Discussion

In this chapter, we have shown that quantum mechanical architectures can indeed implement basic classical logic functions and tackle much more thorny issues such as non-monotonic and counterfactual reasoning. In a sense, this approach to logic -- as a framework for reasoning in general -- is a significant departure from the conventional approach to logic. For one thing, in a conventional framework, logic is basically normative, while in this chapter, the architectures proposed are descriptive and explanatory. As an explanatory framework, it shows how reasoning can be ``boot-strapped'' with quantum computational systems.

Indeed, just like classical grammar is normative while modern linguistics is descriptive and explanatory, the science of thought should also strive to be descriptive. Just as colloquial language was largely unduly ignored in classical linguistics, common sense is all too often ignored or taken as a ``frictional'' or ``impure'' form of classical logical argument. Unfortunately, this latter stance is also taken by most students of artificial intelligence, in that they try to model common sense and non-monotonicity with higher level classical frameworks.

As it is shown in the XOR example, a quantum mechanical approach to a classical question shows many interesting and ``non-classical'' phenomena. The missing points in classical logic are that the ``classical'' region consists of only a fraction of the whole set of possibilities and that classical logic takes the measured outcomes in these regions incorrectly as the underlying ``reasoning mechanism.'' In this way we have shown an alternative computational model that includes and extends a classical one.

However, this is not to say that classical logic is incorrect. On the contrary, classical logic is no doubt a very powerful tool to help us draw conclusions and make decisions. What we want to point out is that even in a more serious context such as a scientific, ethical, and judicial one, we need something more than classical logic. In fact, what we want to show is how a single framework can accommodate both classical logic and common sense.

A quantum computational approach to logic shows that the ``Law of Thought'' as proposed by George Boole reveals only one aspect of our way of reasoning. (Remember classical logic functions can be implemented very accurately if a phase function is introduced to generate input state of affairs. See Section 6.2.) Nevertheless, this picture squares well with classical principles as far as the measurement outcomes are concerned. For one thing, the law of exclusive middle always holds, since either ${\left\vert 0 \right\rangle}$ or ${\left\vert 1 \right\rangle}$ (but not both) may manifest itself as output. There is no other thing in between. In this sense, all quantum assertions are XOR-type assertions, therefore two-valued ($\mathbb{T}$ or $\mathbb{F}$). Strictly speaking, there is no knowingly unknown state in quantum mechanics, only the absence of certain eigenstates. A multi-valued logical approach to common sense has incorrectly asserted the redundant logical value(s) which can be true or false only at a higher level (i.e. from a temporal and / or spatial vantage point)^6.8.

Our ability to consider the necessity and the possibilities of every situations is crucial for us to understand the world. In classical logic, however, only necessity is concerned. So perhaps most importantly, quantum mechanics provides an account of our way of knowing and seeing the world. This comes as no surprise, for in quantum mechanics one has an adequate picture of what is possible and what is realized. Complex numbers and the superposition of eigenstates in quantum mechanics offer the picture.

In the realm of possibilities, contradictory situations can peacefully coexist, as demonstrated in the examples in this chapter. In such schemes, we have a superposition with mutually contradictory states of affairs, each of which has a corresponding complex coefficient. This kind of reasoning is everywhere in our everyday life and indeed in science and any rational activities as well.

To conclude, quantum computation deserves a serious consideration as a general model for common sense logic. Although one can draw this conclusion from a postulated analogy between matter (physics) and mind (cognitive science/linguistics) -- see Chapter 2, it is hoped that this and the following chapter may persuade the reader that this approach could be of practical interest as far as engineering (artificial intelligence or NLP) is concerned.