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Nonlinear Logic (NLL) - Making Sense Out of Logical Self- Reference

The fundamental argument against any kind of faster-than-light (FTL) phenomenon is the issue of temporal paradox. Thanks to Special Relativity, FTL implies time travel and given time travel, the possibility of temporal paradox seems unavoidable. This is an argument from logic, not physics, yet upon closer examination it is discovered that our formal systems of logic studiously avoid self-reference in their foundational premises. This is doubly odd once it is realized that self-reference is unavoidable for any logical system of sufficient power, as demonstrated by Gödel’s Incompleteness Theorem. This paper presents a brief overview of the major attempts to construct logical systems with self-reference built into their foundations. It then covers in some detail Nonlinear Logic (NLL) the selfreferential logical system developed by Novatia Labs as part of their research project into the potential for quantum systems to support superluminal communications. Like most formal systems of logic, statements in NLL consist of typographical strings. The major addition to the symbol set is the label operator so that self-reference can be indicated. There is also a symbolic representation of statements in NLL derived from digital circuits that is isomorphic with the typographical representation. The digital circuit formalism is useful for gaining insight into what self-reference brings to systems of logic. It also suggests that sequential circuits could be designed with greater efficiency using NLL than standard Boolean logic. As with other self-referential logics, NLL supports more truthvalues than just true and false. These imaginary or complex truthvalues can be represented with an excursion into time (NLL) or into space (Kaufman’s Knot Theory). NLL also supports a graphical representation of truthvalue called attractor structures. This representation has its roots in the “attractors” of chaos theory, which has its basis in nonlinear mathematics. The resulting system of logic is closed, complete, and demonstrates that sense can be made of logical paradox. In so doing, it weakens the argument that temporal paradox prohibits time travel and in turn FTL, hinting at possible loopholes in physics for these phenomena.