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Language: Conditionals, Part 3

In part 3 of the series on conditionals, Justin picks up where part 2 leaves off, introducing an alternative theory of conditionals: the strict conditional theory. According to the strict theory, conditionals express necessary connections between their antecedent and consequent. Justin shows how this theory avoids the problems facing the material conditional theory. However, the strict theory turns out to face a similar problem of its own!

Speaker: Dr. Justin Khoo, Assitant Professor, M.I.T.

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  • male robot hal style avatar for user davis
    There is a way that 2+2=4 is false. If we are working modulo 3, then we find that 2+2=1, and not 4, because 4 exists only as 1. I think this is exposing a hole in "must be true", that lots of things come extremely close to being necessarily true and truths that don't have means of being made false under special circumstances.
    Is this anomaly isolated? Are we allowed to take something with an arbitrarily small probability of being false as being true?
    (1 vote)
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    • ohnoes default style avatar for user Tejas
      The definition of 4 is 1+1+1+1. The definition of 2 is 1+1. So 2+2 is literally 1+1+1+1 which is by definition 4. If we were to try and add that we are working in mod 3, then 2+2 would still be 4, but 4 mod 3 would equal 1. But that is not the real problem.

      The problem is that when he says that 2+2=4, we are not in mod 3. Mathematical notation requires that the modulus be stated. So it would be true that 2+2=4, but if we wanted to work in modulo 3, then we need to state that, because otherwise we are not.
      (3 votes)
  • male robot hal style avatar for user Corey Piper
    On the note when he says that i could easily be watching something else and so it is a contingent fact, doesn't it really boil to a debate about freewill vs fate? I am on the side of fate in the sense that it is in fact impossible that I could've watched something else. If I do decide to click on something else it is not true that it was also possible for me to continue watching. It's racking my head! What is the definition of possibility? "a thing that may happen or be the case" but how can you know for sure it was ever a possibility? How can you be sure it may have happened otherwise? I have the power to switch between videos at will, yes, but the moment i do it seems to eliminate all other "possibilities" and retains the unchangeable and nothing else "can be the case"
    (1 vote)
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    • ohnoes default style avatar for user Tejas
      This is still not a freewill-fate argument. No matter how far you push the causes of you watching the video back, they remain contingent. For instance, I could say that it is a contingent fact that the laws of the universe are such that you would be led to watch the video. In fact, it is a contingent fact that the universe exists at all. However, 2+2 always equals 4. There is no way for it not to be, regardless of the laws of the universe, or anything else. Logic dictates that 2+2 must equal 4, and it is not contingent on anything that might be true.
      (1 vote)

Video transcript

(intro music) Hello, my name is Justin Khoo and I'm an assistant professor[br]of philosophy at MIT. This is the third part of[br]our series on conditionals. Last time, we saw a classic problem with the material conditional theory. It predicts that it's just too easy for conditionals to be true. Recall Mitt Romney's conditional, (1): "If the safety net needs repair, "I will fix it." The material conditional theory predicts that Romney says something[br]true in uttering this sentence if in fact the safety[br]net does not need repair. But this seems wrong. For it to be true, the conditional requires[br]that there be some connection between its antecedent (that[br]the safety net needs repair) and its consequent (that[br]Romney will fix it). But how should we[br]understand this connection between the antecedent and[br]consequent of a conditional? One promising theory, due[br]to Charles Sanders Peirce, is that conditionals tell us about more than how things actually are. They tell us about how[br]they must or have to be. Peirce's theory is known as the "strict conditional[br]theory of conditionals." According to this theory,[br]a conditional "If A, then B" is true just if it is[br]impossible for "A" to be true and "B" to be false. Thus, on this theory, (1) means that it is impossible for[br]the safety net to need repair and Romney not fix it. So when Romney utters one, he tells us something both[br]about how things actually are, as well as about how they must be. The words that I used to characterize Pierce's strict conditional[br]theory are known as "modals." "Must," "have to," "impossible," "possible," and so on. These words are used to express claims about how things could have gone, even if they didn't actually go that way. Here's an example. As a matter of fact, you are watching a video on conditionals. But you could have been[br]doing something else, maybe watching a video about cats or having a chat with friends. Since you could've done something else, it's merely a contingent fact that you are now watching[br]a video on conditionals. It's true, but didn't have to be true. It could've easily been false, say if you had decided to click the button for the cat video instead. Contrast the fact that you're now watching[br]a video on conditionals with the fact that two[br]plus two equals four. This is also true, but there is a strong sense[br]in which it has to be true. There's just no way that two[br]plus two could equal five. That's impossible. Now, remember that what's[br]intuitively missing from the material conditional theory is a special connection between[br]antecedent and consequent. According to the strict[br]conditional theory, the special connection is the[br]necessity of the consequent given the assumption of its antecedent. Consider Romney's conditional, (1), again. According to the strict[br]conditional theory, (1) says that it is necessary that Romney will fix the safety net on the assumption that it needs repair. Or, as we put it earlier, in no way can it be that[br]the safety net needs repair and Romney not fix it. Thus, according to the[br]strict conditional theory, (1) tells us that there is[br]a very strong connection between the safety net needing repair and Romney fixing it. Unlike the material conditional theory, it being false that the[br]safety net needs repair is not enough to ensure that[br]the conditional is true. With that in mind, let's return to one of the sentences which we found problematic for the material conditional theory, (2): "If the Earth is flat, I[br]will win the lottery tomorrow." According to the material[br]conditional theory, this sentence is true simply because the Earth isn't flat. But that seems wrong. For (2) to be true, there must be some connection between the Earth being flat[br]and my winning the lottery. Since there is no such connection, that's why it's false, so we think. According to the strict[br]conditional theory, (2) says that it's impossible for the Earth to be flat[br]and I not win the lottery. But intuitively it is possible for the Earth to be flat and I[br]not win the lottery tomorrow. That's because there's[br]no connection at all between flat Earths[br]and my future winnings, so the strict conditional[br]theory predicts, correctly, that (2) is false. It seems then that the[br]strict conditional theory marks a serious advance over the material conditional theory. That's some good news. However, now for the bad news. It turns out that the[br]strict conditional theory is subject to a very similar problem as the material conditional theory. The problem is that it seems like there could be false conditionals that have impossible antecedents. The reason this is a problem for the strict conditional[br]theory is that it says that a conditional "If A, then B" is[br]true just if it is impossible for "A" to be true and "B" to be false. Therefore, if "A" is itself impossible, then of course it's[br]impossible for "A" to be true and "B" to be false, in[br]which case the theory says that "If A, then B" must be true. So, what sort of conditional[br]raises this kind of problem for the strict conditional theory? Consider Goldbach's Conjecture. This is the claim that every[br]even integer greater than two is equal to the sum of two prime numbers. This is a mathematical claim, and hence if it is true,[br]it is necessarily true, and if false, necessarily false. Now, we don't know whether[br]Goldbach's conjecture is true. That's what makes it a conjecture. But suppose for a moment that it is false, and consider the following conditional: "If Goldbach's conjecture is true, "I will win the lottery tomorrow." This conditional seems false, since my winning the lottery tomorrow has nothing whatsoever to do with Goldbach's conjecture[br]being true or false. However, given our assumption that Goldbach's conjecture is false, then it is impossible that[br]Goldbach's conjecture be true, and therefore it is impossible for Goldbach's conjecture be true and I not win the lottery tomorrow. Therefore, according to the[br]strict conditional theory, (3) is true. But this prediction is wrong. (3) is intuitively false. The reason is, again, because[br]there's just no connection between Goldbach's conjecture being true and my winning the lottery tomorrow. Furthermore, we arrive at this conclusion regardless of whether we assume that Goldbach's conjecture[br]is true or false. I leave it as an exercise to the listener to run the argument. Notice the parallel between this problem for the strict conditional theory and the problem facing the[br]material conditional theory. Both theories ran into[br]trouble with conditionals which have false or[br]impossible antecedents. This common core to their problems suggests that perhaps we need a radical new way of[br]thinking about conditionals, on which they are not categorical claims that have well-defined truth values in situations where their[br]antecedents are false. This insight will form the basis of the last theory of conditionals that we'll consider in the[br]final video in this series. Subtitles by the Amara.org community