Network models are given for self-referential expressions in the calculus of indications (of G. Spencer Brown). A precise model is presented for the behavior of such expressions in time. The extension of Brown’s calculus by F. Varela is then shown to describe behavior invariant properties of these networks. Network design is discussed from this viewpoint.

Key words: Self-reference, networks, calculus of indications, calculus for self-reference.

We discuss the relationship of G. Spencer-Brown’s Laws of Form with multiple-valued logic. The calculus of indications is presented as a diagrammatic formal system. This leads to new domains and values by allowing infinite and self-referential expressions that extend the system. We reformulate the Varela/Kauffman calculi for self-reference, and give a new completeness proof for the corresponding three-valued algebra (CSR).

Key words: George Spencer Brown, multiple-valued logic, self-reference, three-valued algebra

The purpose of this essay is to sketch a picture of the connections between the concept self-reference and important aspects of mathematical and cybernetic thinking. In order to accomplish this task, we begin with a very simple discussion of the meaning of self-reference and then let this unfold into many ideas. Not surprisingly, we encounter wholes and parts, distinctions, pointers and indications. local-global, circulation, feedback. recursion, invariance. self-similarity, re-entry of forms, paradox, and strange loops. But we also find topology, knots and weaves. fractal and recursive forms, infinity, curvature and imaginary numbers! A panoply of fundamental mathematical and physical ideas relating directly to the central turn of self-reference.

This is the fifth column in this series on “Virtual Logic.” In this column we shall begin by recalling the “Descent into the Form” that was described in the previous column, and how this descent makes clear how the mark of distinction can be seen as self-referential. We then relate this direct appearance of self-reference to other models and to Godel’s Incompleteness Theorem. The author apologizes beforehand for the ending of Part III of this essay with its modulation on the final lines of “Little Gidding” by T. S. Eliot.

This is the sixth column in this series on “Virtual Logic.” In this column we shall give a short exposition of how symbolic logic is illuminated by the calculus of indications. Columns four and five began an introduction to the calculus of indications. Nevertheless, we shall be self-contained here and recall this construction in section 1.

This is the seventh column in this series on “Virtual Logic.” In this column I will discuss an imaginary machine devised by the logician Raymond Smullyan. Smullyan managed to compress the essence of Gödel’s theorem on the incompleteness of formal systems into the properties of a devilish machine. This column consists in two parts. In the first part we find a story/satire about such a machine, with the Smullyan structure at its core. In this story, the protagonist is bent on detecting a flaw in the machine and he operates with strict two-valued logic. In such logic a statement is either true or false. Thus we call the statement “If unicorns can fly then all numbers are less than pi.” true because it is not definitely false. In general “A implies B” is taken to be false only if A is true and B is false. This is the one significant case where “A implies B” must be false. All other cases, such as A false and B true are taken to be true. This is the classical logical convention. It works quite well in its own domain, but it has its limits. One of these limits occurs when there is a gradation of qualities. For example in statements about tall and short the truth is relative to your idea of this discrimination. Another limit is in the realm of self-referential statements. Certainly the Liar Paradox – “This statement is false.” is neither true nor false in any timeless sense.

This is my eighth column on virtual logic. In this column we shall consider a mode of paradox resolution that I call the “Flagg Resolution” after its inventor James M. Flagg.

This is column number nine in the series. We take as our theme the recent movie “The Matrix.” The Matrix is a cinematic exercise in virtual reality and virtual logic. It is not necessary to have seen the film to read this essay. The Matrix is all around you. It is in the air you breathe, in the ground you walk on, in the sights you see and in the feelings that you feel. You yourself are composed of it just as much as it is composed of you. You imagine yourself to be an observer independent of the Matrix, but the very possibility of your observation, your sense of Self and World is produced by the Matrix.

This is column number 10. In this column we shall discuss a recent relative [1], [2] of the Russell paradox, the Metagame Paradox. This paradox is related to a set theoretic paradox about well-founded sets, the Well-founded Set Paradox. These two paradoxes are both related to the basic nature of any observing system that would include itself in its own observations. I give you these paradoxes and a comment on the nature of the observer. Judge them for yourself. Near the end of the column we show how, by turning the paradox around, there are new proofs of uncountability of certain infinities. These proofs have the remarkable quality that they use a bit of imaginary reasoning, only to have it vanish just as it appears! It is this role of the imagination that is central to our theme. We reason by an imaginary detour. And we obtain a real answer.