Here are the top five best-known (see footnote 1).

Lojban at lojban.org is the better-known successor of Loglan.

Loglan at loglan.org is based on formal logic and is to some extent not unlike a transcription of formal logic (see footnote 2).

Ithkuil at ithkuil.net is like the two above but is as compact as possible, allowing to read, write, say, and hear complex thoughts and ideas in a matter of seconds.

Ceqli at http://ceqli.pbworks.com/w/page/5455970/FrontPage is originally inspired by Loglan but later a bit more naturalist

CycL at http://www.cyc.com/documentation/ontologists-handbook/cyc-basics/syntax-cycl/ is a lot like a programming language intended mainly for neural networks. It's more suitable for expressing knowledge than for communication.

All these have Wikipedia articles under their name, except Ceqli (available on Dutch Wikipedia) to which you can go via the link given.