You open a dictionary and all the words are defined by other words. If concepts have the same circularity as words, ultimately none would have meaning (I suppose that's debatable, but I'm assuming it here). Therefore, it seems like there must be some mental firmware or foundational concepts to ground them all in reality and provide meaning. Would such concepts be undefinable even though we might refer to them? For example, try to define 'time' without using other temporal concepts. Are all concepts definable or are some irreducible?
Analysis of natural language probably belongs to linguistics rather than philosophy. The linked question Since words are defined in terms of other words in dictionaries, leading to infinite loops, does it mean natural languages are meaningless? has typical answers to the general question. Nevertheless the question here is phrased as a problem for directed graphs, and some logical statements can be derived there.
Formally, even if words could be reduced to other words by definition, it is possible that some definitions are circular, and so it is not necessarily true that any words of a language are irreducible.
Even if words could be reduced to other words by definition, and there were words that could not further be reduced, that does not mean those words cannot be defined. Such words could still be "defined without reduction". The relationship "X reduces Y" and "X is used in some definition of Y" should be independent (it seems to me). So irreducible is not necessarily the same as undefinable.
A key point here is that a definition of a word in natural language is typically not unique, so the same word could be defined in two similar ways using very different words. So reducing by means of definition seems inadequate for most purposes. The relationship between concepts and the relationship between the word used for the concepts are not strict enough in natural language to make useful statements.
The comments above are helpful, let me make one addition which may be of interest.
- Definability has been well- treated in the literature. In particular, we have a number of constraints on when a certain concept is definable with respect to a background language.
The high level notions (on a particular notion of definability) are these: that the concepts be "explicable" purely in terms of the ground language, and that they add no additional inferential power to the language.
For a brief introduction, see as always : https://plato.stanford.edu/entries/definitions/.
- Seen thus, your question is one of asking if the concepts of the base language are definable in terms of a more primitive language. It is easy to give examples in which this is true - merely extend any language twice, and take the given base language to be the second extension. What you really wish to know though, is whether given that there is indeed a truly primitive language- whether we can define in it terms other than itself. If we allow the definability relation to be trivial, this is the case, of course, if the language is indeed truly primitive, there is no other language in which we can express its terms (or should be, I haven't worked this out myself).
Thinking of Feynman’s words on ‘what is energy’, we can try as hard as we like to understand the physical world but however thorough or incomplete our understanding we must eventually put up with the idea that it just is what it is, and so ultimately we have no irreducible basis for anything. Even ’cogito ergo sum’ isn’t irreducible because even though we are logically obliged to agree that this must be true, we can still ask questions about thought and consciousness. So our language doesn’t exist in a vacuum but is based on the idea that things are just as they appear to be, which is not irreducible but it’s all we’ve got.