Is it possible that apart from positive and negative charges, there is a third kind of charge that we have somehow, not discovered yet?
If not, is there some proof of the existence of only two types of charges?
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If we did discover some other phenomenon for which the description "charge" was appropriate, we would not call it an electrical charge. For example the gluons which bind an atomic nucleus together carry a triple-state charge which is called "colour charge" after the three primary colours of vision. The theory of electromagnetism is a complete and consistent description of certain natural phenomena. Anything coming outside that set of phenomena would come outside the theory of electromagnetism and we would not call it electrical charge.