# Formalisation Expressing generality

The problem of translating English sentences into the
formal notation is as hard as any of the technical ones
encountered in logic at this level. Part of the reason is
that there are no infallible rules or algorithms to do the
job for us (or at any rate no graspable ones). The best we
can do is to paraphrase, using our command of both formal
system and natural language, and rely on imagination and
intelligence. The entire website
*Logic for
Fun* is aimed at giving you practice in the art of
formalisation, and justifiably so, since this is the
aspect of logic which is the most useful thing you learn
in this course. There now follow illustrations of certain
formalisation techniques.

We know how to say 'Some footballers are hairy' and 'All goats are hairy':

SOME(*x*: *Fx*) *Hx* and
ALL(*x* : *Gx*) *Hx*

Now, how about 'No goats play football'? The earlier suggestion was to use a quantity indicator for 'no':

∅ (*x*: *Gx*) *Fx*

Well, to say that *no* goats behave in this fashion
is to deny that *some* goats do or equivalently to
assert that all goats don't. That is, an alternative would
be

NOTSOME(*x*: *Gx*) *Fx*

or equivalently

ALL(*x*: *Gx*) NOT*Fx*

So the quantity indicator '∅' is definable in terms
of either 'SOME'
or 'ALL' and negation. For that
reason, we do not need a primitive symbol for it: whenever
we need to express it, we can do so with the standard
quantifiers. Moreover, since 'All *F*s are *G*s'
is equivalent to 'There is no *F* which is not a
*G*', we could if we wished define the universal quantifier
in terms of the existential:

ALL(*x*: *Fx*) *Gx* =
NOTSOME(*x*: *Fx*) NOT*Gx*

The converse is also true. Some *F* is *G*
iff not all *F*s are non-*G*:

SOME(*x*: *Fx*) *Gx* =
NOTALL(*x*: *Fx*) NOT*Gx*

The upshot is that any one of these three quantifiers together with negation is sufficient to define the other two. For most purposes, we shall continue to use 'ALL' and 'SOME' but drop '∅' from our vocabulary.

The intended reading of
SOME(*x*: *Fx*) *Gx*
is 'At least one *F* is *G*'. The difference
between the singular 'some *F*' and the plural
'some *F*s' is glossed over at this stage, as it is
irrelevant to most logical purposes. Note that
in saying that some *F* is *G* we are not
thereby asserting that some other *F* is
not *G*, or indeed that there is more than
one *F* in existence. We are asserting only that
there exists at least one *F* which is also
a *G*. Alternative readings of
ALL(*x*: *Fx*) *Gx*
are 'Every *F* is *G*', 'Any *F*
is *G*' or 'Each *F* is *G*'. These are not
all exactly synonymous in English with each other or with
'All *F*s are *G*s', but their truth conditions
are tolerably close to 'There is no *F* which is not
a *G*' for present purposes.

*Logic for Fun*guide.

Formalisation can easily become intricate, especially
where it involves dyadic predicates (i.e. relations) as
well as monadic ones. With a relational
predicate *K* to symbolise '... kicks ...' we can
formalise 'Aristotle kicks Billy'

*Kab*

and hence 'Aristotle kicks goats'

SOME(*y*: *Gy*) *Kay*

Note that it is not necessary that he kick *all*
goats, for he counts as a goat-kicker if there are goats
that he kicks. Then 'All footballers kick hairy goats' is
more of the same:

ALL(*x*: *Fx*)
SOME(*y*: *Gy* AND *Hy*) *Kxy*

: Two readings of 'Only hairy footballers kick goats' |

That is: take any footballer,*x*; there exists a
hairy goat, *y* such that *x*
kicks *y*. Now what about 'Only hairy footballers
kick goats'? This is ambiguous. It could mean either (a)
that the only footballers who indulge in goat-kicking are
hairy ones, or (b) that every goat-kicking episode has a
hairy footballer at its active end. That is, either

ALL(*x*: *Fx* AND
SOME(*y*: *Gy*) *Kxy*) *Hx*

or

ALL(*x*:
SOME(*y*: *Gy*) *Kxy*) (*Fx*
AND *Hx*)

These are not equivalent, for they differ in their claims
about goat-kicking by the rest of the population, although
they agree about that perpetrated by footballers. The
structure of the two versions is worth a second look
(ambiguous). The exposure and
exact explication of ambiguities (multiple senses) is one
of the most important applications of formal logic. It can
often help to clarify difficult issues in philosophy,
mathematics, linguistics or even real life. Logical
notation enables us to say with some precision "Insofar as
it means *this*, these are the consequences; if we
read it as saying *that*, we get those."

One particular ambiguity gives rise to a potential confusion common enough to have been given a name. This is the "quantifier shift fallacy". Consider the sentence 'Every goat kicks a certain footballer'. This could refer to some unfortunate universal kickee, or it might be so read as to allow different kickers different targets. That is, it could translate as either of

SOME(*x*: *Fx*)
ALL(*y*: *Gy*) *Kyx*

ALL(*y*: *Gy*)
SOME(*x*: *Fx*) *Kyx*

The fallacy is to infer the former from the latter. You, of course, would never do such a thing, but it is surprising how many would. Just to make sure you see the formal point, try writing out the structure trees of the above two formulas in the style of ambiguous.

The quantifier-variable notation allows us to treat complicated arguments like this one:

All goats are hairy;

Every footballer loves a goat;

Therefore whoever kicks a footballer kicks someone who loves
something hairy.

The premises are easy:

ALL(*x*: *Gx*) *Hx*

ALL(*x*: *Fx*)
SOME(*y*: *Gy*) *Lxy*

All the fun comes in formalising the conclusion. Let us take it in stages. It is basically universal in form, saying that every footballer-kicker is a hairy-thing-lover-kicker:

ALL(*x*: *x* kicks a
footballer) *x* kicks a hairy-thing-lover

This has fragments which can be treated separately. First
consider '*x* kicks a footballer'. What this means
is that there exists some footballer, say *y*,
which *x* kicks:

SOME(*y*: *Fy*) *Kxy*

Now we turn our attention to '*x* kicks someone who
loves something hairy'. This is also existential in form:
it says there exists some hairy-thing-lover,
which *x* kicks:

SOME(*y*: *y* loves something hairy) *Kxy*

Finally, for *y* to love something hairy is again
existential:

SOME(*z*: *Hz*) *Lyz*

Putting all of the above together, we get

ALL(*x*:
SOME(*y*: *Fy*) *Kxy*)
SOME(*y*:
SOME(*z*: *Hz*) *Lyz*) *Kxy*

The corresponding sequent will be provable in the formal system to be developed next. Such complex argument forms became formally treatable only within the last century or so. Its capacity to handle them is one of the major respects in which twentieth century logic is an improvement on older systems.