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5.4: The Existential Elimination and Universal Introduction Rules: Background in lnformal Argument

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  • Now let's go to work on the two harder rules. To understand these rules, it is especially important to see how they are motivated. Let us begin by looking at some examples of informal deductive arguments which present the kind of reasoning which our new rules will make exact. Let's start with this argument:

    Everyone likes either rock music or country-western.
    Someone does not like rock.
    Someone likes country-western.

    Perhaps this example is not quite as trivial as our previous examples. How can we see that the conclusion follows from the premises? We commonly argue in the following way. We are given the premise that someone does not like rock. To facilitate our argument, let us suppose that this person (or one of them if there are more than one) is called Doe. (Since I don't know this person's name, I'm using 'Doe' as the police do when they book a man with an unknown name as 'John Doe.') Now, since according to the first premise, everyone likes either rock or country-western, this must be true, in particular, of Doe. That is, either Doe likes rock, or he or she likes country-western. But we had already agreed that Doe does not like rock. So Doe must like country-western. Finally, since Doe likes country-western, we see that someone likes country-western. But that was just the conclusion we were trying to derive.

    What you need to focus on in this example is how I used the name 'Doe'. The second premise gives me the assumption that someone does not like rock. So that I can talk about this someone, I give him or her a name: 'Doe'. I don't know anything more that applies to just this person, but I do have a fact, the first premise, which applies to everyone. So I can use this fact in arguing about Doe, even though I really don't know who Doe is. I use this general fact to conclude that Doe, whoever he or she' might be, does like country-western. Finally, before I am done, I acknowledge that I really don't know who Doe is, in essence by saying: Whoever this person Doe might be, I know that he or she likes country-western. That is, what I really can conclude is that there is someone who likes country-western.

    Now let's compare this argument with another:

    (1) Everyone either likes rock or country-western.
    (2) Anyone who likes country-western likes soft music.
    (3) Anyone who doesn't like rock likes soft music.

    This time I have deliberately chosen an example which might not be completely obvious so that you can see the pattern of reasoning doing its work.

    The two premises say something about absolutely everyone. But it's hard to argue about 'everyone'. So let us think of an arbitrary example of a person, named 'Arb', to whom these premises will then apply. My strategy is to carry the argument forward in application to this arbitrarily chosen individual. I have made up the name 'Arb' to emphasize the fact that I have chosen this person (and likewise the name) perfectly arbitrarily. We could just as well have chosen any person named by any name.

    To begin the argument, the first premise tells us that

    (4) Either Arb likes rock, or Arb likes country-western.

    The second premise tells us that

    (5) If Arb does like country-western, then Arb likes soft music.

    Now, let us make a further assumption about Arb:

    (6) (Further Assumption): Arb doesn't like rock.

    From (6) and (4), it follows that

    (7) Arb likes country-western.

    And from (7) and (5), it follows that

    (8) Arb likes soft music.

    Altogether we see that Arb's liking soft music, (8), follows from the further assumption, (6), with the help of the original premises (1) and (2) (as applied through this application to Arb, in (4) and (5)). Consequently, from the original premises it follows that

    (9) If Arb doesn't like rock, then Arb likes soft music.

    All this is old hat. Now comes the new step. The whole argument to this point has been conducted in terms of the person, Arb. But Arb could have been anyone, or equally, we could have conducted the argument with the name of anyone at all. So the argument is perfectly general. What (9) says about Arb will be true of anyone. That is, we can legitimately conclude that

    (3) Anyone who doesn't like rock likes soft music.

    which is exactly the conclusion we were trying to reach.

    We have now seen two arguments which use "stand-in" names, that is, names that are somehow doing the work of "someone" or of "anyone". Insofar as both arguments use stand-in names, they seem to be similar. But they are importantly different, and understanding our new rules turns on understanding how the two arguments are different. In the second argument, Arb could be anyone-absolutely anyone at all. But in the first argument, Doe could not be anyone. Doe could only be the person, or one of the people, who does not like rock. 'Doe' is "partially arbitrary" because we are careful not to assume anything we don't know about Doe. But we do know that Doe is a rock hater and so is not just anyone at all. Arb, however, could have been anyone.

    We must be very careful not to conflate these two ways of using stand in names in arguments. Watch what happens if you do conflate the ways:

    Someone does not like rock.
    Everyone does not like rock. (Invalid)

    The argument is just silly. But confusing the two functions of stand-in names could seem to legitimate the argument, if one were to argue as follows: Someone does not like rock. Let's call this person 'Arb'. So Arb does not like rock. But Arb could be anyone, so everyone does not like rock. In such a simple case, no one is going to blunder in this way. But in more complicated arguments it can happen easily.

    To avoid this kind of mistake, we must find some way to clearly mark the difference between the two kinds of argument. I have tried to bring out the distinction by using one kind of stand-in name, 'Doe', when we are talking about the existence of some particular person, and another kind of stand-in name, 'Arb', when we are talking about absolutely any arbitrary individual. This device works well in explaining that a stand-in name can function in two very different ways. Unfortunately, we cannot incorporate this device in natural deduction in a straightforward way simply by using two different kinds of names to do the two different jobs.

    Let me try to explain the problem. (You don't need to understand the problem in detail right now; detailed understanding will come later. All you need at this point is just a glimmer of what the problem is.) At the beginning of a derivation a name can be arbitrary. But then we might start a subderivation in which the name occurs, and although arbitrary from the point of view of the outer derivation, the name might not be arbitrary from the point of view of the subderivation. This can happen because in the original derivation nothing special, such as hating rock, is assumed about the individual. But inside the subderivation we might make such a further assumption about the individual. While the further assumption is in effect, the name is not arbitrary, although it can become arbitrary again when we discharge the further assumption of the subderivation. In fact, exactly these things happened in our last example. If, while the further assumption (6) was in effect, I had tried to generalize on statements about Arb, saying that what was true of Arb was true of anyone, I could have drawn all sorts of crazy conclusions. Look back at the example and see if you can figure out for yourself what some of these conclusions might be.

    Natural deduction has the job of accurately representing valid reasoning which uses stand-in names, but in a way which won't allow the sort of mistake or confusion I have been pointing out. Because the confusion can be subtle, the natural deduction rules are a little complicated. The better you understand what I have said in this section, the quicker you will grasp the natural deduction rules which set all this straight.


    5-3. For each of the two different uses of stand-in names discussed in this section, give a valid argument of your own, expressed in English, which illustrates the use.

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