Result and analysis

In typical experiments, the training goal can be achieved (see Appendix A for the numerical result used in this section). A quantum mechanical architecture has acquired the ``common sense'' based on its ``experience'' of coherent day-to-day situations.

In the most interesting situation, as described in the second questionable situation above, the absolute square of the assertion-component of the result is 0.24. That is, in a quantum measurement evaluating the symbolic result of $s$ ($\mathbb{T}$ or $\mathbb{F}$), about one fourth of the cases comes up as true. The outcomes jump back and forth between true and false.

Moreover, the phases of $p$ and $q$ may play an important part in this scenario. This can be shown by preparing the input as

$${e^{i\theta_1} \vert p+ \rangle \over \sqrt 3} + {e^{i\theta_... ...t q+ \rangle \over \sqrt 3} + {\vert r+ \rangle \over \sqrt 3}.$$

The corresponding assertion state of $s$ is shown in Figure 6.12. As can be seen in the figure, if the phase of $\vert q+ \rangle$ is somewhere near $\pi$ (relative to the phase of $p$ and $r$), $s$ is almost always asserted. This phenomenon seems enigmatic. However, this situation might indicate something about our intuition regarding such a state of affairs. If we take a phase difference of two, asserting eigenstates as some sort of ``relevance'' measure of two propositions, we may regard $\theta_2=\pi$ as indicating that $q$ is ``irrelevant'' to $s$. Thus we have an intuitive explanation about why $s$ is almost always asserted in this situation, for if $q$ is taken as irrelevant to the state of affairs under discussion, whether $q$ is true (that is, whether Jack is unforgiving) no longer plays a crucial role in determining whether Jack helps Jim in a counterfactual situation (i.e. that Jim is not proud and that they do not have a quarrel). Indeed, this same hypothesis seems to offer an adequate account for the graphics presented in the previous sections. As can be seen in several examples in the previous sections, an irrelevant proposition enables a classical logic operation to deliver outcomes that are not governed by classical logic.

Figure 6.12: The probability of $s$ being asserted based on counterfactual situations where $(p,q,r)=(\mathbb T, \mathbb T, \mathbb T)$. The input states are prepared with different phase ( $\theta _1, \theta _2$)

Another questionable argument is when $(p,q,r)=(\mathbb{T}, \mathbb{F}, \mathbb{T})$. Since the antecedent of the counterfactual conditional is not true, a counterfactual conditional cannot be applied. Specifically, an input state of affairs of this sort is

$${e^{i\theta_1} \vert p+ \rangle \over \sqrt 3} + {e^{i\theta_... ...t q- \rangle \over \sqrt 3} + {\vert r+ \rangle \over \sqrt 3}.$$

The corresponding assertion state of $s$ is shown in Figure 6.13. Not very surprisingly, $s$ is asserted in the vicinity of the origin. The troughs appear when the phase difference is roughly $\pi$, and can be explained as above.

Figure 6.13: The probability of $s$ being asserted based on counterfactual situations where $(p,q,r)=(\mathbb T, \mathbb F, \mathbb T)$. The input states are prepared with different phase ( $\theta _1, \theta _2$)

There are situations where Jim and Jack do have a quarrel, and whether Jim is proud is asserted to a certain degree, and so is the fact that Jack is unforgiving. For example, the input state of affairs can be represented as

$${\rho e^{i\theta} \vert p- \rangle + (1-\rho) e^{i\theta} \ve... ...rangle + \vert r+ \rangle \over \sqrt {2+\rho^2 + (1-\rho)^2}},$$

where $\rho \in \mathbb{R}$ with $0 < \rho \le 1$. The corresponding assertion state of $s$ is shown in Figure 6.14. It seems that in such situations, the ``refutation-degree'' of whether Jim is proud has little influence on proposition $s$. This agrees with our intuition about the state of affairs in these situations, for whether Jim asks for help does not influence whether Jack would help him (Jack is unforgiving, so he will not help Jim anyway).

Figure 6.14: The probability of $s$ being asserted based on counterfactual situations where $p$ is partially asserted.

Alternatively, the input state of affairs can be represented as

$${\rho e^{i\theta} \vert q- \rangle + (1-\rho) e^{i\theta} \ve... ...rangle + \vert r+ \rangle \over \sqrt {2+\rho^2 + (1-\rho)^2}},$$

where $\rho \in \mathbb{R}$ with $0 < \rho \le 1$. The corresponding assertion state of $s$ is shown in Figure 6.15. As can be seen in this figure, $p$ depends heavily on both the ``refutation-degree'' ($\rho$) and phase ($\theta$) of $q$. If the phase difference is kept small, the assertion of $s$ is a monotonously increasing function of $\rho$. This is not surprising. However, if the phase difference is about $\pi$, the degree of assertion behaves very strangely depending on $\rho$. The detailed relation between $\rho$ and the output when $\theta =\pi$ is illustrated in Figure 6.16. When $\rho=0.258609$ there is a minimum. When $\rho=0.343384$ there is a maximum of 0.998. At this moment, there is no intuitive explanation for this enigmatic phenomenon.

Figure 6.15: The probability of proposition $s$ being asserted based on counterfactual situations where $q$ is partially asserted.

Figure 6.16: The detailed relation between $\rho$ and proposition $s$'s being asserted when $\theta =\pi$.