The document provides an overview of knowledge representation and logic. It discusses knowledge-based agents and how they use a knowledge base to represent facts about the world through sentences expressed in a knowledge representation language. It then covers different knowledge representation schemas including propositional logic, first-order logic, rules, networks, and structures. The document also discusses inference, different types of logic, and knowledge representation languages.
This document provides an overview of mathematical logic. It defines key concepts such as propositions, truth values, logical connectives like negation, conjunction, disjunction, implication, biconditional, and quantifiers. Propositions are statements that can be either true or false. Logical connectives combine propositions and quantifiers specify whether statements apply to all or some cases. Truth tables are used to determine the truth values of statements combined with logical connectives. The document also discusses predicates, universal and existential quantification, and DeMorgan's laws relating negation and quantification.
This document provides an overview of the topics that will be covered in a discrete mathematics course for computer science. It includes:
- Administrative details like the instructor's contact information, textbook, exam dates, and policies on cheating and late homework.
- The course objectives which are to learn basic discrete mathematics tools and techniques like propositional logic, set theory, counting, and induction, as well as rigorous mathematical reasoning and proof writing.
- Advice for students on how to study effectively and keep up in the course by practicing problems and examples.
- An outline of the topics to be covered, beginning with propositional logic - including truth tables, logical connectives, tautologies, and translating sentences -
1) The document discusses foundational concepts in propositional logic, including logical form, statements, connectives, and truth tables.
2) It introduces common logical operators such as negation, conjunction, disjunction, implication, equivalence, and explains how to translate sentences between English and symbolic logic.
3) Conditional statements and their contrapositives, converses and inverses are defined. It is shown that a conditional statement is logically equivalent to its contrapositive using truth tables.
The document discusses propositional logic including:
- Propositional logic uses propositions that can be either true or false and logical connectives to connect propositions.
- It introduces syntax of propositional logic including atomic and compound propositions.
- Logical connectives like negation, conjunction, disjunction, implication, and biconditional are explained along with their truth tables and significance.
- Other concepts discussed include precedence of connectives, logical equivalence, properties of operators, and limitations of propositional logic.
- Examples are provided to illustrate propositional logic concepts like truth tables, logical equivalence, and translating English statements to symbolic form.
The document provides an overview of knowledge representation and logic. It discusses knowledge-based agents and how they use a knowledge base to represent facts about the world through sentences expressed in a knowledge representation language. It then covers different knowledge representation schemas including propositional logic, first-order logic, rules, networks, and structures. The document also discusses inference, different types of logic, and knowledge representation languages.
This document provides an overview of mathematical logic. It defines key concepts such as propositions, truth values, logical connectives like negation, conjunction, disjunction, implication, biconditional, and quantifiers. Propositions are statements that can be either true or false. Logical connectives combine propositions and quantifiers specify whether statements apply to all or some cases. Truth tables are used to determine the truth values of statements combined with logical connectives. The document also discusses predicates, universal and existential quantification, and DeMorgan's laws relating negation and quantification.
This document provides an overview of the topics that will be covered in a discrete mathematics course for computer science. It includes:
- Administrative details like the instructor's contact information, textbook, exam dates, and policies on cheating and late homework.
- The course objectives which are to learn basic discrete mathematics tools and techniques like propositional logic, set theory, counting, and induction, as well as rigorous mathematical reasoning and proof writing.
- Advice for students on how to study effectively and keep up in the course by practicing problems and examples.
- An outline of the topics to be covered, beginning with propositional logic - including truth tables, logical connectives, tautologies, and translating sentences -
1) The document discusses foundational concepts in propositional logic, including logical form, statements, connectives, and truth tables.
2) It introduces common logical operators such as negation, conjunction, disjunction, implication, equivalence, and explains how to translate sentences between English and symbolic logic.
3) Conditional statements and their contrapositives, converses and inverses are defined. It is shown that a conditional statement is logically equivalent to its contrapositive using truth tables.
The document discusses propositional logic including:
- Propositional logic uses propositions that can be either true or false and logical connectives to connect propositions.
- It introduces syntax of propositional logic including atomic and compound propositions.
- Logical connectives like negation, conjunction, disjunction, implication, and biconditional are explained along with their truth tables and significance.
- Other concepts discussed include precedence of connectives, logical equivalence, properties of operators, and limitations of propositional logic.
- Examples are provided to illustrate propositional logic concepts like truth tables, logical equivalence, and translating English statements to symbolic form.
The document discusses propositional logic, including:
- The basic components of propositional logic like propositions, connectives, truth tables, and logical equivalences
- Applications such as translating English sentences to propositional logic, system specifications, logic puzzles
- Representing logical relationships using truth tables and showing logical equivalences
- Using propositional logic to represent an electrical system and diagnose faults
This document summarizes Chapter 1, Part I of a textbook on propositional logic. It introduces key concepts in propositional logic, including propositions, logical connectives like negation, conjunction, disjunction, implication, biconditional, and truth tables. It provides examples of applying propositional logic to represent English sentences, system specifications, logic puzzles and logic circuits. It also briefly describes representing knowledge about an electrical system in propositional logic for fault diagnosis in artificial intelligence.
Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكروDr. Khaled Bakro
Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.
Discrete Mathematics covers fundamentals of logic including propositions, truth tables, logical connectives, and propositional equivalences. A proposition is a statement that is either true or false. Logical connectives such as "and", "or", "not" are used to combine propositions into compound statements. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions. Logical equivalences show that two statements are logically equivalent even if written differently. Examples help illustrate key concepts such as tautologies, contradictions and using equivalences to prove statements.
The document discusses a proposed proof by Vinay Deolalikar that P ≠ NP, the famous open problem in computer science. The proof strategy involved showing that satisfiability problems like random k-SAT would have "simple structure" if in P, but some instances do not. However, the proof was found to have flaws, as the "simple structure" property still holds even for problems in P. Multiple objections and counterexamples were found within a week through open online discussion, suggesting the proof is likely unsalvageable. The rapid online peer review process provided both benefits and costs to rigorously evaluating the proposed proof.
GDSC SSN - solution Challenge : Fundamentals of Decision MakingGDSCSSN
This session aims to provide participants with a comprehensive understanding of decision-making fundamentals in AI/ML, covering key concepts like reinforcement learning, different representations, and an exploration of current state-of-the-art methodologies.
This section discusses applications of propositional logic, including translating English sentences to propositional logic, system specifications, and logic puzzles. It provides an example of translating the English sentence "You can access the Internet from campus only if you are a computer science major or you are not a freshman" to the propositional logic statement a→(c ∨ ¬f). It also gives an example of expressing the system specification "The automated reply cannot be sent when the file system is full" in propositional logic as p → ¬q.
This document is the preface to a textbook on discrete mathematics. It introduces the topics that will be covered in the book, including mathematical logic, proofs, set theory, relations, functions, algorithms analysis, counting, probability, and graph theory. It encourages students to work through all the exercises to fully understand the material. It also thanks a professor who provided feedback that improved the book.
Propositional logic represents facts as being either true or false. It defines a syntax of allowed expressions, semantics that map expressions to meanings in terms of truth values, and inference rules for deriving new conclusions from existing statements. The syntax includes propositional symbols, logical constants, and logical connectives. Semantics define the truth conditions for sentences using truth tables. Inference rules like modus ponens, chain rule, substitution, simplification, conjunction, and transposition allow drawing valid conclusions.
This document provides an introduction to propositional logic and logical connectives. Some key points:
- Propositional logic deals with propositions that can be either true or false. Common logical connectives are negation, conjunction, disjunction, implication, biconditional.
- Truth tables are used to define the semantics and truth values of logical connectives and compound propositions.
- Logical equivalences allow replacing a proposition with an equivalent proposition to simplify expressions or arguments. Equivalences can be shown using truth tables or known equivalence rules.
- Propositional logic and logical reasoning form the basis of mathematical reasoning and are useful in areas like computer science, programming, and satisfiability problems.
This document provides an overview of propositional logic including:
- The basic components of propositional logic like propositions, connectives, truth tables
- Applications such as translating English sentences to logic, system specifications, puzzles
- Logical equivalences and showing equivalence through truth tables
- Sections cover propositions, connectives, truth tables, and applications including translation, specifications, puzzles
Artificial Intelligence (AI) | Prepositional logic (PL)and first order predic...Ashish Duggal
The following are the topics in this presentation Prepositional Logic (PL) and First-order Predicate Logic (FOPL) is used for knowledge representation in artificial intelligence (AI).
There are also sub-topics in this presentation like logical connective, atomic sentence, complex sentence, and quantifiers.
This PPT is very helpful for Computer science and Computer Engineer
(B.C.A., M.C.A., B.TECH. , M.TECH.)
Knowledge representation and Predicate logicAmey Kerkar
1. The document discusses knowledge representation and predicate logic.
2. It explains that knowledge representation involves representing facts through internal representations that can then be manipulated to derive new knowledge. Predicate logic allows representing objects and relationships between them using predicates, quantifiers, and logical connectives.
3. Several examples are provided to demonstrate representing simple facts about individuals as predicates and using quantifiers like "forall" and "there exists" to represent generalized statements.
This document is the preface to a lecture notes book on discrete mathematics. It introduces the topics that will be covered in the book, which include mathematical logic, proofs, set theory, relations, functions, counting, probability, and graph theory. The preface encourages students to work through all the exercises to fully understand the material. It also thanks a professor who provided feedback that improved the book.
The document discusses knowledge representation using propositional logic and predicate logic. It begins by explaining the syntax and semantics of propositional logic for representing problems as logical theorems to prove. Predicate logic is then introduced as being more versatile than propositional logic for representing knowledge, as it allows quantifiers and relations between objects. Examples are provided to demonstrate how predicate logic can formally represent statements involving universal and existential quantification.
The document discusses deductive and inductive arguments. It provides examples of valid and invalid deductive arguments using categorical propositions and conditional premises. It also discusses inductive arguments, noting that inductive conclusions generalize from specific premises rather than necessarily following from them. The document then compares deductive and inductive arguments and discusses their uses in everyday life and mathematics. It concludes by introducing some common rules of inference for deductive arguments.
The document provides an overview of topics in discrete mathematics including logic, sets, and functions. It outlines the following content: introduction to logic and logical operators; propositions and logical equivalences; predicates and quantifiers; sets and set operations; and functions. For each topic, it provides definitions, examples, and truth tables to illustrate key concepts in propositional and predicate logic, and sets. It also includes examples, explanations and review questions to help explain the material.
This document provides an overview of propositional logic concepts. It discusses:
- Propositional logic deals with validity and satisfiability of logical formulas using truth tables or deduction systems.
- Natural deduction is a deductive system that mimics natural reasoning using inference rules. Propositional formulas can be derived or proven in this system.
- Axiomatic systems use only axioms and rules of inference to prove formulas, requiring creative use of the minimal structure.
The document gives examples of proving formulas using these logical systems.
1. The document discusses predicate calculus and knowledge representation. It provides examples of forward chaining, backward chaining, and resolution to perform inference in predicate calculus.
2. It also discusses representing knowledge as semantic graphs and in the UNL format. An example knowledge representation of "Ram is reading the newspaper" is shown.
3. The document then presents examples of using predicate calculus to represent and solve problems, including a problem about members of a himalayan club and their preferences to infer if there is a mountain climber who is not a skier. Resolution refutation is applied to solve this problem.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
The document discusses propositional logic, including:
- The basic components of propositional logic like propositions, connectives, truth tables, and logical equivalences
- Applications such as translating English sentences to propositional logic, system specifications, logic puzzles
- Representing logical relationships using truth tables and showing logical equivalences
- Using propositional logic to represent an electrical system and diagnose faults
This document summarizes Chapter 1, Part I of a textbook on propositional logic. It introduces key concepts in propositional logic, including propositions, logical connectives like negation, conjunction, disjunction, implication, biconditional, and truth tables. It provides examples of applying propositional logic to represent English sentences, system specifications, logic puzzles and logic circuits. It also briefly describes representing knowledge about an electrical system in propositional logic for fault diagnosis in artificial intelligence.
Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكروDr. Khaled Bakro
Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.
Discrete Mathematics covers fundamentals of logic including propositions, truth tables, logical connectives, and propositional equivalences. A proposition is a statement that is either true or false. Logical connectives such as "and", "or", "not" are used to combine propositions into compound statements. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions. Logical equivalences show that two statements are logically equivalent even if written differently. Examples help illustrate key concepts such as tautologies, contradictions and using equivalences to prove statements.
The document discusses a proposed proof by Vinay Deolalikar that P ≠ NP, the famous open problem in computer science. The proof strategy involved showing that satisfiability problems like random k-SAT would have "simple structure" if in P, but some instances do not. However, the proof was found to have flaws, as the "simple structure" property still holds even for problems in P. Multiple objections and counterexamples were found within a week through open online discussion, suggesting the proof is likely unsalvageable. The rapid online peer review process provided both benefits and costs to rigorously evaluating the proposed proof.
GDSC SSN - solution Challenge : Fundamentals of Decision MakingGDSCSSN
This session aims to provide participants with a comprehensive understanding of decision-making fundamentals in AI/ML, covering key concepts like reinforcement learning, different representations, and an exploration of current state-of-the-art methodologies.
This section discusses applications of propositional logic, including translating English sentences to propositional logic, system specifications, and logic puzzles. It provides an example of translating the English sentence "You can access the Internet from campus only if you are a computer science major or you are not a freshman" to the propositional logic statement a→(c ∨ ¬f). It also gives an example of expressing the system specification "The automated reply cannot be sent when the file system is full" in propositional logic as p → ¬q.
This document is the preface to a textbook on discrete mathematics. It introduces the topics that will be covered in the book, including mathematical logic, proofs, set theory, relations, functions, algorithms analysis, counting, probability, and graph theory. It encourages students to work through all the exercises to fully understand the material. It also thanks a professor who provided feedback that improved the book.
Propositional logic represents facts as being either true or false. It defines a syntax of allowed expressions, semantics that map expressions to meanings in terms of truth values, and inference rules for deriving new conclusions from existing statements. The syntax includes propositional symbols, logical constants, and logical connectives. Semantics define the truth conditions for sentences using truth tables. Inference rules like modus ponens, chain rule, substitution, simplification, conjunction, and transposition allow drawing valid conclusions.
This document provides an introduction to propositional logic and logical connectives. Some key points:
- Propositional logic deals with propositions that can be either true or false. Common logical connectives are negation, conjunction, disjunction, implication, biconditional.
- Truth tables are used to define the semantics and truth values of logical connectives and compound propositions.
- Logical equivalences allow replacing a proposition with an equivalent proposition to simplify expressions or arguments. Equivalences can be shown using truth tables or known equivalence rules.
- Propositional logic and logical reasoning form the basis of mathematical reasoning and are useful in areas like computer science, programming, and satisfiability problems.
This document provides an overview of propositional logic including:
- The basic components of propositional logic like propositions, connectives, truth tables
- Applications such as translating English sentences to logic, system specifications, puzzles
- Logical equivalences and showing equivalence through truth tables
- Sections cover propositions, connectives, truth tables, and applications including translation, specifications, puzzles
Artificial Intelligence (AI) | Prepositional logic (PL)and first order predic...Ashish Duggal
The following are the topics in this presentation Prepositional Logic (PL) and First-order Predicate Logic (FOPL) is used for knowledge representation in artificial intelligence (AI).
There are also sub-topics in this presentation like logical connective, atomic sentence, complex sentence, and quantifiers.
This PPT is very helpful for Computer science and Computer Engineer
(B.C.A., M.C.A., B.TECH. , M.TECH.)
Knowledge representation and Predicate logicAmey Kerkar
1. The document discusses knowledge representation and predicate logic.
2. It explains that knowledge representation involves representing facts through internal representations that can then be manipulated to derive new knowledge. Predicate logic allows representing objects and relationships between them using predicates, quantifiers, and logical connectives.
3. Several examples are provided to demonstrate representing simple facts about individuals as predicates and using quantifiers like "forall" and "there exists" to represent generalized statements.
This document is the preface to a lecture notes book on discrete mathematics. It introduces the topics that will be covered in the book, which include mathematical logic, proofs, set theory, relations, functions, counting, probability, and graph theory. The preface encourages students to work through all the exercises to fully understand the material. It also thanks a professor who provided feedback that improved the book.
The document discusses knowledge representation using propositional logic and predicate logic. It begins by explaining the syntax and semantics of propositional logic for representing problems as logical theorems to prove. Predicate logic is then introduced as being more versatile than propositional logic for representing knowledge, as it allows quantifiers and relations between objects. Examples are provided to demonstrate how predicate logic can formally represent statements involving universal and existential quantification.
The document discusses deductive and inductive arguments. It provides examples of valid and invalid deductive arguments using categorical propositions and conditional premises. It also discusses inductive arguments, noting that inductive conclusions generalize from specific premises rather than necessarily following from them. The document then compares deductive and inductive arguments and discusses their uses in everyday life and mathematics. It concludes by introducing some common rules of inference for deductive arguments.
The document provides an overview of topics in discrete mathematics including logic, sets, and functions. It outlines the following content: introduction to logic and logical operators; propositions and logical equivalences; predicates and quantifiers; sets and set operations; and functions. For each topic, it provides definitions, examples, and truth tables to illustrate key concepts in propositional and predicate logic, and sets. It also includes examples, explanations and review questions to help explain the material.
This document provides an overview of propositional logic concepts. It discusses:
- Propositional logic deals with validity and satisfiability of logical formulas using truth tables or deduction systems.
- Natural deduction is a deductive system that mimics natural reasoning using inference rules. Propositional formulas can be derived or proven in this system.
- Axiomatic systems use only axioms and rules of inference to prove formulas, requiring creative use of the minimal structure.
The document gives examples of proving formulas using these logical systems.
1. The document discusses predicate calculus and knowledge representation. It provides examples of forward chaining, backward chaining, and resolution to perform inference in predicate calculus.
2. It also discusses representing knowledge as semantic graphs and in the UNL format. An example knowledge representation of "Ram is reading the newspaper" is shown.
3. The document then presents examples of using predicate calculus to represent and solve problems, including a problem about members of a himalayan club and their preferences to infer if there is a mountain climber who is not a skier. Resolution refutation is applied to solve this problem.
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The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
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1. 1
International Islamic University Chittagong
January 22, 2024
Tashin Hossain
Department of Computer Science and Engineering
Lecture#7
Agents that Reason Logically
And Propositional Logic
Adjunct Lecturer
2. Course Information
2
Course Materials:
1. Stuart Russell, Peter Norvig: “Artificial Intelligence A
Modern Approach”, 3rd Edition.
2. Other specific materials and lecture slides
3. Course Information
3
Segment Contents
03
Game playing: Introduction, Perfect Decisions, Imperfect
Decisions, Alpha-Beta Pruning.
Constraints satisfaction problem: Travelling salesman problem,
graph coloring etc.
04
Propositional and First-Order logic: Knowledge
Representation, Reasoning and Logic; Propositional Logic:
Syntax, Semantics, Validity and Inference, Rules of Inference for
Propositional logic; First-Order Logic: Syntax and Semantics,
Using first-order logic.
Inference in first order logic: Inference Rules Involving
Quantifiers, Example Proof, Generalized Modus Ponens, Forward
and Backward Chaining, Completeness, Resolution.
4. Course Information
4
Segment Contents
05
Probabilistic Reasoning: Probability and Bayes’ Theorem,
Certainty Factors and Rule-Based Systems, Bayesian Networks,
Fuzzy Logic; Ds theory, ER
Some Expert Systems: Representation and Using Domain
Knowledge, Expert System Shells, Explanation, And Knowledge
Acquisition.
06
Learning: Introduction to Learning, Inductive Learning, Learning
Decision Trees, Neural Net Learning;
07
Natural language processing: Introduction, Syntactic
Processing, Semantic Analysis, Discourse and Pragmatic
Processing.
6. Knowledge and Reasoning
6
From the previous lectures it has been shown that an agent that
has goals and searches for solutions to the goals can do better
than one that just reacts to its environment (simple reflex agent)
We focused mainly on the question of how to carry out the search
(using different search strategies), leaving aside the question of
general methods for describing states and actions
7. Knowledge and Reasoning
7
It is necessary to powering the agents with the capacity for general
logical reasoning
A logical, knowledge-based agent begins with some knowledge of the
world and of its own actions
It uses logical reasoning to maintain a description of the world as new
precepts arrive and to deduce a course of action that will achieve its goals
In this lecture we introduce the basic design for a knowledge-
based agent, followed by a simple logical language for expressing
knowledge, and show how it can be used to draw conclusions
about the world and to decide what to do
8. A Knowledge Based Agent
8
A knowledge-based agent consists of a knowledge base
(KB) and an inference engine (IE)
A knowledge-base is a set of representations of what one
knows about the world (objects and classes of objects, the
fact about objects, relationships among objects, etc.)
Each individual representation is called a sentence
The sentences are expressed in a knowledge representation
language
9. A Knowledge Based Agent
9
Examples of sentences
– The moon is made of green cheese Fact
– If A is true then B is true Rule
– A is false
– All humans are mortal
– Confucius is a human
The Inference engine derives new sentences from the input
and KB
The inference mechanism depends on representation in KB
10. A Knowledge Based Agent
10
The agent operates as follows:
1. It receives percepts from environment
2. It computes what action it should perform (by IE and KB)
3. It performs the chosen action (some actions are simply
inserting inferred new facts into KB).
12. KB can be Viewed at Different Levels
12
Knowledge Level
– The most abstract level
– describe agent by saying what it knows
– Example: A taxi agent might know that the Golden Gate Bridge
connects San Francisco with the Marin County
Logical Level
– The level at which the knowledge is encoded into sentences
– Example: Links(GoldenGateBridge, SanFrancisco, Marin-County)
Implementation Level
– The physical representation of the sentences in the logical level
– Example: “Links (GoldenGateBridge, SanFrancisco, MarinCounty)”
14. What is Proposition?
14
A proposition is a statement that can be either true or
false; it must be one or the other, and it cannot be both
The sun rises in the east
Today is a very hot day
16. Logic is a Formal Language
16
Propositional Logic
Rahima is intelligent - proposition
Rahima is hardworking – proposition
If Rahima is intelligent and Rahima is hardworking
Then Rahima scores high marks -proposition
17. Elements of PL
17
Rahima is intelligent => intelligent ( Rahima)
Rahima is hardworking
Objects and Relations or functions
Elements
18. Towards the Syntax
18
Intelligent (Rahima) == Rahima is intelligent
Hardworking (Rahima) == Rahima is hardworking
19. Towards more syntax
19
o Let P stand for intelligent (Rahima)
o Let Q stand for Hardworking (Rahima) F
o What does P ˄ Q (P and Q) mean ?
o What does P ˅ Q (P and Q) mean ?
o P ˄ Q, P ˅ Q are compound propositions
20. Syntactic Elements of PL
20
o A set of propositional symbols (P, Q, R etc.)
o each of which can be True or False
o Set of logical operators :˄, ˅, Not, implies
o often parenthesis() is used for grouping
o There are two special symbols
o TRUE (T) and FALSE (F) – these are logical constants
21. How to form propositional sentence?
21
– Each symbol (a proposition or a constant) is a sentence
– If P is a sentence and Q is a sentence, Then
– (p) is a sentence
– P ˄ Q is a sentence
– P or Q is a sentence
– not P is a sentence
– P implies Q is a sentence
– Nothing else is a sentence
– Sentences are called world form formulae (wff)
22. Examples of WFFs
22
P , Q True
– P and Q True
– (P or Q) implies R If the roads get wet, then it rains
– (P and Q) or R implies S
– Not (P or Q)
– Not (P or Q) implies R and S
23. Implication
23
P implies Q P Q
If P is true then Q is true
If it rains then the roads are wet
Sufficient condition
If the roads are wet then it rains ???
24. Equivalence
24
P bidirectional Q
Triangle
If two sides (A,B) of an triangle are equal (P), then
Opposite angles are equal (Q)
If opposite angles of a triangle are equal (Q), then two
sides of an triangle are equal (P)
25. What does a wff mean -Semantics
25
Interpretation in a world
When we interpret a sentence in a world we assign
meaning to it and it evaluates to either True or False
The Child can write
Nursery
Class 2
F
T
World
Semantics
26. Semantic of compound sentence
26
P: Likes ( Rahima, Sanjida)
Q: Knows (Tania, Farzana)
World: Rahima and Sanjida are friends and
Tania and Farzana known to each other
P= T, Q=T
P and Q =T
P and Not Q = F
27. Validity of a sentence
27
If a propositional sentence is true under all possible
interpretation, it is VALID
• Tautology
• Satisfiability
• Inconsistent
P v ¬ P = T
Question:
28. Validity of a sentence
28
• Tautology Compound Proposition is always True
• Satisfiability A compound sentence is satisfiable if there is at least
one true result in its Truth table
• Valid: A compound proposition is valid when it is a tautology
• Invalid: A compound proposition is invalid when it is either a
contradiction or contingency
29. Validity of a sentence
29
Inconsistency in propositional logic refers to a situation
where a set of propositions or statements leads to a
contradiction, meaning that it's impossible for all the
statements in the set to be simultaneously true
Let's consider a simple example:
Statement A: It is raining.
Statement B: It is not raining.
30. Quiz
30
Express the following English sentence in the
language of PL
– It rains in July rains ( July)
– The book is not costly not costly (book)
– If it rains today and one does not carry umbrella, then
he will be drenched
Rains(today) ^ not carry( umbrella) drenched(he)
31. Quiz
31
If P is true and Q is true then are the followings true or
false
– P → Q
– (¬P∨Q) → Q
– (¬P^Q) → P
– P ∨ ¬P
33. Objectives
33
• Infer the truth value of a proposition
• Reason towards new facts, given a set of propositions
• Prove a proposition given a set of propositional facts
34. Procedure to derive Truth Value
34
P Q P^Q PvQ
F F F F
F T F T
T F F T
T T T T
P v Q will be True when any of them is True
P ^ Q will be True when both of them are True
35. Procedure to derive Truth Value
35
P Q P Q
F F T
F T T
T F F
T T T
If you won’t reach the station on time, then you won’t be able to catch
the train T
If P is True, then Q is True, since P is a sufficient condition so that Q is
True
36. De Morgan’s Law
36
1. ¬ ( P ^ Q ) = ¬P v ¬ Q
2. ¬ ( P v Q ) = ¬P ^ ¬ Q
Prove the De morgan’s law using Truth
table
37. Reasoning
37
• P: It is the month of July
• Q: It rains
• R: P→Q
• If it is the month of July then it rains
• It is the month of July
• Conclude: It rains
39. Reasoning –Modus Ponens
39
• Modus Ponens is very much a valid inference
rule prove
P Q ¬P v Q
Prove:
(¬P v Q) ^ P
(¬P v P) v Q
T v Q
Q
40. Valid inference rule
40
• Thus Irrespective of meaning modus ponens allow us to
infer the truth of Q
• Modus Ponens is a inference rule that allow us to
deduce the truth of a consequent depending on the truth
of an antecedents
51. Converting a compound proposition to
the clausal form
51
Example: Consider the Sentence
¬ ( A B) v (C A)
1. Eliminate the Implication Sign
¬ (¬A v B) v (¬ C v A)
2. Eliminate the double negation and reduce scope of “not”
signs
(A ^ ¬ B) v (¬C v A)
3. Convert to conjunctive normal form by using distributive and
associative law
(A ^ ¬ B) v (¬C v A) => Distributive Law: A(B+C) = AB + AC
( A v ¬C v A) ^ (¬B v ¬C v A)
( A v ¬C) ^ (¬B v ¬C v A)
4. Get the set of clauses
( A v ¬C)
(¬B v ¬C v A)