Exercise 3.3 - Answers

1

(i)

We will use the following interpretation:

 

A: Aston are going to be relegated

S: Aston will beat Sutton

W: Aston will beat Weston

L: Weston will lose to Sutton

B: Sutton are a better side than Weston

G: Sutton are at least as good as Aston

 

The conclusion is contained in the first sentence. Leaving out unnecessary verbiage it is: A

The next sentence can be paraphrased as: They will avoid this fate (i.e. they will not be relegated) only if they beat both Sutton and Weston: [A[SW]]. (Note that the sentence does not say that they will avoid the fate if they do beat Weston and Sutton.)

The next sentence can be formalised as: [WL] (or, equivalently as [LW]). Note that we can omit "and even then they are not certain of beating them". The only contribution this makes to the argument is to make it clear that the first part is not saying, "They won't beat Weston unless Weston lose to Sutton, in which case they will beat them."

In the next sentence "Sutton will beat Weston" is equivalent to "Weston will lose to Sutton". So the sentence can be formalised as: [L[BG]].

The last sentence can be paraphrased as, "If Sutton are at least as good as Aston, they won't lose to Aston"; i.e.: [GS].

So the whole argument becomes:

[A[SW]], [WL], [L[BG]], [GS] So, A.

The corresponding (syntactic) sequent is:

[A[SW]], [WL], [L[BG]], [GS] |- A

(We can show that this is a correct sequent by doing a tableau for its counterexample set. We can also see that it is correct by arguing like this. Suppose the premises are true and the conclusion is false. Then A (the negation of the conclusion) must be true. So, given the first premise, [SW] must be true; so S and W must be both true. If S is true, given the last premise, G must be false. And if W is true, given the second premise, L must be true. But if L is true, given the third premise, B and G must both be true. But we have already seen that G must be false; so this is impossible. So it cannot be that the premises are true and the conclusion false. So, given the completeness of the tableau system, the sequent is a correct sequent.)

 
(ii)

We will use the following interpretation:

 

J: John doesn't know much about gardening

C: John will succeed in growing carrots

M: Mary will help John

G: John will get the seeds to germinate

P: John will protect the plants from the carrot fly.

F: John might succeed in getting the seeds to germinate by good fortune.

 

The conclusion is, "John will succeed in growing carrots only if Mary helps him": [CM]. (It should not be a . The sentence says "only if", not "if and only if".) This is indicated by the "Since", which also marks out "John doesn't know much about gardening" as one of the premises. So the first premise is: J.

 

In the second sentence we can treat "grow them to maturity" as meaning the same as "succeed in growing carrots". So the second premise is: [C[GP]].

 

We can't do much about paraphrasing "he might succeed in the former by good fortune" in terms of anything else. Of course, "succeed in the former" means the same as "get the seeds to germinate", but we aren't even told that if he has good fortune he will get the seeds to germinate, only that this might be the case. We can, however, paraphrase "though" in terms of "and", putting any difference of meaning down to difference in implicature. And we can treat "without Mary's help, he is bound to fail in the latter" as equivalent to "if Mary does not help John, John will not protect the plants from the carrot fly". So the whole of the third sentence becomes: [F[MP]].

 

So the whole argument is:

 

J, [C[GP]], [F[MP]] So, [CM]

 

And the corresponding (syntactic) sequent is:

 

J, [C[GP]], [F[MP]] |- [CM]

 

In fact the validity of the argument does not depend on the first premise at all. And (not surprisingly) "John might succeed in getting the seeds to germinate by good fortune", which we have abbreviated as "F", contributes nothing either. So, the sequent which corresponds to the relevant part of the argument is just:

[C[GP]], [MP] |- [CM]

Indeed we might excise the "G" too, leaving ourselves just with:

[CP], [MP] |- [CM].

(This can be shown to be correct by doing a tableau for its counterexample set. We can also see that it is correct by arguing like this. Suppose the premises are true and the conclusion is false. Then C must be true and M false (to make the conclusion false). But if C is true, given first premise, P must be true. And if M is false, given the second premise, P must be true; so P can't be true. So it cannot be that the premises are true and the conclusion false. So, given the completeness of the tableau system, the sequent is a correct sequent.)

 
(iii)

We will use the following interpretation:

 

I: There is an improvement in the orchestra's standards

C: There will be a change in conductor

P: The procedure for selecting orchestral managers will be changed.

M: There will be a major administrative exercise

N: An injection of new funds will be required

G: There will be a substantial grant from central government

 

The conclusion comes at the end. It can be formalised as: [IG]. (We might, perhaps, have used G to abbreviate "A substantial grant from central government will be essential" instead of "There will be a substantial grant from central government".)

The first premise is: [CI]. (Or, [IC]; or, [IC].)

We will treat the next sentence down to ";" as the second premise though we could have gone on to the full stop, treating the "but" as "and". We will treat "A new conductor is unthinkable" as close enough for our purposes to "There will not be a new conductor". (In fact it is rather stronger than that, but we shall be safe to translate it with something weaker here.) The sentence can now be formalised as: [PC]. (Or [CP]; or [CP].)

We can formalise the next premise as: [[PM][MN]].

And the final premise is: [NG].

So the argument is:

[CI], [PC], [[PM][MN]], [NG] So, [IG]

And the corresponding syntactic sequent is:

[CI], [PC], [[PM][MN]], [NG] |- [IG].

 

(This can be shown to be correct by doing a tableau for its counterexample set. We can also see that it is correct by arguing like this. Suppose the premises are true and the conclusion is false. Then I must be true and G false (to make the conclusion false). But if I is true, given first premise, C must be true. And if C is true, given the second premise, P must be true. But then given the third premise, [PM] is true, so M must be. But also, given the third premise, [MN] is true, so N is. Finally, given the last premise, if N is true, G must be. But, to make the conclusion false, G had to be false. So it cannot be that the premises are true and the conclusion false. So, given the completeness of the tableau system, the sequent is a correct sequent.)

 
(iv)

We will use the following interpretation:

 

P: The Prime Minister will intervene

C: The Chancellor will cut taxes before the General Election.

R: There will be a run on the pound

I: The Government will increase the national debt.

 

The conclusion is, "unless the Government increase the national debt, there will be a run on the pound", which we can formalise as: [IR]. (Or as [IR]). Do not be tempted to think that the second sentence is equivalent to "Unless the Government increase the national debt, there will be a run on the pound, but if they do, there won't be."

The first premise can be formalised as: [PC]. (Or, [PC].) Again one should be wary of treating this premise as saying, "Unless the Prime Minister intervenes, the Chancellor will cut taxes, but if he does intervene the Chancellor won't". This may be implied, but it is surely a matter of implicature, rather than logical implication.

In the second premise "he does" can evidently be paraphrased as "the Chancellor cuts taxes before the General Election". So the premise becomes: [C[RI]]. (Should the "or" be treated as exclusive? We have no reason to suppose so.)

The third premise is: P.

So the whole argument becomes:

[PC], [C[RI]], P So, [IR]

And the corresponding syntactic sequent is:

[PC], [C[RI]], P |- [IR]

(This can be shown to be correct by doing a tableau for its counterexample set. We can also see that it is correct by arguing like this. Suppose the premises are true. Then, given the first and third premises, C must be true. But then, given the second premise, [RI] must be true. So the conclusion must be true. So, given the completeness of the tableau system, the sequent is a correct sequent.)

What if, in spite of the warning not to be tempted, we had treated the conclusion as equivalent to "Unless the Government increase the national debt, there will be a run on the pound, but if they do, there won't be." It would then have been, (e.g.) [IR]. (To treat it like this, is to treat "unless" as if it were equivalent to an exclusive "or".) Then the resulting sequent would not have been correct unless we had treated the "or" in the second premise as exclusive too, in which case the case the sequent would have been (say):

[PC], [C[RI]], P |- [IR].

And that sequent is correct, because [RI] is equivalent to [IR].

 
(v)

We will use the following interpretation:

 

F: There will be a fight at the club tonight

J: John will be at the club

H: Henry will be at the club

A: John will be accompanied by Jenny

G: Jenny will go with Henry

 

The conclusion is "There will be no fight at the club tonight": F (This is signalled by the "because".)

The first premise follows the "because". We can formalise it as: [F[JH]].

In the next premise we will need to treat "John will not be allowed at the club" as equivalent to "John will not be at the club", if we want to end up with a correct sequent. It isn't, in fact, equivalent it would be possible for "John will not be at the club" to be true, even though it was not true that he was not allowed there. However, if he is not allowed there, presumably he will not be there. So the liberty we shall be taking is not very great. (We might have been asked to formalise the argument as a correct sequent, supplying an extra premise. In that case we would have used as our extra premise, "If John is not allowed into the club, John will not be at the club".) So the premise will become: [[JA][AG]]. (Or [[AJ][AG]].) (Notice that we have not treated "John will not be allowed into the club unless accompanied by Jenny" as being equivalent to "John will not be allowed into the club unless accompanied by Jenny, in which case he will.")

In the final premise we can omit the "I know", which serves merely to say that what follows is true. Presumably "Henry won't go" means "Henry won't go to the club"; and "Henry will go with Jenny" is equivalent to "Jenny will go with Henry". So we can formalise the premise as: [HG].

The argument is:

[F[JH]], [[JA][AG]], [HG] So, F

And the corresponding syntactic sequent is:

[F[JH]], [[JA][AG]], [HG] |- F

(This can be shown to be correct by doing a tableau for its counterexample set. We can also see that it is correct by arguing like this. Suppose the premises are true and the conclusion is false. Then F must be true. So, given the first premise, J and H must both be true. But given the second premise [JA] is true, so, if J is true, A must be. And also, given the second premise, [AG] is true; so, if A is true, G is false. But then, given the last premise, H must be true; so H must be false. But we have already said that H must be true. So it cannot be that the premises are true and the conclusion false. So, given the completeness of the tableau system, the sequent is a correct sequent.)

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