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1.3: Truth and Its Role in Argumentation - Certainty, Probability, and Monty Hall

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    Only certain sorts of sentences can be used in arguments. We call these sentences propositions, statements or claims.

    Statements or claims have the following characteristics:

    • They are either true or false.
    • They are declarative (that is to say, they are not questions or commands; they are sentences that describe how things are, were, will be, would be, could be or should be.)
    • They are clearly written or stated such that there is no ambiguity as to their meaning (i.e. they don’t have two or more highly distinct interpretations, as in sentences like “I saw the waiter with the glasses”) and they are not so vague as to make it impossible to say under what conditions they would be true.
    • There are lots of ways for a sentence to fail to be a claim, just as there are a lot of way for a sentence to fail to be a question, or description of a dog, or a command.

    Example \(\PageIndex{1}\)

    Which of the following are claims:

    1. All dogs have four legs.
    2. John F. Kennedy was the 35th president of the United States.
    3. Don’t go into Central Park at night.
    4. Why do people always talk on their cell phones on the J train?
    5. Barack Obama is very tall.
    6. There is life on other planets.


    1. Is a claim. It’s false (there are dogs that have lost a leg, thus three-legged dogs.) The fact that it’s false means that it must be a claim. ONLY claims can be false.
    2. Is a claim. It’s true. The fact that it’s true means that it must be a claim. ONLY claims can be true.
    3. Not a claim. It is a command; it is neither true nor false (though it might be good advice.)
    4. Not a claim. It is a question. Questions are never true or false, though sometimes they imply claims.
    5. Not a claim. “Very tall” is too vague. We have no standard, agreed upon method for determining if someone is “very tall.” By Danish standards (where the average male height is 5’11”) Obama is probably not “very” tall. By Vietnamese standards (where the average male is 5’4”) Obama might well be considered “very” tall. We could turn this sentence into a claim by changing it to “President Obama is 6’2” tall.”
    6. This is a claim, but we don’t know if it’s true or not. Still, it’s clearly either true or false, so it must be a claim.

    Subjective and Objective claims

    Claims are either subjective or objective. These words have a special, technical sense in philosophy. A claim is subjective if it is about thoughts, feelings, or other internal states of the mind. A claim is objective if it is about something that is not dependent on a state of the mind. It’s important to keep in mind that you might have used “subjective” and “objective” differently from this, and like all words these have multiple senses. For our purposes, though, we’ll be using subjective and objective only of claims, because only claims are true or false, and here subjective and objective describe something about the truth conditions for a claim.

    evidence, that is, reason to believe it is snowing. But it’s the actual falling snow that makes the claim true, and is the claim’s truth condition. One way to think of this is on the analogy of a court of law. You could be on a jury and hear testimony and see evidence that convinces you that the defendant is guilty of the crime. But your belief that “Mr. Johnson murdered Mr. Ono” comes from the evidence, the truth of the claim comes from Mr. Johnson actually having murdered Mr. Ono. You could, in other words, have compelling evidence but be wrong. In fact, it’s possible that no one will ever know if Mr. Johnson murdered Mr. Ono (perhaps Mr. Johnson blacked out and lost memory of what occurred during Mr. Ono’s death), but the claim is true or false regardless of who believes it. The claims “Mr. Johnson murdered Mr. Ono,” and “it is snowing right now outside my window” are objective claims because their truth conditions are not found in anyone’s thoughts of feelings or mental content; they exist independently of what anyone thinks or feels.

    Whereas, for example, “I'm itchy,” is a subjective claim. “I feel hot,” “John is tired,” “Anita loves Keyshawn,” “Thomas believes in God,” and “The Black Keys are my favorite band,” are all subjective claims. This is because their truth or falsity depends upon what someone thinks or feels, or, we can say, the truth conditions for these claims are found in someone’s mental content. If I say, “I feel nauseated,” that claim is true if, and only if, I actually have the feeling of nausea, and false if, and only if, I do not have that feeling. So if I’m trying to get out of doing something, I might say “I feel nauseated” when I had no such feeling, that is to say, when the truth condition does not exist.

    The following claims are objective: “Dan is six feet tall,” “The Empire state building is made of cheese,” “New York is the largest city in the United States,” and “God exists.” In each of these cases, though I might have a subjective belief about the claim, the actual truth condition is external to my thoughts and feelings.

    Note that the second claim is a false objective claim. The Empire State Building is not made of cheese. But the truth or falsity of the claim is independent of what anyone thinks or feels. It's a fact about the world outside of our minds. Similarly, “God exists” is an objective claim. Some people believe it to be true, some people believe it to be false, but their beliefs do not make the claim true or false any more than one's belief that New York is the largest city in the world would make that claim true. God exists, or fails to exist, whether or not we believe or think that God exists. However, if I said, “I believe that God exists,” that would be a subjective claim. In fact, any objective claim can be turned into a subjective claim by prefixing the words “I think that…” or “I believe that…” to it. That’s because the truth conditions for “I believe there is butter in the refrigerator” are found in my (and only my) beliefs, regardless of whether there is butter in the refrigerator, whereas the claim “there is butter in the refrigerator” is true if, and only if, there is butter in the refrigerator, regardless of what I believe.

    Notably, truth is a very complex philosophical topic, and there are interesting disputes about its nature. But at the basic level, pretty much everyone working on the topic agrees that the claim “there is a dog on my bed,” is true if and only if there is a dog on my bed. That is, there is general agreement about the need for truth conditions (which, minimally, means that there is always some difference between a true claim and a false claim.) For our purposes, then, we’ll divide claims up as subjective or objective depending on the nature of their truth conditions.

    For the following claims, say if they are subjective or objective:

    1. There are over 1200 species of beetles in the world today.
    2. The Yankees will win the World Series in 2034.
    3. Alissa's head hurts.
    4. I'm tired of hearing about the economy.
    5. There is no God but Allah and Mohammed is his prophet.
    6. There are over 9 billion people living in Brooklyn.

    Answers: 1 is an objective claim. 2 is also objective: though it refers to a future event, it's not the case that our thoughts or feelings can make it true or false; we just have to wait to see if it's true or false. Its truth conditions will be independent of thought or feeling. Some hold that it is temporarily neither true nor false; most philosophers, though, hold that claims about the future are true or false but that the truth conditions are simply placed at a different point in time from the claim’s utterance. 3 is subjective: it refers to a feeling that Alissa has. 4 is subjective; it refers to a feeling or thought had by the speaker. 5 is objective: many people believe it to be true, many others do not, but it's true if and only if there is, in fact, one God, that God's name is Allah, and Mohammed is the prophet of that God. My thoughts or feelings on this cannot alter its truth value. 6 is objective: there are not, in fact, 9 billion people living in Brooklyn, and we can ascertain that by counting, looking at the census, or just noting the impossibility of getting 9 billion people into the existing housing in Brooklyn.

    Note: we distinguish subjective from objective claims to aid in argumentation and conversation.

    Generally, we have to be very careful about giving subjective premises for an objective conclusion. “I feel like God exists” or “I feel like Sarmatians are sneaky people” are probably not good premises for the conclusions “God exists” or “Sarmatians are sneaky people.

    It’s also important to understand that, if someone makes an objective claim, we can’t respond with “that’s true for you but not for me.” Objective claims, by their nature, are not true relative to some person. An objective claim can be false, but it can’t be simply relative to a person’s beliefs—if it is, it’s not an objective claim. Further, just because a claim is controversial does not make it subjective! Most of the truly controversial claims are objective. We don’t develop a lot of controversy over claims like “I feel tired,” but there is a great deal of debate over claims like “There is only one God and He is the creator of the world.”

    Finally, when someone makes a subjective claim but states it as though it were an objective claim, this can cause needless disagreement. If Tammy says “Beyonce is the best singer in the world,” she probably just means that Beyonce is her favorite singer. If Lamar responds with “no way, Taylor Swift is the best singer!” they could be on the verge of a pointless disagreement. There is no standard for “best singer in the world,” so there’s no settling this by argument. Instead, recognizing that Tammy was actually making a subjective claim, Lamar might ask, “really, what do you like about Beyonce’s singing?”

    Probability, Certainty, and Monty Hall7

    People often say they know things “for certain,” but they’re certainly wrong. Certainty has a connotation that means there is no doubt: you are absolutely, 100% positive that your claim is true. Skepticism is a theory that claims certainty and truth are impossibilities, and while we can have very good justified beliefs and claims backed by solid reasoning and evidence, nothing is ever certain. If you’re not a skeptic, then finding certainty is possible, but very hard. What do you think you would know for certain? In this class, the closest we will get to uncontroversial certainty certain logical deductions, like inferences and proofs. Proofs are aimed at telling us something for certain, while inferences are simply following something simple we know and figuring out what else it tells us. For example, if I say that “No bananas are underwear” (and we assume it’s true), then we also know that “No underwear are bananas.” Or if I say that “If you eat a banana, then you will sit on its peel, and you ate a banana…” what else could I say? That you’re sitting on its peel. If we assume that the first two claims (about eating bananas and sitting on peels) are true, then the conclusion must follow certainly. We’ll cover these concepts more later on the course, but for now, keep in mind that almost nothing is technically certain. If we’re not certain about things, then what can we say? Just that they’re more or less likely. And that’s where probability comes in to help us.

    Probability is how likely something is to happen.

    Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

    Tossing a Coin

    When a coin is tossed, there are two possible outcomes:

    • heads (H) or
    • tails (T)

    We say that the probability of the coin landing H is ½

    And the probability of the coin landing T is ½

    Throwing Dice

    When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

    The probability of any one of them is 1 in 6

    In general:

    Probability of an event happening = Number of ways it can happen divided by Total number of outcomes

    Example: the chances of rolling a "4" with a die

    Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

    Total number of outcomes: 6 (there are 6 faces altogether)

    So the probability = 1 in 6

    Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

    Number of ways it can happen: 4 (there are 4 blues)

    Total number of outcomes: 5 (there are 5 marbles in total)

    So the probability = 4 in 5 = 0.8 (or 80%)

    Probability is Just a Guide

    Probability does not tell us exactly what will happen, it is just a guide

    Example: toss a coin 100 times, how many Heads will come up?

    Probability says that heads have a ½ chance, so we can expect 50 Heads.

    But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.


    Some words have special meaning in Probability:

    Experiment or Trial: an action where the result is uncertain.

    Tossing a coin, throwing dice, seeing what pizza people choose are all examples of experiments.

    Example: choosing a card from a deck

    There are 52 cards in a deck (not including Jokers)

    So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... }

    Sample Point: just one of the possible outcomes

    Example: Deck of Cards

    • the 5 of Clubs is a sample point
    • the King of Hearts is a sample point

    "King" is not a sample point. As there are 4 Kings that is 4 different sample points.

    Event: a single result of an experiment

    Example Events:

    • Getting a Tail when tossing a coin is an event
    • Rolling a "5" is an event.

    An event can include one or more possible outcomes:

    • Choosing a "King" from a deck of cards (any of the 4 Kings) is an event
    • Rolling an "even number" (2, 4 or 6) is also an event


    • The Sample Space is all possible outcomes.
    • A Sample Point is just one possible outcome.
    • And an Event can be one or more of the possible outcomes.

    Hey, let's use those words, so you get used to them:

    Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

    Each time Alex throws the 2 dice is an Experiment.

    It is an Experiment because the result is uncertain.

    The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points:

    {1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

    The Sample Space is all possible outcomes (36 Sample Points):

    {1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6}

    These are Alex's Results:


    Is it a Double?











    After 100 Experiments, Alex has 19 "double" Events ... is that close to what you would expect?

    Monty Hall

    While we think we have a good understanding of how chance and probability work, our instincts often mislead us. A prime example of problems we have in understanding and identifying probabilities and certainties is “The Monty Hall problem.”

    The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:

    Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

    Is it to your advantage to switch? Does it matter? Think about what you would do and why. Try it out yourself in a little experiment where you use cards to represent the goats and car. Pretend to be the contestant and host by picking a door and then either switching or staying with your door. Try sticking with your pick the first 10 times and then switching the next 10 times (remember that Monty Hall will ALWAYS show you a goat when he reveals a door). Was there any difference in outcome? Why do you think you got these results? You can look online for further discussions of this “puzzle.”

    1.3: Truth and Its Role in Argumentation - Certainty, Probability, and Monty Hall is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Noah Levin.