I was wondering how in English we say "I can" and "I can not" the negative is the longer one, in terms of morphemes, but is there any language where the negative is the default and the positive is the longer construct?
This happens in some Dravidian languages. Specifically, in these languages, positive verbs have a tense marker and a person marker; negative verbs differ from them only in lacking the tense marker.
A set of examples from Old Kannada (from Miestamo 2010):
no:ḍ-uv-eṃ see-FUT-1SG ‘I will see’
no:ḍ-id-eṃ see-PST-1SG ‘I saw’
no:ḍ-eṃ see-1SG ‘I do / did / will not see’
As you can see, the negative is formed simply by omitting the tense marker (with the result that it's ambiguous for tense); there's no explicit marker of negation.
I don't think such a language exists. It seems to be a language universal, that positive is default. It's for the same reason that a language may lack a word for "bad", but may not lack a word for "good".