State of affairs as a representation in a Hilbert space

In classical frameworks, language and logic are modeled by algebra. For instance, a lattice is employed by many as an efficient tool to model intuitionistic logic and certain aspects of natural language (e.g. grammar). Generally speaking, algebra largely pertains to something qualitative. In logic, the main concern is ``truth'' and ``falsehood'' and the soundness of an argument; in (Chomskyan) language, the main concern is the grammaticality and well-formedness of an utterance. All these are qualitative concepts. In the West, these have been the main-stream (classical) thoughts.

In a quantum mechanical approach to logic and linguistics, however, the underlying structure must instead be Hilbert space. A shift from a classical to a quantum mechanical approach to linguistic and logic studies is to be accompanied by a shift in the underlying structure from algebra to Hilbert space.

Definition 2  

A Hilbert space is vector space $\mathfrak H$ with an inner product $\left\langle {f,g} \right\rangle$ such that the norm defined by

$$\left\vert f \right\vert\equiv \sqrt {\left\langle {f,f} \right\rangle }$$

turns it into a complete metric space.

In quantum mechanics, the inner product is usually denoted as $\left\langle f \mathrel\vert g \right\rangle$. $\left\vert f \right\rangle$ is called a ket vector, whereas $\left\langle f \right\vert$ is called a bra vector with $\left\langle f \right\vert=\left\vert f \right\rangle^\dag =(\left\vert f \right\rangle^{*})^{t}$, where $^*$ is the complex conjugate and $^t$ is the transpose operator, respectively. Generally speaking, the dimensions of a Hilbert space can be infinite. As far as our treatment is concerned, we will first start with a Hilbert space with finite dimensions defined on $\mathbb{C}$ with $\left\langle f \mathrel\vert g \right\rangle$ defined as the usual complex inner product. A state of affairs is to be represented by a complex-valued vector.

As we have discussed in Section 4.2.3, quantum mechanics suggests that symbols be considered eigenstates pertaining to an observable operator in a brain quantum system. Thus we have the following definition.

Definition 3   A linguistic formulation operator (or a representationing) $\mathbb{S}$ is a quantum measurement operator:

$$\mathbb{S}: B \to S$$

where $B$ is the space of quantum states (the wave function) of a brain called mental states; $S$ is a set of symbols in a language called the vocabulary. Moreover, the elements in $S$ are shorthands of eigenstates of $\mathbb{S}$. That is,

$$S=\left\{ {s \mathrel\vert {\mathbb{S}\left\vert s \right\ran... ...gle} } , \left\langle s \mathrel\vert s \right\rangle=1\right\}$$

where $\lambda \in \mathbb{R}$ is an eigenvalue of $\mathbb{S}$.

The restriction of $\lambda \in \mathbb{R}$ is implied by the physical consideration of $\mathbb{S}$. In quantum mechanics, physical operators are always Hermitian. An Hermitian operator has real eigenvalues.

Moreover, a brain can legitimately choose its linguistic formulation operator $\mathbb{S}$ and can therefore choose its preferred vocabularies. Generally speaking, the vocabulary of every individual is not static. Whenever we learn a new word or a new meaning, our vocabulary changes. Consequently, the same state of affair may be formulated in many different ways. Furthermore, a state of affairs can be formulated in spoken words, written language, graphics, gestures, and non-linguistic sound. All these are patterned systems of general language. In this sense, every person is multilingual.

To explore multilingual properties, let us take a look at the ``multilingualness'' in verbal languages. For example, a person may speak both English and German. In this case, very often two formulation operators do not commute. Thus a pure state (an eigenstate) in a language (pertaining to $\mathbb{S}_1$) is not necessarily a pure state in another language (pertaining to $\mathbb{S}_2$). For instance, the German symbol ``Taube'' has to be represented in English by a superposition of ``dove'' and ``pigeon''. In this regard, it is clear that a German-operator and an English-operator do not commute.

In quantum mechanical terms, multilingualness manifests itself in that there are multiple ways of decomposing a Hilbert space into bases. An important implication of quantum multilingualness is that a formulation can be active and holistic, because the operator must have access to the whole universe. One should note that the capacity of the brain is not limited by representationing, only the capacity of formulation itself is.

Since $\left\vert s \right\rangle$ is an eigenstate pertaining to $\mathbb{S}$, we have,

Corollary 1   For each $s_i, s_j \in S$ and $s_i \ne s_j$, ${\left\langle {s_i \mathrel\vert s_j} \right\rangle }=0$.

In other words, all symbols in $S$ are orthogonal to each other. Intuitively, this implies that the symbols do not mingle with each other. This is straight-forward: a symbol $x$ is not any symbol which is not $x$. This manifests itself as the Principle of Excluded Middle of symbols. Furthermore, since $S$ is an eigenbasis of $B$, any states in the brain, as far as a linguistic formulation is concerned, can be considered a superposed state of these symbols. So we have,

Corollary 2   For any $\left\vert m \right\rangle \in B$, $\left\vert m \right\rangle$ can always be decomposed into a projection on each member of $S$. That is,

$$\left\vert m \right\rangle =\sum\limits_n {c_n\left\vert {s_n} \right\rangle }$$

where $c_n \in \mathbb{C}$ and $c_n=\left\langle s_n \mathrel\vert m \right\rangle$ is the projection of $\left\vert m \right\rangle$ on $\left\vert s_n \right\rangle$.

An interesting fact is that any complete set of symbols can serve as a basis for a Hilbert space. These symbols are symbols proper -- they can not have any concept attached to them. So they are quite similar to the signs in Saussurean linguistics.

As for how mental states evolve, notice that a mental state, as a physical system, must follow the law of quantum mechanics. Moreover, thinking or reasoning is very likely a closed process (i.e. energy is conserved). Consequently, the evolution of a mental state is a unitary operator and can be called thinking or reasoning.

Definition 4   Thinking or reasoning is a unitary operator $U$ operating on a mental state with $U^{\dag }=U^{-1}$.

A symbol can be superposed with other symbols and form a composite mental state. Indeed, most mental states are composite. A sentence, for example, can be considered as a composition of multiple symbols each of which is an eigenstate. For the purpose of reasoning, a composite mental state can then be subject to unitary operations. This is what one might call flow of thoughts. While the outcome of a flow of thoughts may be fruitful, it has to be measured in order to be stored in the physico-chemical substrate in the brain or somewhere else (in computer files, for example) for later perusal or communication. This can be considered classicizing a quantum states. A classical object is all that is stable and well defined^4.4.

In short, quantum mechanics has something profound to say about the quality of mental objects. If thought can only be mediated by symbols as stated in Postulate 2 and Postulate 3, realization (grasping a particular concept or idea) is to be understood in its literal sense. It is a process of ``real''-izing a complex mental state by collapsing a superposed (impure) mental state into an eigenstate of $\mathbb{S}$. Recall that a mental state is a vector of complex numbers which has both real and imaginary parts. The collapse of a mental state suffers from, roughly speaking, loss of information. There is an exception though: if the mental state is in one of the eigenstates of $\mathbb{S}$, realization can reconstruct the original state. Mental states in the latter situation behave exactly the same as what is deemed as ``true'' or ``well-defined'' in a classical logic or linguistic framework. In other words, eigenstates, being invariant under measurement, are not to be distinguished from true and/or well defined properties in a classical framework. In this sense, the quantum computational cognitive framework extends the classical one.

In fact, a quantum computational framework of cognition includes also those mental situations which are ``not true'' or ``ill-defined'' according to the classical view. In the classical view, only objects of qualitative upper-hand (i.e. what is ``true'' or what is ``well-formed'') can have quantitative properties (and vice versa). In quantum mechanics, however, one has the tool to discuss classically qualitatively inferior objects quantitatively. Specifically, qualitative is a special case of quantitative manifestation (the pureness of a mental state).

One should note a limitation of this approach: viz. what can be described is that which is measured, that is, anything that can be classicized. Since description depends on the arrangement of a measuring device, the true mental state remains in an area forever inaccessible to language. Any measurement (for the sake of representation of the mental state) will destroy the original state.

Nevertheless, the state of affairs is still something that can be discussed, for it is of practical interest, especially in aggregate behavior. Thus,

Corollary 3   A state of affairs is a vector (among many alternatives) in a Hilbert space.

Indeed, the representation of states of affairs is crucial in order for us to build up reality. After all, what an individual has access to is only his/her memories of the past and his/her ephemeral eidetic experience at present. The semi-stability of memories and our competence to access it (by continually measuring memory and reconstructing the corresponding eidetic experience of the past) contributes to our understanding of reality. Since memory is largely classical, we construct a ``reality'' conforming to classical physics and call it substance (matter in Cartesian sense). We go even further to substantialize many other active processes, such as thinking and mind, and treat them as something like matter, only to realize that the mind is different because it is difficult to be substantialized.