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5.2: Subderivations

Many of you have probably been thinking: So far, we have an "introduction" and an "elimination" rule for disjunction and just an "elimination" rule for the conditional. I bet that by the time we're done we will have exactly one introduction and one elimination rule for each connective. That's exactly right. Our next job is to present the introduction rule for the conditional, which involves a new idea.

   How can we license a conclusion of the form X⊃Y? Although we could do this in many ways, we want to stick closely to argument forms from everyday life. And most commonly we establish a conclusion of the form X⊃Y by presenting an argument with X as the premise and Y as the conclusion. For example, I might be trying to convince you that if Adam loves Eve, then Adam will marry Eve. I could do this by starting from the assumption that Adam loves Eve and arguing, on that assumption, that matrimony will ensue. Altogether, I will not have shown that Adam and Eve will get married, because in my argument I used the unargued assumption that Adam loves Eve. But I will have shown that if Adam loves Eve, then Adam will marry Eve.

   Let's fill out this example a bit. Suppose that you are willing to grant, as premises, that if Adam loves Eve, Adam will propose to Eve ('A⊃B'), and that if Adam proposes, marriage will ensue ('B⊃C'). But neither you nor I have any idea whether or not Adam does love Eve (whether 'A' is true). For the sake of argument, let's add to our premises the temporary assumption, 'A', which says that Adam loves Eve, and see what follows. Assuming 'A', that Adam loves Eve, we can conclude 'B' which says that Adam will propose (by conditional elimination, since we have as a premise 'A⊃B', that if Adam loves Eve, he will propose). And from the conclusion 'B', that Adam will propose, we can further conclude 'C', that marriage will ensue (again by conditional elimination, this time appealing to the premise 'B⊃C', that proposal will be followed by marriage). So, on the temporary assumption 'A', that Adam loves Eve, we can conclude 'C', that marriage will ensue. But the assumption was only temporary. We are not at all sure that it is true, and we just wanted to see what would follow from it. So we need to discharge the temporary assumption, that is, restate what we can conclude from our permanent premises without making the temporary assumption. What is this? Simply 'A⊃C', that if Adam loves Eve, marriage will ensue.

   Presenting this example in English takes a lot of words, but the idea is in fact quite simple. Again, we badly need a streamlined means of representing what is going on. In outline, we have shown that we can establish a conditional of the form X⊃Y not on the basis of some premises (or not from premises alone), but on the strength of an argument. We need to write down the argument we used, and, after the whole argument, write down the sentence which the argument establishes. We do it like this:

3 |       | A     
4 |       | A⊃B 
5 |       | B⊃C 
6 |       | B       3, 4, ⊃E
7 |       | V       5, 6, ⊃E
   |       | 
8 | A⊃C           3-7, Conditional Introduction (31)

   For right now, don't worry about where lines 4 and 5 came from. Focus on the idea that lines 3 through 7 constitute an entire argument, which we call a Subderivation, and the conclusion on line 8 follows from the fact that we have validly derived 'C' from 'A'. A subderivation is always an integral part of a larger, or Outer Derivation. Now you can see why I have been using the vertical scope lines. We must keep outer derivations and subderivations separated. A continuous line to the left of a series of sentences indicates to you what pieces hang together as a derivation. A derivation may have premises, conclusions, and subderivations, which are fullfledged derivations in their own right.

   A subderivation can provide the justification for a new line in the outer derivation. For the other rules we have learned, a new line was justified by applying a rule to one or two prior lines. Our new rule, conditional introduction (>I), justifies a new line, 8 in our example, by appealing to a whole subderivation, 3-7 in our example. When a rule applies to two prior lines, we list the line numbers separated by commas-in the example line 6 is licensed by applying ⊃E to lines 3 and 4. But when we justify a new line (8 in our example) by applying a rule (here, ⊃I) to a whole subderivation, we cite the whole subderivation by writing down its inclusive lines numbers (3-7 in our example).

   Now, where did lines 4 and 5 come from in the example, and why did I start numbering lines with 3? I am trying to represent the informal example about Adam and Eve, which started with the real premises that if Adam loves Eve, Adam will propose (A⊃B), and that if Adam proposes, they will marry (B⊃C). These are premises in the original, outer derivation, and I am free to use them anywhere in the following argument, including in any subderivation which forms part of the main argument. Thus the whole derivation looks like this:

1 | A⊃B               P
2 | B⊃C               P
3 |        | A          Assumption (A)
4 |        | A⊃B      1, Reiteration (R)
5 |        | B⊃C      2, Reiteration (R)
6 |        | B          3, 4, ⊃E
7 |        | C          5, 6, ⊃E
8 |  A⊃C              3-7, Conditional Introduction (⊃1)

   I am licensed to enter lines 4 and 5 in the subderivation by the rule:

Reiteration: If a sentence occurs, either as a premise or as a conclusion in a derivation, that sentence may be copied (reiterated) in any of that derivation's lower subderivations, or lower down in the same derivation.

In the present example, 'A⊃B' and 'B⊃C' are assumed as premises of the whole argument, which means that everything that is supposed to follow is shown to be true only on the assumption that these original premises are true. Thus we are free to assume the truth of the original premises anywhere in our total argument. Furthermore, if we have already shown that something follows from our original premises, this conclusion will be true whenever the original premises are true. Thus, in any following subderivation, we are free to use any conclusions already drawn.

   At last I can give you the full statement of what got us started on this long example: the rule of Conditional Introduction. We have been looking only at a very special example. The same line of thought applies whatever the details of the subderivation. In the following schematic presentation, what you see in the box is what you must have in order to apply the rule of conditional introduction. You are licensed to apply the rule when you see something which has the form of what is in the box. What you see in the circle is the conclusion which the rule licenses you to draw.

Conditional Introduction 

In words: If you have, as part of an outer derivation, a subderivation with assumption X and final conclusion Y, then X⊃Y may be entered below the subderivation as a further conclusion of the outer derivation. The subderivation may use any previous premise or conclusion of the outer derivation, entering these with the reiteration rule.

   You will have noticed that the initial sentences being assumed in an outer, or main, derivation get called "premises," while the initially assumed sentence in a subderivation gets called an "assumption." This is because the point of introducing premises and assumptions is slightly different. While we are arguing, we appeal to premises and assumptions in exactly the same way. But premise? always stay there. The final conclusion of the outer derivation is guaranteed to be true only in those cases in which the premises are true. But an assumption introduced in a subderivation gets Discharged.

   This is just a new word for what we have been illustrating. The point of the subderivation, beginning with assumption X and ending with final conclusion Y, is to establish X⊃Y as part of the outer derivation. Once the conclusion, X⊃Y, has been established and the subderivation has been ended, we say that the assumption, X, has been discharged. In other words, the scope line which marks the subderivation signals that we may use the subderivation's special assumption only within that subderivation. Once we have ended the subderivation (indicated with the small stroke at the bottom of the vertical line), we are not, in the outer derivation, subject to the restriction that is assumed to be true. If the premises of the original derivation are true, X⊃Y will be true whether X is true or not.

   It's very important that you understand why this last statement is correct, for understanding this amounts to understanding why the rule for conditional introduction works. Before reading on, see if you can state for yourself why, if the premises of the original derivation are true, and there is a subderivation from X as assumption to Y as conclusion, X⊃Y will be true whether or not X is true.

   The key is the truth table definition of X⊃Y. If X is false, X⊃Y is, by definition, true, whatever the truth value of Y. So we only have to worry about cases in which X is true. If X is true, then for X⊃Y to be true, we need Y to be true also. But this is just what the subderivation shows: that for cases in which X is true, Y is also true. Of course, if the subderivation used premises from the outer derivation or used conclusions that followed from those premises, the subderivation only shows that in all cases in which X and the original premises are true, Y will also be true. But then we have shown that X⊃Y is true, not in absolutely all cases, but in at least those cases in which the original premises are true. But that's just right, since we are entering X⊃Y as a conclusion of the outer derivation, subject to the truth of the original premises.

Exercise 

5-2. Again, for each of the following arguments, provide a derivation which shows the argument to be valid. Be sure to number and annotate each step to show its justification. All of these exercises will 68 Natural Lkdvcfia for Smtmce Logic require you to use conditional introduction and possibly other of the rules you have already learned. You may find the use of conditional introduction difficult until 'you get accustomed to it. If so, don't be alarmed, we're going to work on it a lot. For these problems you will find the following strategy very helpful: If the final conclusion which you are trying to derive (the "target conclusion") is a conditional, set up a subderivation which has as its assumption the antecedent of the target conclusion. That is, start your outer derivation by listing the initial premises. Then start a subderivation with the target conclusion's antecedent as its assumption. Then reiterate your original premises in the subderivation and use them, together with the subderivation's assumptions, to derive the consequent of the target conclusion. If you succeed in doing this, the rule of conditional introduction licenses drawing the target conclusion as your final conclusion of the outer derivation. 

a) A⊃B        b) NvP           c) B             d) ~B            e) K⊃~D
    B⊃C            ~N⊃P           A⊃B             (BvC)⊃C        DvH      
    C⊃D                                                                       K⊃H
    A⊃D

f) A⊃B          g) F⊃(CvM)   h) (DvB)⊃J    i) A⊃K
   A⊃(BvC)      ~C                D⊃J             (KvP)⊃L
                     F⊃M                                 A⊃L

 j) Q⊃~S       k) P
    Q⊃(SvF)        (~DvK)⊃B
    Q⊃F              P⊃(Kv~F)         
                        (Fv~D)⊃(Bv~P)  

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