Purposely means both "on purpose, by design; intentionally, deliberately".
Why use a semi-colon here rather than a comma?
Purposely means both "on purpose, by design; intentionally, deliberately".
Why use a semi-colon here rather than a comma?
Though I can't be sure until I know your context, I think this is a shorthand notation for multiple dictionary definitions, i.e.
The dictionary definition of purposely is as follows:
On purpose; by design
Intentionally; deliberately
Instead of numbering the definitions in an example like the above, whoever wrote this sentence decided to simply separate them in shorthand:
Purposely means "on purpose, by design; intentionally, deliberately"
The both in your example is incorrect and unneccessary. Both would need to be followed by and in order to be clear and technically correct.
Purposely means both "On purpose; by design" and "intentionally; deliberately."
In addition, as illustrated above, the commas would have to be replaced by either semicolons or hyphens in order to clarify that the 'and' separates the two definitions. I have also used separate quotations on the two definitions for further clarity.