logic - How do definitions of words imbue meaning?

How do definitions of words imbue meaning?

To give you a gist of what I try to discover, I'll define a collection of sets of words and show that their intersection contains all 'circular defined' words. How do these words gain their meaning? And how can definitions using these words be considered meaningful?

Let W be the set of words in a dictionary. Denote by w any word in W. The dictionary contains lemmas of each word it lists; for each word w, the lemma L(w) for word w is a subset of words in W. Let's say that the lemma is also the definition of the word w; for our purpose, we ignore examples of usage (otherwise most if not all words would have circular definitions).

Define the following collection of sets:

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  D(1) := {w in W | There is a w' in W such that w in L(w')}, the set of words used in definitions of other words.

  
  


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  for n>=1, D(n+1) := {w in D(n)| There is w' in D(n) such that w in L(w')}, the subset of D(n) of words used in definitions of words in D(n).

  
  


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  D := {w in W | For all n>=1 we have w in D(n)}, the intersection of all D(n).

  
  

We have constructed a nested collection of sets: if n>m then D(n) is a subset of D(m) by definition, and D is their intersection. D is the largest set of words which is such that each of its words occurs in the definition of (at least) one of its words.

Lemma. Since W is finite, D = D(n) for some finite n.

Proof. If not, it means that for each n, there is a D(m) such that D(m) is a strict subset of D(n) and we can construct an infinite collection of strict subsets. But there are only a finite number of words in W. Hence, there is an n such that D = D(n).

It follows that we can effectively determine D for any dictionary W by going through W, D(1), D(2), etc. creating the next D(n) until D(n+1) = D(n) = D.

If W is the English language, then D is not empty.

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  It contains the word 'an' (see http://dictionary.reference.com/browse/an?s=t, its lemma contains the phrase "an initial vowel").

  
  


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  It also contains cycles of length 2 or longer, such as {oak, acorn} and {foot, ankle}; these words are used in their mutual definitions.

  
  

The basic question is about the set D. How do words in D gain meaning from their definitions?

They seem to be examples of words with 'circular definitions'. Worse, words in D can occur in any definition (as they belong to D(1)), so how do they help define other words if their own meaning is ambiguous?

Finally, if words' meanings are ambiguous (for example if they have definitions including words in D), how can definitions succeed in taking away this ambiguity?