Vector logic allows counterfactual virtualization by The Square Root of NOT
This work addresses a theoretical problem in logic and AI by providing a mathematical formalism for counterfactuals, but it appears incremental as it builds on existing vector logic concepts.
The paper tackles the representation of counterfactual conditionals using vector logic, showing that this approach yields an uncertain evaluation in the complex domain as a superposition of truth values. The result demonstrates that applying the square root of NOT matrix allows shifting the decision towards acceptance or refusal of a counterfactual.
In this work we investigate the representation of counterfactual conditionals using the vector logic, a matrix-vectors formalism for logical functions and truth values. Inside this formalism, the counterfactuals can be transformed in complex matrices preprocessing an implication matrix with one of the square roots of NOT, a complex matrix. This mathematical approach puts in evidence the virtual character of the counterfactuals. This happens because this representation produces a valuation of a counterfactual that is the superposition of the two opposite truth values weighted, respectively, by two complex conjugated coefficients. This result shows that this procedure gives an uncertain evaluation projected on the complex domain. After this basic representation, the judgment of the plausibility of a given counterfactual allows us to shift the decision towards an acceptance or a refusal. This shift is the result of applying for a second time one of the two square roots of NOT.