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Reflexivity is, quite simply, self-reference.

Hofstadter beats the subject to death (1979) by discussing at length how sentences can be self-referencial. For example, "This sentence is short."

People are reflexive when they make references to themselves. "I am happy."

Researchers speak of reflexivity when they openly consider their personal biases and consequent influence in their science. Agency, of the structure and agency duality, in its most active form is clearly attributable to reflexivity. And semiotics, the science of symbols, depends profoundly on reflexivity in a process known as semiosis. In social science, reflexivity is not a subject in itself, only an acknowledged attribute of certain behaviours. Have a look at Ryan for a nice little discussion and some references that go past what I want to here.

Reflexivity also operates at a higher level than immediate self-reference. I consider software to be reflexive, reflexiveness being an attribute of its nature. Language is similar in this regard, being highly capable of expressing reflexivity. (See linguistic reflexivity.) Tools can operate the same way.

So, why go on about all this? Reflexivity is the hallmark of consciousness.

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