Experimental setup

In this section we consider an example of non-monotonic reasoning as follows,

$$(p \to q); (p\wedge r \to \neg q).$$

where $p$, $r$ are propositions functioning as ``antecedents'' and $q$ is the ``conclusion.'' An everyday reasoning of this sort states that whenever $p$ is true, $q$ is true. However, if one asserts additionally that $r$ is also true, then $q$ can no longer be true. This scheme captures a prototypical non-monotonic reasoning.

In the same vein as described in the previous experiment, reasoning like this is treated as a quantum mechanical experimental setup. In this scenario, there are, in total, 6 eigenstates ($\vert p- \rangle$, $\vert p+ \rangle$, $\vert q- \rangle$, $\vert q+ \rangle$, $\vert r- \rangle$, and $\vert r+ \rangle$), (for these, the plus sign $+$ following a proposition symbol indicates that the proposition is asserted while a minus sign $-$ indicates that it is refuted.) They are eigenstates corresponding to an operator $S$ that asserts or refutes the state of affairs. A true proposition is therefore represented by an assertion eigenstate alone. And a false proposition is represented by a refutation eigenstate. In common non-monotonic reasoning, for each proposition, there can be a third situation in which the proposition is neither asserted nor refuted. This situation is usually called unknown and will be symbolized by X in a classical account.

In fact, an unknown status of a situation $\sigma$ is nevertheless a state known at a higher level. By this we mean a reasoner knows that he does not know $\sigma$, so he asserts the unknown status of $\sigma$. In this case, he can consider the consequence based on the unknown status. Our training data consists of this higher level knowledge. Specifically, the training data employing the ``unknown status'' used in this section is from the vantage point of the other observer at a higher level, which should not be confused with the ``knowingly unknown'' status of the reasoner.

In non-monotonic reasoning, however, if the reasoner does not even know that he does not know $\sigma$, he can not consider the consequence of the unknown status. It can be argued that this genuine unknown status is a very important source of non-monotonicity. In a quantum computational approach, an eigenstate is an unknown status per se. If it is not measured, it is genuinely unknown. We use this feature of quantum mechanics to account for non-monotonicity. If the ``knowingly unknown'' is to be included in non-monotonic reasoning, one has to introduce another qubit that asserts or refutes the knowing status of a situation. This is quite another question and is beyond the discussion of this section.

We start with what can be regarded as intuitively valid arguments (i.e. classically). That is, the starting point is the sound conclusions one can draw from a temporal and spatial vantage point. These arguments are listed in the following table,

p	r	q	p	r	q
$\mathbb{T}$	$\mathbb{T}$	$\mathbb{F}$	$\mathbb{T}$	$\mathbb{F}$	$\mathbb{T}$
$\mathbb{T}$	X	$\mathbb{T}$	$\mathbb{F}$	$\mathbb{T}$	X
$\mathbb{F}$	$\mathbb{F}$	X	$\mathbb{F}$	X	X
X	$\mathbb{T}$	X	X	$\mathbb{F}$	X

More specifically, the table can be regarded as a formulation representing a person's reflection about her previous non-monotonic reasoning. In the example given in Section 5.3, non-monotonicity appeared only after I learned that it was the first Sunday of daylight savings time and I had not noticed that fact. In essence, a set of rules such as the above table may eliminate non-monotonicity. But this is true only if one sees the issue from a temporal vantage point or from the spatial vantage point of a third observer. Notice that this is exactly the perspective to which the reasoner cannot have access, at the moment of non-monotonic reasoning. Thus the above table should be regarded as a ``classicization'' of non-monotonic reasoning. As I have argued in Section 5.3, a classical scheme like this cannot accommodate non-monotonicity of strong kind.

In a quantum mechanical framework, however, it is easy to express the unknown status of a proposition (an unknown status for the reasoner) without introducing artificial unknown status. This can be done by simply leaving out both the eigenstates pertaining to this proposition. Technically speaking, the components of the eigenstates corresponding to this proposition are set to zero. Therefore an input state is prepared as

$${c_{p-} \left\vert p- \right\rangle}+{c_{p+} \left\vert p+ \... ...t\vert r- \right\rangle}+{c_{r+} \left\vert r+ \right\rangle},$$ (6.5)

where $c_{xy} \in \mathbb{C}$ is the coefficient of the eigenstate corresponding to the $y$-state of proposition $x$. As a concrete example, the input state of affairs corresponding to $(p,r)=(\mathbb{T},\mathbb{F})$ can be written as

$${e^{i\theta_1}\over {\sqrt 2}} {\left\vert p+ \right\rangle} + {e^{i\theta_2}\over {\sqrt 2}} {\left\vert r- \right\rangle}.$$ (6.6)

The ninth possibility $(p,r)=(X,X)$ is excluded from the table because this situation is represented by a zero vector that always is a null output (it is correct, though), and thus will not contribute to training.

In a reasoning process, an input state of affairs is subject to a unitary reasoning operator $U$. The architecture is trained with the states of affairs as shown in the table of valid arguments. The training algorithm is the same as described in the previous experiment. Specifically, these states of affairs are prepared with phases (arguments of complex components) being zero (e.g. $\theta_1=0$ and $\theta_2=0$ in Equation 6.6).