Symbolic Logic: Lecture Notes: University of West Georgia

[10.] Relational Predicate Logic with Identity.

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

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

Bruce Wayne is wealthy.

Bruce Wayne is Batman.

In the first sentence, “is” is used to predicate a property, viz. being wealthy, of Bruce Wayne. This sentence is correctly symbolized:

Ww [w: Bruce Wayne; Wx: x is wealthy]

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

numerical (quantitative) identity: x and y are numerically identical if and only if x is one and the same thing as y, e.g., Bruce Wayne is numerically identical to Batman.

qualitative identity: 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 “Bx” as “x is Batman” and then symbolize the second sentence as

In order to symbolize the second sentence, we must introduce a new symbol for numerical identity: =

With this symbol, we can symbolize “Bruce Wayne is Batman” as follows:

w = b [w: Bruce Wayne; b: Batman]

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 can also use “=” to deny that entities are numerically identical. But instead of writing

~(k = b) [k: Clark Kent; b: Batman]

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

k ¹ b

[10.2.] 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. Ww

2. w = b / \ Wb

(Bruce Wayne is wealthy; Bruce Wayne is Batman; so, Batman is wealthy.)

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

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

1. (...u...)

2. u = w / \ (...w...)

1. (...u...)

2. w = u / \ (...w...)

With this rule, we can complete the proof:

1. Ww p

2. w = b p / \ Wb

3. Wb 1, 2 ID

A second example, this one with a relational predicate:

1. Mbr p [Mxy: x is mentor to y; b: Batman; r: Robin]

2. b = w p /\ Mwr [w: Bruce Wayne]

3. Mwr 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

totally reflexive: 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: 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.285:

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

4. h = h 3 UI

5. Gh 2, 4 MP

Exercise 13-1 (pp.285-86)

· do all of these for next time

[10.3.] 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”

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 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 identity claims. For example, “There are at least three students” can be symbolized:

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

“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 at least 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.

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 “Only Tom didn’t pass the exam.”

This is not correctly symbolized as

~Pt [t: Tom; Px: x passed the exam]

This expression could be true if no one passed the exam, but “Only Tom didn’t pass the exam” implies that everyone besides Tom did pass.

The correct symbolization is:

~Pt · (x){[Sx · (x ¹ t)] É Px}

“Everyone But”

Now consider “Every student but Tom passed.” 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:

~Pt · (x){[Sx · (x ¹ t)] É Px}

See the example box on pp.287-88 for uses of the identity symbol together with relational predicates in statements expressing quantities other than all, some and none.

Exercise 13-2 (p.288)