I find those two words a little confusing, so what are some contexts in which it’s appropriate to use the word “incoherence/incoherent” rather than “inconsistency/inconsistent”?
Also, can the word “incoherence” be used like this “The Incoherence of X” to imply X not having a solid logical ground?
The words do have somewhat overlapping meanings, and a quick dictionary search shows that the definitions tend to reference each other. But they tend to refer to different things and the connotations depend on context.
In general, inconsistent tends to refer to clear contradiction (often a specific contradiction) on some level, while incoherent can be a more general state of confusion.
For example, if measurements are inconsistent, that means some explicitly disagree with others (e.g., some are high, others are low). If measurements are incoherent, there is generally a greater level of confusion that does not permit a clear interpretation (e.g., the meter can’t be read clearly to even determine the measurements).
When referring to logical proofs or philosophical systems, the implication is similar. A system which is logically inconsistent may not work in all cases, but the failure is likely to be a very specific point of contradiction. A system which is logically incoherent has more fundamental flaws which would perhaps cause the entire system to fall apart. Philosophers do, however, sometimes use the word incoherent to refer to a system with specific contradictions of logic (i.e., inconsistencies). Basically, I’d say inconsistent can refer to a specific flaw (an inconsistency), while there’s no such thing as a single incoherency — the system either passes and is coherent or fails completely and is incoherent.