PRACTICAL LOGIC

Bryan Rennie



GENERAL NOTES ON THIS CLASS

THE SET TEXTS

Introduction to Logic, by Irving Copi, Carl Cohen, and Kenneth McMahon (14th edition) = ITL.

COURSE OUTCOMES

This course is an introduction to the basics of logic as an academic discipline. We will consider what logic is. It is the study of the distinction between valid and invalid reasoning. Having established our working attitude to logic we will investigate the basic terms, forms, types, and style of argument and the uses of language in argument. To that end the basic vocabulary of logic and argument must be learned.

Our most extensive analysis will be of deductive logic, that is to say, arguments which produce logically necessary conclusions once their premises are accepted. The standard forms of such arguments will be analyzed, and their accompanying fallacies noted. The symbol systems used to express and analyze these forms will be practiced. However, inductive arguments, the conclusions of which follow with a certain degree of probability rather than being logically necessary, will also be briefly considered.

The overall outcome of this is two-fold: first, it will inform students of the precise and formal nature of logical proof (and its relative rarity); second, exposure to and practice with arguments and their identification as valid or invalid should greatly sharpen the students' natural skill at validating arguments and constructing their own valid arguments. This last is in many ways the final outcome of this course.

The course will include a detailed consideration of informal fallacies, those common, and often unidentified, errors which yield sometimes convincing, but always invalid, arguments in ordinary language. Once again the difficulties of irrefutable argument and final proof will be encountered.

So the course outcomes can be stated to be:

  1. To learn what an argument is. What components does it contain, what assumptions does it make?
  2. To learn what makes a good argument. Why does a given conclusion follow from certain assumptions?
  3. To learn what makes a bad argument. Why are certain conclusions not entailed by certain propositions?
  4. To practice and become more adept at the use of argumentation

COURSE REQUIREMENTS

Attendance
Attendance is crucial. Missed classes will be penalized.

Learning good logic skills is like learning both language and manual skill--they require practice, both physical and mental. To that end all students will be required to answer questions, solve problems, and do exercises from the textbook in class. Note that these exercises are not graded. All you need to do is to demonstrate to the instructor and to the class that you have made an effort to complete them. It is your responsibility to complete enough exercises to understand the points and to raise questions about points which you have not understood. This is your opportunity to seek clarification on difficult passages. You are NOT automatically expected to fully understand everything you read.

IF STUDENTS DO NOT MAKE AN EFFORT TO PERFORM THESE EXERCISES POINTS WILL BE SUBTRACTED FROM THEIR TOTAL.

Homework
There will be a certain amount of reading homework after every class to ensure a constant and ongoing effort to master each section before moving on to the next. This homework will not be handed in and will not be graded as such. It is for your own good rather than for the grading process. However, questions will be asked about the homework at the start of each class, and if it is apparent that you have not done it POINTS WILL BE SUBTRACTED. Time will be allowed in class to attempt the exercises but they should be studied beforehand as homework.


GRADING

Grading will be done on a points system up to a maximum of 400 total possible points:

  1. Quizzes (7 @ 30 points) These quizzes are the most important element of the course and are meant to ensure steady effort and ongoing understanding. WARNING: Failure to score a passing grade on any quiz will result in the loss of all points for that quiz. 210 points = c.52%.
  2. Computer Exercises. Although these exercises will not be graded for performance 100 points will be given for simply completing them. Up to one quarter of your total grade points can be lost by not doing the required exercises. 100 points = 25%
  3. Final examination This will review the whole course so the start is not forgotten at the end. WARNING: Failure to pass the final is failure to pass the course. 90 points = c.23%
The written quizzes will take a form similar (but not identical) to these sample quizzes.


SCHEDULE OF CLASSES

The class will meet Monday, Wednesday, and Friday from 10:30 to 11:30 in Patterson Hall 105.
I will be available in my office in Patterson 336 from 9:30 to 10:30 everyday, and by arrangement.

Week: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
Click the number to see the week.

Week 1.

Wednesday, 8/28
Introduction to the course, the textbook (don't forget the Glossary/Index!), and the class webpage.
ITL Chapter One: Basic Logical Concepts.
ITL 1.1 - What Logic Is.
ITL 1.2 - Propositions and Arguments.

Exercises in class from pages 9-11.
Homework: Read ITL Chapter One to page 24. Do exercises from pages 20-24.

Friday, 8/30
ITL Chapter One: Basic Logical Concepts.
ITL 1.3 - Recognizing Arguments.
ITL 1.4 - Arguments and Explanations.

Exercises in class from pages 20-24.
Homework: Read ITL Chapter One. Do exercises from page 32.


Week 2.

Monday, 9/2
ITL 1.5 - Deductive and Inductive Arguments.
ITL 1.6 - Validity and Truth.
Exercises in class from page 32.

Wednesday, 9/4
Summary of Chapter 1. Preparation for the quiz.

Friday, 9/6
QUIZ ON CHAPTER ONE (Sample)

Homework: Read Chapter Two. Do exercises from pages 36-38 and 43-48.


Week 3.

Monday, 9/9
Chapter Two - Analyzing Arguments.
ITL 2.1 - Paraphrasing Arguments; 2.2 - Diagramming Arguments.

Exercises in class from pages 36-38 and 43-48.
Homework: Read ITL 2.3 and 2.4. Do exercises from pages 52-53 and 59-61.

Wednesday, 9/11
ITL 2.3 - Complex Argumentative Passages; 2.4 - Problems in Reasoning.

Exercises in Class from pages 52-53 and 59-61.
Homework: Re-read ITL Chapter Two. Prepare for Quiz #2.

Friday, 9/13
QUIZ ON CHAPTER TWO (Sample).

Homework: Read ITL 3.1, 3.2, and 3.3. Do exercises from pages 66-71, 73-75, and pages 76-79.


Week 4.

Monday, 9/16
Chapter Three: Language and Definitions.
ITL 3.1 - Language Functions; 3.2 - Emotive and Neutral Language and Disputes; 3.3 - Disputes and Ambiguity.

Exercises in class from pages 66-71, 73-75, and pages 76-79.
Homework: Read ITL 3.4. Do exercises from page 86.

Wednesday 9/18
ITL 3.4 - Definitions and their Uses

Exercises in class from page 86.
Homework: Read ITL 3.5 and 3.6. Do exercises from pages 89-91.

Friday 9/20
ITL 3.5 - The Structure of Definitions: Extension and Intension; 3.6 - Definition by Genus and Difference.

Exercises in class from pages 89-91.
Homework: Read ITL 4.1, 4.2, and 4.3. Do exercises from pages 121-126.


Week 5.

Monday, 9/23
ITL 4.1 - What is a Fallacy?; 4.2 - Classification of Fallacies; 4.3 Fallacies of Relevance.

Exercises in class from pages 121-126.
Homework: Read ITL 4.4, 4.5, and 4.6. Do exercises from pages 138-140 and pages 148-154.

Wednesday, 9/25
ITL 4.4 - Fallacies of Defective Induction; 4.5 - Fallacies of Presumption; 4.6 - Fallacies of Ambiguity.

exercises in class from pages 138-140 and pages 148-154.
Homework: prepare for Quiz #3.

Friday, 9/27
QUIZ ON CHAPTERS THREE AND FOUR (Sample).

Homework: Re-read ITL Chapters Three and Four.


Week 6.

Monday, 9/30
Review of QUIZ #3. Review of Chapter Three: Language and Definitions.

Exercises in class from pages 66-71; 73-75; 76-79; 98.
Homework: Re-read ITL Chapter 4. Do exercises from pages 121-126; 138-140; 148-154; 156-159.

Wednesday, 10/2
Review of ITL Chapter 4 – Informal Fallacies.

Exercises in class from pages 121-126; 138-140; 148-154; 156-159.
Homework: prepare to re-take Quiz #3.

Friday, 10/4
REPEAT QUIZ ON CHAPTERS THREE AND FOUR (Sample).

Homework: Read ITL 5.1, 5.2, and 5.3. Do exercises from page 170.


Week 7.

Monday, 10/7
ITL 5.1 - The Theory of Deduction; 5.2 - Classes and Categorical Propositions; 5.3 - The Four Kinds of Categorical Propositions.

Exercises in class from page 170.
Homework: Read ITL 5.4. Do exercises from pages 175-176.

Wednesday, 10/9
ITL 5.4 - Quality, Quantity, and Distribution.

Exercises in class from pages 175-176.
Homework: Read ITL 5.5. Do exercises from page 180.

Friday, 10/11
ITL 5.5 - The Traditional Square of Opposition.

Exercises in class from page 180.
Homework: Read 5.6. Do exercises from pages 186-188.


Week 8.

Monday, 10/14
ITL 5.6 - Further Immediate Inferences.

Exercises in class from pages 186-188.
Homework: Read ITL 5.7. Do exercises from pages 196-197 and 202-203.

Wednesday, 10/16
ITL 5.7 - Existential Import and the Interpretation of Categorical Propositions; 5.8 - Symbolism and Diagrams for Categorical Propositions.

Exercises in class from pages 196-197 and 202-203.
Homework: Prepare for Quiz #4.

Friday, 10/18
QUIZ #4 (Sample).

Homework: read ITL 6.1. Do exercises from pages 209-210.


Week 9.

Monday 10/21
ITL 6.1 - Standard Form Categorical Syllogisms.

Exercises in class from pages 209-210.
Homework: Read ITL 6.2. Do exercises from pages 212-213.

Wednesday 10/23
ITL 6.2 - The Formal Nature of Syllogistic Argument.

Exercises in class from pages 212-213.
Homework: Read ITL 6.3. Do exercises from pages 222-224.

Friday 10/25
ITL 6.3 - The Venn Diagram Technique for Testing Syllogisms.

Exercises in class from pages 222-224.
Homework: Read ITL 6.4. Do exercises from pages 231-234.
Learn the rules and fallacies for syllogisms.


Mid-Term Break Monday October 28th.

Week 10.

Monday Classes meet Tuesday 10/29
6.4 - Syllogistic Rules and Syllogistic Fallacies.

Exercises in class from 231-234.
Homework: Read ITL 6.5 and Appendix. Do exercises from pages 238 and 242.

Wednesday, 10/30
ITL 6.5 - Exposition and Deduction the 15 Valid Forms of Categorical Syllogism.

Exercises in class from page 238 and 242.
Homework: Prepare for Quiz #5.

Friday, 11/1
QUIZ #5 (Sample).

Homework: Read ITL 7.1 and 7.2. Do exercises from pages 248-249.


Week 11.

Monday, 11/4
ITL Chapter Seven - Syllogisms in Ordinary Language: 7.1 - Syllogistic Arguments; 7.2 - Reducing the Number of Terms to Three.

Exercises in class from pages 248-249.
Homework: Read ITL 7.3. Do exercises from pages 257-258.

Wednesday, 11/6
ITL 7.3 - Translating Categorical Propositions into Standard Form.

Exercises in class from page 257-258.
Homework: Read ITL 7.4. Do exercises from pages 260-263.

Friday, 11/8
ITL 7.4 - Uniform Translation.

Exercises in class from pages 260-263.
Homework: Read ITL 7.5 and 7.6. Do exercises from pages 266-269 and 270-272.


Week 12.

Monday 11/11
ITL 7.5 - Enthymemes; 7.6 - Sorites.

Exercises in class from pages 266-269 and 270-272.
Homework: Read ITL 7.7 and 7.8. Do exercises from 276-278 and 282-285.

Wednesday 11/13
ITL 7.7 - Disjunctive and Hypothetical Syllogisms; 7.8 - The Dilemma.

Exercises in class from pages 276-278 and 282-285.
Homework: Prepare for Quiz #6.

Friday 11/15
QUIZ #6 (Sample).

Homework: Read ITL 8.1 and 8.2. Do exercises from pages 297-300.


Week 13.

Monday, 11/18
ITL 8.1 - Modern Logic and its Symbolic Language; 8.2 - The Symbols for Conjunction, Negation, and Disjunction.

Exercises in class from pages 297-300.
Homework: Read ITL 8.3 and 8.4. Do exercises from pages 308-310 and 312-313.

Wednesday, 11/20
ITL 8.3 - Conditional Statements and Material Implication; 8.4 - Argument Forms and Refutation by Logical Analogy.

Exercises in class from pages 308-310 and 312-313.
Homework: Read ITL 8.5, 8.6, and 8.7. Do exercises from pages 322-323.

Friday, 11/22
On-line Class: Prof Rennie will be attending the American Academy of Religion Conference in Baltimore.
ITL 8.5 - The Precise meaning of "valid" and "invalid"; 8.6 - Testing Argument Validity using Truth Tables; 8.7 - Some Common Argument Forms.

Exercises from pages 322-323.
Homework: Read ITL 8.8. Do exercises from pages 328-329.
Prepare for Quiz #7.


Week 14.

Monday, 11/25
QUIZ #7 (Sample)


Thanksgiving Break: Wednesday 27th through Sunday 1st.


Week 15.

Monday, 12/2
ITL 8.8 - Statement Forms and Material Equivalence.

Exercises in class from page 328-329.
Homework: Read ITL 8.9 and 8.10. Be prepared to discuss in class.

Wednesday, 12/4
ITL 8.9 - Logical Equivalence, including De Morgan's Theorems; 8.10 - The Three "Laws of Thought."

Homework: Revise everything.

Friday, 12/6
The Last Class. Student Assessments of the Course. Preparation for Final Examination.


Week 16.

Monday, 12/9

Final examination period Dec. 9th & 10th; Dec. 12th & 13th. Reading Day, Wednesday, Dec. 11th. Term ends Friday 13th.


SAMPLE QUIZ QUESTIONS

Quiz #1 |Quiz #2 |Quiz #3 |Quiz #4 |Quiz #5 |Quiz #6 |Quiz #7 |

SAMPLE QUIZ #1

(ITL Chapter One)
  1. What is "Logic"?
  2. Explain the difference between deduction and induction.
  3. Define "argument."
  4. What is "inference?"
  5. In the following argument mark the premises with "p," and the conclusion with a "c."
    See ITL pages 9-11 for examples.
  6. Is the preceding passage an argument or an explanation?
    See ITL pages 20-24 for examples.
  7. Indicate which of the following sentences (1-3) are correctly described by the following terms (a-c).
    1) If I understood all this I'd be a genius. 2) What am I, a genius? 3) I'm a genius
    a) rhetorical question, b) hypothetical statement, c) proposition
  8. Sort the following terms into conclusion indicators and premise indicators:
    therefore . ., for the reason that . ., as a result . ., which implies that . ., . . may be deduced from, inasmuch as . ., hence, accordingly, proves that . ., for . ., so . ., I conclude that . ., because . ., since . ., follows from . . .
  9. What is "validity" in Logic?
  10. Construct a deductive argument of the form given below.
    See ITL page 32 for examples.

SAMPLE QUIZ #2

(ITL Chapter Two)
  1. Paraphrase the following passages:
    See ITL pages 36-38 for examples.
  2. Diagram the following passages:
    See Itl pages 43-48 for examples.
  3. Diagram the following complex argumentative passage:
    See ITL pages 52-53 for examples.
  4. From the following passages construct a valid argument that both answers the question and proves that answer to be correct:
    See ITL pages 59-61 for examples.

SAMPLE QUIZ #3

(ITL Chapters Three and Four)
  1. Which of the various functions of language are exemplified by the following passage(s)? (Examples from ITL pages 66-71.)
  2. Identify the kinds of agreement or disagreement exhibited by the following pairs. (Examples from pages ITL 73-75.)
  3. Is the following dispute genuine or merely verbal? (Examples from ITL pages 76-79.)
  4. Name the five ways in which definitions may be used. (ITL 3.4)
  5. Arrange the following group of terms in order of increasing/decreasing intension. (Examples from ITL page 89.)
  6. Define the following terms by genus and difference. (Examples from ITL page 98.)
    Name and explain the following fallacies. These have been selected to be clear examples of certain single fallacies.
  7. Select ONE explanation only for each example.
  8. You should avoid four letter words. “Work” is a four letter word. So you should avoid work.
  9. A public lecture was delivered on smoking as a cause of cancer in the Orr Auditorium. Several students have undertaken never to smoke in that building again.
  10. Every player of the Washington Redskins is a better player than his opposite number in the Green Bay Packers. So the Redskins are the better team.
  11. The local chapter of Phi Sigma Tau has a collective I.Q. of 190, so Dr. Rennie, who is a member, must have an I. Q. of 190.
  12. All plants produce chlorophyll, so the GM plant in Detroit produces chlorophyll.
  13. “I think I like hot dogs more than you.”
    “Well, if that’s the way you feel, I never want to see you again.”
  14. A recent poll shows that 75% of the people have changed their voting allegiance. You should change yours, too.
  15. I deserve a B+ on this course. If I don’t get a B+ I won’t be able to graduate next Fall.
  16. You shouldn't take Dan’s arguments about farming subsidies seriously since he manages one of the largest farms in the area..
  17. Yesterday I had a wonderful stroke of luck just after I had seen a black cat. So black cats are lucky.
  18. It is only when it is believed that I could have acted otherwise that I am held to be morally responsible for what I have done. For a man is not thought to be morally responsible for what it was not in his power to avoid.
  19. There is no such thing as a leaderless group. For, though the style of leadership will differ with each group, a leader will always emerge in a task oriented group or the task will never get done.
  20. Do you realize that the majority of painful animal experimentation has no relationship whatever to human survival or the elimination of disease?
  21. Scientists hope that fish treated with new growth hormones will grow bigger, faster than normal fish. Other scientists are developing fish that could be introduced into cold Northern waters where they cannot now survive. The intention is to boost fish production for food. The economic benefits may be obvious, but not the risks. Does this make the risks reasonable? No, they are not.

SAMPLE QUIZ #4

(ITL Chapter Five)
  1. Name the form of the given propositions. Give both the name (eg. universal affirmative) and letter (eg. "A") identification.
  2. Identify the subject and predicate terms in the following propositions. Are they distributed or undistributed. (egs.)
    • Some television presenters are not responsible citizens.
    • No poets are aggressive.
    (More egs. ITL pages 175-76.)
  3. Draw the traditional/Aristotelian square of oppositions.
  4. What immediate inferences can be made from the following propositions? (Assuming these propositions to be true.) (egs.)
    • All successful business executives are intelligent people.
    • No college professors are entertaining lecturers.
    (More egs. ITL page 180.)
  5. Define the following immediate inferences (egs.):
    • conversion
    • obversion
    • contraposition
  6. Given a certain proposition determine whether following propositions are its converse/obverse/contrapositive/subalternate. If the first given proposition is true, are the following propositions true? That is to say, are these valid immediate inferences?
  7. Given a sequence of immediate inferences which are valid by Aristotelian logic, identify where an existential fallacy occurs.
  8. Express the following propositions as equations (eg. SP = 0, etc.) and as Venn diagrams for propositions.

SAMPLE QUIZ #5

(ITL Chapter Six)

  • Use a Venn diagram to test the validity of the arguments in #1. Write out the argument using S, P, and M, both in standard form (No S is P etc.) and in logical notation (SP = 0 etc.)

  • Fill in the blanks in the Six Rules and Fallacies for categorical syllogisms.

  • Name the fallacies committed or the rules broken by syllogisms of the following forms.
    AII--2EEE--1
    OAO--2EAO--3

  • Name the fallacies committed or the rules broken by the following syllogisms.

  • Answer the following questions with reference to the six rules. Explain how you reach your conclusion.

    SAMPLE QUIZ #6

    (ITL Chapter Seven)
    1. Rewrite the following in standard form, indicating its mood and figure. Is it valid or invalid? What makes it invalid?
      • All those who are neither members nor guest of members are excluded; therefore, no nonconformists are either members or guest of members, because all who are included are conformists.
      • Translate the following propositions into standard-form categorical propositions.
        • Only power-hungry people become politicians.
        • If he isn't rich he isn't successful.
        • There are also positive reasons to vote.

      • Translate the following syllogistic argument into standard form and then
        • Name the mood and figure of the standard form translation.
        • Test its validity using the six syllogistic rules.
        • If it is invalid name the fallacy committed.
        Everyone who smokes marijuana goes on to try heroin.
        Everyone who tries heroin becomes addicted to it.
        So everyone who smokes marijuana becomes addicted to it.

      • Translate the following syllogistic argument into standard form and then
        • Name its mood and figure.
        • Test its validity using a Venn diagram.
        • If it is invalid name the fallacy committed.
        There are plants growing here, and since vegetation requires water, water must be present.

    2. Translate the following propositions into standard form with the help of appropriate parameters.
      • Politicians always criticize other politicians when they want to conceal their own shortcomings.
      • The use of violence is sometimes beneficial.
      • People in subordinate positions do not complain unless provoked.

    3. Explain briefly what (a) Enthymemes and (b) Sorites are.

    4. Are the following arguments (I, II, and III)
      (a) disjunctive, pure hypothetical, or mixed hypothetical?
      (b) Are they valid or invalid?
      (c) If they are invalid what fallacy do they commit? Identify modus ponens or modus tollens forms where applicable.

      I. If the one-eyed prisoner does not know the color of the hat on his own head, then the blind prisoner cannot have on a red hat. The one-eyed prisoner does not know the color of the hat on his own head. Therefore the blind prisoner cannot have on a red hat.

      II. If this syllogism commits the fallacy of affirming the consequent then it is invalid. It does not affirm the consequent, therefore it is valid.

      III. The stranger is either a knave or a fool. The stranger is a knave, therefore he is no fool.

    5. Discuss the various arguments that might be used to refute the following:
      • If the conclusion of a deductive argument goes beyond the premises, then the argument is invalid, while if the conclusion of a deductive argument does not go beyond the premises, then the argument brings nothing new to light. The conclusion of a deductive argument must either go beyond the premises or not go beyond them. Therefore either deductive arguments are invalid or they bring nothing new to light.
      • If Socrates died, he died either when he was living or when he was dead. But he did not die while he was living; for assuredly he was living, and as living he had not died. Nor when he had died, for then he would be dead twice. Therefore Socrates did not die.

    SAMPLE QUIZ #7

    (ITL Chapter Eight)
    (questions could have multiple examples)
    1. Using the truth tables definitions of conjunction, disjunction, and negation determine which of the given statements are true. (egs. ITL page 297)
    2. Symbolize the following statements using letters for simple statements and the symbols for conjunction, disjunction, and negation. (egs. ITL page 299)
    3. If A and B are true statements and X and Y are false statements which of the following are true? (egs. ITL page 308)
    4. If A and B are true statements and X and Y are false statements and the values of P and Q are unknown, which of the given statements can be determined to be true or false?
    5. Match the given specific forms with the given arguments. (ITL page 312)
    6. Use truth Tables to identify which of the given statement forms are tautologous, self-contradictory, or contingent. (egs. ITL 322)
    7. Which of the given biconditionals are tautologies? (egs. ITL page 331)

    The Six Rules of Standard-Form Categorical Syllogisms

    (and corresponding fallacies)

    1. A valid standard-form categorical syllogism must contain exactly three terms, each of which is used consistently in the same sense throughout the argument. (if not--fallacy of four terms, quaternio terminorum)

    2. In a valid standard-form categorical syllogism, the middle term must be distributed in at least one premise. (if not--fallacy of the undistributed middle)

    3. In a valid standard-form categorical syllogism, if either term is distributed in the conclusion, then it must be distributed in the premises. (if not--illicit major or illicit minor depending on where the undistributed term occurs)

    4. No standard-form categorical syllogism having two negative premises is valid. (if it does--fallacy of exclusive premises)
      Note that this rule renders all standard-form categorical syllogisms of EO*, EE*, OE*, AND OO* moods INVALID, regardless of their figure.

    5. If either premise of a valid standard-form categorical syllogism is negative, the conclusion must be negative. (drawing an affirmative conclusion from a negative premise is a fallacy)
      Note that this rule renders all standard-form categorical syllogisms which have a positive conclusion (**A or **I) and any negative premises (E or O) INVALID regardless of their figure.

    6. No valid standard-form categorical syllogism having a particular conclusion can have two universal premises. (If it does it commits the existential fallacy.) Note that this rule renders all standard-form categorical syllogisms with particular conclusions (**I or **O) and two universal premises (AA, EE, AE, EA) INVALID regardless of their figure.


    Symbols used in Basic Logic.
    See The Wikipedia Site

     
    Symbol used in class Alternative form of Symbol Name of symbol in class Alternative Name(s) of symbol Corresponding Verbal expressions
    S ⊃ R S ⇒ R material implication consequence S implies R. 
    R is a consequence of S. 
    S has R as a consequence. 
    If S then R. 
    R given S. 
    S only if R. 
    S is a sufficient condition for R. 
    R is a necessary condition for S.
    S ≡ R S ⇔ R equivalence   S is equivalent to R. 
    R is equivalent to S. 
    S if and only if R. 
    R if and only if S.
    S ∨ R   disjunction   S or R. 
    Either S or R. 
    S unless R. 
    Note that all these can be reversed.
    ~S   negation   not S
    It is not the case that S
    H and E H • E conjunction   H and E 
    the conjunction of H and E 
    both H and E 
    H but E


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    brennie@westminster.edu