Logic 15: Identity

Question 1

Why we need identity

Answer 1

Want to say more than one thing is F
Are limited with the existential quantifer, which says at least one thing is F
And the Universal quantifier that says everything is F

Question 2

Identity and Distinctness

Answer 2

a = b says a is identical to b

a≠b says a is distinct from b

Question 3

Other definition for distinctness

Answer 3

a≠b is the same as ¬(a=b)

In terms of identity and negation

Question 4

Relations of identity and distinctness

Answer 4

They are dyadic predicates, like a “loves” b
- but behave in a special way
The behavior of “loves” can vary from model to model
- on some models Romeo loves Juliet, on others he doesn’t
- on some models love is always requited, on others it is not

The behavior of identity and distinctness is FIXED from model to model

• no matter the model, everything is identical to itself
• nothing in the domain is identical to any other thing in the domain
• nothing in the domain is distinct from itself
• everything in the domain is distinct from every other thing in the domain
• relations of identity and distinctness are held to be logically constant

Question 5

There is more than one thing of a certain kind

Answer 5

Ex. There is more than one thing that is F
∃x∃y(Fx ∧ Fy ∧ x≠y)

there is some x and there is some y, and x is F, and y is F, and x and y are distinct from each other
- third conjunct is crucial

Question 6

Want to say that there

are exactly two Fs

Answer 6

∃x∃y(Fx ∧ Fy ∧ x≠y)
says there are at least 2 F’s, but want to add another conjunct that says that there are no more Fs

So
… ∧ ∀z(Fz → (x=z ∨ y=z)))
and universal z, if F is z then (x is z OR y is z)

THUS:
∃x∃y(Fx ∧ Fy ∧ x≠y ∧ ∀z(Fz → (x=z ∨ y=z)))

Question 7

there are exactly three Fs

Answer 7

∃x∃y∃v(Fx ∧ Fy ∧ Fv ∧ x≠y ∧ x≠v ∧ y≠v ∧ ∀z(Fz → (x=z ∨ y=z ∨ v=z))))

where the exclusion clause is
∧ ∀z(Fz → (x=z ∨ y=z ∨ v=z))))

Question 8

Rules for Identity and Distinctness

Answer 8

1) a sentence a=b is true on a model M iff a and b name the same thing on M
- recall that 2 names can refer to the same thing
- but one name cannot refer to multiple things
- Pat & Pat Pat can refer to me
- but unlike real life, Pat Pat can only refer to me and not someone else

thus, iff I(a) = I(b)

2) a≠b is true on a model M iff a and b name different things on M
iff I(a) ≠ I(b)

Question 9

Validity for Identity and Distinctness

Answer 9

The Indiscernibility of Identicals
aka LEIBNIZ’s Law
“if a and b are one and the same thing, every thing that is true of a is true of b, and vice versa”
if a and b are the very same thing, neither can be some way that the other is not, because it’s not really an “other”

Question 10

Indiscernibility of Identicals Validity example

Answer 10

1) Samuel Clemens is Mark Twain
2) Mark Twain created Huckleberry Finn

therefore
C) Samuel Clemens created Huckleberry Finn

• valid because identity obeys the indiscernibility of identical
• if something is true of Mark Twain, and he just IS Samuel Clemens, then it is true of Clemens

Question 11

Indiscernibility of Identicals

Answer 11

Leibniz’s Law

Question 12

Indiscernibility of Identicals as Distinctness of Discernibles

Answer 12

Distinctness of Discernibles
“if there is some DIFFERENCE between a and b, then a and b are distinct”

where the indiscernibility of identicals say
“if P then Q” is equivalent to “If not Q then not P”

Question 13

Validity example as distinctness of discernibles

Answer 13

1) Michelangelo created the statue David
2) Michelangelo did not create the marble from which David is formed

therefore
C) David is distinct from the marble from which David is formed