Build Case Through Logic Essays

Many of the important points of this section are covered in the section on writing Argumentative Essays: Being Logical. You might want to review that section first and then come back here for a more thorough review of the principles of logic.

This document is part of a collection of instructional materials used in the Purdue University Writing Lab. The on-line version is part of OWL (On-line Writing Lab), a project of the Purdue University Writing Lab, funded by the School of Liberal Arts at Purdue.

We use logic every day to figure out test questions, plan our budgets, and decide who to date. We borrow from the vocabulary of logic when we say, "Brilliant deduction" or even "I don't want to argue about it." In the study of logic, however, each of these terms has a specific definition, and we must be clear on these if we are to communicate.

Vocabulary

Proposition --
T or F in an argument, but not alone. Can be a premise or conclusion. Is not equal to a sentence.
Premise --
Proposition used as evidence in an argument.
Conclusion --
Proposition used as a thesis in an argument.
Argument --
A group of propositions of which one is claimed to follow from the others.
Induction --
A process through which the premises provide some basis for the conclusion
Deduction --
A process through which the premises provide conclusive proof for the conclusion.

Argument Indicators:Premise Indicators:Conclusion Indicators:
  • should
  • must
  • ought
  • necessarily
  • since
  • because
  • for
  • as
  • inasmuch as
  • for the reason that
  • first ...
  • therefore
  • hence
  • thus
  • so
  • consequently
  • it follows that
  • one may infer
  • one may conclude
  • When dealing with persuasive writing, it will be helpful for you to outline the argument by premises and conclusions. By looking at the structure of the argument, it is easy to spot logical error.


    Example 1

    "Universities are full of knowledge. The freshmen bring a little in, and the seniors take none away, and knowledge accumulates.

    -- Harvard President A. L. Lowell

    Premise 1
    Premise 2
    Premise 3
    Conclusion
    Freshmen bring a little (knowledge) in
    Seniors take none away
    Knowledge accumulates
    Universities are full of knowledge

    Example 2

    (Here, the conclusion of one argument is used as a premise in another. This is very common.)

    Even though there may be a deceiver of some sort, very powerful and very tricky, who bends all his efforts to keep me perpetually deceived, there can be no slightest doubt that I exist, since he deceives me; and let him deceive me as much as he will, he can never make me be nothing as long as I think I am something. Thus, after having thought well on this matter, and after examining all things with care, I must finally conclude and maintain that this proposition: I am, I exist, is necessarily true every time that I pronounce it or conceive it in my mind.

    -- Rene Descartes, *Meditations*

    Argument 1 Premise 1:

    Conclusion of Argument 1
    Argument 2 Premise 1:

    Conclusion:

    To be deceived ... I must exist

    When I think that I exist I cannot be
    deceived about that

    I am, I exist, is necessarily true ... .


    Exercises

    Find the Arguments and Outline them in These Statements:

    1. Ask the same for me, for friends should have all things in common.

    -- Plato, Phaedrus

    2. Matter is activity, and therefore a body is where it acts; and because every particle of matter acts all over the universe, every body is everywhere.

    -- Collingwood, The Idea of Nature

    3. The citizen who so values his "independence" that he will not enroll in a political party is really forfeiting independence, because he abandons a share in decision©making at the primary level: the choice of the candidate.

    -- Felknor, Dirty Politics


    Reaching Logical Conclusions

    This article is reprinted from pages 78-79 of Pearson-Allen: Modern Algebra , Book One. In the book it is one of several between-chapter articles that add interest and provike thought on subjects related to the topics discussed in the text.

    Consider the two statements:

      1. Any member of a varsity squad is excused from physical education.
      2. Henry is a member of the varsity football squad.

    Our common sense tells us that if we accept these two statement as true, then we must accept the following third statement as true:

      3. Henry is excused from physical education.

    We say that the third statement follows logically from the other two.

    In drawing logical conclusions it does not matter whether the statements we accept as true are reasonable or sensible. This is because we depend entirely upon the form of the statements and not upon what we are talking about. Thus, if we accept the following statements as true:

      1. All whales are mammals;
      2. All mammals are warm-blooded animals;
      3. All warm-blooded animals are subject to colds;

    then we must conclude that

      4. All whales are subject to colds. Do you see that statements 1, 2, and 3 are arranged in logical order ?

      In the diagram at the right the set of whales is represented
      by W, the set of mammals by M, the set of warm-blooded
      animals by B, the set of animals by B, the set of animals
      subject to colds by C, and the set of all animals by A. The
      diagram shows that W is a subset of M as required by
      statement 1, that M is a subset of B as required by statement
      2, and that B is a subset of C as required by statement 3. The
      only conclusion that uses all of our given statements is that
      W is a subset of C, as asserted by statement 4.
      Had our third statement been "no warm-blooded animals are subject to colds," our diagram would have been the one shown at the right and our conclusion would have been "no whales are subject to colds."

      If you have read Alice's Adventures in Wonderland or Through the Looking-Glass , you know that their author, Lewis Carroll, delighted in giving sets of nonsense statements which lead to logical conclusions. One such set is the following:

      1. Babies are illogical;
      2. Nobody is despised who can manage a crocodile;
      3. Illogical persons are despised.

      When these statements are arranged in logical order we have:

        1. Babies are illogical;
        2. Illogical persons are despised;
        3. Nobody is despised who can manage a crocodile.

      From these we can draw the logical conclusion:

        4. Babies cannot manage crocodiles.

      Other sets of statements written by this author follow. To draw a conclusion from each set of statements, first arrange the statements in logical order. A diagram such as those in the preceding column may help you. The correct conclusions are given at the bottom of the page, but do not look at them until you have written your conclusion.

        I.

          1. Everyone who is sane can do Logic;
          2. No lunatics are fit to serve on a jury;
          3. None of your sons can do Logic.

        II.

          1. No ducks waltz;
          2. No officers ever decline to waltz;
          3. All my poultry are ducks.

        III.

          1. No kitten that loves fish is unteachable;
          2. No kitten without a tail will play with a gorilla;
          3. Kittens with whiskers always love fish;
          4. No teachable kitten has green eyes;
          5. No kittens have tails unless they have whiskers.

        IV.

          1. There is no box of mine here that I dare open;
          2. My writing-desk is a box made of rose-wood;
          3. All my boxes are painted except what are here;
          4. There is no box of mine that I dare not open, unless
          it is full of live scorpions;
          5. All my rose-wood boxes are unpainted.


      Conclusions:

        I. None of your sons are fit to serve on the jury.
        II. My poultry are not officers.
        III. No kitten with green eyes will play with a gorilla.
        IV. My writing-desk is full of live scorpions.

      With this brief introduction to Lewis Carroll type problems, you will find it worthwhile and interesting to construct your own problems of this type.


      Fallacies

      (The fun part)

      A fallacy is an error of reasoning. It can be used against you in an argument, but if you are familiar with them, you will be able to refute the fallacious argument. Likewise, if you are clever, you can use them to convince others.

      Fallacies fall into two major categories:

      Fallacies of Relevance
      -- Premises are irrelevant to the conclusion.
      Fallacies of Ambiguity
      -- Ambiguous, changeable wording in the propositions

      Here are examples of each of the major fallacies. You figure out and write in a definition which makes sense to you.


      Fallacies of Relevance

      1. Argumentum ad Bacculum (appeal to force) --
      "Pay back the loan and 10 % daily interest by Thursday, or be sure that you have you hospital insurance paid up."
      2. Argumentum ad Hominem (abusive) --
      "Don't believe anything John says; he's a nerd."
      3. Argumentum ad Hominem (circumstantial) -- "Of course he thinks fraternities are great. He's a Phi Delta."
      4. Argumentum ad Ignorantiam (argument from ignorance) --
      There is no proof that witches exist; therefore, they do not.
      5. Argumentum ad Misericordiam (appeal to pity) --
      "Your honor, how can the prosecution dare try to send this poor, defenseless child to jail for the murder of his father and mother. Have a heart; the boy is now an orphan."
      6. Argumentum ad Populum --
      "Don't be left out! Buy your Chevette today!"
      7. Argumentum ad Vericundiam (appeal to authority) --
      Joe Namath selling pantyhose; Joe DiMaggio selling Mr. Coffee.
      8. Accident --
      "What you bought yesterday, you eat today; you bought raw meat yesterday; therefore, you eat raw meat today."
      9. Converse Accident (hasty generalization) --
      "That man is an alcoholic. Liquor should be banned."
      10. False cause (Post hoc ergo propter hoc) (Many of our superstitions stem from use of this fallacy.) --
      "a black cat crossed Joe's path yesterday, and he died last night." or "Put your money where your mouth is. Whiter teeth and fresh breath will win Susie."
      11. Petitio Principii (begging the question) --
      "It's time to come in the house now, Billy."
      "Why?"
      "Because I said so!"
      "Why?"
      "Because it's time, and I said so."
      12. Complex Question --
      "Have you given up cheating on exams?"
      13. Ignoratio Elenchi (irrelevant conclusion) --
      In a law court, in attempt to prove that the accused is guilty of theft, the prosecution may argue that theft is a horrible crime for anyone to commit.

      Fallacies of Ambiguity

      1. Equivocation --
      Some dogs have fuzzy ears. My dog has fuzzy ears. My dog is some dog!
      2. Amphibole (grammatical construction) --
      "Woman without her man would be lost." or "Save Soap and Waste Paper."
      3. Accent --
      "We should not speak ill of our friends."
      4. Composition--
      "Each part of this stereo weighs under one pound. This is a very light stereo."
      or " ... ONLY $1.97 plus processing and postage."
      5. Division--
      "Purdue is a great engineering school. Mike went there; he must be a great engineer."

      Listen to your roommate, the T.V., and even your profs. You'll be amazed how many fallacies we encounter each day.

      More important, check your papers. Does your argument have premisses and conclusions stated properly? Have you been guilty of fallacious reasoning?


      Exercises 1-11

      (from Copi, Introduction to Logic pp. 85-87)

      Identify the Fallacies in the Following Passages and Explain how each Specific Passage Involves that Fallacy or Fallacies:

      1. It is necessary to confine criminals and to lock up dangerous lunatics. Therefore there is nothing wrong with depriving people of their liberties.

      2. How much longer are you going to waste your time in school when you might be doing a man's work in the world, and contributing to society? If you had any sense of social responsibility, you would leave immediately.

      3. The army is notoriously inefficient, so we cannot expect Major Smith to do an efficient job.

      4. God exists because the Bible tells us so, and we know that what the Bible tells us must be true because it is the revealed word of God.

      5. Congress shouldn't bother to consult the Joint Chiefs-of-Staff about the military appropriations. As members of the armed forces, they will naturally want as much money for military purposes as they think they can get.

      6. Mr. Brown: I will give no more money to your cause next year.
      Solicitor: That's all right, sir, we'll just put you down for the same amount that you gave this year.

      7. When we had got to this point in the argument, and every one saw that the definition of justice had been completely upset, Thrasymachus, instead of replying to me, said:

        "Tell me, Socrates, have you got a nurse?"
        "Why do you ask such a question," I said, "when you ought rather to be answering?"
        "Because she leaves you to snivel, and never wipes your nose: she has not even taught you to know the shepherd from the sheep."

      -- Plato, Republic

      8. Narcotics are habit-forming. Therefore if you allow your physician to ease your pain with an opiate, you will become a hopeless drug addict.

      9. You can't prove that he was to blame for the misfortune, so it must actually have been someone else who was responsible.

      10. You can't park here. I don't care what the sign says. If you don't drive on, I'll give you a ticket.

      11. But lest you think, that my piety has here got the better of my philosophy, I shall support my opinion, if it needs my support, by a very great authority. I might cite all the divines almost, from the foundation of Christianity, who have ever treated of this or any other theological subjects: but I shall confine myself, at present, to one equally celebrated for piety and philosophy. It is Father Malebranche...

      -- David Hume, Dialogues Concerning Natural Religion

      Exercises 16-30

      (from Copi, *Introduction to Logic* pp. 87-88)

      16. Cooks have been preparing food for generations, so our cook must be a real expert.

      17. More young people are attending high schools and colleges than ever before in the history of our nation. But there is more juvenile delinquency than ever before. This makes it clear that to eliminate delinquency among the youth we must abolish the schools.

      18. You say we ought to discuss whether or not to buy a new car now. All right, I agree. Let's discuss the matter. Which should we get, a Ford or a Chevy?

      19. Our nation is a democracy and dedicated to the proposition that all men are created equal. We believe in equality of opportunity for everyone, so our colleges and universities should admit every applicant, regardless of his economic or educational background.

      20. Anyone who deliberately strikes another person should be punished. Therefore the middleweight boxing champion should be severely punished, for he assaults all of his opponents.

      21. We should reject Mr. Watkins' suggestions for increasing the efficiency of our colleges. As a manufacturer he cannot be expected to realize that our aim is to educate the youth, not to make a profit. His recommendations can have no value for us.

      22. Everyone said that the soup had a very distinctive taste, so they must all have found it very tasty.

      23. If we want to know whether a state is brave we must look at its army, not because the soldiers are the only brave people in the community, but because it is only through their conduct that the courage or cowardice of the community can be manifested.

      -- R. L. Nettleship, Lectures on the Republic of Plato

      24. My client is the sole support of his aged parents. If he is sent to prison, it will break their hearts, and they will be left homeless and penniless. You surely cannot find it in your hearts to reach any other verdict than "not guilty."

      25. There is no proof that the secretary "leaked" the news to the papers, so she can't have done it.

      26. Diamonds are seldom found in this country, so you must be careful not to mislay your engagement ring.

      27. Was it through stupidity of through deliberate dishonesty that the Administration has hopelessly botched its foreign policy? In either case, unless you are in favor of stupidity or dishonesty, you should vote against the incumbents.

      28. Since all men are mortal, the human race must some day come to an end.

    Try these for Fun!

    Exercises in Reasoning

    I. Four men, whom we shall call Robert, Ralph, Ronald, and Rudolph, were playing cards one evening. As a result of a quarrel during the course of the game, one of these men shot and killed another. From the facts below determine the murder and the victim.

      1. Rudolph had known Ronald for only five days prior to the murder.
      2. Robert will not expose his brother's guilt.
      3. Rudolph has been released from jail on the day of the murder, after serving a three day sentence.
      4. Ralph met Robert's father only once.
      5. Robert had wheeled Ralph, a cripple, to the card game at Ronald's home.
      6. The host is about to give evidence against the murderer, whom he dislikes.
      7. The murdered man had eaten dinner on the previous evening with one of the men who did not customarily bowl with Ronald.

    II. Five men are in a poker game: Brown, Perkins, Turner, Jones, and Reilly. Their brands of cigarettes are Luckies, Camels, Kools, Old Golds, and Chesterfields, but not necessarily in that order. At the beginning of the game, the number of cigarettes possessed by each player was 20, 15, 8, 6, and 3, but not necessarily in that order.

    During the game, at a certain time when no one was smoking, the following obtained:

      1. Perkins asked for three cards.
      2. Reilly had smoked half of his original supply, or one less than Turner had smoked.
      3. The Chesterfield man originally had as many more, plus half as many more, plus 2 1/2 more cigarettes than he has now.
      4. The man who was drawing to an inside straight could taste only the menthol in his fifth cigarette, the last one he smoked.
      5. The man who smokes Luckies had smoked at least two more than anyone else, including Perkins.
      6. Brown drew as many aces as he originally had cigarettes.
      7. No one had smoked all his cigarettes.
      8. The Camel man asks Jones to pass Brown's matches.

    How many cigarettes did each man have to begin with, and of what brand?

    Improprieties

    A functional impropriety is the use of a word as the wrong part of speech. The wrong meaning for a word can also be impropriety.

    Mark improprieties in the following phrases and correct them in the blanks at the right. If you find none, write C in the blank. Example: (occupation) hazards -- occupational

      1. reforming institution policies

      2. percentaging students by grades

      3. dead trees as inhabitants for birds

      4. an initiate story about a young girl

      5. a recurrence theme in literature

      6. a wood chisel

      7. a wood baseball bat

      8. a frivolity conversation on the weather

      9. a utopia hideaway of alpine villas

      10. a utilize room complete with workbench

      11. the unstabled chemical compounds

      12. the unschooled labor force

      13. the vandals who rapined Rome

      14. an erupting volcano crevassing the hills

      15. criticism writing which is often abstract

      16. abstracted beyond understanding

      17. classified as an absorbent

      18. a handwriting letter

      19. banjoed their way to the top ten

      20. a meander stream

      21. hoboing across the country

      22. holidayed the time away

      23. the redirective coming from the officer

      24. grain-fed slaughter cattle

      25. ivy tendoned to the walls


    By Mark Zegarelli

    When people say “Let’s be logical” about a given situation or problem, they usually mean “Let’s follow these steps:”

    1. Figure out what we know to be true.

    2. Spend some time thinking about it.

    3. Determine the best course of action.

    In logical terms, this three-step process involves building a logical argument. An argument contains a set of premises at the beginning and a conclusion at the end. In many cases, the premises and the conclusion will be linked by a series of intermediate steps. In the following sections, these steps are discussed in the order that you’re likely to encounter them.

    Generating premises

    The premises are the facts of the matter: The statements that you know (or strongly believe) to be true. In many situations, writing down a set of premises is a great first step to problem solving.

    For example, suppose you’re a school board member trying to decide whether to endorse the construction of a new school that would open in September. Everyone is very excited about the project, but you make some phone calls and piece together your facts, or premises.

    Premises:

    • The funds for the project won’t be available until March.
    • The construction company won’t begin work until they receive payment.
    • The entire project will take at least eight months to complete.

    So far, you only have a set of premises. But when you put them together, you’re closer to the final product — your logical argument. In the next section, you’ll discover how to combine the premises together.

    Bridging the gap with intermediate steps

    Sometimes an argument is just a set of premises followed by a conclusion. In many cases, however, an argument also includes intermediate steps that show how the premises lead incrementally to that conclusion.

    Using the school construction example from the previous section, you may want to spell things out like this:

    According to the premises, we won’t be able to pay the construction company until March, so they won’t be done until at least eight months later, which is November. But, school begins in September. Therefore. . .

    The word therefore indicates a conclusion and is the beginning of the final step.

    Forming a conclusion

    The conclusion is the outcome of your argument. If you’ve written the intermediate steps in a clear progression, the conclusion should be fairly obvious. For the school construction example, here it is:

    Conclusion:

    The building won’t be complete before school begins.

    If the conclusion isn’t obvious or doesn’t make sense, something may be wrong with your argument. In some cases, an argument may not be valid. In others, you may have missing premises that you’ll need to add.

    Deciding if the argument is valid

    After you’ve built an argument, you need to be able to decide if it’s valid, which is to say if it’s a good argument.

    To test an argument’s validity, assume that all of the premises are true and then see if the conclusion follows automatically from them. If the conclusion automatically follows, you know it’s a valid argument. If not, the argument is invalid.

    Understanding enthymemes

    The school construction example argument may seem valid, but you also may have a few doubts. For example, if another source of funding became available, the construction company may start earlier and perhaps finish by September. Thus, the argument has a hidden premise called an enthymeme (pronounced EN-thi-meem), as follows:

    There is no other source of funds for the project.

    Logical arguments about real-world situations (in contrast to mathematical or scientific arguments) almost always have enthymemes. So, the clearer you become about the enthymemes hidden in an argument, the better chance you have of making sure your argument is valid.

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