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.

中文翻译：

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矢量逻辑允许通过 NOT 的平方根进行反事实虚拟化

在这项工作中，我们使用向量逻辑来研究反事实条件的表示，向量逻辑是一种用于逻辑函数和真值的矩阵向量形式。在这种形式主义中，反事实可以在复杂矩阵中转换，预处理一个蕴涵矩阵，其中一个是复杂矩阵 NOT 的平方根。这种数学方法证明了反事实的虚拟特征。发生这种情况是因为这种表示产生了一个反事实的估值，它是两个相反的真值的叠加，分别由两个复共轭系数加权。该结果表明，该过程给出了在复杂域上投影的不确定评估。在这个基本表示之后，对给定反事实的合理性的判断使我们能够将决定转向接受或拒绝。这种转变是第二次应用 NOT 的两个平方根之一的结果。