One can describe hypertext documents as families of interlinked texts. One
can take each text and decompose (parse) it into sub-texts. I presume context
to have provided meaning for various forms of sub-text (ultimately depending on
the syntax and semantics of the language of the text, in the present case
English; but potentially layering some more precise terminology on it); I
presume that meaning to provide a framework for discussing values
(entities potentially external to the text) and statements
(a.k.a. assertions) about values
; I presume that one can thereby
formalise discussion of relations among values; the context of my own
mathematical web pages takes relations themselves as a flavour of value, only
indirectly discussing any other kind of value, and my plaintext denotations
provide means of expression for relations and tools to manipulate them and, via
them, any other values a discourse, using these denotations, may chose to
introduce.
Two parts of the semantics of a text are particularly pertinent: the text
may make an assertion; the text may stand for
a value. I do not define
any denotations which do neither; and see no reason why a given text need not do
both. Assertions add directly to what the text means. Values are generally
anything context choses to describe as such, in so far as context provides a
means for a text to stand for
(a.k.a. denote) the
value. An expression
or denotation
is a text which stands for a
value, optionally also making some assertions.
A denotation's value may be ambiguous (though its semantics won't be, other
than via its value): a square root of 4
may stand for either 2
or -2
, as indeed may either 2 or -2
or ±2
. The
first form of this is a case of some input to a function that gives a
specified output
, using the function square = (: x.x ←x :) and input 2
in this case; this shall indeed be a common form of ambiguity in
expressions. This is just one special case of any value satisfying some
particular set of conditions
; and, indeed, I'll have texts that introduce
(for example) any linear map (U: f |V)
, which may indeed be introducing
not only the linear map f but the linear spaces U and V; in such a case all of
the values are ambiguous, although they are constrained to have the specified
relationship to one another (f is a mapping from all of V into U; all three are
linear). As in this example, assertions made by the denotations (subject to
consistent binding of each name to the same value throughout each sub-text whose
context does not provide it with a value for the given name) are given truths of
the text, so limit the ambiguity.
Such texts denoting ambiguous values serve various purposes; the most obvious (and orthodox) is to provide the means to say something general, that is true for every value the text may denote; the next is to provide the means to quantify over all values that the ambiguous text may denote, for example to collect them up in a set or specify how some function acts on each of them.
Parsing a text breaks it down into its syntactic units and associates assertions with it that are stipulated by the templates used to break it down. Making sense of the text then involves binding names in the text to values that are compatible with the assertions thus made about them; where a subtext quantifies over some of the names introduced in it, each binding of those names compatible with the assertions then goes into building up the meaning of the quantifying sub-text, e.g. as the collection of all values satisfying some condition. Names introduced outside that sub-text and referenced from within it then constrain how the built meaning of that sub-text relates to the rest of its context. For each binding of unquantified names used in a text, we then get a reading of that text, which constitutes one of the things that text says. Thus substituting values for their denotations (potentially) leads to a family of meanings for each text, parameterised by the values of the various ambiguous sub-texts. Indeed, a simple name, used to denote a value, denotes an arbitrary value; its appearance as a sub-text may involve it being the subject of some assertions, so that the text in which it appears only sees those values for the name as are consistent with its context.
Written by Eddy.