Re: [ontolog-forum] Axioms and definitions

[rrovetto@xxxxxxxxxxx]

For a given term, symbol, _expression_, word, one can seek the definition by exploring the thing that may be referred to by the term and form a description based on that, and/or by identifying and understanding the meaning (conceptual or abstract) and use of the term. Whether the meaning of the term overlaps or is identical with the description formed by investigating the thing to which it refers is a question to ask. Context is also important, but maybe more so for nominal definitions.

In some philosophy courses, specifically logic, but also including ontology (applied or philosophical), a few things are commonly presented:
- The biconditional is presented as the symbol used for defining and for expressing logical equivalence,
-- Necessary and sufficient conditions are used for providing the definition of a term. When one identifies the n & s conditions, one has identified the (a?) definition of the term

- Substitution is associated with definitions and the biconditional. The definiendum and the definiens should be interchangeable in statements using the term

But, generally speaking, not all definitions are (or need to be) in that form.

When it comes to symbolic logic and forming a definition with axioms (axiomatic definitions), the full meaning of a term or concept is rarely represented in axiomatic form. One can list numerous axioms, trying to get at the complete sense or meaning of the term, but might fall short due to the expressive limitations of the formal/artificial language, fail to account for something, miss a counterexample, etc.

And although it might be common to present or use genus-species definition or Aristotelian definitions in philosophy or ontology, there are other types of definition (as others in the thread and elsewhere have mentioned). And needless to say, inquiry into definition or defining predated Aristotle.

Indeed, I asked related questions at the ICBO 2013 Definitions in Ontology workshop:
Q1: What are other types of definitions that can be used in applied ontology (and philosophy)?
and
Q2: Why is the Aristotelian type commonly used and being presented?

(perhaps it's not so common?)

Some answers to Q2 might be for computational efficiency, or ease-of-organization (as in biology), etc.
I encourage readers of this thread to answer both questions here (and in more detail if possible). I'd like to hear responses from others (philosophical and comp sci answers). 

Definitions can be distinguished in various ways, but some types of definition include (if others have been mentioned any of following already, pardon the repetition. But this is also an exercise in self-study, memory and learning):

- Extensional definition(def): citing the (collection of) objects the term applies to or denotes. Ostensive definition as a sub-type, perhaps.

- Intensional def: giving the meaning of a term
- Ostensive def: definitions by giving examples

- Stipulative def: one prescribes the definition to a term/symbol

- Lexical def: states the meaning of words presently used by speakers of the natural language. As in dictionaries.

- Citing synonyms alone
- ...

Respectfully,
Robert Rovetto

rrovetto[at]buffalo[dot]edu -- For general communication
ontologos[at]yahoo[dot]com -- For ontology-specific communication only
http://ontolog.cim3.net/cgi-bin/wiki.pl?RobertRovetto

On Mon, Dec 2, 2013 at 9:39 PM, Patrick Browne <patrick.browne@xxxxxx> wrote:

Hi,
I am trying to understand the distinction between terms 'axiom' and 'definition' in an ontological and logical context.
Do the terms have the same meanings in logic and as they do in an ontological context?
Are the terms really distinct in their respective contexts?
Below is my background research and a fragment of BFO that I am currently working on.
I appreciate that the knowledge level of contributors to this forum is extremely high.
So, if my above questions and the research below are naive perhaps there is a beginners level forum that I could use.

Regards,
Pat Browne

== From philosophy ==
According Joseph[1] a definition “makes explicit the intension of a term, the essence it represents”.
 Joseph lists two types of logical;
 1)logical definition expresses the essence of a species in terms of its proximate genus and its specific differentia,
 2)definitions definition by property. 
Other types of definition described by Joseph are: causal definitions,descriptive definition, definition by example.

An axiom is a statement for which no proof is required Flew[2]
A definition is a process or _expression_ that provides the precise meaning of a word or phrase. Flew[2].
An axiom may be thought to constitute an implicit definition of the terms it contains or to contribute to such definitions. Flew[2]
 
== From Mathematics ==
Definitions are only required to be understood, they do not assert the existence or non-existence of anything. (Heath[3] page 119)
According Hunter[3] formal system S is a formal language L with deductive apparatus given by
1) laying down by fiat that certain formulas of L are to be axioms.
2) laying down by fiat a set of transformation or inference rules

Necessary features from of a good mathematical definition from VanDormolen & Zaslavsky[8] are as follows
Criterion of hierarchy: According to Aristotle, any new concept must be described as a special case of a more
general concept a square is a quadrilateral (general concept) with four congruent sides and one right angle
(special case).
Criterion of existence: Also required by Aristotle this criterion demands proof that at least one instance of the newly defined concept exists.
Criterion of equivalence: If one gives more than one definition for the same concept, one must prove that they are equivalent.
Criterion of acclimatization: A definition must fit into and be part of a deductive system.


== From Ontology/Knowledge Representation ==
In the OWL language the term defined class means a class that has necessary and sufficient conditions for membership.
The  “necessary” part alone is required for the semantics of IS-A that permits generalization or specialization.
Definitions in terms of primitives ultimately derive from Aristotle's mode of definition of genus and differentiae Sowa[5]
Three views on definition: classical, probabilistic, prototype, definitions can specify type(104) Sowa[5]
Types of definitions constructive, non-constructive, implicit, explicit, extensional, intensional, recursive Sowa [6]

== From my own research ==
This example represents my efforts to represent parts of BFO[7] in equational logic using loose semantics
Equations labelled with A indicates that the formula is an axiom, ‘D’ that it is a definition.
Mereology equations, variables A5-D8 are universally quantified, D9 has one existential quantifier
eq  [A5] : part(x, x) = true .
ceq [A6] : part(x, z) = true if (part(x, y) and part(y, z))  .
eq  [A7] : (x = y) = if (part(x, y) and part(y, x)) then true else false fi .
eq  [D8] : properPart(x, y) = (part(x, y) and  not(x = y)) .
-- Skolem functions are named using the variable in the BFO manual and axiom number
op  z9 : Entity Entity -> Entity
eq  [D9] :  overlap(x, y) = (part(z9(x,y), x) and part(z9(x,y), y)) .



 References
[1] Joseph, S. M. (1937). The Trivium: The Liberal Arts of Logic, Grammar, and Rhetoric, Dry Books.
[2] Flew, A. (1979). A Dictionary of Philosophy, Pan Books.
[3] Heath, T., Ed. (1925). Euclid: The thirteen books of the elements., Dover (1956).
[4] Hunter Geoffrey Metalogic: An Introduction to the Metatheory of Standard First-Order Logic,, University of California Press, 1971
[5] Sowa, J. F. (1984). Conceptual structures: information processing in mind and machine, Addison-Wesley Longman Publishing Co.
[6] Sowa, J. F. (2000). Knowledge Representation : Logical, Philosophical and Computational Foundations Brooks/Cole.
[7] Grenon and Smith SNAP and SPAN : Towards Dynamic Spatial Ontology
[8]Van Dormolen, J., & Zaslavsky, 0. (2003). The many facets of a definition: The case of periodicity. Journal of Mathematical Behavior, 22, 91-196.

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