Predicate Logic

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 thatthe 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. 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

8. 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…

9. Individual Constants & Predicates Socratesis 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

10. Translation Socratesis a man. Hs Socrates is mortal. Ms Athensis 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 is brown { brown things }

13. 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.

14. 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.

15. About sets… • 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

16. Intersection & Conjunction • This says that something is in both A and B. X A B

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

18. About sets… • 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

19. Union and Disjunction • If we say something is in the union of A and B we’re saying that it’s either in A orin B. A B

20. 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. ε Fido { brown things }

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. Sets and Ordered N-Tuples • 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>

23. Relations • So now we can explain what relations are: they’re 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>…}

24. Standing in a relation • 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>…}

25. 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>…}

26. 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}

27. 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.

28. 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.

29. General Sentences To get a handle on this we go back in the time machine to consider what the Greeks said…

30. Aristotelian Logic Aristotle: student of Plato; grandstudent of Socrates

31. Categorical Propositions Aristotle got the idea that we could display the logical relations propositions bear to one another by displaying the form of each as one of the following: A: All S are P E: No S are P I: Some S are P O: Some S are not P “S” stands for “subject” and “P” stands for “predicate” “All,” “No” and “Some” in Aristotelian logic are quantifiers. “Are” and “are not” are copulas.

32. Traditional Square of Opposition All S are P No S are P contrariety subalternation subalternation contradiction subcontrariety Some S are P Some S are not P

33. Existential Presupposition But all this assumes that there are S’s—and maybe there aren’t! Suppose “S” stands for “unicorns”: No unicorns live in Chula Vista because there are no unicorns; but we don’t want to infer from this that there are some unicorns that don’t live in Chula Vista--because there are no unicorns living anywhere. We prefer therefore to interpret A and E propositions without the existential presuppostion which gives us the “Boolean Square of Opposition”

34. Boolean Square of Opposition All S are P No S are P contradiction Some S are P Some S are not P

35. Venn Diagrams • To see how, and why, the Boolean Square of Opposition works we need to consider how the 4 categorical proposition forms are represented in 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, so…

36. Venn Diagrams Unicorns There are no unicorns

37. Venn Diagrams X horses There are horses

38. Venn Diagrams X horses brown things Some horses are brown

39. Venn Diagrams X horses brown things Some horses are not brown

40. Venn Diagrams horses mammals All horses are mammals

41. Venn Diagrams horses carnivores No horses are carnivores

42. Boolean Square of Opposition P S P S All S are P No S are P contradiction Some S are P Some S are not P X X P S P S

43. 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}

44. 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. immortal men: nothing here! men mortals

45. 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

46. Therefore, Socrates is a mortal. • So Socrates automatically ends up in the mortals circle: the argument is, therefore, valid! men mortals

47. Shortcomings of Aristotelian Logic • We want logic to explain the validity of inferences • But Aristotelian logic doesn’t explain the validity of inferences from propositional logic and • Doesn’t explain the validity of inferences that depend on relations • Because it treats all predicates as one-place predicates

48. AL can’t handle 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

49. Predicate Logic to the Rescue! Predicate logic does what we want! • Includes connectives so we can explain the validity of arguments that are valid by propositional logic. • Adds quantifiers so we can translate sentences like “All men are mortal” and explain the validity of the Socrates Argument. • Includes multi-place predicates that designate relations so we can explain the validity of the Portuguese Water Dog Argument.

50. Predicate Logic Vocabulary • Connectives:  ,  , • ,  , and  • Individual Constants: lower case letters of the alphabet (a, b, c,…, u, v, w) • Predicates: upper case letters (A, B, C,…, X, Y, Z) • Variables: x, y and z • Quantifiers: • Existential ( [variable]), e.g. (x), (y)… • Universal ([variable]), e.g. (x), (y)…