Some Notes on Exercise 1.3

Solutions for all of these are given in Allen/Hand. Here are a few notes on how those solutions are arrived at.

FINDING THE MAIN CONNECTIVE. When English sentences contain more than one connective, the trick to translating them into our formal language is determining which connective is the main connective, that is, the one that operates on the largest pieces (or piece) of the sentence. Our formal language is so defined that every non-atomic sentence has one and only one main connective. In English, it's sometimes not quite so nice. However, if you can succeed in finding the main one, all you have to do is translate it, then translate the components it operates on. If those components are atomic, then all you need to to is assign them letters. If they're non-atomic, then you translate them in exactly the same way. To use a word I've used before, this process is recursive: the process of looking for the main connective in one of the components is just like the process of looking for it in the whole sentence. It's also guaranteed to end, eventually, since every time you translate a connective, you wind up with smaller components; eventually, you will reach components with no connectives at all, at which point you're done.

Here's an illustration. Consider the sentence

If it's raining but it's not pouring, then unless it's Thursday, this is Harlingen.

A good first step in translating this is to go through and mark all the connectives:

If it's raining but it's not pouring, then unless it's Thursday, this is Harlingen.

Now, which one of these is the main connective? A test that often works is this: if you take out anything that's not the main connective and divide the sentence at that point, the pieces left behind are not (both) complete sentences. To illustrate, here's what you get if you split the above example at 'but':

If it's raining

it's not pouring, then unless it's Thursday, this is Harlingen.

Neither of these is a coherent sentence, and so 'but' is not the main connective of the example. What about 'unless'? We get:

If it's raining but it's not pouring, then

and

it's Thursday, this is Harlingen.

The first of these obviously isn't a coherent sentence, and the second is at least a little rocky. What, then, about 'if'?

Before we look at the result, here's a little note about 'if'. Frequently, 'if' is associated with 'then' to form a two-piece connective. You need to regard the 'then' as part of an 'if...then', not as a separate connective, in such cases. To apply our test, what we do is delete the entire 'if...then' pair, to get:

it's raining but it's not pouring,

and

unless it's Thursday, this is Harlingen.

Notice that these are two perfectly coherent sentences? That tells you that the main connective here is 'if...then'. Now, 'if A then B' translates into 'A->B'. Let's do that much translation here:

(it's raining but it's not pouring) -> (unless it's Thursday, this is Harlingen)

To continue translating this, we translate each of the parts in the same way.

(it's raining but it's not pouring)

becomes

(it's raining) & (it's not pouring)

And

(it's not pouring)

becomes

~(it's pouring)

So, the whole left side is

(it's raining & ~(it's pouring)

Now, let's turn to the right hand side. This has 'unless' out front. It's frequently easier if we rearrange the sentence to put the second component after 'unless' in front of it:

(This is Harlingenunless it's Thursday)

And we can translate that as

(This is Harlingen) v ( it's Thursday)

So, the whole sentence looks like this:

(it's raining & ~(it's pouring)->((this is Harlingen) v ( it's Thursday))

Just turn the atomic sentences into letters and this becomes:

((P&~Q)->(RvS))

(Of course, you can drop unnecessary parentheses from this.)

Some notes on Exercise 1.3

1. John is dancing but Mary is not dancing.

P&~Q. This is quite straightforward.

2. If John does not dance, then Mary will not be happy.

~P->~T. Note that the 'if' has a 'then' with it.

3. John's dancing is sufficient to make Mary happy.

P->T. Also straightforward.

4. John's dancing is necessary to make Mary happy.

T->P. Watch this one: the order of antecedent and consequent is vital.

5. John will not dance unless Mary is happy.

~TvP. Also straightforward.

6. If John's dancing is necessary for Mary to be happy, Bill will be happy.

(T->P)->~U). Finding the main connective is a little tricky here. Note that you could add 'then' where the comma is without changing the meaning of the sentence. Rewrite it like that, and analyzing is straightforward.

7. If Mary dances although John is not happy, Bill will dance.

(Q&~S)->R. Same note as before.

8. If neither John nor Bill is dancing, Mary is not happy.

~(PvR)->~T. Notice that 'neither...nor' also functions as a single unit. This is an important help in finding the main connective. Also notice that this is partly abbreviated: 'neither John nor Bill is dancing' is a short way of saying 'neither John is dancing nor Bill is dancing.'

9. Mary is not happy unless either John or Bill is dancing.

~Tv(PvR). Notice again that 'either...or' is a single unit.

10. Mary will be happy if both John and Bill dance.

(P&R)->T. Abbreviation again: 'both John and Bill dance' = 'Both John dances and Bill dances.'

11. Although neither John nor Bill is dancing, Mary is happy.

T&~(PvR). Here, the parts were rearranged to put 'although' in the middle.

12. If Bill dances, then if Mary dances John will too.

R->(Q->P). The structure is 'If (Bill dances), then (if Mary dances John will too)'. An 'if' with a 'then' often provides clues about what the main connective is. Notice also that 'John will too' is shorthand for John will dance.'

13. Mary will be happy only if Bill is happy.

T->U. Straightforward. Notice that 'only if' translates into '->' and keeps the antecedent and consequent in the same order. AND REMEMBER THAT 'ONLY IF' IS NOT THE SAME THING AS 'IF'.

14. Neither John nor Bill will dance if Mary is not happy.

~T->~(PvR). The structure is '(Neither John nor Bill will dance) if (Mary is not happy),' and that's translatable as '(Mary is not happy)->(neither John nor Bill will dance)'. AS I WAS JUST SAYING, 'IF' AND 'ONLY IF' ARE NOT THE SAME CREATURE.

15. If Mary dances only if Bill dances and John dances only if Mary dances, then John dances only if Bill dances.

(Q->R)&(P->Q)->(P->R). This is a real exercise for finding the main connective. It helps to add back some unnecessary parentheses: ((Q->R)&(P->Q))->(P->R). The main connective is 'if...then': the first 'if' goes with the 'then'.

16. Mary will dance if John or Bill but not both dance.

(PvR)&~(P&R)->Q. Well, I don't know whether I think this example is quite English. I would have preferred something like 'Mary will dance if John or Bill dances, but not both.' However, the main connective is 'if': (Mary will dance) if (John or Bill but not both dance) is the structure. So, that becomes: (John or Bill but not both dance)->(Mary will dance). The cumbersome 'John or Bill but not both dance' might be rewritten as '(John or Bill dances) but (John and Bill do not both dance).' John or Bill dances' = 'John dances or Bill dances.' 'John and Bill do not both dance' = 'not (John dances and Bill dances)'; the 'not both' construction is a way of indicating that the 'not' governs the entire conjunction.

17. If John dances and so does Mary, but Bill does not, then Mary will not be happy but John and Bill will.

(P&Q)&~R->(~T&(S&U)). Here's a really good example on which to try out tests for the main connective. Adding a word or two to spell out abbreviations, and marking off the main connective and its components, we get:
If (John dances and Mary dances, but Bill does not dance), then (Mary will
not be happy but John will be happy and Bill will be happy)
That is:
(John dances and Mary dances, but Bill does not dance)->(Mary
will not be happy but John will be happy and Bill will be happy)
From here on out, it's easy.

18. Mary will be happy if and only if John is happy.

T<->S. That's what double-arrows are for. AND NOTICE THAT 'IF AND ONLY IF' IS NOT THE SAME THING EITHER AS 'IF' OR AS 'ONLY IF'.

19. Provided that Bill is unhappy, John will not dance unless Mary is dancing.

~U->(~PvQ). Just be careful here to notice the order of antecedent and consequent.

20. If John dances on the condition that if he dances Mary dances, then he dances.

((P->Q)->P)->P. Sounds hard to follow, but the rules suggested for finding the main connective work very well. The connectives are as marked here:
If John dances on the condition that if he dances Mary dances, then he dances
The only trick is telling which 'if' goes with the 'then.' But if you take out the second 'if' and the 'then,' you get these three pieces
If John dances on the condition that
he dances Mary dances
and
he dances
This is gibberish. However, if you take out the first 'if' and the 'then', you get:
John dances on the condition that if he
dances Mary dances,
and
he dances
And those are perfectly coherent sentences. So, the structure is
If (John dances on the condition that if he
dances Mary dances,) then (he dances)
which is:
(John dances on the condition that if he
dances Mary dances,) -> (he dances)
Analyzing the left side, we get:
(if (he dances) (Mary dances))->(John dances)
That is, ((P->Q)->P)
So, the whole thing is ((P->Q)->P)->P.

21. If a purpose of punishment is deterrence and capital punishment is an effective deterrent, then capital punishment should be continued.

Structure:
If ((a purpose of punishment is deterrence) and (capital punishment is an effective deterrent)), then (capital punishment should be continued).
Adding connectives:
((a purpose of punishment is deterrence) & (capital punishment is an effective deterrent)) -> (capital punishment should be continued).
Replacing atomic constituents and deleting one pair of parentheses:
(P&Q) -> R

22. Capital punishment is not an effective deterrent although it is used in the United States.

Structure (and spelling out a little abbreviation):
(Capital punishment is not an effective deterrent) although (capital punishment is used in the United States).
Substituting connectives:
~(Capital punishment is an effective deterrent) & (capital punishment is used in the United States).What we get:
~Q&S

23. Capital punishment should not be continued if it is not an effective deterrent, unless deterrence is not a purpose of punishment.

Structure:
((Capital punishment should not be continued) if (capital punishment is not an effective deterrent), unless (deterrence is not a purpose of punishment).
Part of the way there:
(~(Capital punishment should be continued) if ~(capital punishment is an effective deterrent), v ~(deterrence is a purpose of punishment).
That is to say:
(~(capital punishment is an effective deterrent) -> ~(Capital punishment should be continued) v ~(deterrence is a purpose of punishment).
NOTE THE ORDER HERE!!
(~Q -> ~R) v ~P

24. If retribution is a purpose of punishment but deterrence isn't, then capital punishment should not be continued.

Structure:
If ((retribution is a purpose of punishment) but (deterrence isn't a purpose of punishment)), then (capital punishment should not be continued).
((retribution is a purpose of punishment) & ~(deterrence is a purpose of punishment)) -> ~(capital punishment should be continued).
(T&~P) -> ~R

25. Capital punishment should be continued even though capital punishment is not an effective deterrent, provided that a purpose of punishment is retribution in addition to deterrence.

Structure, and a little rewriting into something not quite English:((Capital punishment should be continued) even though (capital punishment
is not an effective deterrent)), provided that ((a purpose of punishment is
retribution) in addition to (a purpose of punishment is deterrence)).
((Capital punishment should be continued) & ~(capital punishment
is an effective deterrent)), provided that ((a purpose of punishment is
retribution) & (a purpose of punishment is deterrence)).
So (watching the order again--careful with 'provided that'!):
(T&P) -> (R & ~Q)