Also beware the temptation to design a grand hierarchy of all meanings. There have been many attempts in the last 300 years: [Eco95] often as part of the design of a universal or perfect language. None of them have stuck. On the other hand you can use these attempts as a source of ideas to model specific application domains and user concepts. There is a well tested classification system for English called Roget's Thesaurus.
Each application domain has its own languages and these need to be understood when developing software. There is some evidence that constructing a clear and precise model of such a domain is a step in the direction of better software. Mathematics and logic can be very helpful with eliminating the "goo and dribble" and boiling things down[Asimov].
What follows here are some examples drawn from the twentieth century. By the end of the twentieth century and start of the twenty-first century there were several logical models of natural language placed on the web in the form of ontologies. Below are some examples of one way the MATHS notation could be used to express these hierarchies but they are not intended to be a complete and finished description. Professionals have been working on these, see [ Ontologies in index ] [ lookup.php?search=ontolog ] for more.
This page gives examples of how to use the logical fragment of MATHS to construct models of natural world phenomena like dogs, cats, rocks, and parents(STUFF). This is the best approach when we want to express properties of the objects or entities in a domain.
An alternative would be to use Sets
and Relations instead of Nets of variables and axioms. This is
close to the way that abstract algebras like groups are modeled
in MATHS
[ notn_6_Algebra.html ]
[ math_43_Algebras.html ]
[ notn_7_OO_vs_Algebra.html ]
for example. The SUMO ontology is also best modeled
this way. See
[ Suggested Upper Merged Ontology ]
at the end of this page.
Abbreviation.
Natural languages have large numbers of terms indicating objects of
different sets and types. This are typically organized in a hierarchy. It
is useful to have a simple way of expressing such a hierarchy:
A statement like
either A or B.asserts the existence of two Boolean variables A and B and the property that precisely one of them is true. The above is short hand for:
A::@,
B::@,
|- A or B.
|- not(A and B).Similarly,
either A, B, or C.asserts the three conditions, but only one is true at a time. If A is true then B and C are false.
.RoadWorksAhead
I may be moving the following sample into my directory of samples:
[ http://www.csci.csusb.edu/dick/samples/ ]
before September 2004.
Sampler of Ontologies
Mistakes in learning science:
.RoadWorksAhead
.RoadWorksAhead
Leech preents the idea (from anthropology) that terms in natural language can be given meaning in terms of components:
man +HUMAN+ADULT+MALE
woman +HUMAN+ADULT-MALE
boy +HUMAN-ADULT+MALE
girl +HUMAN-ADULT-MALEComponents match the idea of attributes in a MATHS Net.
The following Nets translate Leech's diagram on page 121 of his book.
The above creates a net with Boolean attributes: countable, mass, concrete, abstract, singular, plural, ... and so on.
We can now express Leech's example above:
. . . . . . . . . ( end of section Stuff) <<Contents | End>>
}
For more on natural spatial relations see [ math_93_Graphics.html ]
Tenses, Time, and Temporal modalities
Also see [ logic_9_Modalities.html ]
. . . . . . . . . ( end of section Tenses, Time, and Temporal modalities) <<Contents | End>>
Colors
Leech pp234-237.
Anthropology??
For R: {father, mother, son,daughter, child_of, sister, brother, ....}, x R y::=x is a R of y.
ownership:@OWNERSHIP. for x,y:$ STUFF, x owns y ::= x /owner;ownership;owned y.
Let
Here is my second draft at transmogrifying the top of SUMO into MATHS. Because it is expressed in terms of sets of objects and the properties of the sets (not the properties of the objects in the sets) it best to express it as a net of set definitions and axioms about these sets.
. . . . . . . . . ( end of section Sampler of Ontologies) <<Contents | End>>
Systems English -- a subset of Basic English
P3Basic English - a subset of Basic English that uses the present tense(P) and third person only(3).
Example Queries
A semantic approach is to use natural predicates.
Or infix operators with structures
. . . . . . . . . ( end of section Systems English -- a subset of Basic English) <<Contents | End>>
. . . . . . . . . ( end of section Semantics of English) <<Contents | End>>
Notes on MATHS Notation
Special characters are defined in
[ intro_characters.html ]
that also outlines the syntax of expressions and a document.
Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.
The notation also allows you to create a new network of variables and constraints, and give them a name. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents.
For a complete listing of pages in this part of my site by topic see [ home.html ]
Notes on the Underlying Logic of MATHS
The notation used here is a formal language with syntax
and a semantics described using traditional formal logic
[ logic_0_Intro.html ]
plus sets, functions, relations, and other mathematical extensions.
For a more rigorous description of the standard notations see