Propositional and Predicate Calculus - Carleton University

Outline and Introduction 

Propositional Calculus 

Predicate Calculus 

Inference Rules 

Unification 

Resolution Theorem Proving 

Propositional and Predicate Calculus 1 

Instructor: Dr. B. John Oommen 

Chancellor’s Professor 

Fellow: IEEE; Fellow: IAPR 

School of Computer Science, Carleton University, Canada. 

1 The primary source of these notes are the slides of Dr. Ebaa Fayyoumi 

from Jordan. I sincerely thank her for this. 

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Outline 





Unification 


Outline 

Introduction 

Introduction 



Using Inference Rules to Produce Predicate Calculus Expressions 

Unification 

Clause Form 

Resolution and Theorem Proving

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Introduction 





Unification 


Outline 

Introduction 

We will introduce the Propositional calculus and the language by which 

it is presented in AI 

We will explain the syntax and semantics of the Propositional and 

Predicate calculus languages and the rules of inference

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Introduction 





Unification 


Introduction 

Propositional Syntax 

Propositional Semantic 

Constructing New Logical Equivalence 

Propositional calculus uses Words, Phrases and Sentences to 

represent and reason about properties and relationships in the world 

Every Proposition has a truth value 

true (1) 

false (0) 

Examples of Propositional calculus: 

The moon is made of green cheese 

Open the door −→ X; Imperative. 

“What time is it” −→ X; Interrogative 

Propositional calculus consists of 

Syntax {Symbols and Sentences} 

Semantics

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Introduction 





Unification 


Introduction 







true (1) 

false (0) 







Semantics

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Introduction 





Unification 


Introduction 







true (1) 

false (0) 







Semantics

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Introduction 





Unification 


Introduction 







true (1) 

false (0) 







Semantics

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Unification 



Introduction 




DEFINITION 

PROPOSITIONAL CALCULUS SYMBOLS 

The symbols of propositional calculus are the Propositional symbols: 

P, Q, R, S,... 

The Truth symbols: 

true, false 

And the Connectives: 

∧, ∨, ¬, →, ≡

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Unification 


Propositional Syntax 1 

Introduction 




DEFINITION 

PROPOSITIONAL CALCULUS SENTENCES 

Every propositional symbol and truth symbol is a sentence 

For example: true, P, Q, and R are sentences 

The negation of a sentence is a sentence 

For example: ¬P and ¬false are sentences 

The conjunction, or and, of two sentences is a sentence 

For example: P ∧¬P is a sentence

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Unification 



Introduction 




The disjunction, or or, of two sentences is a sentence 

For example: P ∨¬P is a sentence 

the implication of one sentence from another is a sentence 

For example: P → Q is a sentence 

The equivalence of two sentences is a sentence 

For example: P ∨ Q ≡ R is a sentence 

Legal sentences are also called well-formed formulae or WFFs

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Unification 



Introduction 




The expression of the form (P ∧ Q) (P and Q) is called a “Conjunct” 

The expression of the form (P ∨ Q) (P or Q) is called a “Disjunct” 

The expression of the form (P ⇒ Q) (P implies Q), 

P is the “premise” or “antecedent” 

Q is a “conclusion” or “consequent”

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Unification 


Propositional Semantics 1 

Introduction 




DEFINITION 

PROPOSITIONAL CALCULUS SEMANTICS 

An interpretation of a set of propositions is the assignment of a truth value, 

either T or F, to each propositional symbol 

The symbol true is always assigned the value T, and the symbol false is 

assigned F 

The interpretation or truth value for sentences is determined by: 

The truth assignment of negation,¬P, where P is any propositional 

symbol, is F if the assignment to P is T, and T if the assignment to P is F 

The truth assignment of conjunction,∧, is T only when both conjuncts 

have truth value T; Otherwise it is F

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Unification 


Propositional Semantics 2 

Introduction 




The truth assignment of disjunction,∨, is F only when both disjuncts 

have truth value F; Otherwise it is T 

The truth assignment of implication,→, is F only when the premise or 

symbol before the implication is T and the truth value of the consequent 

or symbol after the implication is F; Otherwise it is T 

The truth assignment of equivalence,≡, is T only when both 

expressions have the same truth assignment for all possible 

interpretations; Otherwise it is F

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Unification 


Propositional Semantic: Negation 

Introduction 




Negation: “The Not” symbol “¬” 

P : I am going to town 

¬P : I am not going to town 

That is: It is not the case that I am going to town. 

P ¬P 

T F 

F T

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Unification 


Introduction 




Propositional Semantics: Conjunction 

Conjunction: The “and” symbol “ ∧ ” 


Q : It is going to rain 

P ∧ Q : I am going to town and it is going to rain. 

P Q P ∧ Q 

T T T 

T F F 

F T F 

F F F

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Unification 


Introduction 




Propositional Semantics: Disjunction 

Disjunction: The “or” symbol “ ∨ ” 



P ∨ Q : I am going to town or it is going to rain. 

P Q P ∨ Q 

T T T 

T F T 

F T T 

F F F

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Unification 


Introduction 




Propositional Semantics: Implication 

Implication: “If...then...” symbol → 



P → Q : If I am going to town then it is going to rain. 

P Q P → Q 

T T T 

T F F 

F T T 

F F T

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Unification 


Introduction 




Propositional Semantics: Biconditional 

Biconditional: The “if and only if” symbol “↔” 



P ↔ Q : I am going to town if and only if it is going to rain. 

P Q P ↔ Q 

T T T 

T F F 

F T F 

F F T

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Propositional Laws 





Unification 


Introduction 



Constructing New Logical Equivalence

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Unification 


Propositional Example 

Introduction 



Constructing New Logical Equivalence

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Unification 


Introduction 





Proof: ¬(p → q) ≡ p∧¬q 

¬(¬p ∨ q) 

¬(¬p)∧¬q 

p ∧¬q 

Using Implication 

Using De Morgan’s Law 

Using Double Negation

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Unification 


Introduction 





Proof: ¬(p∨(¬p∧ q)) ≡ ¬p∧¬q 

¬p ∧¬(¬p ∧ q) Using De Morgan’s Law 

¬p ∧(p ∨¬q) Using De Morgan’s Law 

and Double Negation 

(¬p ∧ p)∨(¬p ∧¬q) Using Distribution Law 

F ∨(¬p∧¬q) Using Negation Law 

¬p ∧¬q Using Identity Law

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Introduction 





Proof: p ∧ q → p ∨ q ≡ T 

¬(p ∧ q)∨(p∨q) 

(¬p ∨¬q)∨(p∨q) 

(¬p ∨ p)∨(¬q ∨ q) 

T ∨ T 

T 

Using Implication 

Using De Morgan’s Law 

Using Commutative Law 

Using Domination Law 

Using Tautology

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Unification 


Predicate Calculus: Introduction 

Introduction 

Predicates and Sentences: Syntax 

General Important Notes 

Atomic Sentences 

Predicate Calculus: Semantics 

Predicate Equivalences 

Some Examples 

In Propositional calculus: 

Each atomic symbol P, Q or R represents a single proposition 

There is no way to access the component of an individual assertion 

Predicate calculus exists for the following reasons: 

We want to Access the component of an individual assertion 

We want to Manipulate predicate calculus expressions 

We want to Infer new predicate sentences

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Unification 



Introduction 






Some Examples 

Instead of saying P = “It rains on Tuesday”. 

We could create a “predicate” weather in order to describe the relationship 

between the weekday and the weather. 

weather(tuesday, rainy)

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Introduction 






Some Examples 

We could generalize this. Describe the weather for any day in the week as: 

weather(sunday,rainy) 

weather(monday,rainy) 

weather(tuesday,rainy) 

. 

weather(X,rainy) 

weather(X,sunny) 

weather(X,windy) 

. 

weather(X,Y) 

X is a variable representing the day of the week 

Y is a variable representing the weather on a day

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Introduction 






Some Examples 

DEFINITION 

PREDICATE CALCULUS SYMBOLS 

Alphabet of the symbols of the predicate calculus are: 

1. The set of letters, both upper and lowercase, of the English alphabet 

2. The set of digits, 0, 1, 2,..., 9 

3. The underscore, _ 

Symbols in the predicate calculus begin with a letter and are followed by any 

sequence of these legal characters

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Introduction 






Some Examples 

Legitimate characters in the alphabet of predicate calculus symbols include 

a R 6 9 p _ z 

Examples of characters not in the alphabet include 

# % / & ¨" 

Legitimate predicate calculus symbols include 

George fire3 tom_and_jerry bill XXXX friends_of 

Examples of strings that are not legal symbols are 

3jack "no blanks allowed" ab%cd ***71 duck!!!

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Important Notes 





Unification 


Introduction 






Some Examples 

l(j,k) ≡ like(john,kate). 

These two expressions are equivalent 

The second expression can be of great help 

Indicates what relationship the expression presents 

Descriptive names: Improves the expression’s readability 

Improper Symbols are 

Parentheses “()” 

Comma “,” 

Period “.” 

These are used to construct a WFF






Unification 


Introduction 






Some Examples 

Predicate calculus may represent 

Constants: 

Specifying certain object/property in the world 

Must start with a lowercase letter 

Examples: blue, tree and tall. 

“true” and “false” are reserved truth symbols 

Variables: 

Specifying general class of objects/properties in the world 

Must start with an uppercase letter 

Kate and George are legal variables. 

bill and george are not legal variables. 

Functions: 

Maps one or more elements in the domain 

To a unique elements in the range 

Must start with a lowercase letter. 

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Unification 


Introduction 






Some Examples 


Constants: 





Variables: 





Functions: 




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Unification 


Introduction 






Some Examples 


Constants: 





Variables: 





Functions: 




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Unification 


Introduction 






Some Examples 

Each function has an associated arity 

Indicates the number of elements in the domain mapped 

Which maps onto each element of the range 

father(sam) → arity = 1. 

plus(x1, x2)→ arity = 2. 

Function expression: Function symbol followed by its arguments 

Arguments: Elements from the function’s domain 

Number of the arguments equals the function’s arity 

Arguments: Enclosed in parentheses; Separated by commas

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Unification 


Introduction 






Some Examples 

Value of the function: 

Single object in the range that the arguments are mapped to. 

Examples: 

The value of the function “father(sam)” is george. 

The value of the function “plus(one, two)” is three. 

Evaluation involves replacing the function with its value

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Unification 


The Syntax of Atomic Sentences 

Introduction 






Some Examples 

DEFINITION 

SYMBOLS and TERMS 

Predicate calculus symbols include: 

1. Truth Symbols true and false (these are reserved symbols) 

2. Constant Symbols are symbol expressions beginning with an lowercase 

character 

3. Variable Symbols are symbol expressions beginning with an uppercase 

character 

4. Function Symbols are symbol expressions having the first character 

lowercase. Functions have an attached arity indicating the number of 

elements of the domain mapped onto each element of the range

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Unification 


Introduction 






Some Examples 

A function expression consists of a function constant of arity n followed by n 

terms, t 1 , t 2 , ..., t n, enclosed in parentheses and separated by commas. 

A predicate calculus term is either a constant, varaible or function expression.

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Unification 


Predicates and Atomic Sentences 

Introduction 






Some Examples 

An Atomic sentence is a primitive unit of the predicate language. 

An Atomic sentence is a predicate with arity followed by n terms 

Enclosed between parentheses and separated by commas. 

Examples: 

friends(george, kate) 

friends(george, X) 

friends(george, sara, sam) 

friends(father(david), father(sara)) → friends(mike, john)

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Introduction 






Some Examples 

DEFINITION 

PREDICATES and ATOMIC SENTENCES 

Predicate symbols are symbols beginning with a lowercase letter 

Predicates have an associated positive integer referred to as the arity or 

“argument number” for the predicate Predicates with the same name but 

different arities are considered distinct. 

An atomic sentence is a predicate constant of arity n followed by n terms, t 1 , 

t 2 , ..., t n, enclosed in parentheses and separated by commas 

The truth values, true and false, are also atomic sentences

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Introduction 






Some Examples 

Atomic sentences might be combined by using the logical operations: 

not: ¬ 

and: ∧ 

or: ∨ 

if: → 

equivalence: ≡ 

universal quantifier: ∀ 

existential quantifier: ∃

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Unification 



Introduction 






Some Examples 

Consider: ∀X likes(X, cake) 

This sentence will be true when all of the values of the variable (that 

belong to the disclosure domain) satisfy this predicate 

Consider: ∃X friends(X, sara) 

This sentence will be true when there is at least one value of the 

variable (that belongs to the disclosure domain) that satisfies this 

predicate

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Unification 


Predicates Calculus Sentences 

DEFINITION 

PREDICATE CALCULUS SENTENCES 

Every atomic sentence is a sentence. 

1. If s is a sentence, so is its negation, ¬s 

Introduction 






Some Examples 

2. If s 1 and s 2 are sentences, then so is their conjunction s 1 ∧ s 2 

3. If s 1 and s 2 are sentences, then so is their disjunction s 1 ∨ s 2 

4. If s 1 and s 2 are sentences, then so is their implication s 1 → s 2 

5. If s 1 and s 2 are sentences, then so is their equivalence s 1 ≡ s 2 

6. If X is a variable and s a sentence, then ∀X s is a sentence. 

7. If X is a varaible and s a sentence, then ∃X s is a sentence.





Unification 


Predicates Calculus Sentences 

verify_sentence algorithm 

Introduction 






Some Examples 

function verify_sentence(expression); 

begin 

case 

expression is an atomic sentence: return SUCCESS 

expression is of the form QXs where Q is either∀ or ∃, X is a variable, and s is an expression; 

if verify_sentence(s) returns SUCCESS 

then return SUCCESS 

else return FAIL 

expression is of the form ¬s: 

if verify_sentence(s) returns SUCCESS 



if verify_sentence(s)returns SUCCESS 



expression is of the form s 1 op s 2 , where op is a binary logical operator: 

if verify_sentence(s_1)returns SUCCESS and 

verify_sentence(s_2)returns SUCCESS 


else return FAIL; 

otherwise: return FAIL 

end 

end 

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Examples 





Unification 


Introduction 






Some Examples 

Examples: Determine whether the following is a sentence or not. 

plus(two,three) 

No 

equal(plus(two, three), five) 

Yes 

equal(plus(two, three), seven) 

Yes 

∃X likes(X, ice-cream) ∧ equal(plus(two, three), five) Yes

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Examples 





Unification 


Introduction 






Some Examples 

Write a predicate calculus expression in order to capture the following 

relationships by using the provided facts. 

parent(mike, sam) 

parent(sara, sam) 

parent(mike, john) 

parent(sara, john) 

female(sara) 

male(mike) 

male(john) 

male(sam)

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Examples 





Unification 


Introduction 






Some Examples 

Mother relationship: ∀X∀Y parent(X,Y)∧ female(X) 

Son relationship: ∀X∀Y parent(X,Y)∧ male(Y) 

Sibling relationship: ∀X∀Y∀Z parent(X,Y)∧ parent(X,Z) 

Brother relationship: ∀X∀Y∀Z parent(X,Y)∧ parent(X,Z)∧ male(Y)

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Unification 


Semantics for Predicate Calculus 

Introduction 






Some Examples 

DEFINITION: INTERPRETATION 

Let the domain D be a nonempty set. 

An interpretation over D is an assignment of the entities of D to each of the constant, 

variable, predicate, and function symbols of a predicate calculus expression, such that: 

1. Each constant is assigned an element of D 

2. Each variable is assigned to a nonempty subset of D; these are the allowable 

substitutions for that variable 

3. Each function f or arity m is defined on m arguments of D and defines a mapping 

from D m into D 

4. Each predicate p of arity n is defined on n arguments from D and defines a 

mapping from D n into {T, F}

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Unification 



Introduction 






Some Examples 

DEFINITION:TRUTH VALUES OF PRED. CALC. EXPRESSIONS 

Assume an expression E and an interpretation I for E over a nonempty domain D 

The truth value for E is determined by: 

1. The value of a constant is the element of D it is assigned to by I 

2. The value of a variable is the set of elements of D it is assigned to by I 

3. The value of a function expression is that element of D obtained by evaluating the 

function for the parameter values assigned by the interpretation. 

4. The value of truth symbol “true” is T and “false” is F. 

5. The value of an atomic sentence is either T or F 

6. The value of the negation of a sentence is T if the value of the sentence is F and is 

F if the value of the sentence is T

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Unification 



Introduction 






Some Examples 

1. The value of the conjunction of two sentences is T if the value of both the sentences 

is T and is F otherwise 

2. The truth value of expressions using ∨, →, ≡ is determined from the value of their 

operands as defined earlier 

Finally, for a variable X and a sentence S containing X: 

3. The value of ∀X S is T if S is T for all assignments to X under I, and it is F otherwise 

4. The value ∃X S is T if there is an assignment to X in the interpretation under which 

S is T; otherwise it is F

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Definitions 





Unification 


Introduction 






Some Examples 

friends(sara, X) XεD = {sam, george, lura, mike, louis, nel} 

X can be replaced by any other dummy variable such as Y 

Any variable: Quantified by its universal or existential quantifier 

∀X∃Y(parent(X,Y)∧female(Y))∧ like(X, Z) 

Variable X in parent(X, Y): 

Bounded by the universal quantifier 

Variable Y in parent(X,Y) and female(Y): 

Bounded by the existential quantifier 

Variables X and Z in like(X,Z) are free variables

Definitions 





Unification 


Introduction 






Some Examples 

Closed expression: 

An expression is “closed” if all of its variables are quantified 

i.e., ∀X∃Yparent(X,Y) 

Ground expression: 

An expression is “ground” if it does not contain any variable 

i.e., i.e., P ∧(Q ∨(R → S)). 

Parentheses are used to indicate the scope of quantification 

∀X∃Y(parent(X,Y) 

X and Y belong to certain domain 

If the domain of an interpretation is infinite: 

Exhaustive testing of all substitutions to a universally and 

existential quantified variable is computationally impossible 

The algorithm may never halt 

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Definitions 





Unification 


Introduction 






Some Examples 






i.e., i.e., P ∧(Q ∨(R → S)). 








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Definitions 





Unification 


Introduction 






Some Examples 






i.e., i.e., P ∧(Q ∨(R → S)). 








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Unification 



Introduction 






Some Examples 

Here are some predicate equivalences 

1 ¬∃X p(X) = ∀X ¬p(X) 

2 ¬∀X p(X) = ∃X ¬p(X) 

3 ∃X p(X) = ∃Y P(Y) 

4 ∀X p(X) = ∀Y P(Y) 

5 ∀X(p(X)∧q(X)) = ∀X p(X)∧∀Y q(Y) 

6 ∃X(p(X)∨q(X)) = ∃X p(X)∨∃Y q(Y)

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Unification 


First Order Predicate Calculus 

Introduction 






Some Examples 

DEFINITION: FIRST-ORDER PREDICATE CALCULUS 

First-order predicate calculus allows quantified variables to refer to 

objects in the domain of discourse and not to predicates or functions.

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Example 





Unification 


Introduction 






Some Examples 

Translate the following English sentence to predicate calculus: 

If it does not rain on Thursday, Sara will go to the beach. 

¬weather(thursday,rain) → goes(sara, beach) 

All basketball players are tall. 

∀X basketball_player(X) → tall(X) 

Nobody likes taxes. 

¬∃X likes(X, taxes)

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Example 





Unification 


Introduction 






Some Examples 








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Example 





Unification 


Introduction 






Some Examples 








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Unification 


Introduction 






Some Examples 

Example: Block World Relationship

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Introduction 





Unification 


Introduction 

Some Definitions 


Example 

An expression X logically follows from a set of expressions S 

If every interpretation that satisfies S also satisfies X 

An interpretation that makes the sentence true 

Is said to satisfy that sentence 

Question: Why Inference Rules for Predicate Systems (PS) 

Question: What is the meaning of Inference Rule 

PS: Can have an infinite number of possible interpretations 

These are very hard to be traced or tested 

Inference Rules: Used to determine when an expression is 

logically followed

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Unification 


Introduction 



Example 

Introduction 

Inference Rule (IR) 

Is a “mechanical” means 

Can produce new predicate sentence from other sentences 

Inference Rules: Classified as being Sound or Complete. 

Sound Inference Rule: 

Expression produced by the IR logically follows from S 

Complete Inference Rule: 

It has the ability to produce every expression that logically 

follows from S

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Definitions 





Unification 


Introduction 



Example 

DEFINITION: SATISFY, MODEL, VALID, INCONSISTENT 

For a predicate calculus expression X and interpretation I, 

If X has a value of T under I and a particular variable assignment, then I 

said to satisfy X 

If I satisfies X for all variable assignments, then I is a model of X 

X is satisfiable if and only of there exist an interpretation and variable 

assignment that satisfy it; otherwise, it is unsatisfiable 

A set of expressions is satisfiable if and only if there exist an 

interpretation and variable assignment that satisfy ever element 

If a set of expressions is not satisfiable, it is said to be inconsistent 

If X has a value T for all possible interpretations, X is said to be valid

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Unification 


Introduction 



Example 

Examples of Inconsistent and Valid predicate sentences 

∃X p(X)∧¬p(X) ⇒ “Inconsistence”. 

∀X p(X)∨¬p(X) ⇒ “Valid”.

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Unification 1 





Unification 


Introduction 



Example 

DEFINITION: LOGICALLY FOLLOWS, SOUND and COMPLETE 

A predicate calculus expression X logically follows from a set S of predicate calculus 

expressions if every interpretation and variable assignment that satisfies S also 

satisfies X 

An inference rule is sound if every predicate calculus expression produced by the rule 

from a set S of predicate calculus expressions also logically follows from S 

An inference rule is complete if, given a set S of predicate calculus expressions, the 

rule can infer every expression that logically follows from S

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Unification 2 





Unification 


Introduction 



Example 

DEFINITION: 

MODUS PONENS, MODUS TOLLENS, AND ELIMINATION, AND 

INTRODUCTION, and UNIVERSAL INSTANTIATION 

If the sentences P and P → Q are known to be true, then modus ponens lets us infer Q 

Under the inference rule modus tollens, if P → Q known to be true and Q is known to 

be false, we can infer ¬P 

And Elimination allows us to infer the truth of either of the conjuncts from the truth of a 

conjunctive sentence For instance, P∧Q lets us conclude P and Q are true 

And Introduction allows us to infer the truth of the conjunction from the truth of the 

conjuncts For instance, if P and Q are true, then P∧Q is true 

Universal Instantiation states that if any universally quantified variable in a true 

sentence is replaced by any appropriate term from the domain, the result is a true 

sentence. Thus, if a is from the domain of X, then ∀X p(X) lets us infer p(a).

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Unification 


Introduction 



Example 

Modus Ponens 

Modus Tollens 

P → Q 

P → Q 

P ¬Q 

−−−−− −−−−− 

Q ¬P 

And Elimination 

P ∧ Q 

−−−−− 

P 

Q 

And Introduction 

P 

Q 

−−−−− 

P ∧ Q

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Unification 


Introduction 



Example 

Universal Instantiation 

∀X tall(X) 

−−−−− 

tall(sara)

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Example 1 





Unification 


Introduction 



Example 

Determine (Justify) whether the following arguments are valid or invalid 

All men are mortal. Socrates is man. Therefore, Socrates is mortal 

Solution 

It is valid using Universal Instantiation and Modes Ponens IRs. 

∀X man(X) → mortal(X) 

man(socrates) 

man(socrates) → mortal(socrates) 

man(socrates) 

∴ mortal(socrates)

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Example 2 





Unification 


Introduction 



Example 

Determine (Justify) whether the following arguments are valid or invalid 

Sara is a student in COMP1805. 

Sara does not read the book. 

Everyone in COMP1805 passed the midterm exam. 

Therefore, Sara passed the midterm and she did not read the book. 

Solution 

S(X): X is a student in class COMP1805. 

R(X): X is a student who does not read the book. 

P(X): X is a student who passes the midterm exam.

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Example 2 





Unification 


Introduction 



Example 

S(Sara) 

¬R(Sara) 

∀X S(X) → P(X) 

S(Sara) → P(Sara) 

S(Sara) 

P(Sara) 

¬R(Sara) 

∴ P(Sara)∧¬R(Sara) 

Univ. Instantiation 

Modus Ponens 

Conjunction 

The Assertion is Valid

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Unification 





Unification 


Introduction 

Skolemization 

Example 

Unification: Determine if the following two sentences are similar or not 

Propositional calculus→ Very trivial 

Predicate calculus → Very difficult (because of variables) 

X/Y means X is substituted for the variable Y 

The substitution→ Binding

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Unification 





Unification 


Introduction 

Skolemization 

Example 

Examples: What is the final value of Y and Z 

{X/Y, W/Z}, {V/X}, {a/V, f(b)/W} 

Predicate calculus expression should be transformed to list syntax. 

p(a,b) → (p a b). 

p(f(a),g(X,Y))→ (p (f(a)) (g(X,Y))). 

equal(eva, mother(sara))→ (equal eva (mother(sara))).

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Skolemization 





Unification 


Introduction 

Skolemization 

Example 

Skolemization is a process of replacing each existentially quantified 

variable with a function that returns the appropriate constant 

Eliminating existentially quantified variable is complicated due to the 

dependency of this variable on another variables 

∀X ∃Y mother(X, Y) It is equivalent to ∀X mother(X, f(X)) 

∀X ∀Y ∃Z m(X, Y, Z) ⇒ ∀X ∀Y m(X, Y, f(X, Y))

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Example 1 





Unification 


Introduction 

Skolemization 

Example

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Example 1 





Unification 


Introduction 

Skolemization 

Example

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Example 2 





Unification 


Introduction 

Skolemization 

Example 

Question:- A logical_Based Financial Advisor

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Clause Form 





Unification 


Clause Form 

Resolution Theorem 

Examples 

Clause form: The logical databases as a set of disjunctions of literals 

∀X ([a(X)∧ b(X)] → [c(X, I)∧∃Y (∃Z c(Y, Z) → d(X, Y))])∨∀X e(X) 

Eliminate the if conditional→ by using disjunction a → b ≡ ¬a∨b 

∀X (¬[a(X)∧b(X)]∨[c(X, I)∧∃Y (∃Z ¬c(Y, Z)∨d(X, Y))])∨∀X e(X) 

Reduce the scope of the negation by using one of the following 

1 ¬(¬a) ≡ a 

2 ¬∃X a(X) ≡ ∀X ¬a(X) 

3 ¬∀X a(X) ≡ ∃X ¬a(X) 

4 ¬(a∧b) ≡ ¬a ∨ ¬b 

5 ¬(a∨b) ≡ ¬a ∧ ¬b 

∀X([¬a(X)∨¬b(X)]∨[c(X, I)∧∃Y(∃Z¬c(Y, Z)∨d(X, Y))])∨∀X e(X)

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Unification 


Clause Form 


Examples 

Standardize all variables by renaming them so that variables bound by 

different quantifiers have unique names 

∀X([¬a(X)∨¬b(X)]∨[c(X, I)∧∃Y(∃Z¬c(Y, Z)∨d(X, Y))])∨∀W e(W) 

Move all quantifiers to the left without changing their order 

∀X ∃Y ∃Z ∀W([¬a(X)∨¬b(X)]∨[c(X, I)∧(¬c(Y, Z)∨d(X, Y))]∨e(W)) 

All existential quantifiers are eliminated by Skolemization 

∀X ∀W([¬a(X)∨¬b(X)]∨[c(X, I)∧(¬c(f(X), g(X))∨d(X, f(X)))]∨ 

e(W)) 

Drop all universal quantifiers 

[¬a(X)∨¬b(X)]∨[c(X, I)∧(¬c(f(X), g(X))∨ d(X, f(X)))]∨e(W)

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Examples 

Convert the expression to the conjunct of disjuncts form 

¬a(X)∨¬b(X)∨c(X, I)∨e(W)∧ 

¬a(X)∨¬b(X)∨¬c(f(X), g(X))∨d(X, f(X))∨ e(W) 

Call each conjunct a separate clause 

¬a(X)∨¬b(X)∨c(X, I)∨e(W) 

¬a(X)∨¬b(X)∨¬c(f(X), g(X))∨d(X, f(X))∨ e(W) 

Standardize each variable again 

¬a(X)∨¬b(X)∨c(X, I)∨e(W) 

¬a(U)∨¬b(U)∨¬c(f(U), g(U))∨d(U, f(U))∨e(V)

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Examples 

It is a technique for proving theorems in propositional and predicate 

calculus 

It describes a way of finding contradictions in the databases with a 

minimum use of substitutions 

Method: 

1 It proves a theorem by negating the statement to be proved 

2 It adds this negated goal to the set of axioms that are 

known to be true 

3 It uses the inference rules to show that this leads to a 

contradiction 

4 If the negated goal is inconsistent with the given set of 

axioms, it follows that the original goal is consistent

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Examples 

Resolution refutation proofs involve the following steps: 

1. Put the premises or axioms into clause form 

2. Add the negation of what is to be proved, in clause form, to the set of 

axioms 

3. Resolve these clauses together, producing new clauses that logically 

follow from them 

4. Produce a contradiction by generating the empty clause 

5. The substitutions used to produce the empty clause are those under 

which the opposite of the negated goal is true

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Examples 

Resolution requires the axioms and the negation of the goal to be in a 

normal form called clause from 

Clause From represents the logical databases as a set of disjunctions 

of literals 

Literal is an atomic expression or the negation of an atomic expression. 

In general, binary resolution is the common form of resolution 

It is applied to two clauses: the first one contains the literal 

and the second one contains its negation 

The literals must be unified to make them equivalent 

The new clause that is produced consisting of the disjuncts 

of all the predicates in the two clauses minus the literal and 

its negative instance

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Example 1 





Unification 


Clause Form 


Examples 

1 All dogs are animals 

2 Fido is a dog 

3 All animals will die 

4 We want to prove that “Fido will die” 


∀X dog(X) → animal(X) 

dog(fido) 

∀Y animal(Y) → die(Y) 

Negate the conclusion 

¬die(fido) 

Clause Form 

¬dog(X)∨animal(X). 

dog(fido). 

¬animal(Y)∨die(Y). 

¬die(fido).

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Example 1 





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Clause Form 


Examples

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Example 2 





Unification 


Clause Form 


Examples 


Clause Form 

1 a ← b ∧ c 1. a∨¬b ∨¬c 

2 b 2. b 

3 c ← d ∧ e 3. c ∨¬d ∨¬e 

4 e∨f 4. e ∨ f 

5 d ∧¬f 5.1 d 

5.2 ¬f 

6 We want to prove that “a” 6. ¬a

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Example 2 





Unification 


Clause Form 


Examples

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Example 3 





Unification 


Clause Form 


Examples

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Example 4 





Unification 


Clause Form 


Examples

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Example 4 





Unification 


Clause Form 


Examples

Propositional and Predicate Calculus - Carleton University

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