Here is the proposed interpretation of a row in a table. It is a proposition of a certain form, called “EPI” for Existential quantifications (“EXISTS”), Predicate (giving a criterion of application), and Identities; in its most general form:

(E)

$x₁ $x₂ ... $x_m

(P)

P (x₁, x₂, ... x_m)

(I)

& x₁ =₁ v₁ & x₂ =₂ v₂ ... & x_p =_p v_p

Notice that the value, v_j, is a common name: it names each thing that is identical (in the sense, “=_j”) to anything that it names (just as “dog” names each thing that is identical, in the sense, “is the same species as”, to anything that it names).

To take a very simple example, a PERSON record with columns PersonId, Surname, and Religion:

$x x is a person & x is the same person as 47273 & x has the same surname as Robinson & x has the same faith as Buddhism.

Notice that the only constraint placed on the column identities (=_j) is that each must be symmetric and transitive:

An entity like PERSON, where the EPI interpretation uses a single quantified variable, is (usually) an independent entity. On an ER diagram with the many-to-one dependency links running down the page from “parent” (one) to “child” (many) entities (the so-called “live crows” convention), such an independent entity has no parents. Suppose that both PERSON and HOUSE are independent entities. An EPI interpretation of HOUSE is left to the reader; we assume here a key of HouseId. Suppose we want a table representing the relationship of (possibly shared) current ownership:

$x $y x owns y & x is the same person as 47273 & y is the same house as H2371
 

where “H2371” is a HouseId value.

Now we must provide an interpretation of the operations of our DML. Again, we treat restrictions and Cartesian products as conjunctions. A restriction is a predicate, say:

Q(u₁, u₂, ... u_q )

(E)

$x₁ $x₂ ... $x_m

(P)

P (x₁, x₂, ... x_m) & Q(u₁, u₂, ... u_q )

(I)

& x₁ =₁ v₁ & x₂ =₂ v₂ ... & x_p =_p v_p

(E)

$y₁ $y₂ ... $y_n

(P)

Q (y₁, y₂, ... y_n)

(I)

& u₁ =_m+1 v_m+1 & u₂ =_m+2 v_m+2 ... & u_q =_m+q v_m+q

We simply conjoin the interpretations, and shuffle the “E”, “P”, and “I” components to restore the standard sequence:

(E)

$x₁ $x₂ ... $x_m $y₁ $y₂ ... $y_n

(P)

P (x₁, x₂, ... x_m) & Q (y₁, y₂, ... y_n)

(I)

& x₁ =₁ v₁ & x₂ =₂ v₂ ... & x_p =_p v_p
& u₁ =_m+1 v_m+1 & u₂ =_m+2 v_m+2 ... & u_q =_m+q v_m+q

(E)

$x₁ $x₂ ... $x_m

(P)

P (x₁, x₂, ... x_m)

(I)

& x₁ =₁ v₁ & x₂ =₂ v₂ ... & x_g =_g v_g

& x_g+1 =_g+1 v_g+1 & x_g+2 =_g+2 v_g+2 ... & x_p =_p v_p

We can now show how natural join and composition are interpreted. Take the SUPPLY and USE records, with the EPI interpretations:

(SUPPLY) $x $y x supplies y
& x is the same supplier as S1 & y is the same part as P1
 

(USE) $y´ $z y´ is used in z
& y´ is the same part as P1 & z is the same project as J1

$x $y $y´ $z x supplies y & y´ is used in z & y is the same part as y´
& x is the same supplier as S1 & y is the same part as P1
& y´ is the same part as P1 & z is the same project as J1

$x $y $y´ $z x supplies y & y´ is used in z & y is the same part as y´
& x is the same supplier as S1 & z is the same project as J1
 

In English: S1 supplies something that is the same part as something used in J1.

There can, of course, be no knock-down argument to demonstrate the value of the EPI interpretation, or of any other foundation. What we are looking for are a variety of gains throughout the ER model:

The EPI interpretation is formal, as is that of Relational DMLs; indeed it is the very same logic, the FOPC, but somewhat differently applied. In particular, it avoids the use of absolute identity, and this enables values to be interpreted as common rather than as proper names. Consequently, although it covers data manipulations, and therefore derived records as well as base records, it does not give rise to join traps.
 

A method of defining (rather than, more vaguely, “describing”) entities is provided: criteria of definition, i.e. of application and identity. Attributes also are defined using a criterion of identity. The links (non-attribute bearing relationships) between entities are describable in terms of the criteria of definition of the connected entities.
 

We can say precisely what we mean by a record “representing” an entity (instance): the interpretation of the record is a proposition asserting (1) that something exists (hence “entity” – an existent), (2) that it is a certain sort of thing (falls under a given criterion of application), and (3) that it meets certain criteria of identity (one for each attribute).
 

Notice that if an attribute or attributes form a candidate key of a record, then its criterion of identity (or the conjunction of their criteria of identity) is the criterion of identity of the entity represented by the record. The foundation of higher normalisation theory, from 2nd to Boyce/Codd normal form is determinacy. It will be appreciated that attribute c determines attribute c´ under this condition: for all x, x´: IF x =_c x´ THEN x =_c´ x´
 

We can distinguish between independent entities (and their subtypes) on the one hand, and relationships or dependent entities on the other, because the interpretations of the former use only one existentially quantified variable. We have a formal interpretation of the resolution of many-to-many relationships into entities: the introduction of an additional variable to which are applied attribute identities that do not hold of any of the originally related entities.
 

Metaphysics and metaphysical correctness are replaced by logic and logical correctness: instead of asking “What is an entity?”, or “Are there such things in reality?”, we ask “What is asserted to exist, with what criteria of definition?” Whether the thing asserted to exist is a record-keeping entity (like an invoice, where the thing asserted to exist by the record is the record itself), or a real world entity (like a person), is no part of the formal theory. This avoids any problems with dubious entities (be they vacancies, intimations or mortality, or anything else): once provided with criteria of definition, logical propriety is assured.
 

Finally, we can even use the EPI approach to distinguish “types” from “instances” without resort to the complexities of set theory. A person instance is something that falls under the criterion of application, “... is a person”, and has the criterion of identity, “... is the same person as something” (or even, “as itself”). The person type is something that falls under the same criterion of application, but has the criterion of identity, “... is a person, and so is ...”: the type is an entity with a very coarse classification (a single equivalence class into which each person falls).

The ER model has proved extremely useful in practice, especially in the analysis of data, the design of databases, and the informal interpretation of records. The interpretation proposed here (“EPI”) is that a record be interpreted as a proposition asserting the existence of one or more things (entities), a predicate giving their criterion of application and/or their relationship, and a number of identities (attributes) that hold of them. This interpretation is formal (it is expressed in the first order logic), and it enables the interpretations of derived records to be found effectively and safely (without join traps).

The EPI interpretation appears promising as an aid to understanding and teaching the analysis of data; it clarifies and systematises the definitions of entities and attributes; it provides explanations of the terms of art of the ER model (entity, attribute, relationship); and it gives insight into the nature and importance of higher normalisation.