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8.3: Three Shortcuts

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  • In general, it is very dangerous to do two or more steps at the same time, omitting explicitly to write down one or more steps which the rules require. When you fail to write down all the steps, it becomes too easy to make mistakes and too hard to find mistakes once you do make them. Also, omitting steps makes it extremely hard for anyone to correct your papers. However, there are three step-skipping shortcuts which are sufficiently clear-cut that, once you are proficient, you may safely use. You should begin to use these shortcuts only if and when your instructor says it is alright to do so.

    Suppose you encounter the sentence -(Vx)(Vy)Lxy on a tree. The rules as I have given them require you to proceed as follows: J1 -(Vx)(Vy)Lxy J2 (3x1-(Vy)Lxy 1, -V J3 -(Vy)Lay 2, 3 J4 Qy)-Lay 3, -V 5 -Lab 4, 3, New name

    Now look at line 2. You may be tempted to apply the rule -V inside the sentence of line 2. In most cases, applying a rule inside a sentence is disastrous. For example, if you should try to instantiate '(Vx)Bx' inside ' -[(Vx)Bx v A]', you will make hash of your answer. But in the special case of the rule for negated quantifiers, one can justify such internal application of the rule.

    In the example we have started, an internal application of the rule -V gives the following first three lines of the tree:

    In fact, we can sensibly skip line 2 and simply "push" the negation sign through both quantifiers, changing them both. Our tree now looks like this:

    We can do the same with two consecutive existential quantifiers or a mixture of quantifiers:

    Indeed, if a sentence is the negation of a triply quantified sentence, you could push the negation sign through all three quantifiers, changing each quantifier as you go.

    Why is this shortcut justified? To give the reason in a very sketchy way, the subsentences to which we apply the negated quantifier rule are logically equivalent to the sentences which result from applying the rule. In short, we are appealing to the substitution of logical equivalents. To make all this rigorous actually takes a little bit of work (for the reasons explained in exercise 34, and I will leave such niceties to your instructor or to your work on logic in a future class.

    Here is another shortcut: Suppose you have a multiple universally quantified sentence, such as '(Vx)(Vy)Lxy', on a tree that already has several names, say, 'a' and 'b'. Following the rules explicitly and instantiating with all the names is going to take a lot of writing:

    a, b, 1 (Vx)(tly)Lxy a, b, 2 Wy)Lay 1, V 3 Laa 2,V 4 Lab 2, V a, b, 5 Wy)Lby 1, V 6 Lba 5, V 7 Lbb 5, V

    In general, it is not a good idea to skip steps, because if we need to look for mistakes it is often hard to reconstruct what steps we skipped. But we won't get into trouble if we skip steps 2 and 5 in the above tree:

    1 Wx)Wy)Lxy (a, a), (a, b), (b, a), (a, b) 2 Laa 1, V, V 3 Lab 1, V, V 4 Lba 1, V, V 5 Lbb 1, V, V

    (In noting on line 1 what names I have used in instantiating the doubly universally quantified sentence '(Vx)(Vy)Lxy', I have written down the pain of names I have used, being careful to distinguish the order in which they occurred, and I wrote them to the right of the line simply because I did not have room on the left.)

    In fact, if you think you can get all branches to close by writing down just - some of the lines 2 to 5, write down only what you think you will need. But if in doing so you do not get all branches to close, you must be sure to come back and write down the instances you did not include on all open branches on which line 1 occurs.

    What about using the same shortcut for a doubly existentially quantified sentence? That is, is it all right to proceed as in this mini-tree?

    1 (3x)(3y)Lxy 2 Lab 1, 3, 3, New names

    This is acceptable if you are very sure that the names you use to instantiate both existential quantifiers are both new names, that is, names that have not yet appeared anywhere on the branch.

    Our last shortcut does not really save much work, but everyone is tempted by it, and it is perfectly legitimate: You may drop double negations anywhere they occur, as main connectives or within sentences. This step is fully justified by the law of substitution of logical equivalents from sentence logic.

    A final reminder: You may use these shortcuts only if and when your instructor judges that your proficiency is sufficient to allow you to use them safely. Also, do not try to omit other steps. Other shortcuts are too likely to lead you into errors.

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