On Functions and Composition of Relations

Abstract

The aim of this paper is to put forward a reminder, in a Wittgensteinian way, about the pitfalls that we might fall into if, following the almost standard usage (Quine’s is an exception) of certain notions by logicians and mathematician, within mathematics, we try to adhere to that same usage in ordinary language.
The notions I have in mind are those of function and composition (of functions). Now, the standard definitions are as follows:

(1) R is a function ↔ ∀xyz (xRy & xRy → y = z)

Since the second member of the pair is uniquely determined by the first (we must bear in mind that this is a mere convention which has no mathematical significance and so that it might be substituted by the opposite convention, as we shall see, in which the firs member of the pair is the one which is uniquely determined by the second) this allows us to characterize the R of x as the unique y such that xRy, i.e.:

(2) R (x) = ι y (xRy)
from which it follows that for any function R and argument x, x has R to R (x):

(3) x R R (x)
Finally, from (1), (2) and (3), it follows that for any function R and argument x, x has R to an individual if that individual is precisely R (x):

(4) ∀y (xRy ↔ y = R (x) )

And now we turn to functional composition. The standard definition runs as follows, for R, S relations:

(5) ∀xy (x R ◦ S y ↔ Ǝz (xSz & zRy)

from which mathematicians can obtain, when R, S are functions:

(6) R ◦ S (x) = R (S (x) )

Now let’s see what happens if we take these definitions literally and apply them in our ordinary speech. Let F be the relation father of, and let Antonio (a) be the father of Carlitos (c), his only child. Then, we have that Antonio is Carlitos’ father, i.e.:

a F c;

but now, according to (2), we have that

F (a) = ι y (aFy);

Since ι y (aFy) = c, we therefore have that

F (a) = c

which says that Antonio’s father is Carlitos, his only son! By accepting (2) this is an inescapable conclusion.
If we now take (3) into account and continue with our example, we have that

a F F (a)

i.e., that Antonio is the father of the father of Antonio, which means that Antonio is his own grandfather! Finally, let’s see what other genetical surprises does (4) have in the offing. What we get from it, as applied to our example, is that

a F c ↔ c = F (a) ;

in plain words it says that Antonio is Carlitos’ father if and only if Carlitos is Antonio’s father; an interesting, although impossible, genetical loop! These peculiar and grotesque results can be obtained not only from (1) - (4), as we have just seen, but from (5) and (6) as well. Let’s consider our well know relation of being a paternal uncle of (U) ; this relation is the composition of two other relations: “... is a brother of...” (B) and “... is the father of...” (F). Then:

U = B ◦ F.

Hence, to say that x is a (paternal) uncle of w amounts to saying that x is a brother of someone who is w’s father. By applying (5), we get:

x U w ↔ Ǝz (x F z & z B w)

which is to say that x is an uncle of w if and only if x is the father of someone’s brother, i.e. x is an uncle of w if and only if x is w’s father. If we want to obtain results which do not violate so catastrophically our ordinary usage, we would like to have a definition of composition like the following one:

(7) x R ◦ S y ↔Ǝz (x R z & z S y).

But if we have (7) and (1) and (2), we would then obtain:

R ◦ S (x) = S (R (x))

which would go against our intuitions. Instead we would like to obtain something more acceptable, like

R ◦ S (x) = R (S (x) )

i.e. our (6) above. But to attain this result, (1) and (2) have to be modified; hence, instead of (1) we might have (following Quine):

(8) R is a function ↔ ∀xyz (x R z & R z→ x = y)

where now the second element of the pair, instead of the first one as in (1), would uniquely determine the first, instead of the second. And so, (2) would become

(9) R (x) = ι y (y R x).

With these simple changes all our worries come to an end as the reader can easily check. But then, if this is so, why is the usage we have been here criticizing so widespread?
A possible answer is that this is so in a large measure owing to the great use which nowadays is made of functions as mappings from a set —that of the arguments— into another (or the same) set —that of the values of images. The fact that the domain of F (the set of arguments; D F) be A and its range (the set of values; R F) be included in B; i.e. that F is a mapping from A into B is expressed by

F : A ↔ B

and the usual definition is:

(12) F : A → B ↔ F is a function & D F = A & R F ⊂ B.

Now since this is a very widespread, firmly rooted and intuitive way of representing functions, it’s no use going against it; but if we define, as is commonly done, the domain and range of F by means of

(13) D F = {x : Ǝy xRy}
R F = {y: Ǝx xRy}

then (12) is an adequate definition only if we assume our counterintuitive (1) and (2) above. So, to obtain again a more pleasing result, leaving (6) and (12) as they are —which are the intuitively good mathematical propositions we want to preserve—, a slight change in (13) will be in order and so we would have, instead of it,

(14) D F = {y : Ǝx xRy}
R F = {x: Ǝy xRy}

Hence, if we do have propositions (8), (9), (7) and (14) instead of (1), (2), (5) and (13) respectively, we can avoid those undesirable results concerning ordinary language and still retain propositions (6) and (12) which are mathematically desirable. But will this proposal be heeded? Maybe it’s already too late to reverse the tide. Whit time, formal language gets hardened, as it so happens with natural language, and retains features which depend just on usage and tradition and not on conscious rational design. If this is so, the best we can do then is to be conscious of the traps hidden in formal languages so as not to be caught in them.

[J.A. Robles]