FORMAL LOGIC
PREDICATE LOGIC REVIEW
LESSON PLAN
PROFESSOR JULIE YOO
Single Quantifiers – The Use of Monadic Predicates
Basic Universal and Existential Statements – A, E, I O
Longer A, E, I, O Sentences
‘All’ versus ‘Only’
Using More than One Quantifier in a Sentence
‘Any’ versus ‘Every’
Selection of Predicates
Overlapping Quantifiers – The Use of Polyadic (Relational or n-adic) Predicates
The Case of Loving: UD Restricted to People
The Case of Loving: UD Unrestricted
Ambiguities and Scope Distinctions
Polyadic Predicate Formulas with Negation
Other Cases: UD Unrestricted
SINGLE QUANTIFIERS – THE USE OF MONADIC PREDICATES
Basic Universal and Existential Statements – A, E, I O
Subjects/Constants
(none)
Predicates
Sx = x is a swan
Wx = x is white
A-sentence
All swans are white.
(∀x)(Sx → Wx)
UD: unrestricted
E-sentence
No swans are white.
(∀x)(Sx → ~ Wx)
UD: unrestricted
I-sentence
Some swans are white.
(∃x)(Sx ∧ Wx)
UD: unrestricted
O-sentence
Some swans are not white.
(∃x)(Sx ∧ ~ Wx)
UD: unrestricted
Using Negations with Basic A, E, I, O Sentence (Relevant Negations in RED)
A-sentence
All swans are white.
~(∃x)(Sx ∧ ~Wx)
UD: unrestricted
E-sentence
No swans are white.
~(∃x)(Sx ∧ Wx)
UD: unrestricted
I-sentence
Some swans are white.
~(∀x)(Sx ∧ ~Wx)
UD: unrestricted
O-sentence
Some swans are not white.
~(∀x)(Sx ∧ Wx)
UD: unrestricted
Longer A, E, I, O Sentences
Subjects/Constants
a = Alice
b = Bob
c = Charlie
Predicates
Lxy = x likes y (y is liked by x)
Longer A-sentence
Whoever Charlie likes also likes either Bob or Alice.
(∀x)[Lcx → (Lxb ∨ Lxa)]
UD: people
Longer E-sentence
No one who Charlie and Bob like is liked by Alice.
(∀x)[(Lcx ∧ Lbx) → ~ Lax]
UD: people
Longer I-sentence
There is someone who likes Charlie who also likes Bob and Alice.
(∃x)[Lxc ∧ (Lxb ∧ Lxa)]
UD: people
Longer O-sentence
There is someone who Charlie likes who doesn’t like either Bob or Alice.
(∃x)[Lcx ∧ ~ (Lxb ∨ Lxa)]
UD: people
‘All’ versus ‘Only’
Subjects/Constants
(none)
Predicates
Cx = x fails the course
Lx = x is lazy
All lazy students fail the course.
(∀x)(Lx → Fx)
UD: students
Only lazy students fail the course.
(∀x)(Fx → Lx) : : (∀x)(~ Lx → ~ Fx)
UD: students
Using More than One Quantifier in a Sentence
Subjects/Constants
(none)
Predicates
Bx = x is black
Wx = x is white
Rx = x is a raven
Sx = x is a swan
Multiple Universal Quantifiers
Every stripe is black or white.
(∀x)(Bx ∨ Wx)
NOT: (∀x)Bx ∨ (∀x)Wx
Everyone failed the final or no one did.
(∀x)Fx ∨ (∀x) ~ Fx
UD: zebra stripes
UD: students
Multiple Existential Quantifiers
There are swans that are white and there are swans that are black.
(∃x)(Sx ∧ Wx) ∧ (∃x)(Sx ∧ Bx)
UD: unrestricted
There are no ravens that are white but there are swans that are black.
~ (∃x)(Rx ∧ Wx) ∧ (∃x)(Sx ∧ Bx)
UD: unrestricted
Using Both Universal and Existential Quantifiers in a Single Formula
Some, but not all, swans are white.
(∃x)(Sw ∧ Wx) ∧ ~ (∀x)(Sw → Wx)
UD: urestricted
‘Any’ versus ‘Every’
If anyone likes Alice, then Charlie does.
(∃x)Lxa → Lca
UD: people
If everyone likes Alice, then Charlie does.
(∀x)Lxa → Lca
UD: people
If anyone likes Alice, then he or she likes Charlie.
(∀x)(Lxa → Lxc)
UD: people
Anyone who fails the final exam fails the course.
(∀x)(Fx → Cx)
UD: people
If anyone fails the exam, then everyone will fail the course.
(∃x)Fx → (∀x)Cx
UD: people
Selection of Predicates
Charlie is a good murderer.
Kc
NOT: (Gc ∧ Mc)
Gx = x is good
Mx = x is a murderer
Kx = x is a good murderer
Ponce de Leon searched for the fountain of youth. Sxy = x searched for y
Yp
NOT: Spf
Yx = x searched for the f.o.y.
p = Ponce de Leon
f = f.o.y
Ponce de Leon searched for a good harbor.
Hp
NOT: (∃x)(Gx ∧ Spx)
NOT: (∀x)(Gx → Spx)
Sxy = x searched for y
Gx = x is a good harbor
OVERLAPPING QUANTIFIERS – THE USE OF RELATIONAL/n-ADIC PREDICATES
Monadic Predicates
Hx = x is a house
Dx = x is a day
Px = x is a person
Polyadic Predicates Predicates
Lxy = x likes y; y is liked by x
Bxy = x is born on y
Vxy = x lives in y
The Case of Loving: UD Restricted to People (Important changes in variable order noted in BLUE.)
Everyone loves everyone.
(∀x)(∀y)Lxy
Everyone is loved by everyone.
(∀x)(∀y)Lyx
There is someone who loves someone or other.
(∃x)(∃y)Lxy
There is someone who is loved by someone or other.
(∃x)(∃y)Lyx
There is someone who loves everyone.
(∃x)(∀y)Lxy
Everyone is loved by someone or other.
(∀x)(∃y)Lyx
There is someone who is loved by everyone.
(∃x)(∀y)Lyx
Everyone loves someone or other.
(∀x)(∃y)Lxy
The Case of Loving: UD Unrestricted (Important changes in variable order noted in BLUE; prenex
normal form is given on the right.)
Everyone loves everyone.
(∀x)[Px → (∀y)(Py → Lxy)] : : (∀x)(∀y)[(Px → (Py → Lxy)]
Everyone is loved by everyone.
(∀x)[Px → (∀y)(Py → Lyx)] : : (∀x)(∀y)[(Px → (Py → Lyx)]
There is someone who loves someone or other.
(∃x)[Px ∧ (∃y)(Py ∧ Lxy)] : : (∃x)(∃y)[(Px ∧ (Py ∧ Lxy)]
There is someone who is loved by someone or other.
(∃x)[Px ∧ (∃y)(Py ∧ Lyx)] : : (∃x)(∃y)[(Px ∧ (Py ∧ Lyx)]
There is someone who loves everyone.
(∃x)[Px ∧ (∀y)(Py → Lxy)] : : (∃x)(∀y)[Px ∧ (Py → Lxy)]
Everyone is loved by someone or other.
(∀x)[Px → (∃y)(Py ∧ Lyx)] : : (∀x)(∃y) [Px → (Py ∧ Lyx)]
There is someone who is loved by everyone.
(∃x)[Px ∧ (∀y)(Py → Lyx)] : : (∃x)(∀y)[Px ∧ (Py → Lyx)]
Everyone loves someone or other.
(∀x)[Px → (∃y)(Py ∧ Lxy)] : : (∀x)(∃y)[(Px → (Py ∧ Lxy)]
Ambiguities and Scope Distinctions
Someone is born everyday.
(∃x)[Px ∧ (∀y)(Dy → Bxy)] Some one person is reborn every day.
(∀x)[Dx → (∃y)(Py ∧ Byx)] Every day is a day when someone or other is born.
Someone lives in every house.
(∃x)(Px ∧ (∀y)(Hy → Lxy)] Some one person lives in every house.
(∀x)(Hx → (∃y)(Py ∧ Lyx)] Every house is lived in by someone or other.
Polyadic Predicate Formulas with Negation with Negation
There is no one loves who loves everyone. (No one loves everyone)
~ (∃x)(∀y)Lxy
There is someone who doesn’t love anyone. (Someone doesn’t love everyone.)
(∃x)(∀y) ~ Lxy
There is no one who loves someone or other. (No one loves anyone.)
~ (∃x) ~ (∀y) ~Lxy : :
~ (∃x)(∃y)Lxy
::
(∀x)(∀y) ~Lxy
There is no one who doesn’t love everyone. (Everyone loves everyone.)
~ (∃x) ~ (∀y)Lxy
::
~ (∃x)(∃y) ~ Lxy
::
(∀x)(∀y)Lxy
There is no one who doesn’t love anyone. (Everyone loves someone or other.)
~ (∃x)(∀y) ~ Lxy
::
~ (∃x) ~ (∃y)Lxy
::
(∀x)(∃y)Lxy
There is someone who doesn’t not love everyone. (Someone loves someone or other.)
(∃x) ~ (∀y) ~ Lxy : :
~ (∀x) ~ (∃y)Lxy
::
(∃x)(∃y)Lxy
Other Cases: UD Unrestricted
There is no day on which everyone is born.
~ (∃x)[(Dx ∧ (∀y)(Py → Byx)]
Not everyone lives in a house.
~ (∀x)(Px → (∃y)(Hy ∧ Lxy)]