General Synonyms:
* Assume
* Presume
* Suppose
* Hypothesize
* Propose
* Assert
* Declare
* State
More specific synonyms:
* Axiomatize (for mathematical postulates)
* Demand (for a strong statement of need)
* Request (for a polite statement of need)
* Require (for a necessary condition)
Antonyms:
* Disprove
* Refute
* Contradict
The best synonym will depend on the specific meaning you want to convey. For example, if you are talking about a mathematical postulate, then "axiomatize" would be a good synonym. If you are talking about a request, then "demand" or "require" might be more appropriate.