Q:

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:

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.
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).
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.

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

A:

Languages are not formal logic systems, and words do not derive their meanings from the definitions in dictionaries. If they did, then your argument would mean something, but as it is what you have is an amusing but pointless exercise.
Words derive their meanings from their shared usage in a linguistic community, and those meanings are ultimately grounded in shared sensory human experiences. When you were a baby and were learning to speak, at some point someone put you atop a rigid flat surface supported by four roughly cylindrical poles with another rigid surface a long one side, and they told you to stay in your chair. Eventually, through repetition and familiarity, you came to recognize that the object on which you sat was a "chair". Then you learned to generalize the concept of "chair" to other, similar objects, and eventually to distinguish "chair" from "stool" and "table".
At no point did you ever learn the definition of the word "chair" as it appears in the dictionary. Instead, the word begins as a label for an object in your sensory experience, is quickly generalized to other, similar objects, and eventually acquires a set of semantic and linguistic relations with other objects in your experience. You also learn to apply names to less concrete phenomena such as "red" or "fear", and eventually to non-physical constructs such as "mathematics". These words mean things because everyone who speaks English has a roughly commensurate idea of what they mean, an idea which begins in shared sensory experience and is built up over many years by a process of enculturation.
Dictionary definitions are post-facto attempts to describe what words mean to the community of English speakers. Words don't derive their meaning from dictionaries; rather the dictionary attempts to capture the meaning which is ascribed to the word by those who use it. As such, the dictionary is naturally circular, because it does not actually contain the sensory world or the culture of the speakers. If language could only refer to itself, then it would be impossible for it to mean anything in the usual sense. But because language refers to an external world, meaning is possible.

A:

At the risk of oversimplifying:
Your premise that "meaning is imbued through definition" is wrong. Meaning is imbued through usage. Dictionaries merely describe that usage.

