Exercise 5.4  Answers
1

(i)

Correct.
(If everything is F and everything is G, everything must be both.)


(ii)

Correct.
(If everything is both F and G, everything must be F, and everything
must be G.)




(ii)

Correct.
(If everything is both F and G, everything must be F, and everything
must be G.)


(iii)

Correct.


(iv)

(iii)

Correct.


(iv)

Incorrect.
The following is a counterexample:
Domain: Positive
integers.
Fx: x is
odd
Gx: x is
even


(v)

Incorrect.
The following is a counterexample (I expect)
Domain: Human
beings
Fx: x is
female
Gx: x is
male


(vi)

Correct.


(vii)

Incorrect.
The empty domain provides a counterexample – in fact, the only one.


(viii)

Correct.
(The empty domain does not provide a counterexample because we are
not allowed empty names in an interpretation: names must have bearers.)


(ix)

Incorrect.
Not only does the empty domain provide a counterexample, so does
any interpretation of Fx which makes Fx true of nothing. For instance:
Domain: Birds
Fx: x can
understand predicate logic
Gx: x has
a beak.


(x)

Incorrect.
The empty domain does not provide an example, because we are not
allowed empty names. But an interpretation which make Fxa true of
nothing will do. For instance:
Domain: the
natural numbers
a: 0
Fxy: x<y
Gxy: anything
you like!


(xi)

Correct.
Given that something is F, it cannot be both that everything
F is G (i.e. that there is nothing which is F but not G) and that
nothing F is G. 

(xii)

Incorrect.
""x$y[FyÙRxy]" says that in the case of each thing
there is something which is F to which it bears R – but leaves it
open whether this is the same thing in each case. "$y"x[FyÙRxy]", on the other hand, says there
is a thing which is F and to which everything bears R. A counterexample
would be:
Domain: Natural
numbers
Fx: x is
a number.
Rxy: x=y
Or:
Domain: {Queen
Elizabeth, Prince Philip}
Fx: x is
human
Rxy: x is
married to y.


(xiii)

Correct.
Given that there is something, or $xFx would not be true, if "x$y[FxÙGy]
is true, everything must be F and something must be G. But in that
case $y"x[FxÙGy] will be true too. 

(xiv)

Incorrect.
Domain: Humans
Fx: x is
female.
P: anything
false.
(This will
make ["xFx®P]
true, since "xFx is
false. But "x[Fx®P] is false. (Consider the Queen.)


(xv)

Correct.
One way to see this is to notice that ["xFx®P] is equivalent to [¬"xFxÚP], which is equivalent to [$x¬FxÚP],
which, together with $xFx
(which implies that the domain can't be empty) entails $x[¬FxÚP],
which is equivalent to $x[Fx®P]. Another way would be to notice that ["xFx®P] is equivalent to ¬["xFxÙ¬P] and to argue from there. 

(xvi)

Correct.
[$xFx®P] is equivalent to ¬[$xFxÙ¬P], which is equivalent to ¬$x[FxÙ¬P],
which is equivalent to "x¬[FxÙ¬P], which is equivalent to "x[Fx®P]. 

(xvii)

Correct.
"x[P®Fx]
is equivalent to "x[¬PÚFx], which is equivalent to [¬PÚ"xFx],
which is equivalent to [P®"xFx]. 
2

(i)

The conclusion
says that everything F is G. The premise will be true if not everything
is F. It will also be true if something is G. So all we need to
do is to find an interpretation according to which not everything
is F (or one in which something is G), and not everything that is
F is G. We could think of a real life interpretation with the appropriate
features. But another way of proceeding is to tell a story
which has the appropriate characteristics. So suppose we have room
containing just three living things: a dog, a man and a woman. Our
interpretation is:
Domain: living
things in the room
Fx: x is
human
Gx: x is
a woman.
That makes
the premises true (because not everything in the domain is human);
and it makes the conclusion false, because it is not true that all
the humans are women. (As it happen this interpretation also makes
it true that something is G.)


(ii)

It sometimes
helps to do a little relettering of variables, so as to bring out
the key features. If we reletter the conclusion we get the equivalent
formula [$xFx®$y"xGyx].
Now we have done that, it looks as if the key to finding a counterexample
lies in the fact that in the premise the $y comes after the "x,
but in the conclusion they are the other way round. So maybe we
can exploit the same sort of counterexample as would do for:
"x$yGyx =
$y"xGyx.
For that
we could use:
Domain: natural
numbers
Gxy: x>y
All we need
is to find an interpretation for Fx, which will make the premise
true and will also make $xFx
true. Let us add:
Fx: x is
a number.


(iii)

It
often helps when one is looking for a counterexample to replace some
or all the formulae by equivalent formulae. Here it looks as if it
may help to replace the ®s
with something equivalent using Ús.
Remember that [j®y] is equivalent
to [¬jÚy]. So the premise
is equivalent to "x[[¬FxÚGx]Ú[¬¬FxÚHx]].
But then we notice that this is bound to be true, because everything
is bound to be ¬F or ¬¬F, so to speak. So all we need to do is to
find an interpretation which makes the conclusion false and which
assigns some meaning to Fx, and we are there. (Left to you.) 

(iv)

A strategy
which often works is to try to think of whether we can make the
premises true and the conclusion false by taking a very small domain.
(It is worth thinking of the empty domain first – but that won't
do for this sequent, because it will make the conclusion true.)
It is fairly obvious that it won't do to take a domain with just
one thing either. (Suppose we do. Call the thing "a".
Then the conclusion amounts to saying that Faaa . But that is what
both the premises say too. So if they are true, the conclusion will
be.) Let us try a domain with just two things in it, which we will
call "a" and "b". To make the second premise
true, let us say that Faaa and Faba. We want the conclusion to be
false. So let us say that it is not true that Fbab nor that Fbbb.
So far so good. Now can we make the other premise true? Well it
tells us that whatever x and y are $zFxyz. So we need to make $zFbaz
true without making Fbab true after all. Let us say, then, that
Fbaa. And we need to make $zFbbz
true without making Fbbb true. Let us say that Fbba. And now we
need to deal with the other values of x and y in $zFxyz,
and we are there. But we already got Faaa and Faba. So all is well.
Now, at this
stage we haven't yet got an interpretation, but we all we need to
do is (i) to think of a domain with two things, (ii) to think of
something for Fxyz to mean, and (iii) to tell a (consistent) story
which makes Fxyz true just when we want it to be. Here is an example:
Domain: Mary,
John
Fxyz: x and
z like y.
Let's think
of the a we spoke of as being Mary, and the b as being John. Then
the story is that Mary (and Mary) likes Mary, as does John (and
John) and (so) John and Mary like Mary and so do Mary and John.
Neither, however, likes John.
Or we could
take a more mathematical example:
Domain: {1,
2}
Fxyz: <x,y,z>
Î {<1,1,1>, <1,2,1>,
<2,1,1>, <2,2,1>}
("<1,2,1>",
e.g., means "the ordered triple consisting of 1, 2 and 1, in
that order".)


(v)

We want the
conclusion to be false – so we don't want everything to be G. We
want the premises to be true. Suppose we had an interpretation according
to which nothing was G. That would be perfect! Remember that
"x[Gx®Hx] is true if nothing is G. Why? Well, because
[P®Q] is true whenever
P is false. Or, if you prefer, because "x[Gx®Hx]
is equivalent to "x[¬GxÚHx]. So, if nothing is G, the second premise
will be true. So all we need to do is to find an interpretation
which makes it true that nothing is G, and also makes the first
premise true – easy, because what we choose for Fxy won't affect
the truth of the second premise or the falsity of the conclusion.
What if we
hadn't noticed that it would do to make nothing G? Well, we could
take a small domain with just two things, a and b, in it, and say
that a is G but b isn't. Now, we have to make the second premise
true. So [Gx®"y[Fyx«Gy]] has to be true of both a and b. Easy,
in the case of b: the fact that it is not G gives us want we want.
But what about a? It has to be true that [Ga®"y[Fya«Gy]] . Since Ga is true, this means that "y[Fya«Gy].
So we must have both [Faa«Ga] and [Fba«Gb]. To avoid making Gb true we want Fba to
be false. And because Ga is true, Faa must be. But we still have
to make the first premise true. We have a choice between "y[FayÙFya] and "y[FbyÙFyb]. The first won't do, because, although
it will make Faa true, which we want, it will make Fba true too,
which we don't want; and the second won't do because it also will
make Fba true (and won't make Faa true). Try adding a third thing?
Try it, and see. Actually it won't help. There is no counterexample
which makes G true of something. (Try adding $xGx
to the premises, and proving the corresponding syntactic sequent.)


(vi)

Try a small
domain. We can't have just one thing in it, or we will make the
conclusion true, if we make the first premise true. So let us try
two things, a and b. And let us try to make Raa false, so that the
conclusion will be false. If you look at the third premise you will
see that we need to avoid letting Rba be true, because then we shall
have [[RbaÙRba]®Raa], which will make Raa true. But we need
to make the first premise true. We could try Rab, or Rbb. Let us
go for Rbb, which conveniently makes the other premises true without
our having to add anything else. All that remains is to construct
an interpretation with the appropriate features. Here is an example.
Domain: {Mary,
John}
Fxy: x likes
y
The story
is that John likes himself (but not Mary), and Mary likes no one
at all.


(vii)

The
empty domain will do, because all the premises begin with """,
and so are true in the empty domain, whereas the conclusion begins
with "$", and so is false. (It is always worth
thinking of the empty domain, because it sometimes provides an easy
answer, and sometimes, indeed, the only answer. And, as it happens,
this sequent has no counterexample in any other domain.) 

(viii)

Here some
initial manipulation seems as if it may help. Remember that ¬"xj
is equivalent to $x¬j, and that ¬[j®y] is equivalent to [jÙ¬y]. So the first premise is equivalent to "x"y[Fxy®$z[GzÙ¬Hzy]].
But the conclusion is equivalent to "x"y[[Fxy®$z[GzÙ¬Hzy]]Ù[$z[GzÙ¬Hzy]®Fxy]], which is equivalent to ["x"y[Fxy®$z[GzÙ¬Hzy]]Ù"x"y[$z[GzÙ¬Hzy]®Fxy]],
So if we make the first premise true, we will make the first part
of the conclusion true. So we had better concentrate on trying to
make "x"y[$z[GzÙ¬Hzy]®Fxy] false. Let's try a domain with just one
thing, a, in it. Then we need Faa to be false and $z[GzÙ¬Hza] true. So we need [GaÙ¬Haa]
– i.e Ga true and Haa false.
We also want
the second premise to be true. It is easier to see what it amounts
to if we replace it by the equivalent "x"y$z[HzxÚFyx]. But if there is just one thing, a, that
means that [HaaÚFaa] is true, which it can't be, because we've already got Faa
false and Haa false. Let's add another thing, b, and try to keep
[GaÙ¬Haa] true, so as to
keep the conclusion false. We want "x"y$z[HzxÚFyx] to be true, and in particular that means
that $z[HzaÚFaa] must be true. So since Faa needs to be
false and Haa needs to be false, we had better have Hba true. Now
the easiest way to make sure that the first premise true is to have
Fxy true of nothing at all. In which case, to preserve the truth
of the second premise, we need to have Hbb or Hab – it doesn't matter
which. Everything else can be false. So our recipe for a counterexample
is: Ga, Hba, Hab true; everything else false. The following will
do.
Domain: {Mary,
John}
Gx: x is
female
Hxy: x is
the brother or sister of y.
Fxy: x is
the father of y
The story
is that Mary and John are brother and sister, and Mary is female.

