & -be. Only Adam loves Eve: (Vx)(Lxe = x = a), or Lae & (Vx)(Lxe 3 x = a). Cid is Eve's only son: (Vx)(Sxe = x=c), or Sce & (Vx)(Sxe 3 x=c).

    Exercise \(\PageIndex{1}\)

    EXERCISES 9-1. Using Cx: x is a clown, transcribe the following: a) There is at least one clown. b) There is no more than one clown. c) There are at least three clowns. d) There are exactly three clowns. e) There are at most three clowns.

    Exercise \(\PageIndex{2}\)

    9-2. Use the following transcription guide: a: Adam Sxy: x is smarter than y e: Eve Qxy: x is a parent of y Px: x is a person Oxy: x owns y Rx: x is in the classroom Mxy: x is a mother of y Cx: x is a Cat Fx: x is furry Transcribe the following: Three people love Adam. (Three or more) Three people love Adam. (Exactly three) Eve is the only person in the classroom. Everyone except Adam is in the classroom. Only Eve is smarter than Adam. Anyone in the classroom is smarter than Adam. Eve is the smartest person in the classroom. Everyone except Adam is smarter than Eve. Adam's only cat is furry. Everyone has exactly one maternal grandparent. No one has more than two parents.

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