Witnessing in Mathematical Logic
1 Imagine a language (e.g. English, Venusian, Arithmetic, L) which is
capable of describing a certain World (e.g. the Earth, Venus, the Integers,
2 The description of the World, were it to be carried out, would be endless.
But imagine that there is endless time in which to carry it out, and that
the work is under way.
3 The language is used to form sentences and these are stored away in
an Archive, which we can call the Archive of (that) World (Earth, Venus,
the Integers, W).
4 For illustrative purposes let us imagine that the Archive is divided
into two sections which we can call the Encyclopaedia and the Inventory,
without insisting that these be completely distinct.
4.1 The Encyclopaedia contains descriptions, both general or particular,
(e.g. "some trees never lose their leaves", "all books
have pages", "there are five kinds of Venusian wine", "odd
plus odd is even", "(Ex)Px").
4.11 We can imagine that the ‘librarians’ of the Archive seek to ensure
that any sentence which is a (logical) consequence of sentences already
in the Encyclopaedia is added to it.
4.2 The Inventory contains a list of the names of particular things belonging
to the world, ("the lemon tree outside my study window", "the
fourth leaf, counting from the base, on the seventh twig, counting from
the base, of the branch of the lemon tree outside my window on which that
butterfly is resting ... otherwise known as leaf number 3,246,734 Carlton",
"the Venusian wine known as ‘koeisch’", "3", "dkf(7)").
5 There would seem to be problems in knowing: a. what sorts of symbols
can be used as names; b. whether possible names might be redundant; c.
how fine-grained our naming needs to be; d. whether we have named everything
there is to be named.
None of these are treated as problems because:
a. anything can be a name provided once a name it stays a name and
is distinguished, say, from an identical (string of) symbol(s) which
in the Encyclopaedia is treated as a description. The name could be
hyphenated to make this point, (e.g. "there-is-a-cat-in-the-corner"
could be the name of the cat in the corner).
b. It doesn’t matter how many different names turn out to name the
c. Fine grained and coarse-grained naming are compatible without limit,
(e.g. we can name a coast or every grain of sand in a particular mound
on a particular beach, or indeed in the entire coast or beyond).
d. Our inventory is potentially infinite but it doesn’t matter how
far short of exhaustive even this infinity may turn out to be (e.g.
we might devote all of our infinity of time (sempiternity) to naming
the features of a single flower). Exhaustiveness is an issue but in
the end it will be determined internally to the Archive, not by reference
to a World beyond. And besides, since one infinity or sempiternity can
be added to another, by interleaving their moments, we can always make
up for lost time, or even lost eternity.
6 An Archive is called a theory, the Encyclopaedia is called the set
of sentences of the theory and the Inventory its set of constants.
7 A sentence is called existential if its truth, that is, its being included
in the Encyclopaedia, has direct implications for what ought to be in
the Inventory, ("There is a tree outside my study window." raises
the question of whether the Inventory mentions such a thing, "the-lemon-tree-outside-my
study-window" answers to this; it is said to witness that sentence.)
The inventorial implications of an existential sentence are called its
7.1 The existential import of sentences depends in a crucial way upon
what is already in the Inventory. ("All trees have roots.",
by itself is not existential because it could be true even if there
were no trees (e.g. "All unicorns have horns."), but once
there is a tree in the Inventory then this sentence becomes existential
because now we can look for a name in the Inventory, e.g. that for the
roots of the lemon tree outside my study window, (which indeed might
be named by that very phrase, suitably hyphenated.)) If a sentence in
the Encyclopaedia considered with reference to some names in the Inventory
gives rise to an existential sentence, then this latter is an existential
consequence of the former.
The process of generating consequences can continue without limit,
for example, "Every human has two other humans as parents, and
no-one can be an ancestor of themselves." in the Encyclopaedia,
together with "the human, Joshua B." in the Inventory, lead
to an infinite number of existential consequences, and hence would require
an infinite number of witnesses, that is ancestral names in the Inventory.
7.2 If a certain name in the Inventory is said to be (or to bear) a
witness this refers to a relation which it bears not to the World, but
to certain contents of the Encyclopaedia.
8 A theory is said to have witnesses if every sentence in the Encyclopaedia
which has existential consequences can find witnesses for all of these
in the Inventory.
8.1 The fact of having witnesses or not is purely internal to the
Archive, it could be decided by the ‘librarians’, after a simple eternity
of checking files.
8.2 If an Archive does not have witnesses then it can be augmented
so that it has them by expanding the Inventory. We first add names for
witnesses for all the as yet unwitnessed existential sentences in the
Encyclopaedia, making them up as we go along (e.g. hyphenated versions
of the sentences themselves), and then add names for witnesses for all
the new existential consequences that this change in the Inventory has
given rise to, and then do the same again, on and on forever. We interleave
the process of entering this infinite number of new names into the Inventory
with whatever endless tasks of naming we were already engaged upon.
We encounter no problems since sempiternity is infinitely accommodating.
8.21 The librarians looking after the Encyclopaedia would also get
caught up in this expansion process. They loosen the criterion for
adding on new sentences, so that it is not only sentences which follow
from what is already in the Encyclopaedia that they accept, but also
any sentence which is not incompatible with the present state of the
8.211 This represents a lot of extra work. The Archive would seem now
to be not only very large but a veritable hive of activity with vast numbers
of librarians running backwards and forwards through a maze of halls where
the tasks of tabulation and checking are going on, day and night, week
after week, endlessly. Actually, however, this could all be done by one
librarian, even one who was rather slow but immortal, working at one desk
The Archive itself would be infinite, that is, the librarian would have
to be able to activate new sections of it indefinitely. You can imagine
that every so often he goes to the end of the furthest chamber that has
been used so far and opens a great door and turns a switch which illuminates
a new chamber beyond. Here the rows of shelves are full of blank paper
and in the middle of the room there is a desk piled high with fresh pencils,
because all the paper and pencils in the previous chambers have been completely
9 An Archive with witnesses does not need, and never needed, a World.
Its own inventory is its world and its librarians’ motto is il n’y
a pas hors du texte.
9.1 - Witnesses, then, don’t witness facts but statements. The statements
are however statements of facts, are they not?
- There would seem to be a position somewhere standing before (or representing)
the World, for which the arising, the uttering of statements about the
World marks our access, our intimacy with the World, rather than our
separation from it. We would like to be able to say that language is
the house of Being, and that what witnesses in our names, is witness
to the truth of the World. In this imaginary truth-scape of a logical
universe, however, such a position seems impossible; if there were a
privileged reader in the Archive he would be continually deflected by
the officiousness of the librarians on whom we would so much depend.
The Archive is already, and by definition, a kind of house, a house
dedicated to the storage of a reserve which copies and supplants the
World. Here there is an acute awareness of the poignancy of naming,
as if we somehow sense, each time a new name is uttered, or written
down, the silent cry of a piece of the Real which recedes into an infinitely
inaccessible proximity. This is a very different thing from never having
existed at all, the victory of the librarians is as illusory as was
their purely ancillary neutrality.
10 Among Archives having witnesses the key distinction is not between
those with and without worlds, but between those which are consistent
and those which are inconsistent In a consistent Archive the presence
of "A" in the Encyclopaedia implies the absence of any sentence
which would include "not-A" among its consequences. The relation
between witnesses and the sentences that they witness creates a clear
segregation of witnesses. (Two witnesses are indiscernible if they witness
precisely the same set of sentences of the Encyclopaedia.) In an inconsistent
Archive there are no laws of exclusion from the Encyclopaedia, and every
witness witnesses everything.
... and so on
|David Odell 'Mathematics"