At the end of Lecture VI – we were labouring mightily then in our investigations of identity – we discovered an interesting association between a column, PERSONNO in REGISTRATION, and a criterion of identity, “is the same person as”. You will recall that this criterion of identity was peculiarly stable: it persisted even across a change in the primary key of its relation. But there is one other thing to say about it: in stating this criterion of identity we gave, with considerable precision and clarity, the meaning of the column.

Think of any column in a relation, say COLOUR in a CAR relation. One of the things the data analyst, the data base designer, has to do is to explain the meaning of such a column. And, of course, the choice of an appropriate name is half the battle. You would be amazed (though the FOPC wouldn’t twitch a whisker) if I told you that the COLOUR column held the number of cylinders. But, putting aside such a crude error (not a formal error, I stress – the FOPC doesn’t understand English), you would be right to be annoyed if you found that “Shocking Pink and Royal Purple” was an acceptable value of COLOUR. Wouldn’t it be better to have called the column “COLOURS” or “COLOUR COMBINATION”? At least, you would be right unless the actual users – however unadvisedly – really did say that “Shocking Pink and Royal Purple” constituted just one colour.

Now this sort of discussion is, as you’ll appreciate, about the criterion of application of “colour”: what counts as a colour? Suppose that we could get this criterion agreed, and we said, for instance, that Scarlet was a colour and that Primary Red was a colour. Well we wouldn’t quite have finished, would we? I want to ask – and I’m sure you do now: but is Scarlet the same colour as Primary Red? That is, do the users regard them as the same colour?

So you can see that we haven’t really given the interpretation of a column until we have associated with it a criterion of identity. But we now know, because it’s one of the few rays of light that have gleamed in the dark valley that we’ve passed through, that, once we’ve specified this criterion of identity, we have really pinned down the meaning of the column in a very stable way.

Now I ask you to think back to the Classical interpretation and to consider the question: under that interpretation, what are the entities (sense 2) of the system – the things said by the records to exist? They are the things named by the values in fields. Thus, in a CAR record, there would be, say, a registration number (assume that’s the “name” of a car), a colour (the name of a colour), a make and model (the names of a make and of a model), and so on.

The record is a proposition that states:

Such-and-such car has this colour and is of this model etc.

But take the MAKE, say m. What entity does that name? It names x, just as n does, but under the criterion “is the same make as”. x is the same make as anything that has the make m.

So we don’t need an extra entity for m to name. But take the colour, c. Surely that names a colour, an extra entity. Not at all. There’s that charming story about the little lost girl who is asked “Do you know what sort of car your mummy has?” “A green one,” she replies. And quite right too. Any way of classifying things gives sorts, kinds, or types. It’s just prejudice that makes us think colour doesn’t count, whereas model does.[1]

For the doubtful, I define “is the same colour-car as”:

x is the same colour-car as y =_df x is a car, and y is a car, and x is the same colour as y.

x is a colour-car =_df x is the same colour-car as x. I.e. x is a car and has a colour.

As I’ve mentioned broader and narrower classifications, let me say that a relative identity, xIy, is no broader than another, xJy, when FOR EACH x, FOR EACH y, IF BOTH xIy AND xJx THEN xJy. The systemic identity of a theory is its narrowest identity (its finest classification). Thus, if x=y is the systemic identity of a theory, and xIy is an identity of that theory, then FOR EACH x, FOR EACH y, IF BOTH x=y AND xIx THEN xIy. It is, incidentally, well known and easily provable that any theory has only one systemic identity, which is what justifies us in talking of the systemic identity.[2]

We can simplify these “narrowness” definitions by introducing what we might call the completion of an identity. Suppose we have an identity, xIy. As we know, we do not in general demand that FOR EACH x, xIx (total reflexivity). However we can define another, totally reflexive identity on xIy, thus:

xIy =_df xIy OR (NOT xIx AND NOT yIy)

But now we can say that one identity is no broader than another if the one implies the completion of the other: xIy is no broader than xJy, when FOR EACH x, FOR EACH y, IF xIy THEN xJy. And the systemic identity, “=”, is such that FOR EACH x, FOR EACH y, IF x=y THEN xIy.

In our last lecture I gave the definition of “proper” name as: a name given under the systemic identity. This made the notion of proper name system-relative. I should add that there are two other possible definitions. First, “Geoffrey” is the proper name of a person; that is, a name may be said to be proper with respect to a certain criterion of identity (in this case, “is the same person as”). Any name is therefore proper with respect to the criterion of identity under which it was given. Incidentally, the use of “Geoffrey” as the proper name of different persons is just a non-systematic ambiguity: to treat it as a common name, and talk of “Geoffreys”, is a howler (like saying, “There are four banks in our town, two on either side the river, and two financial houses on the main street.”) Second, those who believe in absolute identity would regard a proper name – an absolutely proper name – as a name given under the absolute identity. Naturally, as there is no absolute identity, no name is in that sense proper.

In discussing a natural language we use the term “proper” name, as far as I can see, for a name given under a criterion of identity, when we do not normally use (in the language) any narrower criterion of identity. Logic is, however, blind to what we “normally” do, and the distinction between proper and common names in natural language is blurred indeed. Remember Mr Man in Brer Rabbit, Tommy Atkins, or every Tom, Dick, and Harry. We once Christened the redbreast, the daw, and the pie with “proper” names: “Robin”, “Jack”, and “Mag”. Now, in their turn, “robin”, “jackdaw”, and “magpie” have become common names. And think of the quislings, the little hitlers, and the mute inglorious miltons. Or consider: were “Sun “ and “Moon” once proper names? I guess so. And “occupational” surnames – Smith, Fletcher, or Engineer – started as common names, became proper perhaps – applied to the sole smith or fletcher or engineer in the village – and then became “family” names (a special sort of common name).[3]

Now we can approach our new interpretation, and we shall see that it has features of the Classical interpretation, in that each record is a proposition; but it also has features of the Entity interpretation. Each record will be a proposition to the effect that something exists (or, as we shall see, that some things exist). So, if you wish, you may say that the record “represents” or “corresponds to” the thing (or a relationship between the things) so proposed to exist. In some cases, the thing proposed may be understood as the record itself; in others not. So self-interpretation is neither required nor excluded.

I’ll give you the interpretation in its most general form. A record of type P with field-types (columns) C1, C2, ...Cn, and values respectively V1, V2, ... Vn, will be interpreted as a proposition of the form:

EI

FOR SOME x₁, FOR SOME x₂, ... FOR SOME x_p P(x₁, x₂, ... x_p) AND y₁ =_C1 V1 AND y₂ =_C2 V2 ... AND y_n =_Cn Vn

It might be interpreted as:

FOR SOME x, FOR SOME y, x owns y AND x is the same person as (PERSONNO) 12345 AND x is the same surperson as (SURNAME) Brown AND y is the same car as (CARREGNO) 123ABC

In the above context, “x is the same surperson as Brown” may therefore be replaced by “x has the surname ‘Brown’” (because the previous clause tells us that x is a person).

There are some special cases of the above general interpretation:

SI

FOR SOME x P(x) AND x =_C1 V1 AND x =_C2 V2 ... AND x =_Cn Vn
 

This is, we may say, the Singulary Interpretation: we use just one variable.

TI

FOR SOME x₁, FOR SOME x₂, ... FOR SOME x_p y₁ =_C1 V1 AND y₂ =_C2 V2 ... AND y_n =_Cn Vn
 

In this case, the “Truth Interpretation”, the proposition “P(x₁, x₂, ... x_r) amounts merely to “true” (i.e. all information is conveyed by the expressions “y_v =_Cv Vv”).

II

FOR SOME x, x =_C1 V1 AND x =_C2 V2 ... AND x =_Cn Vn
 

This, the “Independent Interpretation”, is simply the Singulary Truth Interpretation.[5]

CI

P(VI, V2, ...Vn)

(PERSONNO) 12345 has the surname (SURNAME) “Brown” and owns the car (CARREGNO) 123ABC

HI

FOR SOME x₁, FOR SOME x₂, ... FOR SOME x_n P(x₁, x₂, ... x_n) AND x₁ =_C1 V1 AND x₂ =_C2 V2 ... AND x_n =_Cn Vn

FOR SOME x, FOR SOME z, FOR SOME y, x is called z AND x owns y AND x is the same person as (PERSONNO) 12345 AND z is the same surname as (SURNAME) “Brown” AND y is the same car as (CARREGNO) 123ABC

And that’s that! We have our new interpretation: EI with optional variants SI, TI, II, and HI. Silence.

Of course that’s not that! That’s only structure reinterpreted. What about data manipulation? We interpreted restrict, project, and join on CI. But now we have to reinterpret them for EI. Very well, recall EI:

FOR SOME x₁, FOR SOME x₂, ... FOR SOME x_p P(x₁, x₂, ... x_p) AND y₁ =_C1 V1 AND y₂ =_C2 V2 ... AND y_n =_Cn Vn

FOR SOME x₁, FOR SOME x₂, ... FOR SOME x_p P(x₁, x₂, ... x_p) AND Q(z₁, z₂, ... z_q) AND y₁ =_C1 V1 AND y₂ =_C2 V2 ... AND y_n =_Cn Vn

Notice that the form of the interpretation remains the same:

A number of existential quantifications (“FOR SOME x_i”)
 

A predicate (here “P(x₁, x₂, ... x_p) AND Q(z₁, z₂, ... z_q)”)
 

A conjunction (ANDing) of column equivalences (“y_v =_Cv Vv”)

Consider now the Cartesian Product of EI, as shown, with a record interpreted as:

FOR SOME z₁, FOR SOME z₂, ... FOR SOME z_q, Q(z₁, z₂, ... z_q) AND w₁ =_K1 U1 AND w₂ =_K2 U2 ... AND w_m =_Km Um

The result comes out as:

FOR SOME x₁, FOR SOME x₂, ... FOR SOME x_p, FOR SOME z₁, FOR SOME z₂, ... FOR SOME z_q, P(x₁, x₂, ... x_p) AND Q(z₁, z₂, ... z_q) AND y₁ =_C1 V1 AND y₂ =_C2 V2 ... AND y_n =_Cn Vn AND w₁ =_K1 U1 AND w₂ =_K2 U2 ... AND w_m =_Km Um

To get a Natural Join, we can start from the Cartesian Product, as above, and then take each join column, say Cv is the same column as Ku. We then have two equivalences: “y_v =_Cv Vv” and “w_u =_Ku Uu”. We remove one of them (say the latter) and add “AND y_v =_Cv w_u” to the predicate, making it: “P(x₁, x₂, ... x_p) AND Q(z₁, z₂, ... z_q) AND y_v =_Cv w_u”.

Projection is pretty straightforward: we merely drop the equivalences of the columns that are projected away. It can be seen that the natural join amounted to a Cartesian Product, a Restriction (adding the expression, “y_v =_Cv w_u” to the predicate), and a Projection (removing the equivalences, “w_u =_Ku Uu”). It is also obvious, I think, that our primitive operations, Cartesian Product, Restriction, and Projection, are all logical implications; the first two are of the form: from propositions “p” and “q” to derive “p AND q”; no change here from CI. Projection, however, was an existential quantification in CI, i.e. from “Fa” to derive “FOR SOME x, Fx”; this, you will recall, was what required “a” to be a name. Under EI Projection is now an implication of the form: from “p AND q” to derive “p”: still a good implication, of course, though a radical change in interpretation.

But do remember that this inverse of conjunction happens within one or more existential quantifications. Let’s assume just one. We go from:

FOR SOME x, F(x) AND G(x) to
 

FOR SOME x, F(x)

But notice what happens when we try to reverse this (on a join). We have:

FOR SOME x, F(x)

FOR SOME y, G(y)

FOR SOME x, FOR SOME y, F(x) AND G(y)

FOR SOME x, F(x) AND G(x)

Well, I’m sorry about the extra complexity in interpretation: but we were driven to it. It does, however, have some interesting features. One of them is that we can – rather surprisingly – now distinguish (if we choose) between records representing Entities, namely those records whose interpretation is of the form SI – or II of course – the records with a single assertion of existence; and other records that represent Relationships. I’m not inclined to make much of this myself, indeed I’m slightly annoyed that I might have given comfort to the EAR fans by providing a decent underpinning for their theory. Moral: we should follow the argument where it leads, for though none of us is unbiassed, reason is.[7]

Let’s think about what our primitive operators do:

None of them removes an existential quantification, and none of them removes any part of the predicate.
 

Restriction adds to the predicate.
 

Cartesian Product includes everything pertaining to both of its operands.
 

Projection removes equivalences.

So very clearly we can see that Restrictions add information, Cartesian Product combines (but neither adds nor removes) information, and Projection loses information. And this is just as it should be.

As we have a few moments left, let’s look again at our new standard interpretation, (EI):

FOR SOME x₁, FOR SOME x₂, ... FOR SOME x_p P(x₁, x₂, ... x_p) AND y₁ =_C1 V1 AND y₂ =_C2 V2 ... AND y_n =_Cn Vn

y_v =_Cv y_v viz Vv

Well sometimes we might just want to do one of these without the other. Today we’ll consider just identification proper: marking x and y as the same such-and-such, without – so to speak – picking out just what such-and-such they are. It’s a bit like this; suppose you were looking at data about persons’ coats of arms, and you found that two persons had the same colour ground on their shields, say “Vert”. Well, you might not know what “Vert” meant (what colour it was) but you would know about a similarity – an identity – between the shields.

As it happens, it is often helpful to have this sort of data in a data base: values that the system can compare for equality, but which are not shown to users. Dr Codd calls such values (in a limited context) “surrogates”:[8] I will say that they are of a cryptic data type. I won’t go into why we want cryptic data at the moment; we may come across it from time to time in the future. But just remember the idea: it drives a neat wedge between the two senses of “identification”.

And while we’re on the subject of column equivalences, or, as we can properly say, “column criteria of identity”, what does it mean, given a column, C, with criterion of identity, “is the same C as”, to say of some value, V, that:

V is the same C as itself?

You will remember that trick we used to go from an ordinary relative identity, merely reflexive, to a totally reflexive identity: “completion” we called it. Let’s define the completion of our arbitrary column criterion. Taking “=_C” to mean “is the same C as”, we can define its completion:

x º_C y =_df x =_C y OR (NOT x =_C x AND NOT y =_C y)

Then we could say that V is not proper to C when V is the same C-wise as something but not the same C as anything. If we did allow a value in a column that was not proper to that column, we would have to modify EI slightly; it would become:

FOR SOME x₁, FOR SOME x₂, ... FOR SOME x_p P(x₁, x₂, ... x_p) AND y₁ º_C1 V1 AND y₂ º_C2 V2 ... AND y_n º_Cn Vn

(Professor Platoclast was pressed on the question whether records ought to interpreted as assertions.)

You are right. We speak loosely when we say that inserting a record amounts to asserting a proposition. What, for instance, do we wish to say about a record representing a merely planned warehouse? Certainly not that there exists something that is that warehouse.

“Proposition”, you know, doesn’t mean, “something that is asserted”; it means, as you might guess, “something that is proposed”. It’s proposed for our consideration. And that means: so that we can work out what follows from its truth, which is precisely what we want in a planning data base. Of course, a proposition can be asserted: when we propose it we can also indicate, explicitly or implicitly, that we do assert it.

But I’m quite pleased you raised the question. If you think about this new method of interpretation, it consists of a conjunction within one or more existential quantifications. But the manipulations operate only on the conjunction: they don’t affect the prefixed quantifications. However, it’s the quantifications that bear any assertion we intend: they say that there exist this or that. So, if we wanted “planning records” we could give them an interpretation beginning not simply “FOR SOME x” or “There is something such that”, but “We plan that there will be something such that”.

And likewise for things like “fictional persons” or “fabulous beasts”: we don’t have to assert that there is someone who is Mr Pickwick, or something that is a cockatrice; we could have a prefix saying that “It is said that there is someone” or “It is said that there is something”. Our manipulations simply leave such prefixes in place.