Predicate LogicPredicate LogicSplitting the AtomSplitting 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 Individual Constants & PredicatesPredicates
SocratesSocrates is a manis a man.
The name of a person—which we’ll express by an individual constant
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 Individual Constants: lower case letters of the
alphabet (a, b, c,…, u, v, w)alphabet (a, b, c,…, u, v, w)
• Predicates: upper case letters (A, B, C,…, X, Y, Z)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
SocratesSocrates is a man.
Hss
SocratesSocrates is mortal.
Mss
AthensAthens is in GreeceGreece
Iagag
ThomasThomas preferred AristotleAristotle to Plato.Plato.
Ptaptap
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
{ brown things }
is brown
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
Mama GrandmaThe Kids
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
• 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.
Fido { brown things }ε
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}
Singular statementsTranslation
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
A
This says that at least one thing is an A.
There are horses.
horses
X
(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.
This is the intersection of sets A and B
A B
No horses are carnivores
(x) (Hx • Cx)
horses carnivores
Intersection & Conjunction
This region represents the set of things that are both A and B.
A B
X
Some horses are brown
brown things
(x)(Hx • Bx)
X
horses
Set Complement
The complement of a set if everything in the universe that’s not in the set.
The shaded area is the complement of A—the set of things that are not A.
A
Set Complement
This says that something is in both A but not B.
A B
X
Some horses are not brown
brown thingshorses
X
(x)(Hx • Bx)
All horses are mammals
(x)(Hx Mx)
horses mammalshorses
All horses are mammals
horses mammals
There are no non-mammalian horseshorses ∩ 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.
This is the union of sets A and B
A 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
There’s nothing in the set of men outside of the set of mortals;
the set of immortal men is empty.
men mortals
immortal men: nothing here!
Socrates is a man.
• This is the only place in the men circle Socrates can go since we’ve already said that there are no immortal men.
men mortals
Therefore, Socrates is a mortal.
• So Socrates automatically ends up in the mortals circle: the argument is, therefore, valid!
men mortals
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 one-place 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 2-place 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
2. Bo is Obama’s Dog
____________________
3. Bo loves Obama
1. (x)(y)(Dxy Lxy)
2. Dbo______________
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
(x)(y)(Dxy Lxy)
__ is the dog of __ __ loves __
Bo is Obama’s Dog
Dbo
__ is the dog of __ __ loves __
<Bo,Obama>
Bo loves Obama
Lbo
__ is the dog of __ __ loves __
<Bo,Obama>
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-a-
cat
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 EndThe End