A conceptual system is a system of abstract concepts, of various kinds. The abstract concepts can range "from numbers, to emotions, and from social roles, to mental states ..". In humans, a conceptual system may be understood as kind of a metaphor for the world. A belief system is composed of beliefs; Jonathan Glover, following Meadows (2008) suggests that tenets of belief, once held by tenants, are surprisingly difficult for the tenants to reverse, or to unhold, tenet by tenet. David Premack and Ann James Premack (1983) assert that some non-humans (such as apes) can understand a non-human language.
The earliest activities in the description of language have been attributed to the 6th-century-BC Indian grammarian Pāṇini who wrote a formal description of the Sanskrit language in his ' (Devanagari अष्टाध्यायी). Today, modern-day theories on grammar employ many of the principles that were laid down then.
In the formal sciences, formal systems can have an ontological status independent of human thought, which cross across languages. Formal logical systems in a fixed formal language are an object of study. Logical forms can be objects in these formal systems. Abstract rewriting systems can operate on these objects. Axiomatic systems, and logic systems build upon axioms, and upon logical rules respectively, for their rewriting actions. Proof assistants are finding acceptance in the mathematical community. Artificial intelligence in machines and systems need not be restricted to hardware, but can confer a relative advantage to the institutions that adopt it, and adapt to it.