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6.2: Recognizing the Main Connective

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  • Suppose you are asked to provide a derivation which shows the following argument to be valid:

    (1) (A⊃B)&[C≡(A⊃B)]
    (2) C

    The premise is a mess! How do you determine which rule applies to it? After your labor with some of the exercises in the last chapter, you probably can see that the key lies in recognizing the main connective. Even if you got all of those exercises right, the issue is so important that it's worth going over from the beginning.

    Let's present the issue more generally. When 1 stated the rules of inference, I expressed them in general terms, using boldface capital letters, 'X' and 'Y'. For example, the rule for &E is

    (3) | X&Y and | X&Y
    | X | Y

    The idea is that whenever one encounters a sentence of the form X&Y in a derivation, one is licensed to write either the sentence X or the sentence Y (or both on separate lines) further down. Focus on the fact that this is so whatever sentences X and Y might be. This is the point of using boldface capital letters in the presentation. 'X' and 'Y' don't stand for any particular sentences. Rather, the idea is that if you write any sentence you want for 'X' and any sentence you want for 'Y', you will have a correct instance of the rule for &E. This is what I mean by saying that I have expressed the rule "in general terms" and by talking about a sentence "of the form X&Y".

    How will these facts help you to deal with sentence (I)? Here's the technique you should use if a sentence such as (1) confuses you. Ask yourself: "How do I build this sentence up from its parts?" You will be particularly interested in the very last step in putting (1) together from its parts. In this last step you take the sentence

    (4) A⊃B which you can think of as X

    and the sentence

    (5) C=(A⊃B) which you can think of as Y

    and put them on either side of an '&' to get the sentence

    (A⊃B)&[C≡(A⊃B)] which has the form X&Y

    You have just established that (1) has the form of X&Y; that is, it is a conjunction with sentences (4) and (5) as its conjuncts. Consequently, you know that the rule for &E, (3), applies to sentence (l), so that if (1) appears in a derivation you are licensed to write sentence (4) or (5) (or both) below.

    Similarly, if in a derivation you are faced with the sentence

    (6) C≡(A⊃B)

    ask yourself "What is the last thing I do to build this sentence up from its parts?" You take the sentence

    (7) C which you can think of as X

    and the sentence

    (8) A⊃B which you can think of as Y

    and you put them on either side of a biconditional, '≡', giving

    (9) C≡(A⊃B) which thus has the form X≡Y

    Consequently, if you find sentence (6), you can apply the rule of inference for ≡E:

    | X≡Y and | X≡Y
    | X⊃Y | Y⊃X

    which, when we put in sentences (7) and (8) for X and Y, look like

    | C≡(A⊃B) | C≡(A⊃B)
    | C⊃(A⊃B) ≡E and | (A⊃B)⊃C ≡E

    Thus ≡E applies to sentence (6), licensing us to follow (6) on a derivation either with the sentence 'C⊃(A⊃B)' , or the sentence '(A⊃B)⊃C', or both on separate lines.

    In a moment we will apply what we have just done to provide a derivation which shows how (2) follows from (1). But we will need to see how to treat one more compound sentence. This time, try to figure it out for yourself. What is the form of the sentence '(A⊃B)⊃C'?

    The last thing you do in putting this one together from its parts is to put '(A⊃B)' and 'C' on either side of a conditional, '⊃'. So the sentence has the form X⊃Y, with 'A⊃B' as X and 'C' as Y. If we have 'A⊃B' as well as '(A⊃B)⊃C' in a derivation, we can apply ⊃E to the two to derive 'C'.

    Perhaps you can now see how we can write a derivation which shows (2) to follow from (1). In this case, because the desired objective, 'C', is atomic, we can't get it by an introduction rule. So it is useless to try to work backward from the end. The reductio ad absurdum strategy could be made to work, but only by doing more indirectly what we're going to have to do anyway. In this case the best strategy is to use elimination rules to take the premise apart into simpler pieces.

    When we think through what these pieces will look like, we will see that they provide just what we need to derive 'C'. &E applies to the premise. '(A⊃B)&[C≡(A⊃B)]' to give us 'A⊃B' and 'C≡(A⊃B)'. In turn, ≡E applies to 'C≡(A⊃B)' to give us '(A⊃B)⊃C', which, together with the 'A⊃B', immediately gives us 'C' by ⊃E. (≡E applied to 'C=(A⊃B)' also gives 'C⊃(A⊃B)'. But we have no use for 'C⊃(A⊃B)', so although licensed to write it down as a conclusion, we don't bother.) Altogether, the completed derivation looks like this:

    1| (A⊃B)&[C≡(A⊃B)] P
    2| A⊃B 1, &E
    3 | C≡(A⊃B) 1, &E
    4 | (A⊃B)⊃C 3, ≡E
    5 | C 2, 4, ⊃E

    The, key idea you need to untangle a sentence such as (1) is that of a main connective:

    The Main Connective in a sentence is the connective which was used last in building up the sentence from its component or components.

    (A negated sentence, such as '~(Av~B)', has just one component, 'Av~B' in this case. All other connectives use two components in forming a sentence.) Once you see the main connective, you will immediately spot the component or components to which it has been applied (so to speak, the X and the Y), and then you can easily determine which rules of inference apply to the sentence in question.

    Let's practice with a few examples:

    (AvB)⊃~(B&D) AvB and ~(B&D)
    [(A⊃B)v~D)v(~A≡D) v (A⊃B)v~D and ~A≡D
    ~[(A⊃B)⊃~(B⊃~A)] ~ (A⊃B)⊃~(B⊃~A)

    The second and third examples illustrate another fact to which you must pay attention. In the second example, the main connective is a 'v'. But which occurrence of 'v'? Notice that the sentence uses two 'v's, and not both occurrences count as the main connective! Clearly, it is the second occurrence, the one used to put together the components '(A⊃B)v~D' and '~A≡D', to which we must pay attention. Strictly speaking, it is an occurrence of a connective which counts as the main connective. It is the occurrence last used in putting the sentence together from its parts. In the third example, '~' occurs three times! Which occurrence counts as the main connective? The very first.

    In the following exercises you will practice picking out the main connective of a sentence and determining which rule of inference applies. But let's pause first to say more generally how this works:

    The elimination rule for '&' applies to a sentence only when an '&' occurs as the sentence's main connective. The same thing goes for 'v', '⊃', and ≡'. The components used with the main connective are the components to which the elimination rule makes reference.

    The elimination rule for '~' applies only to a doubly negated sentence, ~~X; that is, only when '~' is the sentence's main connective, and the '~' is applied to a component, ~X, which itself has a '~' as its main connective.

    The introduction rule for '&' licenses you to write as a conclusion a sentence, the main connective of which is '&'. The same thing goes for 'v', '⊃', '≡', and '~'

    Exercise 6-2.

    Give derivations which establish the validity of the following arguments:

    a) (AvB)&(AvB)⊃CJ b) A
    C (AvB)[(A⊃K)&(B⊃K)]

    c) [A⊃(Dv~B)]&{[A⊃(Dv~B)]⊃(B⊃A)}

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