If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Language: Conditionals, Part 1

Justin invites us to think about conditional sentences ("if P then Q"). Perhaps surprisingly, the question of what these sentences mean has vexed philosophers for thousands of years. In part one, Justin motivates the question and introduces one of the oldest answers to it, the material conditional theory. 

Speaker: Dr. Justin Khoo, Assistant Professor, MIT.
Created by Gaurav Vazirani.

Want to join the conversation?

  • leaf green style avatar for user rhouts
    Understood as a "material" conditional, is the following conditional true?

    "If Barack Obama is a Republican, then Barack Obama is a Democrat."
    (9 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user lvkas.keller
      yes. as long as the antecedents is wrong, the material implication/conditional is always right. in contrast to a hypothetical statement or a hypothetical conditional with counterfactual statements in it, the material implication is linked neccesarily to the world.
      (4 votes)
  • leafers ultimate style avatar for user Justin
    I don't understand around when he is filling in the truth table is he filling in the T's and F's randomly?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Estevão Lucio Barbosa
      No, he isn't. Sentence 1 is a conditional, so the third column is a little bit confusing. See:
      1st case: The safety net is broken and he'll fix it - The conditional is true.
      2nd case: The safety net is broken, but he WON'T fix it - He is lying. The conditional is false.
      3rd case: The safety net ISN'T broken, but he'll fix it - The conditional is true, because he'll fix it anyway, broken or not.
      4th case: The safety net ISN'T broken and he WON'T fix it - We don't know if he won't fix it just because it's not broken or because he is lying, but, anyway, the conditional is true.
      (5 votes)
  • piceratops ultimate style avatar for user John
    I look forward to seeing how the material conditional theory breaks down. I'm still not sure why there is such an interest in the meaning of conditional sentences. It seems obvious what these conditional statements mean, or have I missed something? Why a video on it? Why bother with the truth tables?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • aqualine seedling style avatar for user Sabeen
    Is this a biconditional statement? If so how would you write it?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Jp Fallon
      This is not a biconditional statement because the truth table shows that Mitt will fix the safety net even if the net does not need repair. At the same time, the truth table shows Mitt could refuse to repair a broken net.

      The truth values do not map to any logic gates, because the hypotheses are unrelated. These two rows on the truth table make a TxF and an FxT have different conditional values which ruin the biconditional-ness.

      "I will fix the net if and only if it is torn" would be a biconditional statement.

      I am new to this, but this is how I understand it. Fun fact: A butterfly net with a hole in it, has fewer holes in it than if it was whole.
      (1 vote)
  • blobby green style avatar for user candyhiggins10
    I am trying to understand the concept of truth tables. Why are there two T's under the box and two F's. At , he is talking about the safety net is broken. Shouldn't under the safety net column have all T's? Because we are saying it is broken, not if it is broken.
    (2 votes)
    Default Khan Academy avatar avatar for user
  • mr pants purple style avatar for user Sandra Maria
    Unless they copy in the exam they won't pass.
    Why unless is used instead of if ?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • winston default style avatar for user James Dixon
    I agree with below comment.
    (1 vote)
    Default Khan Academy avatar avatar for user

Video transcript

(intro music) My name is Justin Khoo, and I am an assistant professor[br]of philosophy at MIT. Today we are going to[br]look at conditionals, which are a class of sentences that have puzzled philosophers[br]for thousands of years. Here's an example of a[br]conditional sentence from a speech by former presidential[br]candidate, Mitt Romney: "If the safety net needs[br]repair, I will fix it." Conditional sentences, like[br](1), consist of two parts: an antecedent ("the[br]safety net needs repair") and a consequent ("I will fix it"). Our question today is "What do conditional[br]sentences, like (1), mean?" In other words, by uttering this sentence, what has Mitt Romney told us? Here's a way to think about[br]questions of meaning like this. When I say, "The cat is on the mat," I tell you that the cat is on the mat, rather than not on the mat. This is because the meaning of (2) is that the cat is on the mat. Okay, that's pretty easy. What about our conditional sentence (1)? What has Mitt Romney[br]told us by uttering it? One way of figuring out[br]what Romney has told us is to get clear on what[br]he has not told us. He hasn't told us that the[br]safety net needs repair, and he also hasn't told us that[br]he will fix the safety net. Rather, what he said is that[br]there is some connection between the safety net needing[br]repair and his fixing it. But what connection? Here's a simple answer. By saying the sentence (1), Romney has told us that it is not the case that the safety net needs[br]repair and he won't fix it. Equivalently, he said that either the safety net doesn't need repair, or that he will fix it. Let's call this theory the[br]"material conditional theory." Philosophers at least as far back as the Hellenistic philosopher Philo of Megara have been[br]attracted to this theory about what conditionals mean. In order to state the[br]material conditional theory more precisely, we will[br]make use of a device from logic called a "truth table." A truth table is a way of representing how the truth of a complex sentence, in this case, the conditional (1), depends on the truth values of its parts, in this case, the antecedent[br]and consequent of (1). Let's start with a simple[br]example of a conjunction. Take this sentence: "The cat is on the mat,[br]and the cat is fat." Naturally, someone who says[br]this tells you two things: that the cat is on the mat,[br]and that the cat is fat. Thus, the sentence (3) is true if both the cat is on the[br]mat and the cat is fat, and false if either the[br]cat isn't on the mat or the cat isn't fat. We draw this dependence of the truth value of the whole sentence on its parts in our truth table as follows, noting "T" for true when[br]the sentence is true, and "F" for false when[br]the sentence is false. Notice that, since we want to represent how the truth value of (3) depends on the truth values of its parts, the first two columns contain[br]every possible combination of assigning either "T" or[br]"F" to the parts of (3). Furthermore, notice (3) only has a "T" in the row where both[br]of its parts have "T"s. This captures the fact[br]that conjunctions are true only if both of their conjuncts are true, and false otherwise. It also captures the fact that (3) tells us both that the cat is on the mat and that the cat is fat, since that is the only condition[br]under which it is true. Okay, so now what about our[br]conditional sentence (1)? According to the material[br]conditional theory, one tells us that it is not the case that the safety net needs[br]repair and Romney won't fix it. So, in our truth table, we assign "F" to the conditional only on row two, where it is true that the[br]safety net needs repair and false that Romney will fix it. We assign "T" to it on all other rows. This assignment of[br]truth values, therefore, entirely captures the meaning[br]of the conditional (1) according to the material[br]conditional theory. Now, although the material[br]conditional theory has been endorsed by many philosophers, it faces several difficult challenges. You might have noticed that,[br]according to the theory, the sentence (1) is true in all rows besides the second. Does this seem right to you? In the next video, we will[br]explore some challenges facing the material conditional theory, and see how some other theories about what conditionals[br]mean may fare better. Subtitles by the Amara.org community