Slides adapted fromMichael P. Frank's course based on the textDiscrete Mathematics & Its Applications

(5th Edition)by Kenneth H. Rosen

University of Aberdeen, Computing Science

CS2013Mathematics for Computing

ScienceAdam Wyner

Predicate Logic

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

Fall 2013 Frank / van Deemter / Wyner 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

Fall 2013 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:((cs sa a) c) is a tautology

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Predicate Logic

• In propositional logic:(((cs) (sa) a) c) is a tautology

• Alternatively, it follows by propositional logic:(cs) (sa) a (premise) that c (conclusion).

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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.

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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.

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Predicate Logic

• Predicate logic extends 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.

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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.

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Applications of Predicate Logic

• Basis for many Artificial Intelligence systems.– Automatic program verification systems– Real world rule engines with NL interfaces

• Oracle Policy Automationhttp://www.youtube.com/OraclePAVideos

• Predicate-logic like statements are supported by some database query engines.

• Limitations (more later)

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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.

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Constants and Variables (informal)

• Individual constants that denote 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 variables that range over objects: x, y, z , …

Variables are like pronouns (e.g. he, she, it). The meaning varies, depending on what is indicated.

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Predicates and Arguments (informal)

• A predicate P applied 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.

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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)

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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

of things 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

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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 iff jill {jill, bill, phil}

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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....

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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'.

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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.

Fall 2013 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.

Fall 2013 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.

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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 z is_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?

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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...

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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.

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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...

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SyntaxQuantified Expressions

• Some well-formed expressions: xP(x) yQ(x) xy 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 xy R(x,y) is a formula (nested quantifiers)

• P(b) is a (atomic) formula, hence xP(b) is a formula (vacuous quantification).

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Formal SemanticsModel

• A model M is an ordered pair <D,I>, where D is a set of entities, and I is an interpretation function. M defines 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) {(,): D and D}• So on for 3-place predicates etc.

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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: bill is 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".

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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....

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Formal SemanticsTruth of Predicate Formulae

• A formula of the form P(a) is true with respect to M iff I(a) I(P).

• A formula of the form R(a,b) is true with respect to M iff (I(a),I(b)) I(R).

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Formal SemanticsTruth for Functional Connectives

• A formula of the form is true wrt M iff is false wrt M.

• 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 wrt M.

• A formula of the form is true wrt M iff is false wrt M or is true wrt M.

Recall the Truth Tables

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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 iff for

every I(a) D, (x:=a) is true wrt M.

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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?

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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?

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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?

Fall 2013 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?

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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