Exercise 3.3 - Answers
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(i)
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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®[SÙW]]. (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: [¬WÚL] (or, equivalently as [¬L®¬W]).
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®[BÙG]].
The
last sentence can be paraphrased as, "If Sutton are at least
as good as Aston, they won't lose to Aston"; i.e.: [G®¬S].
So
the whole argument becomes:
[¬A®[SÙW]], [¬WÚL],
[L®[BÙG]], [G®¬S]
So, A.
The
corresponding (syntactic) sequent is:
[¬A®[SÙW]], [¬WÚL], [L®[BÙG]], [G®¬S]
|- 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, [SÙW]
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.)
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(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": [C®M].
(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®[GÙP]].
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: [FÙ[¬M®¬P]].
So
the whole argument is:
J, [C®[GÙP]], [FÙ[¬M®¬P]] So, [C®M]
And
the corresponding (syntactic) sequent is:
J, [C®[GÙP]], [FÙ[¬M®¬P]] |-
[C®M]
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®[GÙP]], [¬M®¬P]
|- [C®M]
Indeed
we might excise the "G" too, leaving ourselves just with:
[C®P], [¬M®¬P]
|- [C®M].
(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.)
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(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: [I®G]. (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: [¬C®¬I].
(Or, [I®C]; or, ¬[IÙ¬C].)
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: [¬P®¬C]. (Or [C®P];
or ¬[CÙ¬P].)
We
can formalise the next premise as: [[P®M]Ù[M®N]].
And
the final premise is: [N®G].
So
the argument is:
[¬C®¬I], [¬P®¬C],
[[P®M]Ù[M®N]], [N®G]
So, [I®G]
And
the corresponding syntactic sequent is:
[¬C®¬I], [¬P®¬C],
[[P®M]Ù[M®N]], [N®G]
|- [I®G].
(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,
[P®M] is true, so M must
be. But also, given the third premise, [M®N] 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.)
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(iv)
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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: [IÚR]. (Or as [¬I®R]).
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: [PÚC]. (Or, [¬P®C].)
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®[RÚI]]. (Should the "or" be treated as exclusive? We have
no reason to suppose so.)
The
third premise is: ¬P.
So
the whole argument becomes:
[PÚC], [C®[RÚI]], ¬P So, [IÚR]
And
the corresponding syntactic sequent is:
[PÚC], [C®[RÚI]], ¬P |-
[IÚR]
(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, [RÚI]
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.) [¬I«R]. (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):
[PÚC],
[C®[R«¬I]], ¬P |-
[¬I«R].
And
that sequent is correct, because [R«¬I] is equivalent to [¬I«R].
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(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®[JÙH]].
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: [[¬JÚA]Ù[A®¬G]]. (Or [[¬A®¬J]Ù[A®¬G]].) (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: [¬HÚG].
The
argument is:
[F®[JÙH]], [[¬JÚA]Ù[A®¬G]], [¬HÚG]
So, ¬F
And
the corresponding syntactic sequent is:
[F®[JÙH]], [[¬JÚA]Ù[A®¬G]], [¬HÚG]
|- ¬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 [¬JÚA]
is true, so, if J is true, A must be. And also, given the second
premise, [A®¬G] 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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