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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?

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closed as off-topic by Jim, Robusto, Matt E. Эллен, Kristina Lopez, Brian Hooper Feb 7 '14 at 6:37

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I do not accept your premise. First, I do not accept that there are a finite sum of English words (consider the Coastline paradox). Second, words are ambiguous or the lawyers would all starve. Finally, I assume you meant gist not jest. –  Elliott Frisch Feb 5 '14 at 3:45
@Elliott. I do not see the relationship between the Coastline paradox (with which I am familiar) and the fact that there are only a finite number of words in the English Dictionary. I can help you a little bit: for each natural number there is a different word and there are an infinite number of numbers. However, these number names still consist of a finite number of words (units, tens, hundred, thousand, million, billion, etc.) Numbers aside, are there really an infinite number of words? - Your second objection is humorous, but does not help. - Finally, I changed jest in gist. Thanks. –  Cuc Feb 5 '14 at 4:03
"How many dictionaries have ever been published?" is a finite number. "How many dictionaries will ever be published?" is unanswerable. Compare any two dictionaries and you will "discover" new words. Hence the coastline paradox. Still, I'm only commenting; because I think your question is flawed. Definitions do not give words meaning; definitions are words! –  Elliott Frisch Feb 5 '14 at 4:14
This question appears to be off-topic because it applies to any language with a dictionary and not just English. –  Jim Feb 5 '14 at 4:17
@Cuc You're very clever, young man, very clever. But it's words all the way down! –  Elliott Frisch Feb 5 '14 at 13:58

3 Answers 3

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.

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I think some words might derive their meaning from dictionaries. One that springs to mind is the word dictionary. Then again, there's also loglan. –  Elliott Frisch Feb 5 '14 at 3:48
@JSB. Thanks for your explanation. The relation to the external world is indeed an important aspect. However, many words do not relate to the external world. For instance, the "generalization process" occurring when a specific table is used to exemplify the concept of table does not have a counterpart in the external world. It is a process that probably works differently (although arguably similar) in each person. –  Cuc Feb 5 '14 at 4:13
@Cuc: You interpret "the external world" too narrowly. Language as such is also part of the external world. Once we have links between a linguistic phenomenon and a sensory experience, this phenomenon thereby functions as an extension of the outside world. It is true that within this linguistic "realm" of the (extended) external world we sometimes create and interpret meaning in a circular fashion, but ultimately there has to be some indirect original connection to sensory experience. See also en.wikipedia.org/wiki/Hermeneutic_circle , extremely important in the humaniora. –  Cerberus Feb 5 '14 at 16:06

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.

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Thanks, but this does not help me in real discussions. As I found too often, words seem to have different meaning for me as for another person. The suggestion was made to mutually agree on a definition (in a dictionary or through any other means). I question if this is possible, since all words I use, have 'my own' definition, and they are basically structured like I have described, even if I did not put all those definitions in words (yet). In other words, how can I know in any case that I mean the same with a certain word as somebody else? If not through definition, then how? –  Cuc Feb 5 '14 at 4:07
I think you're asking me: "If we can't ever be sure that a word means the same thing to both of us, how can we ever reach a shared understanding of anything?" The short answer is: carefully. The longer answer is: we can perform lots of "tests" to check and re-check that we really do understand one another. (English as a Foreign Language teachers call them "Comprehension Check Questions".) So long as we have some shared understanding then, through dialog and a Bayesian process of checking and re-checking, we can reduce the probability of error to a tolerable level. –  Pitarou Feb 5 '14 at 11:44
The Bayesian approach leads us to the "tolerable level" of probability, which is, even if the term itself is clear, a vague criterion. Yes, I wonder how we can ever agree on anything. Lately this has been a recurring problem. I wish there were a quick fix. –  Cuc Feb 6 '14 at 22:44

What a great question. Here is my answer:

The process is constructive. As others have stated there are some base concepts that are experiential - sort of a foundational consensus reality. From there we combine there concepts and get a definition for something more abstract. This process continues until we can define everything that we do. So the definitions imbue meaning by appealing to the concepts associated that we have to the defining words, which are then fundamentally, after some number of steps based on something foundational and "simple".

Now, because the foundations are subjective you have necessarily subjective definitions all the way up the line. Indeed, you can never know much about anything outside of your own mind c.f. Solipsism/Subjective Idealism.

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Thanks, could you up my question? From your answer I got the idea that the set D as defined above may in fact contain a lot of words which describe real objects in terms of other real objects. I need to check that sometime. Complicating the process of abstraction is that even the word 'table' is already an abstraction (or concept), although instances of it are percepts (not concepts). As noted by another comment, the dictionary does not contain the experiential world (but sometimes it has pictures, a different mode of expressing an instance). I'm looking into your suggestions. Thanks. –  Cuc Feb 5 '14 at 6:49
@Cuc I'm already upping your question :) and I'm still considering the issue as well. –  d'alar'cop Feb 5 '14 at 13:13

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