Identity

1
Identity

• Many ordinary language arguments rely on the
  relation of identity for their validity
• We introduce the predicate to stand for the
  identity relation
• A wide variety of claims can be translated using
  the identity predicate

2
Examples

• Cary Grant is Archibald Leach
• c a
• Cary Grant is not James Stewart
• (c j)
• The only undefeated team is the New England
  Patriots
• Tn Dn (x)(Tx Dx)?x n
• No planet except Saturn has rings
• Ps Rs (x)(Px Rx)?x s

3
Continued

• All teams except the Miami Dolphins have a
  victory
• Tm Vm (x)Tx (x m)?Vx
• Yao Ming is the tallest basketball player
• Bm (x)Bx (x m)?Tmx
• There are at most two students who pass the class
• (x)(y)(z)(Sx Px Sy Py Sz Pz)?(x y v
  x z v y z)
• There are at least two students who pass the
  class
• (?x)(?y)Sx Px Sy Py (x y)
• There are exactly two students who pass the class
• (?x)(?y)Sx Px Sy Py (x y) (z)(Sz
  Pz)?(z x v z y)

4
Definite Descriptions

• A description that identifies an individual
  person, place, thing, etc.
• Examples
• The leader of the free world
• The capital of France
• The author of Candide

5
How to Translate

• The leader of the free world is an American
• (?x)Lxf (y)(Lyf?y x) Ax
• The capital of France is Paris
• (?x)Cxf (y)(Cyf?y x) x p
• The author of Candide is French
• (?x)Axc (y)(Ayc?y x) Fx

6
Translate These

• The only nation having a maple leaf flag is
  Canada (Nx x is a nation Mx x has a maple leaf
  flag c Canada)
• Hydrogen is the lightest element (Ex x is an
  element Lxy x is lighter than y h hydrogen
• Every pitcher except Cy Young has won fewer than
  500 games (Px x is a pitcher Wx x has won
  fewer than 500 games c Cy Young)
• There are at most two cities in Malta (Cx x is a
  city Mx x is in Malta)
• The author of Vanity Fair was born in India (Wxy
  x wrote y Bx x was born in India v Vanity
  Fair)

7
See?

• The only nation having a maple leaf flag is
  Canada
• Nc Mc (x)(Nx Mx)?x c
• Hydrogen is the lightest element
• Eh (x)Ex (x h)?Lhx
• Every pitcher except Cy Young has won fewer than
  500 games
• Pc Wc (x)Px (x c)?Wx
• There are at most two cities in Malta
• (x)(y)(z)(Cx Mx Cy My Cz Mz)?(x y v
  x z v y z)
• The author of Vanity Fair was born in India
• (?x)Wxv (y)(Wyv?x y) Bx

8
Identity Rules

• Prem
• a a
• At any point in a proof we can write down a
  self-identity statement, since anything is
  identical to itself
• (2) a b b a
• Obviously, if a is identical to b, then b is
  identical a
• (3) Fa
• a b
• Fb
• Clearly, if a if F, and a is identical to b, then
  b if F
• Note, the variables a and b range over constants,
  not variables

9
Examples

• (x)(x a)
• (?x)Rx /Ra
• Rb 2, EI
• b a 1, UI
• Ra 3, 4 Id

10
Another

• Ke
• Kn /(e n)
• e n AIP
• Kn 1, 3 Id
• Kn Kn 2, 4 Add
• (e n) 3-5 IP

11
Translate and Prove Valid

• Some of Jane Colliers novels are interesting.
  The only novel Jane Collier wrote is The Cry.
  Therefore, The Cry is interesting.
• Nx x is a novel Wxy x wrote y Ix x is
  interesting j Jane Collier c The Cry

12
Translate

• Some of Jane Colliers novels are interesting.
  The only novel Jane Collier wrote is The Cry.
  Therefore, The Cry is interesting.
• (?x)(Nx Wjx Ix)
• Nc Wjc (x)(Nx Wjx)?x c
• Ic

13
Prove

• (?x)(Nx Wjx Ix)
• Nc Wjc (x)(Nx Wjx)?x c /Ic
• Na Wja Ia 1, EI
• (x)(Nx Wjx)?x c Nc Wjc 2, Com
• (x)(Nx Wjx)?x c 4, Simp
• (Na Wja)?a c 5, UI
• Na Wja 3, Simp
• a c 6, 7 MP
• Ia Na Wja 3, Com
• Ia 9, Simp
• Ic 8, 10 Id

14
Another

• Every member except Ellen sang a song. Every
  member except Nancy gave a speech. Ellen is not
  Nancy. Therefore, Ellen gave a speech and Nancy
  sang a song.
• Mx x is a member Sx x sang a song Gx x gave
  a speech e Ellen n Nancy

15
Translate

• Every member except Ellen sang a song. Every
  member except Nancy gave a speech. Ellen is not
  Nancy. Therefore, Ellen gave a speech and Nancy
  sang a song.
• Me Se (x)Mx (x e)?Sx
• Mn Gn (x)Mx (x n)?Gx
• (e n)
• Therefore, Ge Sn

16
Prove

• Me Se (x)Mx (x e)?Sx
• Mn Gn (x)Mx (x n)?Gx
• (e n) /Ge Sn
• (x)Mx (x e)?Sx Me Se 1, Com
• (x)Mx (x e)?Sx 4, Simp
• Mn (n e)?Sn 5, UI
• Mn 2, Simp
• (n e) 3, Id
• Mn (n e) 7, 8 Conj
• Sn 6, 9 MP
• (x)Mx (x n)?Gx Me Se 2, Com
• (x)Mx (x n)?Gx 11, Simp
• (Me (e n)?Ge 12, UI
• Me 1, Simp
• Me (e n) 3, 14 Conj
• Ge 13, 15 MP
• Ge Sn 10, 16 Conj

17
One More

• The author of King Lear was an English actor.
  John Milton was English but not an actor.
  Therefore, John Milton is not the author of King
  Lear.
• Wxy x wrote y Ex x is English Ax x is an
  actor k King Lear m John Milton

18
Translate

• The author of King Lear was an English actor.
  John Milton was English but not an actor.
  Therefore, John Milton is not the author of King
  Lear.
• (?x)Wxk (y)(Wyk?y x) Ex Ax
• Em Am
• Therefore, Wmk

19
Prove

• (?x)Wxk (y)(Wyk?y x) Ex Ax
• Em Am /Wmk
• Wak (y)(Wyk?y a) Ea Aa 1, EI
• (y)(Wyk?y a) Ea Aa Wak 2, Com
• (y)(Wyk?y a) 4, Simp
• Wmk?m a 5, UI
• Wmk AIP
• m a 6, 7 MP
• Aa Wak (y)(Wyk?y a) Ea 3, Com
• Aa 9, Simp
• a m 8, Id
• Am 10, 11 Id
• Am Em 2, Com
• Am 13, Simp
• Am Am 12, 14 Conj
• Wmk 7-15 IP