1. University of Aberdeen, Computing ScienceCS2013Mathematics for Computing ScienceAdam Wyner Slides adapted from Michael P. Frank's course based on the textDiscrete Mathematics & Its Applications(5th Edition)by Kenneth H. Rosen

2. Predicate Logic Rosen 5th ed., §§1.3-1.4 (but much extended)

3. Agenda • Why Predicate Logic? • Applications • Syntax and semantics – informal to formal • Work thorough examples • Bound and free variables • Complex expressions (quantifiers with logical connectives; nested quantifiers) • Quantifier equivalences • Vacuous quantification and false antecedents • Defining (or not) other quantifiers Frank / van Deemter / Wyner

4. To Predicate Logic • We can use propositional logic to prove that certain real-life inferences are valid. • If it’s cold then it snows. • If it snows there are accidents • There are no accidents. • Therefore, it’s not cold • In propositional logic: ((cssa a) c) is a tautology Frank / van Deemter / Wyner

5. Predicate Logic • In propositional logic: (((cs) (sa) a) c) is a tautology • Alternatively, it follows by propositional logic:(cs) (sa) a (premise) thatc (conclusion). Frank / van Deemter / Wyner

6. Predicate Logic • How to understand why Everyone is happy is false where the people are {bill, jill, mary} and Mary is not happy is true. • How to prove valid inferences that cannot be proven valid in propositional logic • Every human is mortal. Socrates is human. Therefore, Socrates is mortal. • We need a more expressive logic for quantifiers and predicates. Frank / van Deemter / Wyner

7. Predicate Logic • In propositional logic, we have simple propositions(atomic statements) and complex propositions (formed using truth functional operators). • In propositional logic, we have truth tables that represent alternative contexts. Frank / van Deemter / Wyner

8. Predicate Logic • Predicate logicextends propositional logic • Keeps the truth functional operators and truth tables. • Adds constants, variables and quantification over classes of entities. • Adds a distinction between the arguments of a sentence (e.g. subject, object, etc) and the predicates of a sentence (e.g. verbs). • Has models arguments and predicates of a domain. Frank / van Deemter / Wyner

9. Applications of Predicate Logic It is one of the most-used formal notations for writing mathematical definitions, axioms, and theorems. For example, in linear algebra, a partial order is introduced saying that a relation R is reflexive and transitive – and these notions are defined using predicate logic. Frank / van Deemter / Wyner

10. Applications of Predicate Logic • Basis for many Artificial Intelligence systems. • Automatic program verification systems • Real world rule engines with NL interfaces • Oracle Policy Automation http://www.youtube.com/OraclePAVideos • Predicate-logic like statements are supported by some database query engines. • Limitations (more later) Frank / van Deemter / Wyner

11. A Bit of Grammar • In The dog is sleeping: • The dog is an argument, the subject. • is sleeping is a one-place predicate (an intransitive verb), a property that is ascribed to the subject. • In The girl hit the ball: • The girl is the subject and the ball is the object. • hit is a two-place predicate (a transitive verb), a relation that is ascribed between the subject and object. • Predicate logic follows a similar form. Frank / van Deemter / Wyner

12. Constants and Variables (informal) • Individual constantsthatdenote individuals/objects: a,b,c,… Constants are a bit like proper names (e.g. names for things such as Bill, Jill, IBM). The meaning is fixed. • Individual variablesthat range over objects: x, y, z, … Variables are likepronouns (e.g. he, she, it). The meaning varies, depending on what is indicated. Frank / van Deemter / Wyner

13. Predicates and Arguments (informal) • A predicate Papplied to constant a is the proposition P(a). • Meaning: the object denoted by a has the property P. P(a) is T or F. • A predicate P applied to a variable x is a propositional form P(x). • Meaning: given an object that denotes a value of x, the object has the property P. P(x) is T or F depending on the value of x. Frank / van Deemter / Wyner

14. Forms and Evaluations (informal) • Jill is happy (NL) • is_happy'(jill'). • True if in the model the denotation of jill' is an element of the denotation of is_happy' (jill cannot vary). • He is happy (NL) • is_happy'(x). No gender on variables! • True if in the model given a denotation of x, that object is an element of the denotation of is_happy' (x can vary) Frank / van Deemter / Wyner

15. Models and Evaluation (informal) • A model, M, is a "toy" representation of a "world" has • a Domain (Universe) of Discourse D, which is the set ofthings that the predicates of the logic relate to and nothing else. • denotations for constants and predicates. Constants are related to a particular thing in D. The denotation (extension) of a predicate is the set of objects (set of relations) from D for which the predicate holds. • a manipulation on variables Frank / van Deemter / Wyner

16. Models and Evaluation (informal) • M1: • D = {jill, bill, phil, will, mary} • is_happy' denotes {jill, bill, phil} • jill' denotes jill • Evaluation (of Jill is happy) • is_happy'(jill') is T in m1 iffjill {jill, bill, phil} Frank / van Deemter / Wyner

17. Models and Evaluation (informal) • M1: D = {jill, bill, phil, will, mary}, is_happy' denotes {jill, bill, phil}, jill' denotes jill • Evaluation (of It is happy) • is_happy(x) is T in m1 iff x  {jill, bill, phil}?? • Depends on what x is: • is F where x = mary, for mary{jill, bill, phil} • is T where x = phil, for phil {jill, bill, phil} • Say where x is assigned the value mary.... Frank / van Deemter / Wyner

18. Quantifiers (informal) • Someone is happy. • Is True iff for some value of y wrt D, is_happy'(y) is True. • y is_happy'(y) • Everyone is hungry. • Is True iff for every value of x wrt D, is_hungry'(x) is True. • x is_hungry'(x) • Everyone is happy. Quantifiers provide a notation to express 'some' and 'all'. Frank / van Deemter / Wyner

19. Models and Evaluation (informal) • M2: where D = {jill, bill, phil, will, mary}, is_happy' denotes {jill, bill, phil}, is_hungry' denotes {jill, bill, phil, will, mary}, jill' denotes jill (and so on). • Evaluation (of Someone is happy) • y is_happy'(y) is T iff there is an assignment of a value from D to y such that is_happy'(y)is true. • is_happy'(y) is: T where y =jill; F where y =mary; F where y = will; T where y =phil; T where y = bill. • y is_happy'(y) is T for there is an assignment such that is_happy'(y)is true. Frank / van Deemter / Wyner

20. Models and Evaluation (informal) • M2: where D = {jill, bill, phil, will, mary}, is_happy' denotes {jill, bill, phil}, is_hungry' denotes {jill, bill, phil, will, mary}, jill' denotes jill (and so on). • Evaluation (of Everyone is happy) • y is_happy'(y) is T iff for every assignment of a value from D to y, is_happy'(y) is true. • is_happy'(y) is: T where y =jill; F where y =mary; F where y = will; T where y =phil; T where y = bill. • y is_happy'(y) is F for it is false that for every assignment of a value to y, is_happy'(y)is true. Frank / van Deemter / Wyner

21. Models and Evaluation (informal) • M2: where D = {jill, bill, phil, will, mary}, is_happy' denotes {jill, bill, phil}, is_hungry' denotes {jill, bill, phil, will, mary}, jill' denotes jill(and so on). • Evaluation (of Everyone is hungry) • x is_hungry'(x) is T iff for every assignment of a value from D to x, is_hungry'(x) is true. • is_hungry'(x) is: T where x =jill; T where x =mary; T where x = will; T where x =phil; T where x = bill. • y is_hungry'(x)is T for every assignment of a value to x, is_hungry'(x) is true. Frank / van Deemter / Wyner

22. Models and Evaluation (informal) • Questions: • Are x is_hungry'(x) and z is_hungry'(z) equivalent (true/false in the same models)? So, does the selection of variable matter? • Are x is_happy'(x) and z is_happy'(z)equivalent? • Does x is_hungry'(x) entail zis_hungry'(z)? • Are is_happy'(bill') and is_happy'(mary') equivalent? • Are is_happy'(x)and is_happy'(z) equivalent? • x is_happy'(x) andx is_hungry'(x) relate variables? Frank / van Deemter / Wyner

23. Formal SyntaxVariables, Constants, Predicates • Variable: x,y,z,… Constants: a,b,c,… • 1-place predicates: P,Q,… • 2-place predicates: R,S,… • 3-place, ... , n-place predicates: so on... Frank / van Deemter / Wyner

24. Formal SyntaxAtomic Formulae Atomic formulas: • If  is a 1-place predicate and  a variable or constant then () is an atomic formula. • If  is a 2-place predicate and  and  are variables or constants then (,) is an atomic formula. • And so on for 3-place, etc. Frank / van Deemter / Wyner

25. Formal SyntaxWell-formed Formulas (Well-formed) Formulas: • All atomic formulas are formulas. • If  and  are formulas then ,(  ), (  ), (  ) are formulas. • If  is a formula then x and y are formulas. • Lots of comments to make about quantified well-formed formulas – variable binding, vacuous quantification, nested quantifiers... Frank / van Deemter / Wyner

26. SyntaxQuantified Expressions • Some well-formed expressions: xP(x) yQ(x) xy R(x,y) xP(b) • P(x) is a (atomic) formula, hence xP(x) is a formula • Q(x) is a (atomic) formula, hence yQ(x) is a formula • R(x,y) is a (atomic) formula, hence y R(x,y) is a formula, hence xy R(x,y) is a formula (nested quantifiers) • P(b) is a (atomic) formula, hence xP(b) is a formula (vacuous quantification). Frank / van Deemter / Wyner

27. Formal SemanticsModel • A model M is an ordered pair <D,I>, where D is a set of entities, and I is an interpretation function. Mdefines gives a specific ‘meaning’ to the non-logical symbols (logical symbols are ). • If a is an individual constant then I(a)  D. Assume for every entity in D there is a constant. • If P is a 1-place predicate then I(P)  D • If R is a 2-place predicate, then I(R)  {(,):   Dand   D} • So on for 3-place predicates etc. Frank / van Deemter / Wyner

28. An Example of a Model • Suppose M = (D,I) where D={bill, jill, mary} • I(john') = john, I(mary') = mary, I(bill') = bill; I(B)={bill, jill}I(G)={mary}I(A)={(bill, jill) , (mary,bill)} • Note: billis a person (i.e., a part of the world)bill'is an individual constant (i.e., a part of the language of predicate logic). In a sentence, we would use the proper name "Bill". Frank / van Deemter / Wyner

29. Formal SemanticsTruth Definition Given a formula and a model, we want a systematic way to calculate whether (or not) the formula is true (or false) with respect to the model. The Truth Definitions are of the form formula is true iff.... Frank / van Deemter / Wyner

30. Formal SemanticsTruth of Predicate Formulae • A formula of the form P(a) is true with respect to M iffI(a)  I(P). • A formula of the form R(a,b) is true with respect to M iff(I(a),I(b))  I(R). Frank / van Deemter / Wyner

31. FormalSemanticsTruth for Functional Connectives • A formula of the form  is true wrt M iff is false wrtM. • A formula of the form    is true wrt M iff is true wrt M or  is true wrt M or both. • A formula of the form    is true wrt M iff is true wrt M and  is true wrtM. • A formula of the form   is true wrt M iff is false wrt M or  is true wrtM. Recall the Truth Tables Frank / van Deemter / Wyner

32. Formal SemanticsTruth with Variables and Quantifiers • Notation: where  is a formula,(x:=a) is the result of substituting all free occurrences of the variable x in  by the constant a. • More to say about 'free occurrences'. • A formula of the form x is true wrt M iff there is an I(a) D such that (x:=a) is true wrt M. • A formula of the form x is true wrt M ifffor every I(a)  D, (x:=a) is true wrt M. Frank / van Deemter / Wyner

33. Examples - Simple • Suppose M = (D,I) where D={bill, jill, mary} I(jill') = jill, I(mary') = mary, I(bill') = bill, I(B)={bill, jill}I(G)={mary}I(A)={(bill, jill) , (mary,bill)} • Is B(jill') true in M? • Is A(mary',bill') true in M? • Is A(jill',bill') true in M? Frank / van Deemter / Wyner

34. Examples - Connectives • Suppose M = (D,I) where D={bill, jill, mary} I(jill') = jill, I(mary') = mary, I(bill') = bill, I(B)={bill, jill}I(G)={mary}I(A)={(bill, jill) , (mary,bill)} • Is [B(jill')  G(bill')] true in M? • Is [A(mary',bill')  G(mary')] true in M? Frank / van Deemter / Wyner

35. Examples - Quantifiers • Suppose M = (D,I) where D={bill, jill, mary} I(jill') = jill, I(mary') = mary, I(bill') = bill, I(B)={bill, jill}I(G)={mary}I(A)={(bill, jill) , (mary,bill)} • Is y B(y) true in M? • Is y [B(y)  G(y)]true in M? • Is y [B(y)  A(y,bill')] true in M? Frank / van Deemter / Wyner

36. Examples - Quantifiers • Suppose M = (D,I) where D={bill, jill, mary} I(jill') = jill, I(mary') = mary, I(bill') = bill, I(B)={bill, jill}I(G)={mary}I(A)={(bill, jill), (mary,bill), (jill, jill)} • Is y B(y) true in M? • Is y [B(y)  G(y)]true in M? • Is y [B(y)  A(y,bill')] true in M? • Is x [B(x)  A(x,jill)] true in M? Frank / van Deemter / Wyner

37. Next • Work thorough examples • Bound and free variables • Complex expressions (quantifiers with logical connectives; nested quantifiers) • Quantifier equivalences • Vacuous quantification and false antecedents • Defining (or not) other quantifiers • Other Frank / van Deemter / Wyner