Shantila's Inside Logic #26

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.

157 Dm, m=e } De

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

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

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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.

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. How might we say that? Well, consider first that if Cx says x is cold, it is of course consistent to say that Bob is cold, Cb, and also that somebody else, Anne, also is cold, Ca. That is,

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').

*

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.

*

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.

One obvious such area of everyday reasoning is simple arithmetic. Identity is a fundamental notion in arithmetic (as in statements like 2+3=5). 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 Godel 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 can be fun to be involved in doing it.

* Practice

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

26.2 Symbolize each of the following sentences.

(a) Joe is the same as Ben.

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

(c) Only Joe is funny.

(d) Only turtles are funny. (This type of sentence was discussed in chapter 22. Note how this differs from the preceding problem c. In this sentence we are not using any names.)

(e) Only Joe is cold.

(f) Sally is not Joe.

(g) Only turtles are cold.

26.3 Construct derivations to show that each of the following sequents is valid. Use any of the Rules that we have discussed since the beginning of the semester.

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

160 Fa, ~Fc } ~a=c

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

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

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

26.4 Symbolize the following reasoning and construct derivations to show that the reasoning is valid.

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

(b) Only criminals should be in jail. Joe is the only criminal. Sally is not Joe. So Sally should not be in jail. (as above, also s: Sally, j: Joe)