1. Chapter 7. Propositional and Predicate Logic
Comp3710 Artificial Intelligence
Computing Science
Thompson Rivers University

2. Propositional/Predicate Logic
Course Outline
Part I – Introduction to Artificial Intelligence
Part II – Classical Artificial Intelligence, and Searching
Knowledge Representation
Searching
Search Methodologies
Advanced Search
Genetic Algorithms (relatively new study area)
Knowledge Represenation and Automated Reasoning
Propositinoal and Predicate Logic
Inference and Resolution for Problem Solving
Rules and Expert Systems
Part III – Machine Learning
Part IV – Advanced Topics
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3. Propositional/Predicate Logic
Chapter Objectives
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4. Propositional/Predicate Logic
Chapter Outline
Propositional logic
Introduction
What is logic? Why is logic used in Artificial Intelligence?
How to use logical operators
How to translate an English statement with logic notations
Let’s recall complex truth tables
Let’s recall tautology and contradictory
How to use equivalent propositions
How to logically use propositions – propositional logic
Introduction to predicate calculus
Summary
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Propositional/Predicate Logic

5. 1. Introduction [Q] What are expert systems? [Q] What is reasoning?
[Wikipedia] Reason is the capacity for consciously making sense of things, applying logic, for establishing and verifying facts, and changing or justifying practices, institutions, and beliefs based on new or existing information (facts and/or observation).
To form conclusions, inferences, or judgments
[Q] How to automate reasoning?
Need to know
How to represent information, knowledge, facts and beliefs, and
How to apply logic, …

6. [Q] How to formalize/validate our arguments?
Argument = premises (propositions or statements) + conclusion
To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true.
[Q] Any logical forms for valid arguments?

7. Examples
Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate an error message, then the program syntax is correct and program execution does not result in division by zero.
Argument 2:
If x is a real number such that x < -2 or x > 2, then x2 > 4.
For y, such that y2 /> 4,
y /< -2 and y /> 2.
The common logical form of both of the above arguments:
If p or q, then r.
Not r.
Therefore not p and not q.
Is this logical form valid?

8. Propositional/Predicate Logic
Topics
You may recall “Logic and Truth Tables” in MATH 1380.
In this unit,
Boolean logic
Propositional logic
Introduction to predicate calculus – first-order predicate logic
Propositional logic will be used in the following unit to solve some interesting problems.
Based on rules, knowledge, facts, and observation
Decide if a given query is valid.
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9. Propositional/Predicate Logic
Topics
2. What is Logic?
Logic: reasoning about the validity of arguments.
An argument is valid if its conclusions follow logically from its premises (proposition or statement) – even if the argument doesn’t actually reflect the real world:
Mary is a lemon.
All lemons are blue.
Therefore, Mary is blue conclusion
Logic is widely used as a representation method of AI, and allows us to easily reason about negatives (i.e., “NOT”) and disjunctions (,i.e., “OR”)
However, one of the main weaknesses of traditional logic (i.e., Boolean logic) is its inability to deal with uncertainty and ambiguity.
Later probabilistic method and fuzzy logic will be discussed to deal with uncertainty and ambiguity.
premises
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10. 3. How to Use Logical Operators
Definition of statement
A statement (or proposition) is a sentence that is true or false but not both.
Examples
Two plus two equals four.
2 + 2 = 4
I am a TRU student.
x + y > 0 ???
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Compound Statements
Symbols used in complicated logical statements:
~ not ~p negation of p
 and p  q conjunction of p and q
 or p  q disjunction of p and q
 exclusive or p  q
Order of operations: ( ) and ~ have the higher precedence.
~p  q = (~p)  q
~(p  q)
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12. Propositional/Predicate Logic
Topics
And (conjunction) 
Or (disjunction) 
Not (negation)  or ~
Implies (conditional)  (if… then…)
Iff (biconditional)  (if and only if)
Exclusive OR?
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13. 4. Translating between English and Logic
Facts, observations and rules need to be translated into logical notations.
For example:
It is Raining and it is Thursday:
R  T, where
R represents “It is Raining”, T represents “it is Thursday”.
R and T are called propositional symbols.
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More complex sentences need predicates. That part of a proposition that is affirmed or denied about the subject. For example, in the proposition We are mortal, mortal is the predicate.
E.g.,
It is raining in New York:
R(N)
Could also be written N(R), or even just R.
[Q] How to express “It is not raining in New York”???
It is important to select the correct level of detail for the concepts you want to reason about.
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Topics
Example
It is not hot but it is sunny. It is neither hot nor sunny.
-> It is not hot, and it is sunny. It is not hot, and it is not sunny.
Let h = “it is hot” and s = “it is sunny.” Then the above statements can be translated as
~h  s ~h  ~s
Suppose x is a particular real number. Let p, q, and r symbolize
“0 < x,” “x < 3,” and “x = 3.” respectively.
Then the following inequalities
x  3 0 < x < 3 0 < x  3
can be translated as
q  r p  q p  (q  r)
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5. Truth Tables
Tables that show truth values for all possible inputs to a logical operator.
For example:
[Q] Truth table for implication () ???
A  B: A is the antecedent, and B is the consequent.
A  B  A  B [Q] Can you prove it? How?
A  B  B  A [Q] Can you prove it? How?
[Q] Truth table for iff () ???
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Topics
We can produce truth tables for complex logical expressions, which show the overall value of the expression for all possible combinations of variables:
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18. 6. Tautology and Contradictory
Topics
6. Tautology and Contradictory
A tautology is true under any interpretation.
The expression A ˅ ¬A is a tautology.
This means it is always true, regardless of the value of A.
P is a tautology: this is written
╞ P
An expression which is false under any interpretation is contradictory (or unsatisfiable).
A  ¬A
Some expressions are satisfiable, but not always valid. This means that they are true under some interpretation, but not under all interpretations.
A  B
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19. 7. How to Use Equivalent Propositions
Two expressions are equivalent if they always have the same logical value under any interpretation:
A ˄ B  B ˄ A
[Q] How to prove the above equivalence?
Equivalences can be proven by examining truth tables.
[Q] Are there more equivalences?
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A ˅ A  ???
A ˄ A  ???
A ˄ (B ˄ C)  (A ˄ B) ˄ C
A ˅ (B ˅ C)  (A ˅ B) ˅ C
A ˄ (B ˅ C)  (A ˄ B) ˅ (A ˄ C) A ˅ (B ˄ C)  ???
A ˄ (A ˅ B)  ???
A ˅ (A ˄ B)  ???
A ˄ true  ??? A ˄ false  ???
A ˅ true  ??? A ˅ false  ???
[Q] DeMorgans’ Laws ???
[Q] Why do we need these equivalences?
By using the above and other equivalences, logical expressions can be simplified.
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Topics
Some more
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22. 8. How to Use Propositions
Propositional logic in this chapter is a logical system.
It deals with propositions.
Propositional calculus is the language we use to reason about propositional logic.
A sentence in propositional logic is called a well-formed formula (wff) (or sentence).
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Propositional calculus:
BNF (Backus-Naur Form) – The following are wff’s:
P, Q, R… – propositional symbols
true, false
(A) ¬A
A ˄ B
A ˅ B
A → B
A ↔ B
Any combination of wff’s is a wff.
[Q] Is P  Q  (B  C)  A  B  D  (E) a wff ???
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24. Propositional/Predicate Logic
Deduction: the process of deriving a conclusion from a set of assumptions.
Will be discussed in the following unit again to solve some problems.
If we deduce a conclusion C from a set of assumptions, we write:
{A1, A2, …, An} ├ C
If C can be concluded without any assumption
├ C
The inference rule A ├ B is expressed as A
B Given A, B is deduced (or concluded).
The meaning is that if A is true (or valid), then B is true (or valid).
[Q] What if A is false?
B can be true or false.
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25. Propositional/Predicate Logic
 introduction
Given A and B, we can deduce A  B.
A, B {A, B} ├ A ˄ B
A ˄ B
 introduction A_
A  B
 elimination
A  B A  B
A B
 elimination (called Modus Ponens)
A, A  B [Q] Can you prove? B
  elimination
  A A
Some valid inference rules
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Reduction to absurdity (proof by contradiction)
 A .
_ Contradiction, i.e., false A
 Induction (called deduction theorem)
C__
A  C
Some valid inference rules
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Topics
 Introduction
A__
B  A
Modus Tollens
~B, A  B ~A
Some valid inference rules
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28. 9. Introduction to Predicate Calculus
Predicate Calculus extends the syntax of propositional calculus with predicates and quantifiers:
P(X) – P is a predicate.
First Order Predicate Calculus (FOPC) allows predicates to apply to objects or terms, but not functions or predicates.
Just introduction in this unit.
Predicate calculus is used in solving more complex problems.
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29. Propositional/Predicate Logic
Quantifiers  and 
 - For all:
x P(x) is read “For all x’es, P (x) is true”.
E.g., for all TRU COMP students, they are smart.
 - There Exists:
x P(x) is read “there exists an x such that P(x) is true”.
E.g., there is a TRU COMP student who is not smart.
Relationship between the quantifiers:
x P(x)  ¬ ((x)¬P(x))
“If There exists an x for which P holds, then it is not true that for all x P does not hold”.
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30. Propositional/Predicate Logic
Topics
Properties of Logical Systems
Four factors to consider of:
Soundness: Is every theorem valid?
Completeness: Is every tautology a theorem?
Decidability: Does an algorithm exist that will determine if a wff is valid?
Monotonicity: Can a valid logical proof be made invalid by adding additional premises or assumptions?
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31. Propositional/Predicate Logic
Topics
10. Summary
Propositional logic
Propositions
Boolean logic
Deduction; inference rules
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