Predicate Logic
Splitting the Atom
What Propositional Logic can’t do
• It can’t explain the validity of the Socrates Argument
• Because from the perspective of Propositional Logic
its premises and conclusion have no internal
structure.
The Socrates Argument
1. All men are mortal
2. Socrates is a man
_______________________
3. Therefore, Socrates is mortal
Translates as:
1. H
2. S
__
3. M
The Problem
The problem is that the validity of this argument
comes from the internal structure of these
sentences--which Propositional Logic cannot
“see”:
1. All men are mortal
2. Socrates is a man
_______________________
3. Therefore, Socrates is mortal
Solution
• We need to split the atom!
• To display the internal structure of those “atomic”
sentences by adding new categories of vocabulary
items
• To give a semantic account of these new vocabulary
items and
• To introduce rules for operating with them.
Singular statements
Make assertions about persons, places, things or
times
Examples:
• Socrates is a man.
• Athens is in Greece.
• Thomas Aquinas preferred Aristotle to Plato
To display the internal structure of such sentences
we need two new categories of vocabulary items:
• Individual constants (“names”)
• Predicates
Individual Constants & Predicates
Socrates is a man.
The name of a
person—which
we’ll express
by an
individual
constant
A predicate—
assigns a
property
(being-a-man)
to the person
named
More Vocabulary
• So, to the vocabulary of Propositional Logic we add:
• Individual Constants: lower case letters of the
alphabet (a, b, c,…, u, v, w)
• Predicates: upper case letters (A, B, C,…, X, Y, Z)
– These represent predicates like “__ is human,”
“__ the teacher of __,” “__ is between __ and
__,” etc.
– Note: Predicates express properties which a
single individual may have and relations which
may hold on more than one individual
Predicates
• 1-place predicates assign properties to individuals:
__ is a man
__ is red
• 2-place predicates assign relations to pairs of
individuals
__ is the teacher of __
__ is to the north of __
• 3-place predicates assign relations to triples of
individuals
__ preferred __ to __
• And so on…
Translation
Socrates is a man.
Hs
Socrates is mortal.
Ms
Athens is in Greece
Iag
Thomas preferred Aristotle to Plato.
Ptap
We always put the predicate first followed by
the names of the things to which it applies.
What are properties & relations?
Are they ideas in people’s heads?
Are they Forms in Plato’s heaven?
All this is controversial so we resolve
to treat properties and relations as
things that aren’t controversial, viz.
sets.
Predicates designate sets!
• Individual constants name individuals—persons,
places, things, times, etc.
• We resolve to understand properties and relations as
sets—of individuals or ordered n-tuples of individuals
(pairs, triples, quadruples, etc.)
Fido
is brown
{ brown things }
Predicate Logic Vocabulary
• Individual Constants: lower case letters of the alphabet (a,
b, c,…, u, v, w)
• Predicates: upper case letters (A, B, C,…, X, Y, Z)
• Connectives:  ,  , • ,  , and 
• Variables: x, y and z
• Quantifiers:
– Existential ( [variable]), e.g. (x), (y)…
– Universal ( [variable]) or just ([variable] ), e.g. (x),
(y), (x), (y)…
Sets
• A set is a well defined collection of
objects.
• The elements of a set, also called its
members, can be anything: numbers,
people, letters of the alphabet, other
sets, and so on.
• Sets A and B are equal iff they have
precisely the same elements.
• Sets are “abstract objects”: they don’t
occupy time or space, or have causal
powers even if their members do.
Venn Diagrams
• Set theoretical concepts, like union and
intersection can be represented visually by Venn
Diagrams
• Venn Diagrams represent sets as circles and
• Represent the emptiness of a set by shading out
the region represented by it and
• Represent the non-emptiness of a set by putting
an “X” in it.
Set Membership & Set Inclusion
A
B
The Kids
C
D
Mama
E
Grandma
The Babushka Family: C is the daughter of D and D
is the daughter of E, but C is not the daughter of E!
Set Membership & Set Inclusion
• x  S : x is a member of the set S
– 2  {1, 2, 3}
– C  {daughters of D}
• Anything can be a member of a set…including
another set!
– {1, 2, 3}  {{1, 2, 3}, 4}
• But members of members aren’t members!
– 4  {1, 2, 3}, 4}, but 2  {{1, 2, 3}, 4} !
• C is a daughter of D and D is the daughter of E but C
is not the daughter of D!
Set Membership & Set Inclusion
S1 ⊆ S2 : S1 included in (is a subset of) S2
•This means every member of S1 is a member of S2
– {1, 2, 3} ⊆ {1, 2, 3, 4}
The numbers 1, 2, and 3 are members of {1, 2, 3, 4},
but the set {1, 2, 3} is not a member!
– {1, 2, 3}  {1, 2, 3, 4}
•It is however a member of this set:
– {1, 2, 3}  {{1, 2, 3}, 4}
•Sets are identical if they have the same members so
– {1, 2, 3, 4} ≠ {{1, 2, 3}, 4}
Ordered Sets
• We use curly-brackets to designate sets so, e.g. the
set of odd numbers between 1 and 10 is {3, 5, 7, 9}
• Order doesn’t matter for sets so, e.g.
– {3, 5, 7, 9} = {3, 9, 5, 7}
• But sometimes order does matter: to signal that we
use pointy-brackets to designate ordered n-tuples—
ordered pairs, triples, quadruples, quintuples, etc.
– <Adam, Eve> ≠ <Eve, Adam>
– <1, 2, 3> ≠ <3, 2, 1>
The Set of Russian Doll Sets
ABC
DE
• C  {Babushkas}
• {Babushkas}  {Russian Doll Sets}
• C  {Russian Doll Sets}
• C is a doll—not a set of dolls!
Having a property
• An individual’s having a property is understood as its
being a member of a set.
• “Fido is brown” says that Fido is a member of the set
of brown things.
Fid
ε
{ brown things }
What Singular Sentences Say
• Singular sentences are about set membership.
• Ascribing a property to an individual is saying that it’s a
member of the set of individuals that have that property,
e.g.
– “Descartes is a philosopher” says that Descartes is a
member of {Plato, Aristotle, Locke, Berkely, Hume,
Quine…}
• Saying that 2 or more individuals stand in a relation is
saying that some ordered n-tuple is a member of a set so,
e.g.
– “San Diego is north of Chula Vista” says that <San
Diego,Chula Vista> is a member of {<LA,San Diego>,
<Philadelphia,Baltimore>…}
Singular Sentences
1. All men are mortal
2. Socrates is a man
_______________________
3. Therefore, Socrates is mortal
•
We can now understand what 2
and 3 say:
•
2 says that Socrates ε {humans}
•
3 says that Socrates ε {mortals}
Translation
SINGULAR STATEMENTS
Ducati is brown.
Bd
Tweety bopped Sylvester
Bts
GENERAL STATEMENTS
General Sentences
1. All men are mortal
2. Socrates is a man
_______________________
3. Therefore, Socrates is mortal
•
But we still don’t have an account
of what 1 says
•
Because it doesn’t ascribe a
property or relation to any
individual or individuals.
Emptiness
Shading in a region says that the set it represents
is empty.
A
This says that there are no As.
There are no unicorns.
unicorns
 (x) Ux
Non-emptiness
‘X’ in a region says that the set it the region
represents is non-empty, i.e. there’s something there
X
A
This says that at least one thing is an A.
There are horses.
X
horses
(x) Hx
Intersection
Intersection of the sets A and B, denoted A ∩ B, is the
set of all objects that are members of both A and B.
A
B
This is the intersection of sets A and B
No horses are carnivores
horses
carnivores
(x)  (Hx • Cx)
Intersection & Conjunction
This region represents the set of things
that are both A and B.
X
A
B
Some horses are brown
X
horses
brown things
(x)(Hx • Bx)
Set Complement
The complement of a set if everything in the
universe that’s not in the set.
A
The shaded area is the complement of A—the set
of things that are not A.
Set Complement
This says that something is in both A but not B.
X
A
B
Some horses are not brown
X
horses
brown things
(x)(Hx • Bx)
All horses are mammals
horses
mammals
(x)(Hx  Mx)
All horses are mammals
horses
mammals
There are no non-mammalian horses
horses ∩ non-mammals = { } (the empty set)
Set Union
Union of the sets A and B, denoted A ∪ B, is the set of
all objects that are a member of A, or B, or both.
A
B
This is the union of sets A and B
Union and Disjunction
• If we say something is in the union of A and B we’re
saying that it’s either in A or in B.
A
B
General Sentences
1. All men are mortal
2. Socrates is a man
_______________________
3. Therefore, Socrates is mortal
•
We can’t treat “all men” as the
name of an individual as, e.g. the
sum or collection of all humans
•
Consider: “All men weigh under
1000 pounds.”
•
We want to ascribe properties to
every man individually.
The Socrates Argument is Valid!
1. All men are mortal
2. Socrates is a man
_______________________
3. Therefore, Socrates is mortal
•
1 says that there are no men that
don’t belong to the set of mortals
•
2 says that Socrates ε {men}
•
3 says that, therefore, Socrates ε
{mortals}
All men are mortals
immortal
men: nothing
here!
men
mortals
There’s nothing in the set of men outside of the set of
mortals; the set of immortal men is empty.
Socrates is a man.
men
mortals
• This is the only place in the men circle Socrates can go
since we’ve already said that there are no immortal men.
Therefore, Socrates is a mortal.
men
mortals
• So Socrates automatically ends up in the mortals circle:
the argument is, therefore, valid!
Arguments Involving Relations
1. All dogs love their people
2. Bo is Obama’s Dog
____________________
3. Bo loves Obama
If we treat the predicates in
these 3 sentences as oneplace predicates, we can’t
explain the validity of this
argument!
It looks like we’re ascribing 3
different properties that don’t
have anything to do with one
another: people-loving,
being-Obama’s-dog and
Obama-loving
Relations
• We understand relations as sets of ordered n-tuples,
e.g.
• being to the north of is {<Los Angeles, San Diego>,
<Philadelphia, Baltimore>, <San Diego, Chula
Vista>…}
• being the quotient of one number divided by another:
x into y is {<2,2,1>, <2,4,2>, <5,45,9>…}
Singular sentences about relations
• Standing in a relation is also understood in terms of
set membership.
• It’s a matter of being a member of an ordered n-tuple
which is a member of a set.
Michelle is Barak’s wife.
ε
{<Eve, Adam>, <Hillary,
Bill>…}
Validity of the Dog Argument
1. All dogs love their people
2. Bo is Obama’s Dog
____________________
3. Bo loves Obama
We can consider being the
dog of and loves as relations
and designate them by the 2place predicates “D”and “L”
Now the 3 sentences have
something in common--they
don’t involve 3 different
properties; they involve 2
relations: being the dog of
and loves
So we can link them to show
validity
Bo is Obama’s dog: Dbo
All dogs
love their
people
(x)(y)(Dxy  Lxy)
For all x, y, if x is the dog of y then x loves y
Validity of the Dog Argument
1. All dogs love their people
1. (x)(y)(Dxy  Lxy)
2. Bo is Obama’s Dog
____________________
2. Dbo
______________
3. Bo loves Obama
3. Lbo
1 says that for any x and y, if x
is the dog of y then x loves y.
2 says that Bo is the dog of
Obama.
So it follows that Bo loves
Obama!
All dogs love their people
__ is the dog of __
__ loves __
(x)(y)(Dxy  Lxy)
Bo is Obama’s Dog
<Bo,Obama>
__ is the dog of __
__ loves __
Dbo
Bo loves Obama
<Bo,Obama>
__ is the dog of __
__ loves __
Lbo
SCOPE
There are dogs and there are cats.
(x)Dx  (x)Cx : There exists an x such that x is a dog and
there exists an x such that x is a cat
There are dog-cats
(x)(Dx  Cx): There exists an x such that x is a dog-and-acat
OVERLAPPING QUANTIFIERS
Everybody loves somebody
(x)(y)Lxy: for all x, there exists a y such that x
loves y
Everybody is loved by somebody
(x)(y)Lyx: for all x, there exists a y such that y loves x
There’s somebody everybody loves
(x)(y)Lyx: There exists an x such that for all y, y loves x
There’s somebody who loves everybody
(x)(y)Lxy:
there exists an x such that for all y, x loves y
The End