1. Predicate Logic
Splitting the Atom

2. 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.

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 person—which we’ll express by an individual constant
A predicate—assigns 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 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.

12. 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

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 don’t 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 InclusionA B C E D
Mama
Grandma
The 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!

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 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!

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 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>

20. The Set of Russian Doll SetsB C E D
C  {Babushkas}
{Babushkas}  {Russian Doll Sets}
C  {Russian Doll Sets}
C is a doll—not 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 doesn’t 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. ε Having a property Fido { brown things }
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. ε
Fido
{ brown things }

23. 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>…}

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 ε {humans}
3 says that Socrates ε {mortals}

25. Translation
Singular statements

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 don’t have an account of what 1 says
Because it doesn’t 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. This says that at least one thing is an A.
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.

33. There are horses. X
horses
(x) Hx

34. This is the intersection of sets A and B
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

35. No horses are 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
horses
brown things
(x)(Hx • Bx)

38. 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.

39. This says that something is in both A but not B.
Set Complement
This says that something is in both A but not B. X A B

40. Some horses are not brownX
horses
brown things
(x)(Hx • Bx)

41. All horses are mammals
horses
horses
mammals
(x)(Hx  Mx)

42. All horses are mammals There are no non-mammalian horses
horses ∩ 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 A B
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

45. General Sentences All men are mortal
Socrates is a man _______________________
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.

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 don’t belong to the set of mortals
2 says that Socrates ε {men}
3 says that, therefore, Socrates ε {mortals}

47. All men are mortals men 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.

48. 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.

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
All dogs love their people
Bo is Obama’s Dog ____________________
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

51. 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>…}

52. 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>…}

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 “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
All dogs love their people
Bo is Obama’s Dog ____________________
Bo loves Obama

54. Bo is Obama’s dog: Dbo

55. For all x, y, if x is the dog of y then x loves y
(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
All dogs love their people
Bo is Obama’s Dog ____________________
Bo loves Obama
(x)(y)(Dxy  Lxy)
Dbo ______________
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 Obama’s Dog
<Bo,Obama>
__ is the dog of __
__ loves __
Dbo

59. Bo loves Obama
<Bo,Obama>
__ 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. There’s somebody everybody loves
(x)(y)Lyx: There exists an x such that for all y, y loves x

67. There’s somebody who loves everybody
(x)(y)Lxy: there exists an x such that for all y, x loves y

68. The End