THE LOGIC NOTES

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) NOTFx

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 Fs are Gs' 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) NOTGx

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

SOME(x: Fx) Gx   =   NOTALL(x: Fx) NOTGx

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 Fs' 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 Fs are Gs', but their truth conditions are tolerably close to 'There is no F which is not a G' for present purposes.

Expressing plurals is treated below. See also the page on numerical quantifiers in the 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

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