[9.] Relational Predicate Logic with Identity.

 

[9.1.] Identity. (Ch. 13:1)

 

The English word “is” has different senses in the following two sentences:

 

Clark Kent is a reporter.

Clark Kent is Superman.

 

In the first sentence, “is” is used to predicate a property, viz. being a reporter, of Clark Kent. This sentence is correctly symbolized:

 

Rk                    [k: Clark Kent; Rx: x is a reporter]

 

But in the second sentence, “is” is doing something very different. It is being used to assert that Clark Kent is one and the same individual as Superman. In other words, “is” is used to assert that Clark Kent and Superman are numerically identical.

 

numerical (quantitative) identity (df.): x and y are numerically identical if and only if x is one and the same thing as y, e.g., Clark Kent is numerically identical to Superman.

 

Numerical identity is not the same thing as…

 

qualitative identity (df.): x and y are qualitatively identical if and only if x and y have all of their properties in common (except for exact location in space-time), e.g., these two shirts are qualitatively identical (both are green polo shirts, made in China in 2007, sold at the Gap, etc.).

 

It would be incorrect to define a property constant “Sx” as “x is Superman” and then symbolize the second sentence as

 

Sk

 

This is because being numerically identical to something is not a property that an individual can possess on its own. When we say that Clark Kent is identical to Superman, we are not ascribing a property (being identical to Superman) to Clark Kent. We are instead saying that there is a relational property in which something stands to something (and in this case, the two somethings are the same thing!).

 

In order to symbolize “Clark Kent is Superman,” we need a new symbol for numerical identity:

 

=

 

With this symbol, we can symbolize “Clark Kent is Superman” as follows:

 

k=s       [k: Clark Kent; s: Superman]

 

 

And similarly:  Mark Twain is Samuel Clemens.                      t=c

George Eliot is Mary Ann Evans.                     e=m

Bruce Banner is the Hulk.                                b=h

Ted Kaczynski is the Unabomber.                   k=u

 

We could also use “=” to deny that entities are numerically identical. But in our system, instead of writing using “=” and writing

 

~(k=s)              [k: Clark Kent; s: Superman]

 

we instead use the symbol for non-identity,  ¹   , and write

 

k¹s

 

 

[9.2.] Rules for Identity. (Ch. 13.1).

 

It is essential that we include the identity symbol in our system of logic, because without it, some valid arguments cannot be proved to be valid within that system. For example:

 

1.      Rk

2.      k=s             /\ Rs

 

(Clark Kent is a reporter; Clark Kent is Superman; so, Superman is a reporter.)

 

To prove this argument valid within our system, we also need a new rule:

 

Identity (ID): you may substitute identicals for identicals; where “u” and “w” are individual constants or quasi-variables, the following is valid:

 

1.      (...u...)

2.      u=w   / \ (...w...)

 

or

 

1.      (...u...)

2.      w=u   / \ (...w...)

 

With this rule, we can complete the proof:

 

1.      Rk                          p

2.      k=s                         p     / \ Rs

3.      Rs                          1, 2 ID

 

A second example, this one with a relational predicate:

 

1. Lkl                           p                      [Lxy: x loves y; k: Clark Kent; l: Lois Lane]

2. k=s                           p   /\ Lsl         [s: Superman]

3. Lsl                           1, 2 ID

 

 

A second rule we will sometimes (although infrequently) need is

 

Identity Reflexivity (IR): you may introduce “(x)(x = x)” into a proof at any time.

 

The name of the rule comes from reflexivity, which is a property possessed by some properties:

 

totally reflexive (df.): a relation Fxy is totally reflexive if and only if everything must bear that relation to itself, i.e., if and only if it is true that (x)Fxx. [Numerical identity is a totally reflexive relation.]

 

reflexive (df.): a relation Fxy is reflexive if and only if the following is true: if something stands in that relation to something (or other), then it stands in that relation to itself. [E.g., belonging to the same political party as is a reflexive relation (p.285), but it is not totally reflexive, since some things do not belong to any political party at all, and thus do not belong to the same party as themselves.]

 

Typically, the strategy for using IR will be to introduce “(x)(x = x)” into a proof and then to apply UI to eliminate the universal quantifier and replace the bound variable with a constant or a quasivariable. An example of the use of IR, based on the proof on p.288:

 

h: the Hulk

Gx: x is green

 

1.      (x)[(x=h) É Gx]     / \ Gh

2.      (h=h) É Gh                        1 UI

3.      (x)(x=x)                 IR        [note that you don’t cite a line number when using IR]

4.      h=h                                    3 UI

5.      Gh                         2, 4 MP

 

 

Exercise 13-1 (pp.288-89)

·         complete the rest for next time—they are relatively long—except for #1, they will all take more than 10 lines to solve

 

 

[9.3.] Symbolizing Quantities Other Than “All,” “Some,” and “None.” (Ch. 13:1).

 

The identity symbol allows us to express in symbolism statements concerning quantities other than all, some and none.

 

“At Least n” (n > 1)

 

We can already symbolize “There is at least one student”:      ($x)Sx

 

But up to now, we haven’t been able to symbolize “There are at least two students.” This expression:

 

($x)($y)(Sx · Sy)

 

does not capture the meaning of “at least two,” because the bound “x” and the bound “y” might refer to the same student. So this formula is true even if there is one and only one student.

 

 

Using the non-identity symbol, we can translate the sentence as:

 

($x)($y)[(Sx · Sy) · (x ¹ y)]

 

The added clause indicates that x and y are not identical; so this expression is false if there is only one student and is true only if there are two or more different students.

 

Larger quantities can be represented by adding further quantifiers and non-identity claims. For example, “There are at least three students” can be symbolized:

 

($x)($y)($z){[(Sx · Sy) · Sz] · {[( x¹y ) · ( x¹z )] · ( y¹z )}}

 

 

“At Most”

 

Consider “There is at most one student.”

 

First notice that this sentence would be true if there were no students at all. This sentence does not imply that there is actually even one student.

 

Rather, it means that if there are any students at all, then there is no more than one of them. This indicates that to symbolize it, we need a universal quantifier rather than an existential quantifier.

 

It is symbolized as:

 

(x){Sx É (y)[Sy É (y = x)]}

 

which is read: for all x, if x is a student, then, for all y, if y is a student, then y is identical to x.

 

The following would be incorrect…

 

($x)Sx É (y)[Sy É (y = x)]

 

… because the “x” in the consequent is not bound to the existential quantifier. It would also be correct to write:

 

($x){Sx É (y)[Sy É (y = x)]}

 

What this says is: there is at least one thing in the domain of discourse about whom the following is true: if that one thing is a student, then it is the only student. But this can be true even if lots of things in the domain of discourse are students and that one thing is not. [Notice the suspicious pairing of the existential quantifier with the horseshoe.]

 

As with “at least”, we can increase the quantity by adding quantifiers and identity claims. For example, “There are at most two students”:

 

(x)(y)([(Sx · Sy) · (x ¹ y)] É (z){Sz É [(z = x) v (z = y)]})

 

In other words: if there are two students (x and y), then any student is identical either to x or to y.

 

As with the first “at most” example, this sentence does not imply the existence of students, and so it would be incorrect to use existential quantifiers.

 

“Exactly”

 

The statement “There is exactly one student” implies two things:

·         that there is at least one student;

·         that there is at most one student.

 

So to symbolize it, we need to symbolize both of these claims:

 

($x){Sx · (y)[Sy É (y = x)]}

 

And again, we can increase the number by adding quantifiers and identity claims. For example, “There are exactly two students”:

 

($x)($y){[(Sx · Sy) · (x ¹ y)] · (z){Sz É [(z = x) v (z = y)]}}

 

 

“Only”

 

Consider the statement “Tom is the only student who passed the exam.”

 

This is not correctly symbolized as

 

St · Pte             [t: Tom; e: the exam; Sx: x is a student Pxy: x passed y]

 

This expression could be true if everyone passed the exam, but “Only Tom passed the exam” implies that everyone besides Tom did not pass.

 

The correct symbolization is:

 

(St · Pte) · (x){[Sx · (x ¹ t)] É ~Pxe}

 

 

“Everyone But”

 

Now consider “Every student but Tom passed the exam.” In other words, Tom did not pass, but every student not identical to Tom did pass. So this is symbolized the same as the previous example:

 

(St · ~Pte) · (x){[Sx · (x ¹ t)] É Pxe}

 

See the example box on pp.290-91 for uses of the identity symbol together with relational predicates in statements expressing quantities other than all, some and none.

 

Exercise 13-2 (p.291)

·         complete all of these for next time

 

 

Stopping point for Tuesday April 17. For next time:

·         complete exercises 13-1 and 13-2.

 

 

 

 

THIS IS THE END OF THE NEW MATERIAL FOR WHICH YOU WILL BE RESPONSIBLE ON YOUR FINAL EXAM. BELOW I PROVIDE THE REMAINDER OF THE LECTURE NOTES THAT I PREPARED FOR THIS SEMESTER, IN CASE YOU WANT TO WORK THROUGH THEM, EITHER NOW OR LATER. PLEASE FEEL FREE TO ASK ME ANY QUESTIONS THAT YOU HAVE ABOUT THIS MATERIAL.

 

 

 

[9.4.] Definite Descriptions.

 

definite description (df.): a descriptive phrase that uniquely picks out a single individual or at least appears to do so, e.g., “the tallest student in this class,” “the first dog in space,” “the president of the United States,” “the winner of the 2012 Oscar for Best Motion Picture,” “the artist formerly known as Prince,” etc.

 

We have to use the identity symbol to symbolize statements containing definite descriptions.[1] For example: “The CEO of Apple Computer, Inc. is wealthy.”

 

Part of what this asserts is that there is one and only one CEO of Apple. So we should symbolize it as follows:

 

Cxy: x is CEO of y

Wx: x is wealthy

a: Apple Computer, Inc.

 

($x){{Cxa · (y)[Cya É (y = x)]} · Wx}

 

 

Another example: “The author of Gone with the Wind is deceased.” In part, this asserts that there is one and only one author of Gone with the Wind:

 

Axy: x is the author of y

g: Gone with the Wind

Dx: x is deceased

 

($x){{Axg · (y)[Ayg É (y = x)]} · Dx}

 

 

And another: “The tallest student in this class is blond.” This asserts, in part, that there is one and only one tallest student in this class--

 

Txy: x is taller than y

Bx: x is blond

Sx: x is a student in y

C: this class

 

($x){{Sxc · (y){[Syc · (y ¹ x)] É Txy}} · Bx}

 

 

And yet another: “Everyone hates the tallest student in this class.”

 

Hxy: x hates y

Px: x is a person

 

($x){{Sxc · (y){[Syc · (y ¹ x)] É Txy}} · (z)(Pz É Hzx)}

 

See further, more complicated examples in the Examples box, pp.292-93.[2]

 

 

[9.5.] “Properties of Relations.” [Ch. 13:3]

 

In this section, we examine several properties that two-place relations can possess:

·         reflexivity, irreflexivity, non-reflexivity

·         symmetry, asymmetry, non-symmetry

·         transitivity, intransitivity, non-transitivity

 

 

[9.5.1.] Reflexivity.

 

A relation Fxy is totally reflexive if and only if the following is true: for all x, x is F to itself. I.e., 

 

(x)Fxx

 

Examples:

·         x is numerically identical to x

·         x is qualitatively identical to x

 

 

A relation Fxy is (not totally reflexive, but nonetheless) reflexive if and only if the following is true: if x is F to something, then x is F to itself, i.e.,

 

(x)[($y)Fxy É Fxx]

 

or, in an equivalent formulation used by your textbook:

 

(x)(y)[Fxy É (Fxx · Fyy)]

 

Examples:

·         x is in the same political party as y

·         x uses the same barber as y

·         x likes the same thing as y

·         x is the same size as y

 

 

A relation Fxy is irreflexive if and only if the following is true: for all x, x is not F to itself. I.e.:

 

(x)~Fxx

 

Examples:

·         x is the father of y

·         x is the child of y

·         x is taller than y

 

 

A relation Fxy is non-reflexive if and only if it is neither reflexive nor irreflexive.

 

Examples:

·         x loves y [the fact that x loves y implies neither that x loves x nor that x does not love x]

·         x hates y

·         x admires y

 

 

[9.5.2.] Symmetry.

 

A relation Fxy is symmetrical if and only if the following is true: if x is F to y, then y is F to x. I.e., a relation is symmetrical if the following is true:

 

(x)(y)(Fxy É Fyx)

 

Examples:

·         x is married to y

·         x is a sibling of y

·         x belongs to the same political party as y

 

A relation Fxy is asymmetrical if and only if the following is true: if x is F to y, then y is not F to x. I.e., a relation is asymmetrical if the following is true:

 

(x)(y)(Fxy É ~Fyx)

 

Examples:

·         x is a child of y

·         x is the mother of y

·         x is underneath y

·         x is taller than y

 

A relation Fxy is non-symmetrical if and only if it is neither symmetrical nor asymmetrical.

 

Examples:

·         x loves y [the fact that x loves y implies neither that y loves x nor that y does not love x]

·         x hates y

·         x admires y

 

 

[9.5.3.] Transitivity.

 

A relation Fxy is transitive if and only if the following is true: if x is F to y, and y is F to z, then x is F to z. I.e., a relation is transitive if the following is true:

 

(x)(y)(z)[(Fxy · Fyz) É Fxz]

 

Examples:

·         x is taller than y

·         x is wealthier than y

·         x is underneath y

 

See the proof on p.296: in order to show that “Art is taller than Betsy” and “Betsy is taller than Charles” deductively imply that “Art is taller than Charles” in our system of logic, you have to add a premise that states that being taller than is a transitive relation:

 

(x)(y)(z)[(Txy · Tyz) É Txz]

 

 

A relation Fxy is intransitive if and only if the following is true: if x is F to y, and y is F to z, then x is not F to z. I.e., a relation is intransitive if the following is true:

 

(x)(y)(z)[(Fxy · Fyz) É ~Fxz]

 

Examples:

·         x is a child of y

·         x is the father of y

 

 

A relation is non-transitive if and only if it is neither transitive nor intransitive.

 

Examples:

·         x loves y [the fact that x loves y and that y loves z implies neither that x loves z nor that x does not love z]

·         x hates y

·         x admires y