Re: [ontolog-forum] Constructs, primitives, terms

[William Frank <williamf.frank@xxxxxxxxx>]

All,

Below is the message I previously sent, with the privacy message removed, and completed (it was mistakenly sent before completion -- sorry). 


It seems to me that beyond their being a *parallel* in mathematics,
the concepts of term, primitive term, and construct (or open
_expression_) come from and are standard in formal mathematics and
model theory, since Tarski's "introduction to logic and scientific
method", continuing today, and applied carefully in formal renditions
of everything from arithmetic to set theory to category theory, and in
such collections as "Explanation and Proof in Mathematics."

For example, in an axiomatic arithmetic for the natural numbers, "0"
and "successor" are usually primitive terms, while "+" and "1" are usually introduced as defined terms,

1 might have the definition 
 1 = successor (0)

and   the open sentence

x+successor(y) = successor (x +y)

might be one of the constructs used in the definition of "+"



definitions are expressions of the form:

definedTerm (x1, xn arguments of term)   means construct not containing definedTerm and containing no free variables except x1,... xn

or

for all x1,,,. xN definedTerm (x1, xn arguments of term) IF AND ONLY IF construct containing (x1., xn) but not containing defined term. 

or 

definedTerm (x, ... xn)  = construct (not containing definedTerm. )

(terms like "1" have no arguments, terms like "+" have two arguments.)

in the definition, the definedTerm is in the definiendum (the left hand side of the definition - that which is to be defined), and the defining construct giving the meaning, on the right hand side,  is called the definiens (that which defines) 

there will be a dependency partial order of defined and primitive terms such that one term depends on another for its meaning if the second term is used in the definiens of the definition of the first term. The root nodes in this partial order will be the primitive terms. . 

so, the meaning of defined terms is relative to the meanings of the primitive terms, and the meaning of the primitive terms is constrained by the axioms that are asserted about them (for example, "there is no number such that its successor is zero", constrains the meaning of zero and successor, but does not fully specify their meanings). 



-- 
William Frank

413/376-8167

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