Predicate%20Logic

1
Predicate Logic

• Splitting the Atom

2
What Propositional Logic cant do

• It cant explain the validity of the Socrates
  Argument
• Because from the perspective of Propositional
  Logic its premises and conclusion have no
  internal structure.

3
The Socrates Argument

• All men are mortal
• Socrates is a man_______________________
• Therefore, Socrates is mortal
• Translates as
• H
• S__
• M

4
The Problem

• The problem is that the validity of this
  argument comes from the internal structure of
  these sentences--which Propositional Logic cannot
  see
• All men are mortal
• Socrates is a man_______________________
• Therefore, Socrates is mortal

5
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.

6
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

7
Individual Constants Predicates
Socrates is a man.
The name of a personwhich well express by an
individual constant
A predicateassigns a property (being-a-man) to
the person named
8
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

9
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

10
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.
11
What are properties relations?
Are they ideas in peoples heads? Are they Forms
in Platos heaven? All this is controversial so
we resolve to treat properties and relations as
things that arent controversial, viz. sets.
12
Predicates designate sets!

• Individual constants name individualspersons,
  places, things, times, etc.
• We resolve to understand properties and relations
  as setsof individuals or ordered n-tuples of
  individuals (pairs, triples, quadruples, etc.)

13
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)

14
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 dont occupy
  time or space, or have causal powers even if
  their members do.

15
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.

16
Set Membership Set Inclusion
The Babushka Family C is the daughter of D and D
is the daughter of E, but C is not the daughter
of E!
17
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 setincluding
  another set!
• 1, 2, 3 ? 1, 2, 3, 4
• But members of members arent 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!

18
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

19
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 doesnt 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-tuplesordered pairs, triples, quadruples,
  quintuples, etc.
• ltAdam, Evegt ? ltEve, Adamgt
• lt1, 2, 3gt ? lt3, 2, 1gt

20
The Set of Russian Doll Sets

• C ? Babushkas
• Babushkas ? Russian Doll Sets
• C ? Russian Doll Sets
• C is a dollnot a set of dolls!

21
Set membership set inclusion

• A ? B says that A is a member of B
• A ? B says that A is included in B, i.e. that all
  members of A are members of B.
• That doesnt mean A itself is a member of B!
• Example S1 1, 2, 3, S2 1, 2, 3
• S1 ? S2!
• 2 is a member of S1 but 2 is NOT a member of S2
• 2, 3 is a member of S2 but not of S1

22
Having a property

• An individuals 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.

e
Fido
brown things
23
What Singular Sentences Say

• Singular sentences are about set membership.
• Ascribing a property to an individual is saying
  that its 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
  ltSan Diego,Chula Vistagt is a member of ltLA,San
  Diegogt, ltPhiladelphia,Baltimoregt

24
Singular Sentences

• All men are mortal
• Socrates is a man_______________________
• Therefore, Socrates is mortal
• We can now understand what 2 and 3 say
• 2 says that Socrates e humans
• 3 says that Socrates e mortals

25
Singular statements

• Translation

26
Ducati is brown.
Bd
27
Tweety bopped Sylvester

• Bts

28
General Statements
29
General Sentences

• All men are mortal
• Socrates is a man_______________________
• Therefore, Socrates is mortal
• But we still dont have an account of what 1 says
• Because it doesnt ascribe a property or relation
  to any individual or individuals.

30
Emptiness
Shading in a region says that the set it
represents is empty.
A
This says that there are no As.
31
There are no unicorns.
unicorns
? (?x) Ux
32
Non-emptiness
X in a region says that the set it the region
represents is non-empty, i.e. theres something
there
A
This says that at least one thing is an A.
33
There are horses.
X
horses
(?x) Hx
34
Intersection

• Intersection of the sets A and B, denoted A n 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
35
No horses are carnivores
horses
carnivores
(x) ? (Hx Cx)
36
Intersection Conjunction

• This region represents the set of things that
  are both A and B.

X
A
B
37
Some horses are brown
X
brown things
horses
(?x)(Hx Bx)
38
Set Complement
The complement of a set if everything in the
universe thats not in the set.
A
The shaded area is the complement of Athe set of
things that are not A.
39
Set Complement

• This says that something is in both A but not B.

X
A
B
40
Some horses are not brown
X
brown things
horses
(?x)(Hx ?Bx)
41
All horses are mammals
horses
mammals
horses
(x)(Hx ? Mx)
42
All horses are mammals
horses
mammals
There are no non-mammalian horses horses n
non-mammals (the empty set)
43
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
44
Union and Disjunction

• If we say something is in the union of A and B
  were saying that its either in A or in B.

A
B
45
General Sentences

• All men are mortal
• Socrates is a man_______________________
• Therefore, Socrates is mortal
• We cant 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.

46
The Socrates Argument is Valid!

• All men are mortal
• Socrates is a man_______________________
• Therefore, Socrates is mortal
• 1 says that there are no men that dont belong to
  the set of mortals
• 2 says that Socrates e men
• 3 says that, therefore, Socrates e mortals

47
All men are mortals
immortal men nothing here!
men
mortals

• Theres nothing in the set of men outside of the
  set of mortals the set of immortal men is empty.

48
Socrates is a man.
men
mortals

• This is the only place in the men circle Socrates
  can go since weve already said that there are no
  immortal men.

49
Therefore, Socrates is a mortal.
men
mortals

• So Socrates automatically ends up in the mortals
  circle the argument is, therefore, valid!

50
Arguments Involving Relations

1. All dogs love their people
2. Bo is Obamas Dog____________________
3. Bo loves Obama

If we treat the predicates in these 3 sentences
as one-place predicates, we cant explain the
validity of this argument! It looks like were
ascribing 3 different properties that dont have
anything to do with one another people-loving,
being-Obamas-dog and Obama-loving
51
Relations

• We understand relations as sets of ordered
  n-tuples, e.g.
• being to the north of is ltLos Angeles, San
  Diegogt, ltPhiladelphia, Baltimoregt, ltSan Diego,
  Chula Vistagt
• being the quotient of one number divided by
  another x into y is lt2,2,1gt, lt2,4,2gt, lt5,45,9gt

52
Singular sentences about relations

• Standing in a relation is also understood in
  terms of set membership.
• Its a matter of being a member of an ordered
  n-tuple which is a member of a set.

Michelle is Baraks wife.
e
ltEve, Adamgt, ltHillary, Billgt
53
Validity of the Dog Argument
We can consider being the dog of and loves as
relations and designate them by the 2-place
predicates Dand L Now the 3 sentences have
something in common--they dont involve 3
different properties they involve 2 relations
being the dog of and loves So we can link them to
show validity

1. All dogs love their people
2. Bo is Obamas Dog____________________
3. Bo loves Obama

54
Bo is Obamas dog Dbo
55
(x)(y)(Dxy ? Lxy) For all x, y, if x is the dog
of y then x loves y
All dogs love their people
56
Validity of the Dog Argument

1. All dogs love their people
2. Bo is Obamas 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!
57
All dogs love their people
__ is the dog of __
__ loves __
(x)(y)(Dxy ? Lxy)
58
Bo is Obamas Dog
ltBo,Obamagt
__ is the dog of __
__ loves __
Dbo
59
Bo loves Obama
ltBo,Obamagt
__ is the dog of __
__ loves __
Lbo
60
Scope
61
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

62
There are dog-cats

• (?x)(Dx ? Cx) There exists an x such that x is a
  dog-and-a-cat

63
Overlapping quantifiers
64
Everybody loves somebody

• (x)(?y)Lxy for all x, there exists a y such
  that x loves y

65
Everybody is loved by somebody

• (x)(?y)Lyx for all x, there exists a y such
  that y loves x

66
Theres somebody everybody loves

• (?x)(y)Lyx There exists an x such that for all
  y, y loves x

67
Theres somebody who loves everybody

• (?x)(y)Lxy there exists an x such that for all
  y, x loves y

68
The End