Representations may be said to be compositional insofar as they retain the same meaning across diverse contexts. Thus, "kick" means the same thing in the context of "-the ball", "- a rock", and "- a dog", although it changes meaning in the context of "- the bucket". One might say that according to the principle of the compositionality of representations atomic representations make the same semantic contribution in every context in which they occur. See systematicity, productivity, symbolicism.

The term "compositionality of representations" is also used to refer to a putative psychological regularity that is supposed to support the view that there exists a syntactically and semantically combinatorial language of thought. The locus classicus for this argument is Fodor and Pylyshyn, 1988.
According to Fodor and Pylyshyn, in normal cognitive agents, there exist intrinsic connections between some thoughts and others. Thoughts come in clumps. This putative fact is the systematicity of cognitive representations. The compositionality of representations says something about the nature of the thoughts that are intrinsically connected. It says something about the nature of the clumps of mental thoughts: the thoughts in the clumps are semantically related. The thoughts in the clumps have common terms and predicates, for example. Thus, the thoughts John loves Mary and Mary loves John are compositional sets of representations since they both represent John, loving, and Mary.
According to Fodor and Pylyshyn, the reason that cognitive representations are compositional, as well as systematic is that there exists a syntactically and semantically combinatorial language of thought that respects the principle of compositionality, that atomic representations mean the same thing in all contexts in which they occur. The existence of a syntactically and semantically combinatorial language of thought respecting the principle of compositionality explains the compositionality of representations.
A theory that admits mental representations, but rejects combinatorial structure, lacks a genuine explanation of the compositionality of representations. Even if a normal cognitive agent has thoughts that are intrinsically connected to each other, there is no principled reason why it should be the case that these thoughts should be semantically related. Suppose that one thought involves the mental representation that is intrinsically connected to the mental representation . Even if means John loves Mary, why should mean Mary loves John, rather than say, Alfred likes pizza? There seems to be no principled answer to this question.
Ken Aizawa