This document covers many topics related to logic, including:
- Greek Trivium and the history of logic
- Logical sentences, definitions, and proofs
- Applications of logic such as email filters and e-commerce systems
- Logic programming examples including Tic-Tac-Toe and a "Sorority World"
- Different types of logical entailment and provability
- Using symbolic logic to represent statements
- Challenges with natural language versus formal logic
The document discusses concepts related to thinking and language. It covers topics such as concepts, problem solving, decision making, belief bias, language structure, language development in children, and thinking in different animal species. The relationship between thinking and language is also explored, with language said to influence how people think.
This document discusses various concepts in geometry including conditional statements, counterexamples, definitions, perpendicular lines, bi-conditional statements, deductive reasoning, the laws of logic, algebraic proofs using properties of equality, segment and angle properties, writing two-column proofs, the linear pair postulate, congruent and supplementary angle theorems, and the vertical angles theorem. Examples are provided to illustrate each concept.
1Week 3 Section 1.4 Predicates and Quantifiers As.docxjoyjonna282
1
Week 3: Section 1.4 Predicates and Quantifiers
Assume that the universe of discourse is all the people who are participating in
this course. Also, let us assume that we know each person in the course. Consider the
following statement: “She/he is over 6 feet tall”. This statement is not a proposition
since we cannot say that it either true or false until we replace the variable (she/he) by a
person’s name. The statement “She/he is over 6 feet tall” may be denoted by the symbol
P(n) where n stands for the variable and P, the predicate, “is over six feet tall”. The
symbol P (or lower case p) is used because once the variable is replaced (by a person’s
name in this case) the above statement becomes a proposition.
For example, if we know that Jim is over 6 feet tall, the statement “Jim is over six
feet tall” is a (true) proposition. The truth set of a predicate is all values in the domain
that make it a true statement. Another example, consider the statement, “for all real
numbers x, x2 –5x + 6 = (x - 2) (x – 3)”. We could let Q(x) stand for x2 –5x + 6 = (x - 2)
(x – 3). Also, we note that the truth values of Q(x) are indeed all real numbers.
Quantifiers:
There are two quantifiers used in mathematics: “for all” and “there exists”. The
symbol used “for all” is an upside down A, namely, . The symbol used for “there
exists” is a backwards E, namely, . We realize that the standard, every day usage of the
English language does not necessarily coincide with the Mathematical usage of English,
so we have to clarify what we mean by the two quantifiers.
For all For every For each For any
There exists at least one There exists There is Some
The table indicates that the mathematical meaning of the universal quantifier, for
all, coincides with our everyday usage of this term. However, the mathematical meaning
of the existential quantifier does not. When we use the word “some” in everyday
language we ordinarily mean two or more; yet, in mathematics the word “some” means at
least one, which is true when there is exactly one.
The Negation of the “For all “Quantifier:
Consider the statement “All people in this course are over 6 feet tall.” Assume it
is false (I am not over six feet tall). How do we prove it is false? All we have to do is to
point to one person to prove the statement is false. That is, all we need to do is give one
counterexample. We need only show that there exists at least one person in this class
who is not over 6 feet tall. Here is a more formal procedure.
Example 1:
Let P(n)stand for “people in this course are over 6 feet tall”, then the sentence
“All people in this course are over 6 feet tall” can be written as: “ n P(n)”. The negative,
“ ( n P(n))”, is equivalent to: “ n( P(n))”. So, in English the negative is, “There is
(there is at least one/ there exists/ some) a person in this room who is not over 6 feet tall.”
2
Example 2:
How w ...
The document summarizes key concepts in geometry including conditional statements, counter-examples, definitions, bi-conditionals, deductive reasoning, laws of logic, algebraic proofs, segment and angle properties, two-column proofs, the linear pair postulate, congruent complement and supplement theorems, the vertical angles theorem, and the common segments theorem. Examples are provided for each concept.
This document outlines Marzano's six-step process for teaching vocabulary. The six steps are: 1) teacher provides explicit instruction of the term, 2) students restate the term in their own words, 3) students create a nonlinguistic representation of the term, 4) students engage in activities involving the term, 5) students discuss the term, and 6) students participate in games involving the term. The document provides examples for each step and recommends that teachers select essential vocabulary to teach, use strategies like think-pair-share and academic notebooks, and schedule vocabulary games regularly to reinforce learning.
This document provides information on deductive reasoning and the laws of detachment and syllogism. It defines deductive reasoning as using facts and rules to reach a logical conclusion. The law of detachment states that if p implies q and p is true, then q must be true. The law of syllogism states that if p implies q and q implies r, then p implies r. The document includes examples applying these concepts and determining whether conclusions are valid deductive reasoning or not.
The document discusses logic and propositions. It begins by defining a proposition as a statement that is either true or false. It then provides examples of propositions and non-propositions. The document also discusses arguments and their validity. An argument is valid if the premises guarantee the conclusion. It discusses logical operators like conjunction, disjunction, negation and implication. Truth tables are used to determine the truth values of compound propositions formed using logical operators. Laws of algebra are also discussed for propositional logic.
This book is written by LOIBANGUTI, BM, it is just an online copy provided for free. No part of this book mya be republished. but can be used and stored as a softcopy book, can be shared accordingly.
The document discusses concepts related to thinking and language. It covers topics such as concepts, problem solving, decision making, belief bias, language structure, language development in children, and thinking in different animal species. The relationship between thinking and language is also explored, with language said to influence how people think.
This document discusses various concepts in geometry including conditional statements, counterexamples, definitions, perpendicular lines, bi-conditional statements, deductive reasoning, the laws of logic, algebraic proofs using properties of equality, segment and angle properties, writing two-column proofs, the linear pair postulate, congruent and supplementary angle theorems, and the vertical angles theorem. Examples are provided to illustrate each concept.
1Week 3 Section 1.4 Predicates and Quantifiers As.docxjoyjonna282
1
Week 3: Section 1.4 Predicates and Quantifiers
Assume that the universe of discourse is all the people who are participating in
this course. Also, let us assume that we know each person in the course. Consider the
following statement: “She/he is over 6 feet tall”. This statement is not a proposition
since we cannot say that it either true or false until we replace the variable (she/he) by a
person’s name. The statement “She/he is over 6 feet tall” may be denoted by the symbol
P(n) where n stands for the variable and P, the predicate, “is over six feet tall”. The
symbol P (or lower case p) is used because once the variable is replaced (by a person’s
name in this case) the above statement becomes a proposition.
For example, if we know that Jim is over 6 feet tall, the statement “Jim is over six
feet tall” is a (true) proposition. The truth set of a predicate is all values in the domain
that make it a true statement. Another example, consider the statement, “for all real
numbers x, x2 –5x + 6 = (x - 2) (x – 3)”. We could let Q(x) stand for x2 –5x + 6 = (x - 2)
(x – 3). Also, we note that the truth values of Q(x) are indeed all real numbers.
Quantifiers:
There are two quantifiers used in mathematics: “for all” and “there exists”. The
symbol used “for all” is an upside down A, namely, . The symbol used for “there
exists” is a backwards E, namely, . We realize that the standard, every day usage of the
English language does not necessarily coincide with the Mathematical usage of English,
so we have to clarify what we mean by the two quantifiers.
For all For every For each For any
There exists at least one There exists There is Some
The table indicates that the mathematical meaning of the universal quantifier, for
all, coincides with our everyday usage of this term. However, the mathematical meaning
of the existential quantifier does not. When we use the word “some” in everyday
language we ordinarily mean two or more; yet, in mathematics the word “some” means at
least one, which is true when there is exactly one.
The Negation of the “For all “Quantifier:
Consider the statement “All people in this course are over 6 feet tall.” Assume it
is false (I am not over six feet tall). How do we prove it is false? All we have to do is to
point to one person to prove the statement is false. That is, all we need to do is give one
counterexample. We need only show that there exists at least one person in this class
who is not over 6 feet tall. Here is a more formal procedure.
Example 1:
Let P(n)stand for “people in this course are over 6 feet tall”, then the sentence
“All people in this course are over 6 feet tall” can be written as: “ n P(n)”. The negative,
“ ( n P(n))”, is equivalent to: “ n( P(n))”. So, in English the negative is, “There is
(there is at least one/ there exists/ some) a person in this room who is not over 6 feet tall.”
2
Example 2:
How w ...
The document summarizes key concepts in geometry including conditional statements, counter-examples, definitions, bi-conditionals, deductive reasoning, laws of logic, algebraic proofs, segment and angle properties, two-column proofs, the linear pair postulate, congruent complement and supplement theorems, the vertical angles theorem, and the common segments theorem. Examples are provided for each concept.
This document outlines Marzano's six-step process for teaching vocabulary. The six steps are: 1) teacher provides explicit instruction of the term, 2) students restate the term in their own words, 3) students create a nonlinguistic representation of the term, 4) students engage in activities involving the term, 5) students discuss the term, and 6) students participate in games involving the term. The document provides examples for each step and recommends that teachers select essential vocabulary to teach, use strategies like think-pair-share and academic notebooks, and schedule vocabulary games regularly to reinforce learning.
This document provides information on deductive reasoning and the laws of detachment and syllogism. It defines deductive reasoning as using facts and rules to reach a logical conclusion. The law of detachment states that if p implies q and p is true, then q must be true. The law of syllogism states that if p implies q and q implies r, then p implies r. The document includes examples applying these concepts and determining whether conclusions are valid deductive reasoning or not.
The document discusses logic and propositions. It begins by defining a proposition as a statement that is either true or false. It then provides examples of propositions and non-propositions. The document also discusses arguments and their validity. An argument is valid if the premises guarantee the conclusion. It discusses logical operators like conjunction, disjunction, negation and implication. Truth tables are used to determine the truth values of compound propositions formed using logical operators. Laws of algebra are also discussed for propositional logic.
This book is written by LOIBANGUTI, BM, it is just an online copy provided for free. No part of this book mya be republished. but can be used and stored as a softcopy book, can be shared accordingly.
This document provides an overview of deductive reasoning, including the Law of Detachment and Law of Syllogism. It defines deductive reasoning as using facts, rules, and definitions to reach a logical conclusion. The Law of Detachment states that if p implies q is true and p is true, then q must be true. The Law of Syllogism states that if p implies q and q implies r are both true, then p implies r must be true. The document provides examples and problems applying these laws of deductive reasoning.
Critical Task 3 Rubric Critical Elements Exemplary (1.docxannettsparrow
Critical Task 3 Rubric
Critical Elements Exemplary (100%) Proficient (85%) Needs Improvement (55%) Not Evident (0%) Value
Main Elements Includes all of the main
elements and requirements
and cites ample appropriate
support to illustrate each
element
Includes most of the main
elements and requirements
and cites appropriate support
to illustrate each element
Includes some of the main
elements and requirements
Does not include any of the
main elements and
requirements
25
Inquiry and Analysis Explores multiple reasons and
offers accurate and in-depth
analysis of the argument in its
structural form
Explores some reasons and
offers somewhat accurate and
somewhat in-depth analysis of
the argument in its structural
form
Explores minimal reasons and
offers minimally accurate
analysis of the argument in its
structural form
Does not explore reasons and
analysis of evidence and does
not offer accurate analysis of
the argument in its structural
form
25
Integration and
Application
All of the course concepts are
correctly applied
Most of the course concepts
are correctly applied
Some of the course concepts
are correctly applied
Does not correctly apply any of
the course concepts
10
Critical Thinking Demonstrates comprehensive
exploration of issues and ideas
before accepting or forming an
opinion or conclusion about
the argument
Demonstrates moderate
exploration of issues and ideas
before accepting or forming an
opinion or conclusion about
the argument
Demonstrates minimal
exploration of issues and ideas
before accepting or forming an
opinion or conclusion about
the argument
Does not demonstrate
exploration of issues and ideas
before accepting or forming an
opinion or conclusion about
the argument
20
Reflection and
Research
Incorporates a highly pertinent
life goal or issue of significant
importance
Incorporates a life goal or issue
of somewhat significant
importance
Incorporates a life goal or issue
of minimally significant
importance
Does not incorporate a life goal
or issue of significant
importance
10
Writing
(Mechanics/Citations)
No errors related to
organization, grammar and
style, and citations
Minor errors related to
organization, grammar and
style, and citations
Some errors related to
organization, grammar and
style, and citations
Major errors related to
organization, grammar and
style, and citations
10
Earned Total 100%
HYPOTHETICAL SYLLOGISMS
Hypothetical thinking involves “If . . . then . . .” reasoning. According to some psychologists, the mental model for hypothetical thinking is built into our brain and enables us to understand rules and predict the consequences of our actions. We’ll be looking at the use of hypothetical reasoning in ethics in greater depth in Chapter 9. Hypothetical arguments are also a basic building block of computer programs.
A hypothetical syllogism is a form of deductive.
The document discusses predicate logic and its use in representing knowledge in artificial intelligence. It introduces several key concepts:
- Predicate logic uses predicates, constants, variables, functions and quantifiers to represent objects and their relations in a knowledge base.
- Well-formed formulas in predicate logic can be used to represent facts about the world. Logical inference rules like resolution and unification can be used to derive new facts or answer queries.
- Knowledge bases can be represented as sets of clauses in conjunctive normal form to apply inference rules like resolution and forward/backward chaining. Converting to clausal form involves techniques like Skolemization.
This is a short free early version GAMSAT Practice test.
For the most current version please go to the PagingDr Forum.
If you use this test, please assist by contributing further questions or suggestions.
The document discusses thinking and language. It provides details about concepts, categories, problem solving using algorithms and heuristics, and language development in children. Language involves structures like phonemes, morphemes, and grammar. While animals can communicate, there is no conclusive evidence they have a true language comparable to human language.
This object analysis paper summarizes the benefits of a dictionary for an international student studying in the United States. The student faced challenges with communication in both academic and social settings due to differences in language and culture. The dictionary helped the student learn new vocabulary, understand cultural norms, and socialize more confidently. While a simple object, the dictionary played an important role in transforming the student by boosting self-confidence and enhancing communication skills. It supported the student's cultural adaptation and helped overcome language barriers.
ASSIGNMENT 1 7.4 EXERCISES Using the predicates listed for each.docxtrippettjettie
This document contains assignments from a logic textbook. Assignment 1 involves translating statements into quantified logical form using given predicates. Assignment 2 involves translating arguments into quantified form and proving their validity using natural deduction. Assignment 3 involves proving various syllogisms valid using natural deduction and tableaux methods. Assignment 4 involves constructing formal proofs for arguments. Assignment 5 involves using tableaux to show arguments are invalid by assigning predicate values.
How to Write an Argumentative Essay Step By Step - Gudwriter. Sample Essay Outlines - 34+ Examples, Format, Pdf | Examples. Argumentative Essay Outline - 9+ Examples, Format, Pdf | Examples. A Sample Argumentative Essay.
Here are phrase structure trees for the sentences:
1.
S
NP VP
Det N VP
The puppy V NP
found Det N
the child
2.
S
NP VP
Det N VP
The ice V
melted
3.
S
NP VP
Det N VP
The hot sun V NP
melted Det N
the ice
4.
S
NP VP
Det N VP
The house PP
on Det N PP
the hill V PP
collapsed P NP
in Det N
the wind
5.
S
NP VP
Det N VP
The boat V PP
sailed P NP
up
The document describes a numeric pattern where each row contains consecutive odd integers centered around 1. It asks students to conjecture the pattern and sum of terms in each row. It also provides homework questions on conditional statements, deductive reasoning, and analyzing the truth value of related conditional statements.
This document provides information about parts of speech including nouns, pronouns, adjectives, verbs, adverbs, prepositions, conjunctions, and interjections. It includes diagnostic tests to identify parts of speech in sentences as well as lessons and activities about specific parts of speech. For nouns, it defines common and proper nouns and lists types of nouns. For pronouns, it defines personal, reflexive, relative, interrogative, demonstrative, and indefinite pronouns. For adjectives, it explains how adjectives modify nouns and lists types of adjectives.
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
This document is the introduction to a book titled "501 Word Analogy Questions" that is designed to help readers prepare for verbal and reasoning sections of assessments and entrance exams through completing analogy question exercises. The introduction explains what analogy questions are, the different types of relationships tested in analogies, strategies for solving them, and tips for using the book effectively as a study tool.
This document provides an introduction to 501 word analogy questions. It explains that word analogy questions test logic and reasoning skills as well as vocabulary. They involve identifying relationships between pairs of words to determine the missing word that completes the analogy. The introduction describes different types of relationships in analogies, such as part to whole, type and category, degrees of intensity, and others. It advises readers to practice these questions to improve familiarity with the question format and range of analogy types.
This document is the introduction to a book titled "501 Word Analogy Questions" that is designed to help readers prepare for standardized tests and entrance exams through completing analogy question practice exercises. The introduction explains what analogy questions are, the different types of relationships tested in analogies, strategies for solving them, and tips for using the book effectively as a study tool.
This document is the introduction to a book titled "501 Word Analogy Questions" that is designed to help readers prepare for verbal and reasoning sections of assessments and entrance exams through completing analogy question exercises. The introduction explains what analogy questions are, the different types of relationships tested in analogies, strategies for solving them, and tips for using the book effectively as a study tool.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
This document provides an overview of deductive reasoning, including the Law of Detachment and Law of Syllogism. It defines deductive reasoning as using facts, rules, and definitions to reach a logical conclusion. The Law of Detachment states that if p implies q is true and p is true, then q must be true. The Law of Syllogism states that if p implies q and q implies r are both true, then p implies r must be true. The document provides examples and problems applying these laws of deductive reasoning.
Critical Task 3 Rubric Critical Elements Exemplary (1.docxannettsparrow
Critical Task 3 Rubric
Critical Elements Exemplary (100%) Proficient (85%) Needs Improvement (55%) Not Evident (0%) Value
Main Elements Includes all of the main
elements and requirements
and cites ample appropriate
support to illustrate each
element
Includes most of the main
elements and requirements
and cites appropriate support
to illustrate each element
Includes some of the main
elements and requirements
Does not include any of the
main elements and
requirements
25
Inquiry and Analysis Explores multiple reasons and
offers accurate and in-depth
analysis of the argument in its
structural form
Explores some reasons and
offers somewhat accurate and
somewhat in-depth analysis of
the argument in its structural
form
Explores minimal reasons and
offers minimally accurate
analysis of the argument in its
structural form
Does not explore reasons and
analysis of evidence and does
not offer accurate analysis of
the argument in its structural
form
25
Integration and
Application
All of the course concepts are
correctly applied
Most of the course concepts
are correctly applied
Some of the course concepts
are correctly applied
Does not correctly apply any of
the course concepts
10
Critical Thinking Demonstrates comprehensive
exploration of issues and ideas
before accepting or forming an
opinion or conclusion about
the argument
Demonstrates moderate
exploration of issues and ideas
before accepting or forming an
opinion or conclusion about
the argument
Demonstrates minimal
exploration of issues and ideas
before accepting or forming an
opinion or conclusion about
the argument
Does not demonstrate
exploration of issues and ideas
before accepting or forming an
opinion or conclusion about
the argument
20
Reflection and
Research
Incorporates a highly pertinent
life goal or issue of significant
importance
Incorporates a life goal or issue
of somewhat significant
importance
Incorporates a life goal or issue
of minimally significant
importance
Does not incorporate a life goal
or issue of significant
importance
10
Writing
(Mechanics/Citations)
No errors related to
organization, grammar and
style, and citations
Minor errors related to
organization, grammar and
style, and citations
Some errors related to
organization, grammar and
style, and citations
Major errors related to
organization, grammar and
style, and citations
10
Earned Total 100%
HYPOTHETICAL SYLLOGISMS
Hypothetical thinking involves “If . . . then . . .” reasoning. According to some psychologists, the mental model for hypothetical thinking is built into our brain and enables us to understand rules and predict the consequences of our actions. We’ll be looking at the use of hypothetical reasoning in ethics in greater depth in Chapter 9. Hypothetical arguments are also a basic building block of computer programs.
A hypothetical syllogism is a form of deductive.
The document discusses predicate logic and its use in representing knowledge in artificial intelligence. It introduces several key concepts:
- Predicate logic uses predicates, constants, variables, functions and quantifiers to represent objects and their relations in a knowledge base.
- Well-formed formulas in predicate logic can be used to represent facts about the world. Logical inference rules like resolution and unification can be used to derive new facts or answer queries.
- Knowledge bases can be represented as sets of clauses in conjunctive normal form to apply inference rules like resolution and forward/backward chaining. Converting to clausal form involves techniques like Skolemization.
This is a short free early version GAMSAT Practice test.
For the most current version please go to the PagingDr Forum.
If you use this test, please assist by contributing further questions or suggestions.
The document discusses thinking and language. It provides details about concepts, categories, problem solving using algorithms and heuristics, and language development in children. Language involves structures like phonemes, morphemes, and grammar. While animals can communicate, there is no conclusive evidence they have a true language comparable to human language.
This object analysis paper summarizes the benefits of a dictionary for an international student studying in the United States. The student faced challenges with communication in both academic and social settings due to differences in language and culture. The dictionary helped the student learn new vocabulary, understand cultural norms, and socialize more confidently. While a simple object, the dictionary played an important role in transforming the student by boosting self-confidence and enhancing communication skills. It supported the student's cultural adaptation and helped overcome language barriers.
ASSIGNMENT 1 7.4 EXERCISES Using the predicates listed for each.docxtrippettjettie
This document contains assignments from a logic textbook. Assignment 1 involves translating statements into quantified logical form using given predicates. Assignment 2 involves translating arguments into quantified form and proving their validity using natural deduction. Assignment 3 involves proving various syllogisms valid using natural deduction and tableaux methods. Assignment 4 involves constructing formal proofs for arguments. Assignment 5 involves using tableaux to show arguments are invalid by assigning predicate values.
How to Write an Argumentative Essay Step By Step - Gudwriter. Sample Essay Outlines - 34+ Examples, Format, Pdf | Examples. Argumentative Essay Outline - 9+ Examples, Format, Pdf | Examples. A Sample Argumentative Essay.
Here are phrase structure trees for the sentences:
1.
S
NP VP
Det N VP
The puppy V NP
found Det N
the child
2.
S
NP VP
Det N VP
The ice V
melted
3.
S
NP VP
Det N VP
The hot sun V NP
melted Det N
the ice
4.
S
NP VP
Det N VP
The house PP
on Det N PP
the hill V PP
collapsed P NP
in Det N
the wind
5.
S
NP VP
Det N VP
The boat V PP
sailed P NP
up
The document describes a numeric pattern where each row contains consecutive odd integers centered around 1. It asks students to conjecture the pattern and sum of terms in each row. It also provides homework questions on conditional statements, deductive reasoning, and analyzing the truth value of related conditional statements.
This document provides information about parts of speech including nouns, pronouns, adjectives, verbs, adverbs, prepositions, conjunctions, and interjections. It includes diagnostic tests to identify parts of speech in sentences as well as lessons and activities about specific parts of speech. For nouns, it defines common and proper nouns and lists types of nouns. For pronouns, it defines personal, reflexive, relative, interrogative, demonstrative, and indefinite pronouns. For adjectives, it explains how adjectives modify nouns and lists types of adjectives.
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
This document is the introduction to a book titled "501 Word Analogy Questions" that is designed to help readers prepare for verbal and reasoning sections of assessments and entrance exams through completing analogy question exercises. The introduction explains what analogy questions are, the different types of relationships tested in analogies, strategies for solving them, and tips for using the book effectively as a study tool.
This document provides an introduction to 501 word analogy questions. It explains that word analogy questions test logic and reasoning skills as well as vocabulary. They involve identifying relationships between pairs of words to determine the missing word that completes the analogy. The introduction describes different types of relationships in analogies, such as part to whole, type and category, degrees of intensity, and others. It advises readers to practice these questions to improve familiarity with the question format and range of analogy types.
This document is the introduction to a book titled "501 Word Analogy Questions" that is designed to help readers prepare for standardized tests and entrance exams through completing analogy question practice exercises. The introduction explains what analogy questions are, the different types of relationships tested in analogies, strategies for solving them, and tips for using the book effectively as a study tool.
This document is the introduction to a book titled "501 Word Analogy Questions" that is designed to help readers prepare for verbal and reasoning sections of assessments and entrance exams through completing analogy question exercises. The introduction explains what analogy questions are, the different types of relationships tested in analogies, strategies for solving them, and tips for using the book effectively as a study tool.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
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How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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6. Observations
Abby likes Bess but Bess does not like Abby.
Definitions
A triangle is a polygon with three sides.
Physical Laws
Pressure times volume is proportional to temperature.
Logical Sentences
11. Email Readers
If the message is from “genesereth”
and the topic is “logic”,
Then the message goes in the “important” folder
eCommerce Systems
If the product is a notebook
and the customer is a student
and the date is in December,
Then the price is $5.99.
Logic-Enabled Computer Systems
21. Logical Sentences
Dana likes Cody.
Abby does not like Dana.
Dana does not like Abby.
Bess likes Cody or Dana.
Abby likes everyone that Bess likes.
Cody likes everyone who likes her.
Nobody likes herself.
27. A set of premises logically entails a conclusion if and
only if every world that satisfies the premises satisfies
the conclusion.
Premises: Conclusions:
Dana likes Cody. Bess likes Cody.
Abby does not like Dana. Bess does not like Dana.
Dana does not like Abby. Everybody likes somebody.
Bess likes Cody or Dana. Everybody is liked by somebody.
Abby likes everyone that Bess likes.
Cody likes everyone who likes her.
Nobody likes herself.
Logical Entailment
30. Abby likes everyone that Bess likes.
Abby does not like Dana.
Therefore, Bess does not like Dana.
Bess likes Cody or Dana.
Bess does not like Dana
Therefore, Bess likes Cody.
Proofs
31. A rule of inference is a reasoning pattern consisting of
some premises and some conclusions.
If we believe the premises, a rule of inference tell us
that we should also believe the conclusions.
Rules of Inference
32. All Accords are Hondas.
All Hondas are Japanese.
Therefore, all Accords are Japanese.
Sample Rule of Inference
33. Sample Rule of Inference
All borogoves are slithy toves.
All slithy toves are mimsy.
Therefore, all borogoves are mimsy.
Sample Rule of Inference
34. All x are y.
All y are z.
Therefore, x are z.
Which patterns are correct?
How many rules do we need?
Sound Rule of Inference
35. All x are y.
Some y are z.
Therefore, some x are z.
Unsound Rule of Inference
36. All Toyotas are Japanese cars.
Some Japanese cars are made in America.
Therefore, some Toyotas are made in America.
Using Unsound Rule of Inference
37. All Toyotas are cars.
Some cars are Porsches.
Therefore, some Toyotas are Porsches.
Using Unsound Rule of Inference
38. A conclusion is provable for a set of premises if and
only if every there is a finite sequence of sentences in
which every element is either a premise or the result of
applying a sound rule of inference to earlier members in
the sequence.
As we shall see, for well-behaved logics, logical
entailment and provability are identical - a set of
premises logically entails a conclusion if and only if the
conclusion is provable from the premises. This is a very
big deal.
Provability
40. Logical Sentences
Dana likes Cody.
Abby does not like Dana.
Dana does not like Abby.
Bess likes Cody or Dana.
Abby likes everyone that Bess likes.
Cody likes everyone who likes her.
Nobody likes herself.
41. One grammatically correct sentence:
The cherry blossoms in the spring.
Another grammatically correct sentence:
The cherry blossoms in the spring sank.
Complexity of Natural Language
42. The University may terminate this lease when the Lessee, having made
application and executed this lease in advance of enrollment, is not eligible
to enroll or fails to enroll in the University or leaves the University at any
time prior to the expiration of this lease, or for violation of any provisions
of this lease, or for violation of any University regulation relative to
resident Halls, or for health reasons, by providing the student with written
notice of this termination 30 days prior to the effective data of termination;
unless life, limb, or property would be jeopardized, the Lessee engages in
the sales of purchase of controlled substances in violation of federal, state
or local law, or the Lessee is no longer enrolled as a student, or the Lessee
engages in the use or possession of firearms, explosives, inflammable
liquids, fireworks, or other dangerous weapons within the building, or
turns in a false alarm, in which cases a maximum of 24 hours notice would
be sufficient.
Michigan Lease Termination Clause
43. There’s a girl in the room with a telescope.
Grammatical Ambiguity
44. Crowds Rushing to See Pope Trample 6 to Death
British Left Waffles on Falkland Islands
Scientists Grow Frog Eyes and Ears
Food Stamp Recipients Turn to Plastic
Fried Chicken Cooked in Microwave Wins Trip
Indian Ocean Talks
Headlines
45. Champagne is better than beer.
Beer is better than soda.
Therefore, champagne is better than soda.
Good Reasoning
46. Champagne is better than beer.
Beer is better than soda.
Therefore, champagne is better than soda.
Bad sex is better than nothing.
Nothing is better than good sex.
Therefore, bad sex is better than good sex.
Bad Reasoning
47. Xavier is three times as old as Yolanda. Xavier's age
and Yolanda's age add up to twelve. How old are
Xavier and Yolanda?
Algebra
48. If Mary loves Pat, then Mary loves Quincy. If it is
Monday and raining, then Mary loves Pat or Quincy. If
it is Monday and raining, does Mary love Quincy?
p q
m r p q
m r q q
m r q
Symbolic Logic
51. Sets
{a, b, c} {b, c, d} = {a, b, c, d}
a {a, b, c}
{a, b, c} {a, b, c, d}
Functions and Relations
f(a, b) = c
r(a, b, c)
Mathematical Background
52. Materials of the Course
Lectures
Lecture Notes
Additional Readings
Exercises
Discussion Groups
Read discussion
Post questions
Answer questions
Work with others!
Hints on How to Take the Course
53. Propositional Logic
If it is raining, the ground is wet.
Relational Logic
If x is a parent of y, then y is a child of x.
Herbrand Logic
The father of a person is older than the person.
Multiple Logics
54. We frequently write sentences about sentences.
Sentence: When it rains, it pours.
Metasentence: That sentence contains two verbs.
We will frequently prove things about proofs.
Proofs: formal
Metaproofs: informal
Meta
55. Logic
“You are not thinking; you are just being logical.”
- Niels Bohr to Albert Einstein
“Mathematics (Logic) may be defined as the subject in
which we never know what we are talking about nor
whether what we are saying is true.''
- Bertrand Russell
Translation: “Garbage in, garbage out."
Translation: “Treasure in, treasure out.”
56. Logic and Computer Science
Logic is the mathematics of Computer Science as
Calculus is the mathematics of Physics.
Editor's Notes
Logic is one of the oldest intellectual disciplines in human history. It dates back to Aristotle.
In fact, it was part of the Trivium - the three fundamental subjects taught by the ancient Greeks. We still teach kids grammar and to some extent, rhetoric; but, for complicated historical reasons, logic is rarely taught as a standalone course before college, at least in this country.
It has been studied through the centuries by people like Leibniz, Boole, Godel, Russell, Turing, and many others. And it is still a subject of active investigation today.
Gottfried Leibniz (who proposed the feasibility of building an automated reasoning machine), George Boole (who actually built one and invented Boolean algebra along the way), Kurt Godel (who proved completeness and incompleteness theorems), Bertrand Russell and Alfred North Whitehead (authors of Principia Mathematica), John Alan Robinson (the inventor of Resolution).
We do not talk much about history in this course, though it is a fascinating stories, with lots of colorful characters. If you want to learn more about the history, get yourself a copy of Logicomix. Comic book introduction to logic. Entertaining and informative.
Today, we use Logic in everything we do. We use it in our professional lives and our personal affairs (at least some of the time).
We use the language of Logic when we state observations, when we write definitions, when we encode physical laws.
We use logical reasoning in deriving conclusions from these bits of information.
We use logic in writing proofs to convince others of our conclusions.
And we are not alone! Logic is increasingly being used by computers…
… to prove mathematical theorems, to validate engineering designs, to encode and analyze laws and regulations and business rules.
Logic is increasingly being used at the interface between man and machine, especially in logic-enabled computer systems, where the users can view and edit logical sentences. Think, for example, about email readers that allow users to write rules to manage incoming mail messages - deleting some, moving others to various mailboxes, and so forth based on properties of those messages. In the business world, eCommerce systems that allow companies to encode price rules based on the product, the customer, the date, and so forth.
Moreover, Logic is sometimes used not just by users in communicating with computer systems but by software engineers in building those systems (using languages like Prolog and a programming methodology known as logic programming).
In this lecture, we are going to talk about the main elements of logic - language, logical entailment, and proofs; we will talk about informal logic vs formal, symbolic logic; and we will finish with a brief study guide for the course. Let’s start by looking at the concept of possible worlds.
Consider the interpersonal relations of a small sorority. There are just four members - Abby, Bess, Cody, and Dana.
Some of the girls like each other, but some do not. The checkmark in the first row here means that Abby likes Cody, while the absence of a checkmark means that Abby does not like the other girls (including herself). Bess likes Cody too. Cody likes everyone but herself. And Dana also likes the popular Cody.
Of course, that is not the only possible state of affairs. This table shows another possible world. In this world, every girl likes exactly two other girls, and every girl is liked by just two girls.
As it turns out, there are quite a few possibilities. Given four girls, there are sixteen possible instances of the likes relation - Abby likes Abby, Abby likes Bess, Abby likes Cody, Abby likes Dana, Bess likes Abby, and so forth. Each of these sixteen can be either true or false; there are 216 (65,536) combinations of these true-false possibilities; and so there are 216 possible worlds.
Typically, we want to find out exactly how things stand, i.e. we want to know which of these possible worlds is correct.
Let's assume that we do not which world is correct, but we have informants who know the girls and are willing to share what they know. Each informant knows a little about the likes and dislikes of the girls, but no one knows everything. This is where Logic comes in. By writing logical sentences, each informant can express exactly what he or she knows - no more, no less. For our part, we can combine these sentences into a logical theory; and we can use this theory to draw logical conclusions, including some that may not be known to any one of the informants.
Here we see one such collection of sentences. The first sentence is straightforward; it tells us directly that Dana likes Cody. The second and third sentences tell us what is not true,without saying what is true. The fourth sentence says that one condition holds or another but does not say which. The fifth sentence gives a general fact about the girls Abby likes. The sixth sentence expresses a general fact about Cody's likes. The last sentence says something about everyone.
Sentences like these constrain the possible ways the world could be. Each sentence divides the set of possible worlds into two subsets, those in which the sentence is true and those in which the sentence is false. Believing a sentence is tantamount to believing that the world is in the first set.
Given two sentences, we know the world must be in the intersection of the set of worlds in which the first sentence is true and the set of worlds in which the second sentence is true.
Ideally, when we have enough sentences, we know exactly how things stand.
Effective communication requires a language that allows us to express what we know, no more and no less. If we know the state of the world, then we should write enough sentences to communicate this to others. If we do not know which of various ways the world could be, we need a language that allows us to express only what we know. The beauty of Logic is that it gives us a means to express incomplete information when that is all we have and to express complete information when full information is available.
Unfortunately, we do not always have complete information. Sometimes, a collection of sentences only partially constrains the world. For example, there are four different worlds that are consistent with the the Sorority World sentences we saw earlier.
What can we conclude from our set of sentences? Quite a bit, as it turns out. For example, it must be the case that Bess likes Cody. Also, Bess does not like Dana. There are also some general conclusions that must be true. For example, in this world with just four girls, we can conclude that everybody likes somebody. Also, everyone is liked by somebody.
Even though a set of sentences does not determine a unique world, there are some conclusions that are true in every world that satisfies the given sentences. A sentence of this sort is said to be a logical conclusion from the given sentences. Said the other way around, a set of premises logically entails a conclusion if and only if every world that satisfies the premises also satisfies the conclusion.
One way to make this determination is by checking all possible worlds. For example, in our case, we notice that, in every world that satisfies our sentences, Bess likes Cody, so the statement that Bess likes Cody is a logical conclusion from our set of sentences. Also, we see that everyone likes someone. We see that everyone is liked by someone. And we see that nobody likes herself.
Unfortunately, determining logical entailment by checking all possible worlds is impractical in general. There are usually many, many possible worlds; and in some cases there can be infinitely many.
The alternative is logical reasoning, viz. the application of reasoning rules to derive logical conclusions and produce logical proofs, i.e. sequences of reasoning steps that leads from premises to conclusions.
For example, from the fact that Abby likes everyone that Bess likes and the fact that Abby does not like Dana, we can conclude that Bess does not like Dana. (If she did, then, by the first fact, Abby would also like Dana; and, from the second fact, we know that she does not.) Also, we know that Bess likes Cody or Dana and, as we have just seen, she does not like Dana; so she must like Cody.
One of Aristotle's great contributions to philosophy was his recognition that what makes a step of a proof immediately obvious is its form rather than its content. It does not matter whether you are talking about blocks or stocks or sorority girls. What matters is the structure of the facts with which you are working. Such patterns are called rules of inference.
As an example, consider the rule of inference shown here. If we know that all Accords are Hondas and we know that all Hondas are Japanese cars, then we can conclude that all Accords are Japanese cars.
Now consider another example. If we know that all borogoves are slithy toves and we know that all slithy toves are mimsy, then we can conclude that all borogoves are mimsy. What's more, in order to reach this conclusion, we do not need to know anything about borogoves or slithy toves or what it means to be mimsy.
What is interesting about these examples is that they share the same reasoning structure, viz. the pattern shown here.
The existence of such reasoning patterns is fundamental in Logic but raises important questions. Which patterns are correct? Are there many such patterns or just a few?
Let us consider the first of these questions first. Obviously, there are patterns that are just plain wrong in the sense that they can lead to incorrect conclusions. Consider, as an example, the (faulty) reasoning pattern shown here.
Now let us take a look at an instance of this pattern. If we replace x by Toyotas and y by Japanese and z by made in America, we get the following line of argument, leading to a conclusion that happens to be correct.
On the other hand, if we replace x by Toyotas and y by cars and z by Porsches, we get a line of argument leading to a conclusion that is not correct.
What distinguishes a correct pattern from one that is incorrect is that it must always lead to correct conclusions, i.e. conclusions that are logically entailed by the premises. As we will see, this is the defining criterion for what we call deduction. Of all types of reasoning, deductive reasoning is the only one that guarantees its conclusions in all cases. It has some very special properties and holds a unique place in Logic.
The concept of proof, in order to be meaningful, requires that we be able to recognize certain reasoning steps as immediately obvious. Once we have identified a set of rules, we can combine the reasoning "atoms" to form proof “molecules”, thereby obtaining conclusions that cannot be derived in a single step.
Formalizing this, we say that a conclusion is provable from a set of premises if and only if every there is a finite sequence of sentences in which every element is either a premise or the result of applying a sound rule of inference to earlier members in the sequence.
As we shall see, for well-behaved logics, logical entailment and provability are identical - a set of premises logically entails a conclusion if and only if the conclusion is provable from the premises. This is a very big deal.
So far, we have illustrated everything with sentences in English. While natural language works well to express logical information in many circumstances, it is not without its problems. Natural language sentences can be complex; they can be ambiguous; and failing to understand the meaning of a sentence can lead to errors in reasoning.
So far, we have illustrated everything with sentences in English. While natural language works well to express logical information in many circumstances, it is not without its problems. Natural language sentences can be complex; they can be ambiguous; and failing to understand the meaning of a sentence can lead to errors in reasoning.
Even very simple sentences can be complex. Here we see two grammatically legal sentences. They are the same in all but the last word, but there structure is entirely different. In the first, the main verb is blossoms, while in the second blossoms is a noun and the main verb is sank.
As another example of grammatical complexity, consider the following excerpt taken from the University of Michigan lease agreement. The sentence in this case is sufficiently long and the grammatical structure sufficiently complex that people must often read it several times to understand precisely what it says.
As an example of ambiguity, suppose I were to write the sentence There's a girl in the room with a telescope. Here we see two possible meanings of this sentence. Am I saying that there is a girl in a room containing a telescope? Or am I saying that there is a girl in the room and she is holding a telescope?
Such complexities and ambiguities can sometimes be humorous if they lead to interpretations the author did not intend, as in these infamous newspaper headlines with multiple interpretations. Crowds Rushing to See Pope Trample 6 to Death. (Who would have guessed that the pope could be so cruel?). British Left Waffles on Falkland Islands. (Is the British left unclear on what to do or do the Brits simply not like waffles?) Scientists Grow Frog Eyes and Ears. Food Stamp Recipients Turn to Plastic. Fried Chicken Cooked in Microwave Wins Trip. Indian Ocean Talks. (I always imagine it has a deep voice.) Using a formal language eliminates such unintentional ambiguities (and, for better or worse, avoids any unintentional humor as well).
As an illustration of errors that arise in reasoning with sentences in natural language, consider the following two examples. In the first, we use the transitivity of the better relation to derive a conclusion about the relative quality of champagne and soda from the relative quality of champagne and beer and the relative quality or beer and soda. Champagne is better than beer, and beer is better than soda; therefore, champagne is better than soda. So far so good.
Now, consider what happens when we apply the same transitivity rule in the case illustrated here. Bad sex is better than nothing, and nothing is better than good sex; therefore, bad sex is better than good sex. The form of the argument is the same as before, but the conclusion is somewhat less believable. The problem in this case is that the use of nothing here is syntactically similar to the use of beer in the preceding example, but in English it means something entirely different.
Logic eliminates these difficulties through the use of a formal language for encoding information. Given the syntax and semantics of this formal language, we can give a precise definition for the notion of logical conclusion. Moreover, we can establish precise reasoning rules that produce all and only logical conclusions.
In this regard, there is a strong analogy between the methods of Formal Logic and those of high school algebra. To illustrate this analogy, consider the following algebra problem.
Typically, the first step in solving such a problem is to express the information in the form of equations. If we let x represent the age of Xavier and y represent the age of Yolanda, we can capture the essential information of the problem as shown below. Using the methods of algebra, we can then manipulate these expressions to solve the problem. First we subtract the second equation from the first. Next, we divide each side of the resulting equation by -4 to get a value for y. Then substituting back into one of the preceding equations, we get a value for x.
Now, consider the logic problem shown here. As with the algebra problem, the first step is formalization. Let p represent the possibility that Mary loves Pat; let q represent the possibility that Mary loves Quincy; let m represent the possibility that it is Monday; and let r represent the possibility that it is raining.
With these abbreviations, we can represent the essential information of this problem with the following logical sentences. The first says that p implies q, i.e. if Mary loves Pat, then Mary loves Quincy. The second says that m and r implies p or q, i.e. if it is Monday and raining, then Mary loves Pat or Mary loves Quincy.
As with Algebra, Formal Logic defines certain operations that we can use to manipulate expressions. The operation shown here is a variant of what is called Propositional Resolution. The expressions above the line are the premises of the rule, and the expression below is the conclusion.
We can use this operation to solve the problem of Mary's love life. Looking at the two premises above, we notice that p occurs on the left-hand side of one sentence and the right-hand side of the other. Consequently, we can cancel the p and thereby derive the conclusion that, if is Monday and raining, then Mary loves Quincy or Mary loves Quincy. Dropping the repeated symbol on the right hand side, we arrive at the conclusion that, if it is Monday and raining, then Mary loves Quincy.
This example is interesting in that it showcases our formal language for encoding logical information. As with algebra, we use symbols to represent relevant aspects of the world in question, and we use operators to connect these symbols in order to express information about the things those symbols represent.
The example also introduces one of the most important operations in Formal Logic, viz. Resolution (in this case a restricted form of Resolution). Resolution has the property of being complete for an important class of logic problems, i.e. it is the only operation necessary to solve any problem in the class.
First of all, a few words about background. The course presumes that you are comfortable with symbolic manipulation, as used, for example, in solving high-school algebra problems, like the one we saw earlier.
This course also presumes some basic mathematical background. In particular, it assumes that you understand sets, set operations (such as union and intersection), and set relations (such as membership and subset). And it presumes that you are familiar with functions and relations. However, nothing else is required. If you have this background, you should be fine. If not, you might want to brush up on these things before continuing.
Next, a few words about how to proceed through the course. The course consists of these lectures, lecture notes, readings, and exercises. All four are useful. While it may be tempting to do so, do not skip the exercises. They reinforce points made in the lectures and readings; and, in some cases, they motivate subsequent material.
Also, consider taking advantage of communication aids to discuss issues with others taking the course. Post questions that are bothering you; and, if you know the answers to questions posted by others, help out by answering. Explaining material to others is one of the best ways of learning a subject.
Although Logic is a single field of study, there is more than one logic in this field. In the three main units of this course, we look at three different types of logic, each more sophisticated than the last.
Propositional Logic is the logic of propositions. Symbols in the language represent "conditions" in the world, and complex sentences in the language express interrelationships among these conditions. The primary operators are Boolean connectives, such as and, or, and not.
Relational Logic expands upon Propositional Logic by providing a means for explicitly talking about individual objects and their interrelationships (not just monolithic conditions). In order to do so, we expand our language to include object constants, function constants, relation constants, variables, and quantifiers.
Epistemic Logic goes one step further and provides a means for talking about sentences as objects in their own right and provides operators for expressing relationships between expressions and other expressions and relationships between expressions and the things they represent. This extension allows us to encode information about information, and it allows us to define the notions of truth and belief.
Each logic brings new issues and capabilities to light. Despite these differences, there are many commonalities among these logics. In particular, in each case, there is a language with a formal syntax and a precise semantics; there is a notion of logical entailment; and there are legal rules for manipulating expressions in the language. These similarities allow us to compare the various logics and to gain an appreciation of the tradeoff between expressiveness and computational complexity.
One final comment. In the hopes of preventing difficulties, it is worth pointing out a potential source of confusion. This course exists in the meta world. It contains sentences about sentences; it contains proofs about proofs. In some places, we use similar mathematical symbology both for sentences in Logic and sentences about Logic. Wherever possible, we try to be clear about this distinction, but the potential for confusion remains. Unfortunately, this comes with the territory. We are using Logic to study Logic. It is our most powerful intellectual tool.