By bidirectional I mean, if word A is synonymous with word B, does it follow that word B is always synonymous with word A? Are there any common exceptions to this rule?
Extending this - if word A is synonymous with word B, and word B is synonymous with word C, word C may or may not be synonymous with word A?
Answer
Perfect synonyms are equivalent, which means that all you say in your answer is true for them. But perfect synonyms are really unusual; some people say they don't exist at all.
In reality, synonyms are usually just partial, which means that they share a common meaning in some circumstances, but not in all of them. So the symmetric property (if A is a synonym of B, B is a synonym of A) holds true, but not the transitive (if A is a synonym of B and B is a synonym of C, A is a synonym of C).
Example: lead is a synonym of main (lead actor = main actor), graphite (in a pencil) or conductor (in electricity), but these words are not synonyms between them at all.
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