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.
Week 14 April 28 & 30 - Love and Death Castillo, Chap. 9 .docxmelbruce90096
Week 14: April 28 & 30 - Love and Death
Castillo, Chap. 9 “Sofia, Who Would Never Again Let Her Husband Have the Last Word…”
Chap. 10 “Wherein Sofia Discovers La Loca’s Playmate…”
Chap. 11 “The Marriage of Sofia’s Faithful Daughter to her Cousin”
Chap. 12 “Of the Hideous Crime of Francisco el Penitente…”
1. For all chapters, identify the four levels of analysis: 1) metaphoric/symbolic; 2) literary; 3) sociological; and spiritual.
Chapter 9 “Sofia, Who Would Never Again Let Her Husband Have the Last Word…”
2. In this chapter, Sofia begins a transformation of her own. What is this transformation and what role does Esperanza play?Chapter 10
3. In this chapter, we return to La Loca, reading from her point of view. What do we learn from this, the youngest of Sofi’s daughters?
4. As Fe leaves Sofia’s home we realize she has not come to terms with what she went through when Tom broke off the engagement. What is Fe like now? Has she also changed?
5. What about Esperanza, what news about her? And what about “La Llorona, Chicana international astral-traveler”?
Chapter 11
6. Much happens to Fe in this chapter. Be able to recount all of Fe’s experiences and the relationship to big business, the U.S. government, and the medical profession.Chapter 12
7. What to make of this last chapter in Caridad and Francisco’s lives? What are the recurring themes and metaphors/symbolizes, etc.?
In preparation for the Opposing Viewpoints short paper due in Module Five, you will outline a position (thesis) on a topic of your choosing.
Using the Prewriting Template provided, outline two to three of your reasons for supporting your thesis and then also outline the objection’s position. Please note that the main purpose of this assignment is to formulate the strongest possible objection to your own position before responding to it.
You will be required to use at least four outside (i.e., other than the textbook) sources for this paper, two for each side of the issue. You do not need to do extensive reearch before completing the outline.
Possible topics: Affirmative Action, Abortion, State-Financed Health Care, Flat Tax...or anything you want. It is best to choose a position for which you can find reasonable arguments on both sides.
Click on the title above to turn in your outline.
First Paper (Opposing Viewpoints):
Critical Elements
Distinguished
Proficient
Emerging
Not Evident
Value
Main Elements
Includes almost all of the main elements and requirements and cites ample appropriate support to illustrate each element
(23-25)
Includes most of the main elements and requirements and cites appropriate support to illustrate each element
(20-22)
Includes some of the main elements and requirements
(18-19)
Does not include any of the main elements and requirements
(0-17)
25
Inquiry and Analysis
Explores multiple reasons and offers in-depth analysis of evidence to make informed conclusions about the issue
(18-20)
Explores so.
Logic is derived from the Greek word ‘LOGOS’ which means primarily the word by which the inward thought is expresses ‘LOGIKE’ which means the work or what is spoken (but coming to mean thought or reason).
Credited to Math 2A.
Week 14 April 28 & 30 - Love and Death Castillo, Chap. 9 .docxmelbruce90096
Week 14: April 28 & 30 - Love and Death
Castillo, Chap. 9 “Sofia, Who Would Never Again Let Her Husband Have the Last Word…”
Chap. 10 “Wherein Sofia Discovers La Loca’s Playmate…”
Chap. 11 “The Marriage of Sofia’s Faithful Daughter to her Cousin”
Chap. 12 “Of the Hideous Crime of Francisco el Penitente…”
1. For all chapters, identify the four levels of analysis: 1) metaphoric/symbolic; 2) literary; 3) sociological; and spiritual.
Chapter 9 “Sofia, Who Would Never Again Let Her Husband Have the Last Word…”
2. In this chapter, Sofia begins a transformation of her own. What is this transformation and what role does Esperanza play?Chapter 10
3. In this chapter, we return to La Loca, reading from her point of view. What do we learn from this, the youngest of Sofi’s daughters?
4. As Fe leaves Sofia’s home we realize she has not come to terms with what she went through when Tom broke off the engagement. What is Fe like now? Has she also changed?
5. What about Esperanza, what news about her? And what about “La Llorona, Chicana international astral-traveler”?
Chapter 11
6. Much happens to Fe in this chapter. Be able to recount all of Fe’s experiences and the relationship to big business, the U.S. government, and the medical profession.Chapter 12
7. What to make of this last chapter in Caridad and Francisco’s lives? What are the recurring themes and metaphors/symbolizes, etc.?
In preparation for the Opposing Viewpoints short paper due in Module Five, you will outline a position (thesis) on a topic of your choosing.
Using the Prewriting Template provided, outline two to three of your reasons for supporting your thesis and then also outline the objection’s position. Please note that the main purpose of this assignment is to formulate the strongest possible objection to your own position before responding to it.
You will be required to use at least four outside (i.e., other than the textbook) sources for this paper, two for each side of the issue. You do not need to do extensive reearch before completing the outline.
Possible topics: Affirmative Action, Abortion, State-Financed Health Care, Flat Tax...or anything you want. It is best to choose a position for which you can find reasonable arguments on both sides.
Click on the title above to turn in your outline.
First Paper (Opposing Viewpoints):
Critical Elements
Distinguished
Proficient
Emerging
Not Evident
Value
Main Elements
Includes almost all of the main elements and requirements and cites ample appropriate support to illustrate each element
(23-25)
Includes most of the main elements and requirements and cites appropriate support to illustrate each element
(20-22)
Includes some of the main elements and requirements
(18-19)
Does not include any of the main elements and requirements
(0-17)
25
Inquiry and Analysis
Explores multiple reasons and offers in-depth analysis of evidence to make informed conclusions about the issue
(18-20)
Explores so.
Logic is derived from the Greek word ‘LOGOS’ which means primarily the word by which the inward thought is expresses ‘LOGIKE’ which means the work or what is spoken (but coming to mean thought or reason).
Credited to Math 2A.
The following text material and terms defined at the end comprise .docxarnoldmeredith47041
The following text material and terms defined at the end comprise part of what will be asked on the Mid-Term Exam for PHIL 1381.
Logic [excerpt from Stan Baronett, Logic, 2E]
Logic is the study of reasoning. Logic investigates the level of correctness of the reasoning found in arguments. An argument is a group of statements of which one (the conclusion) is claimed to follow from the others (the premises). A statement is a sentence that is either true or false. Every statement is either true or false; these two possibilities are called “truth values.” Premises are statements that contain information intended to provide support or reasons to believe a conclusion. The conclusion is the statement that is claimed to follow from the premises. In order to help recognize arguments, we rely on premise indicator words and phrases, and conclusion indicator words and phrases.
Inference is the term used by logicians to refer to the reasoning process that is expressed by an argument. If a passage expresses a reasoning process—that the conclusion follows from the premises—then we say that it makes an inferential claim. If a passage does not express a reasoning process (explicit or implicit), then it does not make an inferential claim (it is a noninferential passage). One type of noninferential passage is the explanation. An explanation provides reasons for why or how an event occurred. By themselves, explanations are not arguments; however, they can form part of an argument.
There are two types of argument: deductive and inductive. A deductive argument is one in which it is claimed that the conclusion follows necessarily from the premises. In other words, it is claimed that under the assumption that the premises are true it is impossible for the conclusion to be false. An inductive argument is one in which it is claimed that the premises make the conclusion probable. In other words, it is claimed that, under the assumption that the premises are true, it is improbable for the conclusion to be false.
Revealing the logical form of a deductive argument helps with logical analysis and evaluation. When we evaluate deductive arguments, we use the following concepts: valid, invalid, sound, and unsound. A valid argument is one where, assuming the premises are true, it is impossible for the conclusion to be false. In other words, the conclusion follows necessarily from the premises. An invalid argument is one where, assuming the premises are true, it is possible for the conclusion to be false. In other words, a deductive argument in which the conclusion does not follow necessarily from the premises is an invalid argument. When logical analysis shows that a deductive argument is valid, and when truth value analysis of the premises shows that they are all true, then the argument is sound. If a deductive argument is invalid, or if at least one of the premises is false (truth value analysis), then the argument is unsound.
A counterexample to astatement is evidenc.
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 ...
Reflect back on what you have learned in this course about how to .docxlorent8
Reflect back on what you have learned in this course about how to construct high-quality arguments for positions. “refer to the attached reading chapter for help”
· Give an example of how the ability to think logically and to construct good arguments could help you in your career and in your daily life?
· In what ways the skill of being able to evaluate the quality of reasoning on all sides will better enable you to discover what is true and to make better choices?
· Finally, consider the argument you have been developing for your writing assignments. How has fairly considering multiple points of view helped you clarify your own perspective?
· What advice would you give to people to help them understand issues more clearly and objectively while being fair to all sides? Feel free to comment on any other values you have gained from this course so far.
Your journal entry must be at least 250 words. You do not need to follow APA style for this journal entry, but you should proofread your work to eliminate errors of grammar and spelling.
3
Deductive Reasoning
White cups stacked with one red cup in the middle.
moodboard/Thinkstock
Learning Objectives
After reading this chapter, you should be able to:
Define basic key terms and concepts within deductive reasoning.
Use variables to represent an argument’s logical form.
Use the counterexample method to evaluate an argument’s validity.
Categorize different types of deductive arguments.
Analyze the various statements—and the relationships between them—in categorical arguments.
Evaluate categorical syllogisms using the rules of the syllogism and Venn diagrams.
Differentiate between sorites and enthymemes.
By now you should be familiar with how the field of logic views arguments: An argument is just a collection of sentences, one of which is the conclusion and the rest of which, the premises, provide support for the conclusion. You have also learned that not every collection of sentences is an argument. Stories, explanations, questions, and debates are not arguments, for example. The essential feature of an argument is that the premises support, prove, or give evidence for the conclusion. This relationship of support is what makes a collection of sentences an argument and is the special concern of logic. For the next four chapters, we will be taking a closer look at the ways in which premises might support a conclusion. This chapter discusses deductive reasoning, with a specific focus on categorical logic.
3.1 Basic Concepts in Deductive Reasoning
As noted in Chapter 2, at the broadest level there are two types of arguments: deductive and inductive. The difference between these types is largely a matter of the strength of the connection between premises and conclusion. Inductive arguments are defined and discussed in Chapter 5; this chapter focuses on deductive arguments. In this section we will learn about three central concepts: validity, soundness, and deduction.
.
Daniel Hampikian's Power point on arguments and moral skepticism - danielhamp...Daniel Hampikian
Dr. Daniel Hampikian's critical thinking and ethics power point on moral skepticism, logical validity, arguments, logic, morality, evidence, induction, deduction, and much more...
danielhampikian
Proposition with example
Explained details OF Prepositional Variables, Truth Value, Atomic Proposition, Compound Proposition & Propositional Logic with example, definition and truth table
Application of propositional Logic
Propositional calculus (also called propositional logic, sentential calculus, sentential logic, or sometimes zeroth-order logic) is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components. Logical connectives are found in natural languages.
1.1Arguments, Premises, and ConclusionsHow Logical Are You·.docxbraycarissa250
1.1Arguments, Premises, and Conclusions
How Logical Are You?
· After a momentary absence, you return to your table in the library only to find your smartphone is missing. It was there just minutes earlier. You suspect the student sitting next to you took it. After all, she has a guilty look. Also, there is a bulge in her backpack about the size of your phone, and one of the pouches has a loose strap. Then you hear a “ring” come from the backpack—and it’s the same ringtone that you use on your phone. Which of these pieces of evidence best supports your suspicion?
Answer
The best evidence is undoubtedly the “ring” you hear coming from her backpack, which is the same ringtone as the one on your phone. The weakest evidence is probably the “guilty look.” After all, what, exactly, is a guilty look? The bulge in the backpack and the loose strap are of medium value. The loose strap supports the hypothesis that something was quickly inserted into the backpack. In this section of the chapter you will learn that evidentiary statements form the premises of arguments.
Logic may be defined as the organized body of knowledge, or science, that evaluates arguments. All of us encounter arguments in our day-to-day experience. We read them in books and newspapers, hear them on television, and formulate them when communicating with friends and associates. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Among the benefits to be expected from the study of logic is an increase in confidence that we are making sense when we criticize the arguments of others and when we advance arguments of our own.
An argument, in its simplest form, is a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion). Every argument may be placed in either of two basic groups: those in which the premises really do support the conclusion and those in which they do not, even though they are claimed to. The former are said to be good arguments (at least to that extent), the latter bad arguments. The purpose of logic, as the science that evaluates arguments, is thus to develop methods and techniques that allow us to distinguish good arguments from bad.
As is apparent from the given definition, the term argument has a very specific meaning in logic. It does not mean, for example, a mere verbal fight, as one might have with one’s parent, spouse, or friend. Let us examine the features of this definition in greater detail. First of all, an argument is a group of statements. A statement is a sentence that is either true or false—in other words, typically a declarative sentence or a sentence component that could stand as a declarative sentence. The following sentences are statements:
Chocolate truffles are loaded with calories.
Melatonin helps relieve jet lag.
Political can.
1.1Arguments, Premises, and ConclusionsHow Logical Are You·.docxjeremylockett77
1.1Arguments, Premises, and Conclusions
How Logical Are You?
· After a momentary absence, you return to your table in the library only to find your smartphone is missing. It was there just minutes earlier. You suspect the student sitting next to you took it. After all, she has a guilty look. Also, there is a bulge in her backpack about the size of your phone, and one of the pouches has a loose strap. Then you hear a “ring” come from the backpack—and it’s the same ringtone that you use on your phone. Which of these pieces of evidence best supports your suspicion?
Answer
The best evidence is undoubtedly the “ring” you hear coming from her backpack, which is the same ringtone as the one on your phone. The weakest evidence is probably the “guilty look.” After all, what, exactly, is a guilty look? The bulge in the backpack and the loose strap are of medium value. The loose strap supports the hypothesis that something was quickly inserted into the backpack. In this section of the chapter you will learn that evidentiary statements form the premises of arguments.
Logic may be defined as the organized body of knowledge, or science, that evaluates arguments. All of us encounter arguments in our day-to-day experience. We read them in books and newspapers, hear them on television, and formulate them when communicating with friends and associates. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Among the benefits to be expected from the study of logic is an increase in confidence that we are making sense when we criticize the arguments of others and when we advance arguments of our own.
An argument, in its simplest form, is a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion). Every argument may be placed in either of two basic groups: those in which the premises really do support the conclusion and those in which they do not, even though they are claimed to. The former are said to be good arguments (at least to that extent), the latter bad arguments. The purpose of logic, as the science that evaluates arguments, is thus to develop methods and techniques that allow us to distinguish good arguments from bad.
As is apparent from the given definition, the term argument has a very specific meaning in logic. It does not mean, for example, a mere verbal fight, as one might have with one’s parent, spouse, or friend. Let us examine the features of this definition in greater detail. First of all, an argument is a group of statements. A statement is a sentence that is either true or false—in other words, typically a declarative sentence or a sentence component that could stand as a declarative sentence. The following sentences are statements:
Chocolate truffles are loaded with calories.
Melatonin helps relieve jet lag.
Political can ...
Deductive Argument
For a deductive argument, if all its premises are true, its conclusion is necessarily true (or it is logically impossible for the conclusion to be false.)
I.e., the truth of premises guarantees the truth of conclusion.
Example
Either you work hard or you will fail the test.
You do not work hard.
Therefore, you will fail the test.
3 Types of Possibility
Technological possibility:
e.g.
Going to the moon is technological possible, but going to Mercury is not.
Physical possibility:
e.g.
Going to Mercury is physical possible, but making water boil at 95 C under one atmospheric pressure is not.
Logical possibility:
e.g.
Making water boil at 95 C under one atmospheric pressure is logical possible, but drawing a triangle with 4 angles is not.
When we talk about deductive arguments, we have already presupposed that the arguments are successful or valid deductive arguments.
The conclusion of a valid argument is called a valid conclusion.
For an unsuccessful deductive argument (the premises are intended to guarantee the conclusion but fail to do so), we call it an invalid argument.
A deductive argument may be valid or invalid, there is nothing in between.
Whether a deductive argument is valid or invalid depends on its form or structure, not on its content.
The above argument is valid because it has this valid form:
p or q.
Not-p.
Therefore, q.
p and q are statement variables.
A valid argument may have false conclusion if it has false premises.
Example:
CY Leung is either a genius or an idiot.
He is not an idiot.
Therefore, He is a genius.
In order to guarantee the truth of conclusion, we have to make sure all the premises are true.
When all the premises of a valid argument are true, the argument is called a “sound argument”.
And the conclusion of a sound argument is called a sound conclusion.
If an argument is invalid or has false premises, it is unsound.
On the other hand, the fact that an argument is invalid does not entail that its conclusion is false.
• It just means that its conclusion does not follow from its premises.
• You can consider a valid argument structure as a truth-keeping machine:
• When you input T information into it, it will output T information.
• When you input F information into it, it will output T or F information
Inductive Argument:
A typical example of inductive argument:
Swan1 is white.
Swan2 is white.
Swan3 is white.
…
Swann is white.
________________
All swans are white.
Another typical example:
An event of type B follows an event of type A at time t1.
An event of type B follows an event of type A at time t2.
…
An event of type B follows an event of type A at time tn.
___________________________
A causes B.
Many people think that the characteristic of inductive arguments is arguing from particular to general.
However, deductive arguments may also argue from particular to general.
Example:
I have two cats, Fluffy and Garfield.
Fluffy does not eat fish.
Garfield does not eat fish either.
Therefore, All of m
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 following text material and terms defined at the end comprise .docxarnoldmeredith47041
The following text material and terms defined at the end comprise part of what will be asked on the Mid-Term Exam for PHIL 1381.
Logic [excerpt from Stan Baronett, Logic, 2E]
Logic is the study of reasoning. Logic investigates the level of correctness of the reasoning found in arguments. An argument is a group of statements of which one (the conclusion) is claimed to follow from the others (the premises). A statement is a sentence that is either true or false. Every statement is either true or false; these two possibilities are called “truth values.” Premises are statements that contain information intended to provide support or reasons to believe a conclusion. The conclusion is the statement that is claimed to follow from the premises. In order to help recognize arguments, we rely on premise indicator words and phrases, and conclusion indicator words and phrases.
Inference is the term used by logicians to refer to the reasoning process that is expressed by an argument. If a passage expresses a reasoning process—that the conclusion follows from the premises—then we say that it makes an inferential claim. If a passage does not express a reasoning process (explicit or implicit), then it does not make an inferential claim (it is a noninferential passage). One type of noninferential passage is the explanation. An explanation provides reasons for why or how an event occurred. By themselves, explanations are not arguments; however, they can form part of an argument.
There are two types of argument: deductive and inductive. A deductive argument is one in which it is claimed that the conclusion follows necessarily from the premises. In other words, it is claimed that under the assumption that the premises are true it is impossible for the conclusion to be false. An inductive argument is one in which it is claimed that the premises make the conclusion probable. In other words, it is claimed that, under the assumption that the premises are true, it is improbable for the conclusion to be false.
Revealing the logical form of a deductive argument helps with logical analysis and evaluation. When we evaluate deductive arguments, we use the following concepts: valid, invalid, sound, and unsound. A valid argument is one where, assuming the premises are true, it is impossible for the conclusion to be false. In other words, the conclusion follows necessarily from the premises. An invalid argument is one where, assuming the premises are true, it is possible for the conclusion to be false. In other words, a deductive argument in which the conclusion does not follow necessarily from the premises is an invalid argument. When logical analysis shows that a deductive argument is valid, and when truth value analysis of the premises shows that they are all true, then the argument is sound. If a deductive argument is invalid, or if at least one of the premises is false (truth value analysis), then the argument is unsound.
A counterexample to astatement is evidenc.
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 ...
Reflect back on what you have learned in this course about how to .docxlorent8
Reflect back on what you have learned in this course about how to construct high-quality arguments for positions. “refer to the attached reading chapter for help”
· Give an example of how the ability to think logically and to construct good arguments could help you in your career and in your daily life?
· In what ways the skill of being able to evaluate the quality of reasoning on all sides will better enable you to discover what is true and to make better choices?
· Finally, consider the argument you have been developing for your writing assignments. How has fairly considering multiple points of view helped you clarify your own perspective?
· What advice would you give to people to help them understand issues more clearly and objectively while being fair to all sides? Feel free to comment on any other values you have gained from this course so far.
Your journal entry must be at least 250 words. You do not need to follow APA style for this journal entry, but you should proofread your work to eliminate errors of grammar and spelling.
3
Deductive Reasoning
White cups stacked with one red cup in the middle.
moodboard/Thinkstock
Learning Objectives
After reading this chapter, you should be able to:
Define basic key terms and concepts within deductive reasoning.
Use variables to represent an argument’s logical form.
Use the counterexample method to evaluate an argument’s validity.
Categorize different types of deductive arguments.
Analyze the various statements—and the relationships between them—in categorical arguments.
Evaluate categorical syllogisms using the rules of the syllogism and Venn diagrams.
Differentiate between sorites and enthymemes.
By now you should be familiar with how the field of logic views arguments: An argument is just a collection of sentences, one of which is the conclusion and the rest of which, the premises, provide support for the conclusion. You have also learned that not every collection of sentences is an argument. Stories, explanations, questions, and debates are not arguments, for example. The essential feature of an argument is that the premises support, prove, or give evidence for the conclusion. This relationship of support is what makes a collection of sentences an argument and is the special concern of logic. For the next four chapters, we will be taking a closer look at the ways in which premises might support a conclusion. This chapter discusses deductive reasoning, with a specific focus on categorical logic.
3.1 Basic Concepts in Deductive Reasoning
As noted in Chapter 2, at the broadest level there are two types of arguments: deductive and inductive. The difference between these types is largely a matter of the strength of the connection between premises and conclusion. Inductive arguments are defined and discussed in Chapter 5; this chapter focuses on deductive arguments. In this section we will learn about three central concepts: validity, soundness, and deduction.
.
Daniel Hampikian's Power point on arguments and moral skepticism - danielhamp...Daniel Hampikian
Dr. Daniel Hampikian's critical thinking and ethics power point on moral skepticism, logical validity, arguments, logic, morality, evidence, induction, deduction, and much more...
danielhampikian
Proposition with example
Explained details OF Prepositional Variables, Truth Value, Atomic Proposition, Compound Proposition & Propositional Logic with example, definition and truth table
Application of propositional Logic
Propositional calculus (also called propositional logic, sentential calculus, sentential logic, or sometimes zeroth-order logic) is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components. Logical connectives are found in natural languages.
1.1Arguments, Premises, and ConclusionsHow Logical Are You·.docxbraycarissa250
1.1Arguments, Premises, and Conclusions
How Logical Are You?
· After a momentary absence, you return to your table in the library only to find your smartphone is missing. It was there just minutes earlier. You suspect the student sitting next to you took it. After all, she has a guilty look. Also, there is a bulge in her backpack about the size of your phone, and one of the pouches has a loose strap. Then you hear a “ring” come from the backpack—and it’s the same ringtone that you use on your phone. Which of these pieces of evidence best supports your suspicion?
Answer
The best evidence is undoubtedly the “ring” you hear coming from her backpack, which is the same ringtone as the one on your phone. The weakest evidence is probably the “guilty look.” After all, what, exactly, is a guilty look? The bulge in the backpack and the loose strap are of medium value. The loose strap supports the hypothesis that something was quickly inserted into the backpack. In this section of the chapter you will learn that evidentiary statements form the premises of arguments.
Logic may be defined as the organized body of knowledge, or science, that evaluates arguments. All of us encounter arguments in our day-to-day experience. We read them in books and newspapers, hear them on television, and formulate them when communicating with friends and associates. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Among the benefits to be expected from the study of logic is an increase in confidence that we are making sense when we criticize the arguments of others and when we advance arguments of our own.
An argument, in its simplest form, is a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion). Every argument may be placed in either of two basic groups: those in which the premises really do support the conclusion and those in which they do not, even though they are claimed to. The former are said to be good arguments (at least to that extent), the latter bad arguments. The purpose of logic, as the science that evaluates arguments, is thus to develop methods and techniques that allow us to distinguish good arguments from bad.
As is apparent from the given definition, the term argument has a very specific meaning in logic. It does not mean, for example, a mere verbal fight, as one might have with one’s parent, spouse, or friend. Let us examine the features of this definition in greater detail. First of all, an argument is a group of statements. A statement is a sentence that is either true or false—in other words, typically a declarative sentence or a sentence component that could stand as a declarative sentence. The following sentences are statements:
Chocolate truffles are loaded with calories.
Melatonin helps relieve jet lag.
Political can.
1.1Arguments, Premises, and ConclusionsHow Logical Are You·.docxjeremylockett77
1.1Arguments, Premises, and Conclusions
How Logical Are You?
· After a momentary absence, you return to your table in the library only to find your smartphone is missing. It was there just minutes earlier. You suspect the student sitting next to you took it. After all, she has a guilty look. Also, there is a bulge in her backpack about the size of your phone, and one of the pouches has a loose strap. Then you hear a “ring” come from the backpack—and it’s the same ringtone that you use on your phone. Which of these pieces of evidence best supports your suspicion?
Answer
The best evidence is undoubtedly the “ring” you hear coming from her backpack, which is the same ringtone as the one on your phone. The weakest evidence is probably the “guilty look.” After all, what, exactly, is a guilty look? The bulge in the backpack and the loose strap are of medium value. The loose strap supports the hypothesis that something was quickly inserted into the backpack. In this section of the chapter you will learn that evidentiary statements form the premises of arguments.
Logic may be defined as the organized body of knowledge, or science, that evaluates arguments. All of us encounter arguments in our day-to-day experience. We read them in books and newspapers, hear them on television, and formulate them when communicating with friends and associates. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Among the benefits to be expected from the study of logic is an increase in confidence that we are making sense when we criticize the arguments of others and when we advance arguments of our own.
An argument, in its simplest form, is a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion). Every argument may be placed in either of two basic groups: those in which the premises really do support the conclusion and those in which they do not, even though they are claimed to. The former are said to be good arguments (at least to that extent), the latter bad arguments. The purpose of logic, as the science that evaluates arguments, is thus to develop methods and techniques that allow us to distinguish good arguments from bad.
As is apparent from the given definition, the term argument has a very specific meaning in logic. It does not mean, for example, a mere verbal fight, as one might have with one’s parent, spouse, or friend. Let us examine the features of this definition in greater detail. First of all, an argument is a group of statements. A statement is a sentence that is either true or false—in other words, typically a declarative sentence or a sentence component that could stand as a declarative sentence. The following sentences are statements:
Chocolate truffles are loaded with calories.
Melatonin helps relieve jet lag.
Political can ...
Deductive Argument
For a deductive argument, if all its premises are true, its conclusion is necessarily true (or it is logically impossible for the conclusion to be false.)
I.e., the truth of premises guarantees the truth of conclusion.
Example
Either you work hard or you will fail the test.
You do not work hard.
Therefore, you will fail the test.
3 Types of Possibility
Technological possibility:
e.g.
Going to the moon is technological possible, but going to Mercury is not.
Physical possibility:
e.g.
Going to Mercury is physical possible, but making water boil at 95 C under one atmospheric pressure is not.
Logical possibility:
e.g.
Making water boil at 95 C under one atmospheric pressure is logical possible, but drawing a triangle with 4 angles is not.
When we talk about deductive arguments, we have already presupposed that the arguments are successful or valid deductive arguments.
The conclusion of a valid argument is called a valid conclusion.
For an unsuccessful deductive argument (the premises are intended to guarantee the conclusion but fail to do so), we call it an invalid argument.
A deductive argument may be valid or invalid, there is nothing in between.
Whether a deductive argument is valid or invalid depends on its form or structure, not on its content.
The above argument is valid because it has this valid form:
p or q.
Not-p.
Therefore, q.
p and q are statement variables.
A valid argument may have false conclusion if it has false premises.
Example:
CY Leung is either a genius or an idiot.
He is not an idiot.
Therefore, He is a genius.
In order to guarantee the truth of conclusion, we have to make sure all the premises are true.
When all the premises of a valid argument are true, the argument is called a “sound argument”.
And the conclusion of a sound argument is called a sound conclusion.
If an argument is invalid or has false premises, it is unsound.
On the other hand, the fact that an argument is invalid does not entail that its conclusion is false.
• It just means that its conclusion does not follow from its premises.
• You can consider a valid argument structure as a truth-keeping machine:
• When you input T information into it, it will output T information.
• When you input F information into it, it will output T or F information
Inductive Argument:
A typical example of inductive argument:
Swan1 is white.
Swan2 is white.
Swan3 is white.
…
Swann is white.
________________
All swans are white.
Another typical example:
An event of type B follows an event of type A at time t1.
An event of type B follows an event of type A at time t2.
…
An event of type B follows an event of type A at time tn.
___________________________
A causes B.
Many people think that the characteristic of inductive arguments is arguing from particular to general.
However, deductive arguments may also argue from particular to general.
Example:
I have two cats, Fluffy and Garfield.
Fluffy does not eat fish.
Garfield does not eat fish either.
Therefore, All of m
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2. Logic
ii | Baraka Loibanguti
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3. Logic
iii | Baraka Loibanguti
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5. Advanced Secondary Education
Page 5 of 36
LOGIC
The rules of logic give precise measuring to
mathematical statements. These rules are used to
distinguish between valid and invalid mathematical arguments.
In addition, it is importance in understanding mathematical reasoning; logic has
numerous applications in computer science.
These rules are used in design of computer
circuits the construction of computer
programs, the verification of the correctness of
programs and in many other ways.
Logic is all about reasons. Every day we
consider possibilities, we think about what
follows from different assumptions, what
would be the case in different alternatives, and
we weight up competing positions or options.
In all of this, we reason. Logic is the study of
good reasoning, and in particular, what makes
good reasoning good.
To understand good reasoning, we must have an idea of the kinds of things we
reason about. What are the things we give reasons for? We can give reasons for
doing something rather than something else (these are reasons for actions) or for
liking some things above other things, (these are reasons for preferences). In the
study of logic, we do not so much look at these kinds of reasoning: instead, logic
concerns itself with reasons for believing something instead of something else. For
beliefs are special, it includes faith. They function not only as the outcome of
reasoning, but also as the premises in our reasoning. So, we start with the
following question: What are the sorts of things we believe? What are the things
we reason with?
George E. Boole
Chapter
3
6. Advanced Secondary Education
Page 6 of 36
PROPOSITIONS
A proposition is a statement that is either true or false but not both or neither.
Proposition are denoted by letters just as letters used in algebra. The most
common letter used to represent propositions are p, q and r although any other
letter may be used.
Sentence is a group of words to give out the meaning phrase. This book is not
discussing more about sentences but propositions (statements).
Example of propositions
(a) Arusha is a capital city of Tanzania
(b) You are a single girl.
(c) Dar Es Salaam is the city in Eastern Tanzania
(d) Kampala is a capital city of Kenya
(e) 3 + 5 = 8
(f) 2 + 2 = 22
The propositions (a), (d) and (f) are false whereas propositions (c) and (e) are true
statements, (b) is neither true nor false, it is hard to tell (this is not a proposition).
Each sentence in this list is the kind of thing to which we might assent or dissent.
We may agree or disagree, believe or disbelieve, or simply be undecided, about
each of these claims.
Not every sentence is proposition; Sentences such as those expressing emotions,
greeting, asking questions or making requests are not propositions.
Example of sentences, which are not propositions: Questions, emotion expression,
commands, etc.
(a) What is your name?
(b) Hello! How are you?
(c) Where is my book?
(d) 3x + 1 = 7
(e) x + y = z
(f) I am very happy today!
All these are not propositioning. Propositions are the sorts of things that can be
true or false. In the study of logic, we are interested in relationships between
propositions, and the way in which some propositions can act as reasons for other
propositions. These reasons are what we call arguments.
An argument is a list of propositions, called the premises, followed by a word such
as ‘therefore’, or ‘so’, and then another proposition, called the conclusion.
Example of an argument.
7. Advanced Secondary Education
Page 7 of 36
“If I study Mathematics, I will not study physics. I am studying physics; therefore, I
am not studying Mathematics”
The first two sentences
(a) If I study Mathematics, I will not study Physics.
(b) I am studying Physics.
Are premises, while the last sentence
(c) Therefore, I am not studying Mathematics.
Is a conclusion
For an argument to be good is for the premises to guarantee the conclusion. That
is if the premises are true then the conclusion has to be true. These good
arguments are called valid arguments. Generally, an argument is valid if and only
if whenever the premises are true, so is the conclusion. In other words, it is
impossible for the premises to be true while at the same time the conclusion is
false.
Consider the next argument below
“If people are mammals, they are not cold-blooded. People are cold-blooded.
Therefore, people are not mammals”
Note that, the argument may be invalid in real life but valid logically. Example of
the argument that is invalid in real life but valid logically
“If human are cold blooded animals, then they are reptilians, human are cold
blooded animas. Therefore, they are reptilians”.
This argument is invalid in real world; we need to be careful here because logically
this argument is valid.
This is clearly a valid argument, but no one in his or her right mind would believe
the conclusion. That is because the premises are false. Well, the second premise
is false—but that is enough. One premise being false is enough for the argument
to be bad. The bad arguments are called invalid arguments.
Sometimes we do not have enough information in our premises to guarantee the
conclusion, but we might make the conclusion more likely than it might be
otherwise. We say that an argument is inductively strong if, given the truth of the
premises, the conclusion is likely. An argument is inductively weak if the truth of
the premises does not make the conclusion likely. The study of inductive strength
and weakness is the study of inductive logic. We will not explore inductive logic in
this book. We will concentrate on what is called deductive logic — the study of
validity of arguments.
8. Advanced Secondary Education
Page 8 of 36
Consider the argument below once again
If people are mammals, they are not cold-blooded. People are cold-blooded.
Let p be “if people are mammals” and q be “they are cold blooded”
These premises have the following structure
“If p then not q, therefore not p”
• Be warned—an argument can be an instance of an invalid form, while still
being valid.
Example: All doctors are quacks; Smith is a doctor. Therefore, Smith is a quack.
Logic is all about reasoning. Letters are used to denote propositions just as letters
used to denote variables in algebra.
From the book of Thought written by English Mathematician George Boole 1954,
we have methods of producing new propositions from those that we already have.
Many Mathematical statements are constructed by combining one or more
propositions.
Definition: “Compound proposition is formed from existing proposition using logical
operators. These are mathematical statements constructed by combining two or
more propositions”
TRUTH TABLE
The truth table displays the relationship between the truth-values of proposition.
The truth table are especially valuable in the determination of the truth-values of
propositions constructed from simpler propositions. To construct the truth table,
follow steps below: -
(i) First column has 1
2 −
= n
T and 1
2 −
= n
F truth values
(ii) The second column has 2
2 −
= n
T and 2
2 −
= n
F truth values
(iii) The th
r column has r
T −
= 2
2 and r
F −
= 2
2 truth-values, where n is
number of simpler propositions and r is the specified column.
Numbers of T and F occurs alternate in the truth table.
LOGIC OPERATORS
The propositional connectives are negation (, not), conjunction ( , and),
disjunction ( , or), implication ( , if - then) and bi-implication ( , if and only
if). The connectives , , and are designated as binary while is
designated as unary.
9. Advanced Secondary Education
Page 9 of 36
Conjunction (and, )
Let p and q be simple propositions. The proposition “p and q” denoted by q
p is
the proposition that is true when both p and q are true and is false otherwise. The
proposition q
p is called the conjunction of p and q.
Here we have two propositions p and q therefore 2
=
n (number of simple
propositions p and q)
Therefore T = 2 and F = 2 in the first column
p q q
p
T T T
T F F
F T F
F F F
Disjunction (OR, )
Let p and q be simple propositions. The proposition “p or q” denoted by q
p is
the proposition that is false when both p and q are false and is true otherwise. The
proposition q
p is called the disjunction of p and q.
Truth table of q
p is given below
p q q
p
T T T
T F T
F T T
F F F
Exclusive Or ( )
Let p and q be propositions. The exclusive or of p and q is denoted by q
p is the
proposition that is true when exactly one p or q is true and is false otherwise. The
proposition q
p is called the exclusive or of p and q.
Truth table of q
p is given below
p q q
p
T T F
T F T
F T T
F F F
10. Advanced Secondary Education
Page 10 of 36
Negation, (NOT, )
Let p be a proposition. The negation of p is denoted by p
or ~p. p
is false only
when p is true and false otherwise.
Truth table of p
p p
T F
T F
F T
F T
Condition/Implication, (→)
Let p and q be propositions. The implication q
p → is the proposition that is false
when p is true and q is false and is true otherwise. In this implication p is called
hypotheses, antecedent, or premise and q is called the conclusion or
consequence.
Truth table of q
p→
Because implication arise in many places in mathematical reasoning a wide variety
of terminology is used to express q
p→ . Keywords used to show implication are:
(a) If p then q
(b) p is sufficient for q
(c) if p, q
(d) p only if q
(e) p is necessary for q
(f) q if p
(g) q whenever p
Note that q
p→ is false only in the case that p is true but q is false so that
it is true when both p and q are true and when p is false (No matter what
truth-value q has).
Consider the example below about implication “If you make more than
Tshs. 75,000/- per month then you must file a tax return”. This statement
is saying nothing about somebody making less than Tshs. 75,000/- and do
not file a tax return.
Similarly, the statement “If a player hits more than 60 home runs, then a
bonus of 10 million is awarded”, then this contract will be violated only if
p q q
p→
T T T
T F F
F T T
F F T
11. Advanced Secondary Education
Page 10 of 36
the player hits more than 60 home runs and not awarded but it says
nothing if the player hits fewer than 60 home runs.
CONVERSE, INVERSE AND CONTRAPOSITIVE
Let p and q be two propositions. There are some related implications that
can be formed from q
p → . The proposition p
q→ is called the converse
of q
p → and the proposition p
q
→
is called contrapositive of q
p →
. Simply, contrapositive is the inverse of converse.
We can build up compound propositions using the negation operator and
the different connectives defined so far. Parentheses are used to specify
the order in which the various logical operator in compound propositions
are applied. In particular, the logical operator in the innermost
parentheses are applied first.
Write down the converse and contrapositive of the implication “If today is
Thursday, then I have a test today”
Solution
Let p be “today is Thursday” and q be “I have a test today”, then the
compound statement can be written as q
p→ , it is converse p
q→ , In
words is “If I have a test today, then today is Thursday”. Contrapositive of
q
p→ is p
q
→
then it is “If I do not have a test today, then today is not
Thursday”.
Biconditional/double implication, ( )
Let p and q be propositions. The biconditional q
p is the proposition
that is true when p and q have the same truth-values and false otherwise.
Truth table for q
p
Keywords to show biconditional
p q q
p
T T T
T F F
F T F
F F T
Example 1
12. Advanced Secondary Education
Page 11 of 36
(a) “p if and only if q”
(b) “p is necessary and sufficient for q”
(c) “If p then q and conversely”.
LAWS OF ALGEBRA OF PROPOSITION
Let p, q and r be the logical statements, where T and F stands for True
and False values
Note that, the symbol show “Logical equivalent”
(a) Identity law
p
T
p
p
F
p
T
p
p
F
p
p
(b) Domination law
T
T
p
F
F
p
(c) Idempotent law
p
p
p
p
p
p
(d) Double negation law
( ) p
p
(e) Commutative law
p
q
q
p
p
q
q
p
(f) Associative law
( ) ( )
r
q
p
r
q
p
( ) ( )
r
q
p
r
q
p
(g) Distributive law
( ) ( ) ( )
r
p
q
p
r
q
p
( ) ( ) ( )
r
p
q
p
r
q
p
(h) De Morgan’s law
( ) q
p
q
p
( ) q
p
q
p
(i) q
p
q
p
→
13. Advanced Secondary Education
Page 12 of 36
( ) ( )
p
q
q
p
q
p →
→
( ) ( )
p
q
q
p
PRACTICE 1
1. Draw the truth table of the following
(a) ( ) q
q
p →
(b) ( ) p
q
p
→
→
(c) ( ) r
q
p
→
(d) ( ) ( )
q
p
q
p
→
→
(e) ( ) p
q
p
→
(f) ( ) ( )
p
q
r →
→
q
p
(g) ( ) ( )
q
p
q
p
2. Use the algebra laws of proposition simplify the following
(a) ( ) ( )
q
q
p
q
p
→
(b) ( ) ( )
p
q
q
p →
(c) ( ) ( )
p
q
p
q
(d) ( ) ( )
q
p
q
p →
→
(e) ( ) ( )
p
q
q
p
→
→
→
(f) ( ) ( ) ( )
q
p
q
p
q
p
14. Advanced Secondary Education
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ARGUMENTS
In everyday situations, arguments are dialogues between people. Here we
do not study everything of these dialogues. We just stick on the
propositions people express when they give reasons for things. In logic,
argument is a list of propositions called the premises, we saw early,
followed by a word such as “therefore” or “so” and then another
proposition called conclusion.
In simple words; an argument is a sequence of statements.
Premises or assumptions or hypothesis these are statements in an
argument.
Conclusion is the final statement in an argument.
“If I live in Arusha, then I’m a Maasai. I’m a Maasai; therefore, I
live in Arusha”.
Solution
Let p be “I live in Arusha” and q be “I am a Maasai” p and q are premises
while the conclusion is “I live in Arusha”
Logically equivalent
The compound propositions that have the same truth-values in all possible
cases are called logically equivalent. The symbol stands for “Equivalent
to”. One the way to show that the logical propositions are equivalent is by
using truth table and another way is to use laws of algebra of propositions
to resemble either left hand sides to the right-hand side or vice versa.
Propositions p and q are said to be logically equivalent if and only if the
columns giving their truth-values agree otherwise, they are not logically
equivalent.
Use the truth table to show that ( ) q
p
q
p
Solution
p q ( )
q
p
q
p
T T F F
Example 2
Example 3
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T F F F
F T F F
F F T T
Because the columns of ( )
q
p
and q
p
on the truth table have
the same truth values, then they are equivalent, written as
( ) q
p
q
p
Show that the propositions q
p → and q
p
are logically equivalent.
Solution
Using the truth table
p q q
p
q
p→
T T T T
T F F F
F T T T
F F T T
The columns of q
p
and q
p → have the same truth values,
therefore q
p
q
p
→
TAUTOLOGY, FALLACY AND CONTINGENCY
Tautology is a compound proposition that is always true no matter what
the truth-values of the proposition that occur in it.
In other words, Is the proposition which is true (T) in every row of the truth
table.
Use the truth table to show that p
p
is tautology
Solution
p p
p
p
T F T
T F T
F T T
F T T
Example 4
Example 6
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Fallacy or Contradiction is a compound statement/proposition that is
always false. It is the opposite of tautology.
Example of a contradiction proposition is p
p
p p
p
p
T F F
T F F
F T F
F T F
Contingency or invalid proposition: is a compound proposition, which is
neither tautology nor fallacy.
Example of an invalid proposition is ( ) p
q
p
→
p q q
p ( ) p
q
p
→
T T T T
T F T T
F T T F
F F F T
Because the last column ( ) p
q
p
→
has truth-values T and F then
statement is nether tautology nor fallacy. This is contingency or invalid
statement.
Show that ( )
( )
q
p
p
and q
p
are equivalent
Solution
p q ( )
( )
q
p
p
q
p
T T F F
T F F F
F T F F
F F T T
Hence ( )
( ) q
p
q
p
p
Show that ( ) ( )
q
p
q
p
→
is tautology using laws of proposition.
Solution
Given ( ) ( )
q
p
q
p
→
Example 6
Example 7
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( ) ( )
q
p
q
p
- Negation law
q
p
q
p
- De Morgan’s law
( ) ( )
q
q
p
p
- Associative law
T
T
- Identity law
T
- Identity law
Hence ( ) ( )
q
p
q
p
→
is tautology proposition
TESTING VALIDITY OF ARGUMENTS
Valid argument: In logic the property of an argument consisting in the fact
that the truth of the premises logically guarantees the truth of the
conclusion.
The following are steps to test the validity of an argument
(a) Express the compound statement (argument) as a logical symbol
(b) Use the truth table to check the validity of an argument or use
laws of algebra of propositions to simplify the compound
statement
(c) Write the conclusion
Test the validity of the following argument using the truth table “If I love
Mathematics then I will practice Mathematics. I do not practice
Mathematics; therefore, I hate Mathematics”
Solution
Using the truth table
Let p be “I love mathematics” and q be “I will practice Mathematics”
Symbolize, ( )
p
q
q
p
→
→
Truth table
p q ( )
q
p → ( ) q
q
p
→ ( )
p
q
q
p
→
→
T T T F T
Example 8
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T F F F T
F T T F T
F F T T T
Therefore, the argument is valid (tautology)
Using laws of algebra of propositions
( )
p
q
q
p
→
→
( )
p
q
q
p
→
- Negation law
( ) ( )
p
q
q
q
p
→
- Distributive law
( )
p
F
q
p
→
- Identity law
( ) p
q
p
→
- Identity law
( ) p
q
p
- Negation law
( ) p
q
p
- Associative law
( ) q
p
p
- Associative law
q
T - Identity law
T
Hence, the proposition is valid
Test the validity of the argument; “If 6 is even then 2 divide 7, either 5 is
not prime or 2 divide 7, but 5 is Prime. Therefore 6 is odd”.
Solution
Let p be “6 is even” q be “2 divide 7” and r be “5 is prime”
Symbolically
( ) ( )
p
r
q
r
q
p
→
→
Using the truth table
Example 9
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p q r q
p→ q
r
( )
( ) r
q
r
q
p
→ ( )
( )
p
r
q
r
q
p
→
→
T T T T F F T
T T F T T F T
T F T F F F T
T F F F T F T
F T T T T T T
F T F T F F T
F F T T T T T
F F F T F F T
Using the truth table, the conclusion is valid.
Determine the validity of the following statement: “If there is no Law,
there is no justice. There is no Law, there is no justice”
Solution
Let p be “there is no law” and q be “there is no justice”
Symbolically
( ) ( )
q
p
q
p →
→ by the truth table
p q q
p → q
p → ( ) ( )
q
p
q
p →
→
T T T T T
T F F F F
F T T T T
F F T T T
The conclusion from the truth table shows that the statement is invalid
PRACTICE 2
1. (a) Check the validity of the following argument ( )
r
q
p
→
,
p
r
→
, r
q
→ and r
(b) What does the proposition ( )
( )
q
p
r
q
p
→
represent?
2. (a)What is mean by the terms;
(i) Fallacy statement
(ii) Tautology statement
Example 10
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(b) Translate into symbolic form and test the validity of the following
argument: “If Kanzi speaks logically, then He does not contradict
laws of Algebra. Either there is no meaning or He contradicts laws
of Algebra. However, there is meaning. Therefore, Kanzi speaks
logically”.
3. (a) Simplify the following ( ) ( )
p
q
p
q
p
→
(b) Draw the truth table of ( ) p
q
p
and conclude
4. Symbolize the following argument and conclude “If I am headache,
then I have malaria or low blood pressure, I have no malaria nor low
blood pressure, therefore I am not headache”.
5. Test the validity of the statements: If 3 + 6 = 9 then 3x + 2x = 6x,
therefore 4 + 5 = 11 or 9 – 5 = 4.
ELECTRICAL NETWORK
Logic is very useful in daily life of electric. For the bulb to be light up the
complete circuit must be well connected, thus wire, source of power and
bulb itself. Electric circuit can also be used to verify or test the validity of
the logical statements (compound statements).
Consider the circuits below
Parallel and series connection of switches
Source of power
Bulb
Switch
Wire
This circuit is open (off), the current is not
passing therefore the bulb is not light up.
When a switch closed (on) the current will
pass to the bulb, which make it to light up.
Source of power
Bulb
Switch
Wire This circuit is closed (on), the current is
passing therefore the bulb is light up.
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Series switch connection, AND gate
Parallel switch connection, OR gate
To light the bulb all one switch (p or q) need to be closed (on). For this
case, the bulb will light when one or both (all) the parallel switches turned
on (closed).
Using the logic concept, one or both switches should be true (on) but not
both for the bulb to work and it will not work only when both p and q are
false (Open or off)
As seen here only switch p is closed and q is open, this makes the current
to pass through the switch p to connect the source of power and the bulb,
which makes the bulb to light up.
Let p and q be the compound statement represented by the following
circuits, the series switches are connected by the word “and” while the
parallel switches are connected by the word “or”.
Source of power Bulb
S1
Wire
Switches S1 and S2 are series switches.
When S1 switch is closed (on) and S2 is
open (on) the bulb will not light up. For the
bulb to light up both switches must be
closed (on) otherwise the bulb will not
work.
S2
T1 T2
q
p
Source of power Bulb
T1 T2
q
p
Source of power Bulb
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Switch p and q are parallel and both are in series with switch r. Even if
switch p and q will be closed and r left, open there is no way a bulb will
light up. Switch r controls current to both switch p and q, logically this can
be expressed as ( ) r
q
p
.
If switch p and q are closed and switch r is open, still the bulb will not work,
because r disconnect the current to reach the bulb.
Note that, when switches are opposite of the normal switch logically is
interpreted as negation of the switch.
Express the following electric circuit in logical symbols using the switch
letters given
Solution
Switch p and q are parallel and r is in series with p and q, this can be written
as ( ) r
q
p
T1 T2
p
q
r
Source of power Bulb
p
q
r
T1 T2
S B
Example 11
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Construct an electrical network corresponding to the proposition
( )
s
r
q
p
Solution
p and q are parallel, q
p is in series with r and lastly ( ) r
q
p
is parallel
to switch s
The circuit look as follows
T1 T2
p
q
r
s
Battery Bulb
Example 12
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MISCELLANEOUS PRACTICE
1. Use properties of operations in logic to show whether the following
proposition is a tautology ( ) ( )
p
q
p
q
p
2. Make a truth table of ( ) ( )
q
p
q
p
→
→
3. What is a proposition as used in mathematics logic?
4. Show that ( )
p
and p are logically equivalent
5. Use the truth table to show that the following proposition are logically
equivalent.
a) ( )
q
p and ( )
p
q
b) ( )
q
p and ( )
p
q
6. Verify the following associative laws by truth table
( ) ( )
r
q
p
r
q
p
=
7. Use the truth table very that ( ) ( )
q
p
q
p
8. Prove that ( ) ( ) ( )
r
p
q
p
r
q
p
=
truth table.
9. Determine the contrapositive of the statement: “If John is a jobless,
then he is poor”
10. Show that each of the following are tautology using truth table
a) ( ) p
q
p →
b) ( )
q
p
p
→ c) ( )
q
p
p →
→
11. Use laws of algebra of propositions show that the following are
tautology
a) ( )
q
p
p
→ b) ( ) q
q
p
→
→
c) ( )
q
p
p →
→
d) ( ) ( )
q
p
q
p →
→
12. Use laws of algebra simplify the following
a) ( )
p
q
p
p →
b) ( ) ( )
( )
r
p
r
q
q
p →
→
→
→
c) ( )
q
q
p
p →
→
d) ( ) ( ) ( )
r
r
q
r
p
q
p →
→
→
13. Verify the following absorption laws
a) ( )
p
q
p
p
b) ( )
p
q
p
p
14. Determine if the following propositions are tautology or not
25. Advanced Secondary Education
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a) ( ) q
q
p
p
→
→
b) ( ) p
q
p
q
→
→
15. Show that q
p is equivalent to ( ) ( )
q
p
q
p
16. Determine if ( ) r
q
p →
→ and ( )
r
q
p →
→ are not equivalent.
17. Show that q
p → is equivalent to it is contrapositive
18. Use truth table to show that q
p
and q
p
are logically
equivalent
19. Show that ( )
q
p
and q
p are logically equivalent
20. Write the negation of the following statements
(a) If she buys book, she will read
(b) If it rains, he will not go to play
(c) If it is a sunny day, we will wash our clothes
21. Symbolize the following statements and write their inverse, converse
and contrapositive
(a) If Mathematics is interesting then I will not study Physics
(b) If I get grade A in Physics then I will be an engineer
(c) If 3 is even, then 4 is odd
(d) History is interesting if and only if mathematics is boring and tough
22. Construct the truth table for the following statements
(a) If I work hard, I will be rich
(b) A number is even if and only if it is divisible by 2
(c) You will pass your examinations if you study hard
(d) We will play football if there is no rain or extensive sunlight
(e) You like Geography and Advanced Mathematics or you do not like
Geography and Economics’
23. Draw a simple electric network diagram for the statements
(a) ( ) ( )
q
p
q
p
→
(b) ( ) ( )
q
p
q
p
24. Prove that ( ) p
q
p
→ and q
p are equivalent, using
(a) Laws of algebra of sets
(b) The truth tables
25. Draw the electric network for the proposition ( ) ( )
q
p
q
p
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26. Use laws of algebra of sets to show if ( ) ( )
p
q
p
q
p
27. Construct an electrical network corresponding to the proposition
( )
s
r
q
p
28. Differentiate between tautology, contradiction and contingency
29. Simplify the following electrical network and re-draw
(a)
(b)
30. Consider the truth table below
p q X Y
T T F F
T F T F
F T F T
F F F F
Find the compound statements represented by letters X and Y and
hence draw an electric network to represent the simplified
propositions
31. Consider the truth table below
p q r M N L S
T T T F T T T
T T F F T T T
T F T T T T T
T F F F T T T
F T T F F F F
F T F F T T F
F F T T F T F
F F F F T T F
p q
p
p
q
p q
T1 T2
T1 T2
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Find the propositions represented as the letters M, N, L and S and
draw electric network for each statement above.
32. Without simplifying, the propositions ( ) ( )
q
r
q
p
draw an
electric network and construct a truth table.
33. Define the following terminologies as used in logic
(a) Proposition
(b) Disjunction
(c) Contradiction
34. (a) Simply the compound statement ( ) ( ) ( )
q
p
q
p
q
p
and an electric network.
(b) Using the letters and logical connectives, write the following
statement “If q is less than zero then is not positive”
(c) Given the statement “If q is less than zero then is not positive”
write: -
(i) The contrapositive of converse
(ii) Converse of the inverse
(iii) Contrapositive of inverse of converse
35. (a) Show whether ( ) ( )
r
q
q
p
is tautology, invalid or fallacy.
(b) Test the validity of this compound statement “All women are
polite. Irene is a woman. Therefore, Irene is polite”
(c) “If Urmy won the competition, then either Sreya came second or
Samrat came third. Samrat didn’t come third. Thus, if Sreya didn’t
come second, then Urmy didn’t win the competition.”
36. (a) Show that ( ) q
q
p →
is logically equivalent to it is contrapositive
using a truth table.
(b) Given that ( ) ( )
p
q
q
p
S
→
→
→
=
1 and
( ) ( )
q
p
p
q
S →
→
→
=
2 , use the truth table to show
whether 1
S is equivalent to 2
S
(c) Draw a simple network for ( ) ( )
p
r
q
p
q
(d) Construct a compound sentence ( )
r
q
p
X ,
, having the truth table
shown below: -
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p T T T T F F F F
q T T F F T T F F
r T F T F T F T F
( )
r
q
p
X ,
, T F T F F F F F
37. (a) Differentiate between Tautology and contradiction and give one
example for each.
(b) Determine whether or not, the statement ( )
( )
q
p
r
q
p
→
is a tautology.
(c) Given that proposition ( ) r
q
p
→
Write: -
(i) The converse of its contrapositive
(ii) Inverse of its converse
(d) Let p be Sam is rich and q be Sam is unhappy. Write each of the
following in symbolic form.
(i) Sam is poor but unhappy
(ii) Sam is poor or else he is both rich and unhappy
(iii) Sam is either rich or unhappy
(iv) Sam is rich if and only if he is happy
38. (a) Show that q
p does not logically imply q
p →
(b) What is meant by saying that an argument is valid?
(c) Prove the validity of the following argument: “If the country is
democratic and governed by selfish leaders, then the country will
develop. If the leaders are not selfish, the country will not develop
therefore the country is not democratic.”
(d) Construct a truth table and simplified logical circuit of: -
(i) ( )
q
p
p
→
(ii) ( ) p
q
p
→
→
39. Consider two propositions p and q, complete the truth table below.
p q ¬q p ¬q ¬p ¬p q
Decide whether the compound proposition (p ¬q) (¬p q) is
29. Advanced Secondary Education
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a tautology. State the reason for your decision.
40. Complete the truth table shown below.
State whether the compound proposition (p is a
contradiction, a tautology or neither.
41. Consider the following propositions.
p: Isaac finishes her homework
q: Isaac goes to the football match
Write in symbolic form the following proposition. If Isaac does not go
to the football match, then Isaac finishes his homework.
42. (a) Complete the following truth table.
Consider the propositions
p: Cristina understands logic
q: Cristina will do well on the logic test.
(b) Write down the following compound proposition in symbolic form.
“If Cristina understands logic, then she will do well on the logic
test”
(c) Write down in words the contrapositive of the proposition given in
part (b).
43. Consider the two propositions p and q.
p: The sun is shining q: I will go swimming
Write in words the compound propositions
p
q
p
))
(
p q p q p (p q) (p
T T
T F
F T
F F
p q .............................. p q
T T F ................
T F T ................
F T F ................
F F T ................
p
q
p
))
(
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(a) p q;
(b) ¬p q.
(c) The truth table for these compound propositions is given below.
Complete the table
(d) State the relationship between the compound propositions K and
M.
p q K ¬p M
T T T …………. T
T F F …………. F
F T T …………. T
F F T …………. T
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POSSIBLE ANSWERS TO PRACTICE EXERCISES
PRACTICE 1
1. Truth tables
(a) ( ) q
q
p →
p q q
p ( ) q
q
p →
T T T T
T F T F
F T T T
F F F T
(c) ( ) p
q
p
→
→
p q q
p → ( ) p
q
p
→
→
T T T F
T F F T
F T T T
F F T T
(d) ( ) q)
(p
q
p
→
p q q
p q
p )
(
)
( q
p
q
p
→
T T T T T
T F T F F
F T T F T
F F F F T
(e) ( ) ( ) M
q
p
q
p =
→
→
p q q
p q
p
→ M
T T T F F
T F T T T
F T T T T
F F F T T
(f) ( ) N
p
q
p =
→
p q q
p
→ N
T T F T
T F T F
F T T T
F F T T
(b) ( ) p
q
p
→
→
p q q
p→ ( ) p
q
p
→
→
T T T F
T F F T
F T T T
F F T T
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(g) ( ) ( )
q
p
q
p
p q M N This is p
q
p q
p
M N
T T T T T
T F F F T
F T F T F
F F F T F
2. (a) Given ( ) ( )
q
q
p
q
p
→
Simplify ( )
q
q
p
first
( )
( )
q
p
q
q
q
p
→
→
Implication law
( )
q
p
q
q
q
p
negation law
( ) ( ) ( )
q
p
T
q
q
p
Identity law
( ) ( )
p
T
q
q
p
Identity law
( ) ( )
T
q
q
p
Identity law
( ) q
q
p
Distributive law
( ) ( )
q
q
p
q
Distributive law
( ) ( )
p
q
F
p
q
Equivalent
Now return back, ( )
q
p
( ) ( )
p
q
q
p
→
( ) ( )
p
q
q
p
negation law
( )
q
q
p
distributive law
T
p
Identity law
p
(b) Given ( ) ( )
p
q
q
p →
( ) ( )
( ) ( )
q
p
p
q
p
q
q
p
→
→
→
→
Implication law
( ) ( )
( ) ( )
q
p
p
q
p
q
q
p
distributive law
( ) ( )
( ) ( )
q
p
p
q
p
q
q
p
distributive law
( ) ( )
( ) ( )
q
p
p
q
p
q
p
q
q
p
q
p
distr. law
( ) ( )
( ) ( )
q
p
p
T
p
T
q
p
Identity law
q
p
q
p
Identity law
( )
q
p
q
distributive law
F
q Identity law
q
(c) Given ( ) ( )
p
q
p
q
( ) ( )
( ) ( )
q
p
p
q
q
p
p
q
→
→
→
→ Implication law
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( ) ( ) ( ) ( )
q
p
p
q
q
p
p
q
Negation law
( ) ( ) ( ) ( )
q
p
q
p
p
q
p
q
distributive law
( ) ( )
q
q
p
q
q
p
Distributive law
( ) ( )
F
p
F
p
Identity law
p
p
Identity law
F
(d) Given ( ) ( )
p
q
q
p
( ) ( ) ( ) ( )
q
p
p
q
p
q
q
p
→
→
→
→ Implication law
( ) ( ) ( ) ( )
q
p
p
q
p
q
q
p
negation law
( ) ( )
q
q
p
q
q
p
distributive law
( ) ( )
F
p
F
p
Identity law
p
p
Identity law
F
(e) Given ( ) ( )
q
p
q
p
→
→
→
( ) ( )
p
q
q
p
→
Negation law
( ) ( )
p
q
q
p
Implication law
( ) ( )
p
q
q
p
q
p
distributive law
( ) ( )
( ) ( )
p
q
q
q
p
p
q
p
distributive law
( )
( )
p
q
T
T
q
p
Identity law
T
T
T
(f) Given ( ) ( ) ( )
q
p
q
p
q
p
( ) ( )
p
p
q
q
p
Associative law
( ) T
q
q
p
Identity law
( ) q
q
p
Distributive law
( ) ( )
q
q
p
q
Identity law
( ) T
p
q
Identity law
p
q
34. Advanced Secondary Education
Page 33 of 36
PRACTICE 2
1. (a) Using truth table
Let ( )
r
q
p
→
be M
Let ( ) ( )
p
r
r
q
p
→
→
be N
Let ( ) ( ) ( )
r
q
p
r
r
q
p
→
→
→
be A
Let ( ) ( ) ( )
r
r
q
p
r
r
q
p
→
→
→
be B
p q r r
q
M p
r
→
N A B
T T T F T F T F F
T T F T T T F F F
T F T T T T T T F
T F F F T F F F F
F T T F F F F F F
F T F T T T T T T
F F T T T T T T F
F F F F F F F F F
(b) Tautology (show it using truth table or laws of logic)
2. (a) (i) and (ii) refer the definitions in previous pages.
(b)
3. (a) Given ( ) ( )
p
q
p
q
p
→
( ) ( )
p
q
p
q
p
→
De Morgan’s law
( ) ( )
p
q
p
q
p
Implication law
( ) ( ) ( )
p
q
T
p
q
p
Identity law
( ) ( )
T
q
p
q
p
Identity law
( ) ( )
T
p
q
p
De Morgan’s law
( ) q
p
p
Associative law
q
p
(b) Without simplifying ( ) q
q
p
The conclusion is the circuit is Tautology (the current is flowing)
( )
q
p
q
T1 T2
35. Advanced Secondary Education
Page 34 of 36
References
1. Greg Restall (2006) Logic: An Introduction, Madison Ave, New York, NY 10016: Taylor & Francis
e-Library, 2006.
2. Dr. A.R. Mushi Logic: Discrete Mathematics: Course notes
3. Kenneth H. Rosen (1998) Discrete Mathematics and Its applications, fourth edition, Published by
China Machine Press/McGraw-Hill
4. Prof. Mordechai (Moti) Ben-Ari Science teaching department (2012) Mathematical Logic for
Computer science, third edition, Springer London Heidelberg New York Dordrecht.