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9.1: Recognizing Inconsistency and Contradiction

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    22006
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    The topic of inconsistency is at the heart of logic. If you say, "Everyone left the room," and I say, "She is someone who is still in the room," then I've said something inconsistent with what you've said. Noticing an inconsistency is a wake-up call to resolve the conflict. One or both of the conflicting claims must fail to be true.

    Because the study of inconsistency requires you to know what the words "true" and "truth" mean, it might help you to have a definition. Here it is: The truth is a lie that hasn't been found out. I got that definition from my favorite intelligence service (spy organization).

    Just kidding. A truth is a statement of fact, but it is too basic to define.

    A group of statements is inconsistent if it’s not possible for them all be true. What does the word possible mean here? It means something like conceivable or imaginable, assuming words mean what they normally mean.1 A group of sentences (even a group the size of one) that is not inconsistent is consistent. There is no middle ground between consistent and inconsistent.

    Even two false statements can be consistent with each other. These are consistent:

    Abraham Lincoln is my mother.
    Abraham Lincoln is your mother.

    The two are consistent with each other, but not with the facts, such as the fact that Lincoln isn’t the mother of either of us.

    Resolving an inconsistency can be at the heart of deep issues. Theologians recognize that they have a burden of resolving the apparent inconsistency between divine foreknowledge and human free will. Some philosophers of religion argue that the two are inconsistent because God knows what you are going to do, so you are not free to do otherwise than the way God has foreseen. Yet presumably the ability to do otherwise than you do is the essence of your free will. If there's an inconsistency, then you can't have it both ways. Other philosophers of religion say there is no inconsistency, but we won't go further into this thicket of dispute.

    Inconsistency between what we expect and what we get is at the heart of many jokes. Here are some examples:

    "I didn't attend the funeral, but I sent a nice letter saying I approved of it." -- Mark Twain
    "I feel so miserable without you, it's almost like having you here." -- Stephen Bishop

    Let me tell you a story. It is about the second time Candace lost her virginity. While she was on a bridge crossing the stream, walking up the lane toward her was a tall man with a dog.... By now you are suspicious of what I am saying because you were alert to the fact that this remark is inconsistent with our commonsense knowledge that people can lose their virginity only once.

    We have now discussed some different kinds of inconsistencies. They can be put into categories (intellectual boxes). There are logical inconsistencies in which the very meaning of the words requires one of the claims to be false. Example: {Everyone left the room. She is someone who is still in the room.}

    There are inconsistencies with our expectations as in Mark Twain's joke about approving of the funeral.

    There are inconsistencies with facts as when we say she lost her virginity twice. Any false statement is logically inconsistent with the facts.

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    Are these two sentences (or statements) logically inconsistent?

    Almost everyone in the room is an Arab.
    He's in the room, but he's no Arab.

    No, they are consistent. You can image a situation in which they are both true. If you were to change "Almost everyone" to "Everyone," then they'd be inconsistent.

    The notion of logical inconsistency can get more complicated. These two statements can be said to both logically consistent and logically inconsistent:

    Everybody left the room.
    John is still in the room.

    They are inconsistent with the assumption that John is a person, but they aren't consistent as presented, because John could be a teddy bear in the room. However, if you made these two statements to people without them knowing John was a teddy bear, then you'd be tricking them and violating the normal rules of conversation which say that ordinary names of people refer to people and not to other objects unless you say otherwise.

    So, the moral about the complication is that consistency questions can depend crucially on what else you are assuming. To explore this complication a bit more, consider the relationship between these two statements.

    Abraham Lincoln is currently the president of the United States.

    Abraham Lincoln is a Sumo wrestler.

    Would you say the two are

    a. consistent
    b. inconsistent
    c. none of the above

    circle of eighteen standing Sumo wrestlers

    You can't tell whether the answer is a or b. Neither of the two sentences are true. Each one alone is factually inconsistent or inconsistent with the facts, but they are not logically inconsistent with each other and so are logically consistent. If "b" means "factually inconsistent," then the answer is b. If "b" means "logically inconsistent," then the answer is not b. People are notoriously ambiguous when they ask about inconsistency.

    Another way to describe inconsistency is to say that two or more statements are inconsistent with each other if they couldn't all be true. Now the ambiguity is embedded in what the word "could" means. Does it mean "could" as far as the meaning of the words are concerned, or "could" where it is assumed that we are comparing them to all the facts and are not allowed to change any of the current facts of the world? Here's a way to make the point.

    Could eggs grow naturally on trees? They couldn't if they have to obey the laws of biology, but they could so far as what those words mean. That is, the sentence "Eggs could grow naturally on trees" violates biology but not grammar. So, we say the sentence is factually inconsistent but not logically inconsistent.

    The statement that Abraham Lincoln is your mother could be true but in fact is false. Here we are using "could" in the sense of possible so far as grammar and meaning are concerned.

    More on that word "could." Most false statements (sentences) could be true, as far as grammar or meaning is concerned. Similarly, most true statements could be false. But there are exceptions. Here's one. The statement "If it's raining and cold, then it's cold" is true, but it could not be false. Statements like this that can't be false without violating what words mean are said to be analytically true. The statement, "7 + 5 = 13" is analytically false. The statement that there are more than 13 chickens on Earth is true but not analytically true.

    As you deal with problems of consistency in real life, you want to be alert to what people mean rather than just to what they say. For example, suppose Jack says, "Nobody got an A on that test, but she did. Wow, is she smart." What Jack said literally was self-contradictory. If you called him on it, Jack would probably say not to take him so literally because what he really meant was "Nobody (other than her) got an A on that test." What he meant is not self-contradictory. So, to get what Jack intends, you need to overlook his inconsistency.

    Are these three sentences consistent?

    Lincoln is taller than Jones.
    Jones is taller than Shorty.
    Shorty is taller than Lincoln

    The three are logically inconsistent with each other. Understanding this inconsistency is all part of understanding the term "taller than." If a person couldn't see that the three sentences were inconsistent, we'd have to wonder whether they really understood what "taller than" meant.

    Very often, people will use the terms "inconsistency" and "contradiction" as synonyms, but technically they aren't synonyms. A contradiction between two statements is a stronger kind of inconsistency between them. If two sentences are contradictory, then one must be true and one must be false, but if they are inconsistent, then both could be false. Do the following two statements contradict each other?

    The house is all green.
    The house is not all green.

    Yes, these two contradict each other; one of the two must be true and the other must be false. This is so for any house. Do the following two statements contradict each other?

    The house is all green.
    The house is all blue.

    No, both could be false; the house might be white. So, the two statements do not contradict each other, although they are logically inconsistent with each other. This inconsistency is the weaker kind of inconsistency that we call being contrary.

    When you leave the logic classroom and go out onto the street, you'll find that people use our technical terms "contradiction," "inconsistent," and "contrary" in a sloppy manner; sometimes the three terms are meant to be synonyms. Few people are careful to distinguish factual inconsistency from logical inconsistency. So, you have to be alert to this and try to get at what they mean rather than just what they say.

    Exercise \(\PageIndex{1}\)

    Are these two sentences consistent or inconsistent with each other?

    Serena is not taller than Carlos.
    Carlos is not taller than Serena.

    Answer

    This pair is consistent because it is possible that they are both true. They are true in a situation where Samantha and Carlos are the same height. Even if you know that Carlos really is four inches taller we still call the pair logically consistent because it is possible, as far as the meanings of the words are concerned, that there is a situation in which they are the same height.

    Here is a more difficult question to answer. Are the following two statements inconsistent?

    Venice was running in the Boston Marathon at 8 a.m. today.
    Venice was having breakfast at Bob's Restaurant at 8 a.m. today.

    Not quite. Maybe she stopped for breakfast during the marathon.

    Exercise \(\PageIndex{1}\)

    Seal of the Vice President of the United States

    Consider this consistent list of statements:

    i. The president admires the first lady.
    ii. The first lady also admires the president.
    iii. Everybody else admires the president, too.

    These statements are logically consistent. Label the following sentences as being consistent or inconsistent with the above list:

    a. Everybody but the admiral admires the first lady.
    b. The admiral admires the first lady but not the president.
    c. The president admires other people besides the first lady.
    d. The vice-president does not admire the first lady.
    e. The first lady does not admire the vice-president.

    Answer

    (b) is inconsistent with the original three on the list. Each of the others, separately, is consistent with the original three.

    Statements can even be made with body language. A man could say, "Sure, sure, I believe you" as he lifts his eyebrows and rolls his eyes. In doing so, his actions contradict what he says.

    Exercise \(\PageIndex{1}\)

    Are these two sentences inconsistent?

    All real televisions are appliances.
    Some real televisions are appliances.

    Answer

    There might or might not be an inconsistency here because “some” is ambiguous in English. If “some” is meant in the sense of "at least one but definitely not all,” there is a logical inconsistency. But if “some” means "at least one and possibly more," then there is no inconsistency. Because “some” could be meant either way here, you cannot tell whether an inconsistency exists. Speakers who intend to imply with their word “some” that some are and some aren't should stick in the word “only” and say "Only some of the real televisions are appliances." From now on in this book we will make the assumption that “some” means simply “at least one but possibly more.”


    1 When we say it’s not imaginable, we mean we cannot imagine it unless we allow words to change their meanings in mid-sentence or mid-passage—which we do not allow for purposes of assessing possibility. If we were to permit language to go on holiday this way with no restrictions on equivocation, there would never be any inconsistency.


    This page titled 9.1: Recognizing Inconsistency and Contradiction is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Bradley H. Dowden.

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