Tom,
I agree with this, and I would hope that most of us do:
“I identify relational tables as types whose tokens are the rows of those tables. By aligning
this interpretation with the perhaps more familiar one of relational tables as statement schemas and rows of those tables as statements resulting from instantiating each of the variables in those schemas …”
Since the definition of a Table is essentially the rules for the valid structure and content of a row, it seems clear that a table is, or at least conveys,
a type of which the rows are instances/tokens. In a similar way, if a row is seen as a statement, then the table is a grammatical pattern (i.e., type) for the statement.
I am not sure why you say “instantiating each of the variables in those schemas (existentially-quantified
statements)”. If I take the parenthetical phrase to characterize 'schemas', I think it is false. The existence of a table (like the existence of any predicate) does not
assert the existence of any row that satisfies it. The interpretation we impose on the row-as-statement asserts the existence of some state of affairs in the world that the table content models, and the corresponding existence of the things (denoted by values)
that play roles (denoted by variables/columns) in that state of affairs. And I assume that must be what you meant.
'Existence' is always in a particular world/universe, and the average business database these days may cover a complex of actualities, hypotheses, and facts
about types-as-things (as in decision tables) – a world that consists of different kinds of worlds.
-Ed
From: Thomas Johnston [mailto:tmj44p@xxxxxxx]
Sent: Wednesday, October 28, 2015 12:59 AM
To: Edward Barkmeyer; [ontolog-forum]; Pat Hayes
Subject: Re: [ontolog-forum] Prospects made into Customers and Vice Versa
It sounds like I was writing my reply to Pat at the same time you were writing that comment. That is, it seems that our two comments are in substantial agreement.
If you don't agree, let me know.
I repeat here that I withdraw the ill-considered term "time-varying set". But contrary to the impression which Pat's response may have left others with, I was
at worst writing carelessly. Having passed doctoral examinations in set theory and logic (albeit in a Philosophy Department, long ago), I protest that I hardly could have been as ignorant as Pat's comments seemed to imply I was.
However, I think that relational theory and its constructs, and its implementations as relational DBMSs and databases, are based on an unrecognized ontology,
which I attempt to express, and to relate to relational theory and its mathematics, in Chapter 5 of my second book
Bitemporal Data: Theory and Practice. Not surprisingly, the concept of a temporal sequence of sets, which you mention, is an important concept in that book, in my first book on temporal data, and in both my patents
on some new ideas about temporal data.
In developing this ontology, I identify relational tables as types whose tokens are the rows of those tables. By aligning this interpretation with the perhaps
more familiar one of relational tables as statement schemas and rows of those tables as statements resulting from instantiating each of the variables in those schemas (existentially-quantified statements), my own understanding of these concepts was deepened.
This work led me to distinguish a third temporal dimension besides the standard two, my third dimension interpreting a third set of time periods associated with rows in tables as the time during which a "speech-act source" (a person or organization) remained
committed to a speech act associated with those row-represented statements. One such speech act is the speech act of asserting that the statement represented by a row in a table is true; a companion speech act is the speech act of withdrawing such an assertion.
I also clarify my speech-act person/time dimension by extending Grice's rules of conversational implicature to the actions of maintaining and querying databases.
Having made these abbreviated accounts of my work on temporal data in a number of prior comments in this forum, though, I won't carry this comment any further.
A relational table is a thing that exists independently of its content. A 'set' is defined to *be* its content. If you change the
content, you get a *different set* by definition. That is why anyone with a mathematical background objects to terms like "varying set". It is an oxymoron.
The idea "temporal sequence of sets" is exactly the 4-D idea of a 'whole life individual' being realized as a sequence of 'temporal parts'.
What you are talking about is the 'extension' of a rule, which is a set at any given time (for any given universe).
Response to Pat's Comment
Are there time-varying sets, in the sense of sets whose membership varies over time? Of course not. And so, although I never said that the
term "time-varying set" referred to the notion of a set whose membership varied over time, it is certainly easy to read the term that way. So I spoke carelessly, and I apologize for that.
I seem to recall adopting the term "time-varying set" from Chris Date. That doesn't legitimate the term, but it may explain why it came to
mind as I wrote my comment. And the fact that an important author in the field of relational databases also uses the term suggests that there is something that he, like I, think that the term might usefully be referring to.
In my case, as indicated in my reply to John Sowa in this same thread, I think the term is a reasonable one to be used to refer to a specific
temporal sequence of sets. The idea is that each transformation of (update to) a relational table puts that table in a new state; and it is each of those states that is a set, and the temporal sequence of those sets that I used the term "time-varying set"
to refer to.
I expressed this idea much more clearly in my reply to John Sowa in this thread, written at close to the same time as my reply to Pat. Here's
a summary statement of what I said:
(i) A table in a relational database is a time-varying object.
(ii) A transformation to the contents of a table (i.e. an insert, update or delete) replaces the current state of the table with a different
state.
(iii) Each of those states is a set – specifically a subset of the Cartesian Product of the ordered set of sets which makes up the columns
of the table.
(iv) At any point in time in the life-history of a table, it is in one and only one state; it physically realizes one and only one set.
(v) The temporally-ordered sequence of those states, then, constitutes the life-history of that table. (This temporally-ordered sequence is
not itself a set, because the same set may occur in the sequence at multiple non-contiguous periods of time.)
(vi) I used the name "time-varying set" to refer to that temporal sequence of sets.
(vii) What relates the sets in each of these "time-varying sets"? If we use the Principle of Extension, I don't see a straightforward answer
to the question. But if we use the Principle of Abstraction, the answer is that the sets in a "time-varying set" are related by having (a) the same universe of discourse, and (b) the same "predicate" (as Stoll puts it), or "set membership criterion" as I would
prefer to put it.
As an aside I also note (since this topic is one I have worked on for a long time), that the standard (ISO 9075:2011 and/or TSQL2) theories
of bitemporal data include the notion that there are
two "time-varying sets" (in this sense) corresponding to each table, one tracking the temporally-ordered change of state of the objects represented by rows in each table, and the other tracking the temporally-ordered change of state of each table itself.
With this interpretation of the phrase "time-varying set", and with relational database tables being the original topic under discussion, my
original point can be rephrased like this: a relational table is a "time-varying set" (i.e. a temporal sequence) of table states, each state being a specific collection of rows. By the Principle of Abstraction, what each of those states has in common is that
they are defined on the same universe of discourse (a Cartesian Product of sets) and use the same set membership criterion.
And I once again apologize for using a term which originally seemed innocuous enough, but which I now realize is more naturally understood
the way Pat understood it – as an oxymoron. I will henceforth try to use some such term as "temporal sequence of table states (sets)".
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