Shantila's Inside Logic #25

Identity

Two different names can be used to refer the same thing. For example, "the New York Yankees" and "the Bronx Bombers" both refer to a certain baseball team. Likewise, "Marshall Mathers" and "Eminem" are two distinct names that refer to one single person.

In other words (as we naturally say) Marshall Mathers is Enimen; and The New York Yankees are the Bronx Bombers.

Now that we have names as part of our language, it is natural to think about extending the language so as to be able to represent sentences like this. These sentences are true when the two names refer to the same thing. This is the concept of identity. It is a simple concept. We will use the symbol = to express identity.

We use this symbol more generally than in arithmetic, but we use it in a way that also includes its use in arithmetic. It is more general because in arithmetic all the names being used refer to numbers, whereas we also represent names of people, places, things, indeed anything we wish to name.

If a and c are constants, then a=c is a sentence of our language that says that a is the same as c. For example, if we use small letters m and e so that

m: Marshall Mathers

and

e: Eminem

then the sentence m=e says that Marshall Mathers is the same as Eminem.

Reasoning with identity is straightforward. Suppose we know that Marshall Mathers was born in Detroit, which we can represent as Dm (where we are using Dx: x was born in Detroit, m: Marshall Mathers), and suppose we know that m=e, Marshall Mathers is the same as Enimen. Obviously it follows that Eminem was born in Detroit, De. That is, the following sequent is valid.

Dm, m=e } De

This valid sequent is something we can prove given the following new Rule.

134 Dm, m=e } De

Suppose we also have as a premise Rm, where Rx says x is a rapper. This sequent also is valid.

135 Dm, m=e, Rm } De & Re

*

With identity as part of the language, we now can represent statements that we could not represent before. In what follows we will look at some examples of more complex sentences we can represent.

Consider, for example, only Bob is cold. If Cb says Bob is cold, it is of course consistent to say that somebody else, Anne, also is cold, Ca.

Ba & Ca

But, now, to say that only Bob is cold we can say the following:

Cb & (x)(Cx > x=b)

This says Bob is cold, and for each x, if x is cold then x is the same as Bob. That is, nothing else is cold other than Bob, in which case of course Ca would not be true unless b=a (which is possible, notwithstanding that it would be odd for one person to be named both 'Bob' and 'Anne').

We also can say generally (without talking about Bob or using any name at all) that only one thing is cold. This can be represented in our language as follows:

(Ex)(Cx & (y)(Cy > y=x))

Notice that this statement obviously follows from the statement that only Bob is cold, which we can prove in one step using the rule EG introduced earlier, as follows.

136 Cb & (x)(Cx > x=b) } (Ex)(Cx & (y)(Cy > y=x))

It may look complex but all we have done here is to replace each occurrence of name b with the variable y, and added the existential quantifier (Ey). Notice that the sentence we end up with on line 2 has two distinct variables x and y in it.

*

Identity statements like m=e can be negated. The negation of an identity statement, such as Kid Rock is not the same as Eminem, we can represent as follows (for k: Kid Rock, e: Eminem):

~k=e

This says Kid Rock is different from (not the same as) Eminem.

*

We can go on to use the negated identity statement as follows in the representation of a statement like at least two people are cold:

(Ex)(Ey)((Cx & Cy) & ~x=y)

This says that there are things x and y such that Cx and Cy and ~x=y. Notice that we use two different existential quantifiers and two distinct variables here in the same sentence. Earlier we let the sentence (Ex)Fx represent "something is F" and agreed to treat this as true just so long as at least one thing has the property F; but some uses of "some," as in "there are some cookies in the cookie jar" mean there are at least two -- we now can represent this sort of statement as just indicated>

*

We also can represent at most one person is cold:

(x)(Cx > (y)(Cy > y=x))

Notice there here we use two universal quantifiers in the same sentence. (This differs from saying only one person is cold because the sentence here does not guarantee that there is someone who is cold; it says only that no more than one person is cold, and that is consistent with nobody at all being cold.)

*

Definite descriptions are phrases that refer to an individual, using a description, and their use is based on the assumption that the individual who satisfies the description is the only one who satisfies the description. For example, consider the guy who is cold is friendly. If Bob is the only guy who is cold, then this sentence will entail that Bob is friendly. The definite description in this sentence is the phrase, the guy who is cold, and we can begin to represent the sentence by saying

(Ex)(Cx & (y)(Cy > y=x))

-- recall from above that this says only one individual is cold -- and then we conjoin Fx, saying and that individual is friendly, as follows:

(Ex)((Cx & (y)(Cy > y=x)) & Fx)

*

These points about "identity" and the earlier chapters about "some" and "all" are only the beginning of the development of a system for reasoning with the ideas of all, some, no, same, is, and related concepts. Other rules can be added.

We will not look further into the specific development of this system in this course --because we are running out of time. You can explore these points further in a more advanced logic course or on your own. The last few chapters give you an idea of how our symbolic language and our formal system of Rules can be enriched so as to interpret increasingly complex forms of everyday reasoning.

One obvious such area of everyday reasoning is simple arithmetic. Identity is a fundamental notion in arithmetic (as in statements like 2+3=5).

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The basic logic that we have developed in this course can be extended to include all of arithmetic (by adding symbols and Rules to talk about addition, subtraction, and so forth). One interesting result of extending the language and logic to include arithmetic is that it can be shown that it is not possible to develop a consistent extension that captures all the truths of arithmetic! (This result is known as the "Incompleteness of arithmetic".) Recall that earlier we said that the Rules for and, or, if, and not gave us a "Complete" system and so it is -- all the basic logical truths about these notions can be proved given the Rules in the system that we developed. What the Incompleteness of arithmetic means is that no matter what Rules we add to our system to try to capture the truths of arithmetic, we will not be able to capture all them unless we add so much that we would also be able to prove contradictions! This is an astonishing result. The proof of the Incompleteness of arithmetic by Kurt G–del less than 100 years ago is one of the stunning intellectual achievements of the 20th century.

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There are many open ongoing areas of research by philosophers and logicians concerning the interpretation and formalization of ordinary language and reasoning. The process of developing richer languages and more refined logics can deepen our understanding of our own minds as well as of natural languages like English. It also can shed light on the ways in which we communicate successfully with each other and collectively learn about the world. The development of logic requires creativity and patience and it is fun to be involved in doing it.

* Practice

25.1 Give an example in English of a true identity statement concerning one of your friends or a family member.

25.2 Symbolize each of the following sentences.

(a) Joe is the same as Ben.

(b) Madonna is not the same as Kid Rock.

(c) There is only one turtle.

(d) There are at least two turtles.

(e) There is at most one turtle.

(f) The only turtle is lonely.

(g) Madonna is the only pop star in town.

(h) At most one rapper got sued by his mother.

(i) At least two people are friendly.

25.3 Construct derivations to show that each of the following sequents is valid, using our Rules A, MP, DN, MT, CP, &I, &E, vI, DS, RA, UI, EG, and =E.

137 Ka, Fc, a=c } (Ex)(Kx & Fx)

138 Fa, ~Fc } ~a=c

139 Fa & (y)(Fy > y=a), Fc } c=a

140 a=b, Fa } ~(Fc & ~Fb)

141 Fa > a=c, Fc > Hc } Fa > Hc

25.4 Symbolize each of the following and construct derivations to show that the reasoning is valid in each case.

(a) Jim is cold. At most one person is cold. Bob is cold. So Bob is the same as Jim. (Cx: x is cold; j: Jim; b: Bob)

(b) Jim is cold. My fridge is cold. Jim is not the same my fridge. So at least two things are cold. (f: my fridge)

(c) Madonna is a friendly person. Eminem is a friendly person. Madonna is not Eminem. So there are at least two friendly people. (Fx: x is friendly; Px: x is a person; m: Madonna; e: Eminem).

(d) All criminals definitely should be in jail. All thiefs are criminals. Professor X is a thief. Professor X is identical with Mister Blazer. So Mister Blazer definitely should be in jail. (Cx: x is a criminal; Jx: x definitely should be in jail; Tx: x is a thief; c: Professor X; b: Mister Blazer)

25.5 Extra Credit.

Symbolize each of the following and construct derivations to show that the reasoning is valid in each case.

(a) Joe is the only guy who married Sally. Joe is friendly. So the guy who married Sally is friendly. (Let Mx: x is a guy who married Sally; Fx: x is friendly; j: Joe)

(b) Eminem was born in Detroit. Eminem is only the rapper who got sued by his mother. So the rapper who got sued by his mother was born in Detroit. (Dx: x was born in Detroit; Rx: x is a rapper; Sx: x got sued by his mother; e: Eminem)

(c) Bob is the only guy who is nice. Bob just left. So the guy who is nice just left. (Nx: x is nice; Lx: x just left)

25.6 Extra Credit.

Use our rules to prove that this sequent is valid.

142 a=b } Ka v ~Kb