Quantum language is classified by four parts ($(A_1)$, $(A_2)$, $(B_1)$,$(B_2)$) as follows. $$\underset{(=\mbox{quantum language})}{\mbox{ measurement theory}} \left\{\begin{array}{ll} \underset{\mbox{(A)}}{ \mbox{pure type}} \left\{\begin{array}{ll} \!\! (A_1):\mbox{classical system} : \mbox{ Fisher statistics} \\ \!\! (A_2):\mbox{ quantum system} : \mbox{ usual quantum mechanics } \\ \end{array}\right. \\ \\ \underset{\mbox{(B)}} {\mbox{mixed type}} \left\{\begin{array}{ll} \!\! (B_1):\mbox{ classical system} : \text{including Bayesian statistics, Kalman filter}\\ \!\! (B_2):\mbox{ quantum system} : \mbox{quantum decoherence } \\ \end{array}\right. \end{array}\right.$$ In this note,

I mainly devote myself to (A)

Quantum language (A) has the following stucture: \begin{align} & \underset{\mbox{ (=quantum language)}}{\fbox{pure measurement theory (A)}} := \underbrace{ \underset{\mbox{ ($\S$2.7)}}{ \overset{ [\mbox{ (pure) Axiom 1}] }{\fbox{pure measurement}} } + \underset{\mbox{ ($\S$10.3)}}{ \overset{ [{\mbox{ Axiom 2}}] }{\fbox{Causality}} } }_{\mbox{ a kind of incantation (a priori judgment)}} + \underbrace{ \underset{\mbox{ ($\S$3.1) }} { \overset{ {}}{\fbox{Linguistic interpretation}} } }_{\mbox{ the manual on how to use spells}} \tag{1.2} \end{align}

Since quantum language is a language and not physics, the above axioms (i..e., Axioms1 and 2) are not laws (in phyisics ) but kinds of incantations. That is,

Axioms 1 and 2 is almighty spell (=incantation).

Thus, quantum language says:

Learn this incantation (Axioms 1 and 2) by rote!

Here, Axiom 1 (measurement) pure type ( This will be understood in $\S$2.7)
With any system $S$, a basic structure $[ {\mathcal A} \subseteq \overline{\mathcal A}]_{B(H)}$ can be associated in which measurement theory of that system can be formulated. In $[ {\mathcal A} \subseteq \overline{\mathcal A}]_{B(H)}$, consider a $W^*$-measurement ${\mathsf M}_{\overline{\mathcal A}} \big({\mathsf O}{{=}} (X, {\cal F} , F{}), S_{[{}\rho] } \big)$ $\Big($ or, $C^*$-measurement ${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{=}} (X, {\cal F} , F{}), S_{[{}\rho] } \big)$ $\Big)$. Then, the probability that a measured value $x$ $( \in X )$ obtained by the $W^*$-measurement ${\mathsf M}_{\overline{\mathcal A}} \bigl({\mathsf O} , S_{[{}\rho{}] } \bigl)$ $\Big($ or, $C^*$-measurement ${\mathsf M}_{{\mathcal A}} \big({\mathsf O}{{=}} (X, {\cal F} , F{}), S_{[{}\rho] } \big)$ $\Big)$ belongs to $\Xi$ $(\in {\cal F}{})$ is given by \begin{align} \rho( F(\Xi)) (\equiv {}_{{{\mathcal A}^*}}(\rho, F(\Xi) )_{\overline{\mathcal A}} ) \tag{1.3} \end{align}
And Axiom 2 (causality) (This will be understood in $\S$10.3)
Let $T$ be a tree (i.e., semi-ordered tree structure). For each $t (\in T)$, a basic structure $[{\mathcal A}_t \subseteq \overline{\mathcal A}_t]_{ B(H_t)}$ is associated. Then, the causal chain is represented by a $W^*$- sequential causal operator $\{ \Phi_{t_1,t_2}{}:$ ${\overline{\mathcal A}_{t_2}} \to {\overline{\mathcal A}_{t_1}} \}_{(t_1,t_2) \in T^2_{\leqq}}$ $\Big($ or, $C^*$- sequential causal operator $\{ \Phi_{t_1,t_2}{}:$ ${{\mathcal A}_{t_2}} \to {{\mathcal A}_{t_1}} \}_{(t_1,t_2) \in T^2_{\leqq}}$ $\Big)$

If it's learned by rote, repeat it many times. That is,
experiance is best teatcher, or, custum makes all things

This is everything of quantum language. However, if the readers would like to make progress quantum language early, the manual on how to use Aioms 1 and 2 is convenient. Thus, I consider that

Interpretation (This will be explained in Chap. 3)
The manual is just the lingustic interpretation of quantum mechanics

$\fbox{Note 1.3}$If metaphysics did something wrong in the history of science, it is because metaphysics attempted to answer following questions seriously in ordinary language:

 $(\sharp_1)$: What is (the meaning of) the key-words (e.g., measurement, probability, causality, space-time, etc.)?

Although this question $(\sharp_1)$ looks attractive, however, it is not productive. What is important is
to create a language to deal with the keywords.
In fact, it should be noted that successful world descriptin methods ( e.g., Newtonian mechanics, the theory of relativity, etc.) in the right figure are related to the problem "How are the key-words (e.g., force, mass, acceleration) used?".
So we replace the ($\sharp_1$) by the following ($\sharp_2$):

 $(\sharp_2)$: How are the key-words (e.g., measurement, probability, causality, space-time, etc.) used in quantum language?

The problem ($\sharp_1$) will now be solved in the sense of ($\sharp_2$). $\fbox{Note 1.4}$Metaphysics is an academic discipline concerning the propositions in which empirical validation is impossible. Lord Kelvin(1824--1907), who is famous as the absolute temperature $K$, said that
Mathematics is the only good metaphysics. $\qquad \qquad$Fig.1.1: The history of world-descriptions
This is very persuasive saying. However, we step forward:
 $(\sharp)$: quantum language is also another good metaphysics
Lord Kelvin might think that Kant philosophy (Critique of Pure Reason) is not good metaphisics. However, I consider that a priori synthetic judgment (i..e., axiom which can not be examined by experiment) corresponds to Axioms 1 and 2. That is, \begin{align} \underset{\mbox{(Kant philosophy)}}{\fbox{a priori synthetic judgment}} \quad \xrightarrow[\mbox{(quantmization)}]{} \quad \underset{\mbox{(quantum language)}}{\fbox{Axioms 1 and 2}} \end{align} such as ⑥$\rightarrow$⑧$\rightarrow$⑩ in the right figure.
See the following ref.
 $\bullet$: S. Ishikawa, “ Quantum Mechanics and the Philosophy of Language: Reconsideration of Traditional Philosophies," Journal of quantum information science, Vol. 2, No. 1, 2012, pp.2-9.doi: 10.4236/jqis.2012.21002 ( download free)