8.6: Syllogism---Does Socrates die? 8.6.1: Syllogism and its variations: Classical systems

Next, we shall discuss practical syllogism (i.e., measurement theoretical theorem concerning implication (Definition8.6) ). Before the discussion, we note that

 $(\sharp):$ Since Theorem 8.9 ( The existence of the combined observable) does not hold in quantum system, ( cf. Counter Example 8.10 ), syllogism does not hold.

On the other hand, in classical system, we can expect that syllogism holds. This will be proved in the following theorem.

Theorem 8.12 [Practical syllogism in classical systems] Consider the classical basic structure \begin{align} [C_0(\Omega ) \subseteq L^\infty ( \Omega, \nu ) \subseteq B(L^2 ( \Omega, \nu ))] \end{align}

Let ${\mathsf O}_{123}$ $=$ $(X_1 \times X_2 \times X_3,$ ${\cal F}_1 \times {\cal F}_2 \times {\cal F}_3 ,$ $F_{123}{}{{=}} {\mathop{\overset{qp}{\times}}}_{k=1,2,3} F_k)$ be an observable in ${L^\infty (\Omega)}$ Fix $\omega \in \Omega$, $\Xi_1$ $\in {\cal F}_1$, $\Xi_2$ $\in {\cal F}_2$, $\Xi_3$ $\in {\cal F}_3$ Then, we see the following (i) $\mbox{--}$ (iii).

(i).(practical syllogism) \begin{align} [{\mathsf O}_{123}^{(1)};{\Xi_1}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [\omega] }) }{ \Longrightarrow} [{\mathsf O}_{123}^{(2)};{\Xi_2}] , \quad [{\mathsf O}_{123}^{(2)};{\Xi_2}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [\omega] }) }{ \Longrightarrow} [{\mathsf O}_{123}^{(3)};{\Xi_3}] \end{align} implies \begin{align} & \; \; \rm{Rep}_\omega^{\Xi_1 \times \Xi_3}[{}{\mathsf O}^{(13)}_{123}] = \left[\begin{array} [{}F^{(13)}_{123} (\Xi_1 \times \Xi_3)] (\omega) & [{}F^{(13)}_{123} (\Xi_1 \times \Xi_3^c)] (\omega) \\ {} [{}F^{(13)}_{123} (\Xi_1^c \times \Xi_3)] (\omega) & [{}F^{(13)}_{123} (\Xi_1^c \times \Xi_3^c)] (\omega) \end{array}\right] \\ = & \left[\begin{array} [F^{(1)}_{123}(\Xi_1)](\omega) & 0 \\ {} [F^{(3)}_{123}(\Xi_3)](\omega) - [F^{(1)}_{123}(\Xi_1)](\omega) & 1- [F^{(3)}_{123}(\Xi_3)](\omega) \end{array}\right] \end{align} That is, it holds: \begin{align} [{\mathsf O}_{123}^{(1)};{\Xi_1}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [\omega] }) }{ \Longrightarrow} [{\mathsf O}_{123}^{(3)};{\Xi_3}] \tag{8.12} \end{align}

(ii). \begin{align} [{\mathsf O}_{123}^{(1)};{\Xi_1}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [\omega] }) }{ \Longleftarrow} [{\mathsf O}_{123}^{(2)};{\Xi_2}] , \quad [{\mathsf O}_{123}^{(2)};{\Xi_2}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [\omega] }) }{ \Longrightarrow} [{\mathsf O}_{123}^{(3)};{\Xi_3}] \end{align} implies \begin{align} & \; \; \rm{Rep}_\omega^{\Xi_1 \times \Xi_3}[{}{\mathsf O}^{(13)}_{123}] = \left[\begin{array} [{}F^{(13)}_{123} (\Xi_1 \times \Xi_3)] (\omega) & [{}F^{(13)}_{123} (\Xi_1 \times \Xi_3^c)] (\omega) \\ {} [{}F^{(13)}_{123} (\Xi_1^c \times \Xi_3)] (\omega) & [{}F^{(13)}_{123} (\Xi_1^c \times \Xi_3^c)] (\omega) \end{array}\right] \\ = & \left[\begin{array} \alpha_{_{\Xi_1 \times \Xi_3}} &\;\;\;\; [F^{(1)}_{123}(\Xi_1)](\omega)-\alpha_{_{\Xi_1 \times \Xi_3}} \\ {} [F^{(3)}_{123}(\Xi_3)](\omega)-\alpha_{_{\Xi_1 \times \Xi_3}} &\;\;\;\; 1-\alpha_{_{\Xi_1 \times \Xi_3}} - [F^{(1)}_{123}(\Xi_1)] - [F^{(3)}_{123}(\Xi_3)] \end{array}\right] \end{align} where \begin{align} & \hspace{-1cm} \max \{ [F^{(2)}_{123}(\Xi_2)](\omega), [F^{(1)}_{123}(\Xi_1)](\omega)+ [F^{(3)}_{123}(\Xi_3)](\omega) - 1 \} \nonumber \\ & \hspace{0.5cm}{{\; \leqq \;}} \alpha_{_{\Xi_1 \times \Xi_3}} (\omega) {{\; \leqq \;}} \min \{ [F^{(1)}_{123}(\Xi_1)](\omega) , [F^{(3)}_{123}(\Xi_3)](\omega) \} \tag{8.13} \end{align}

(iii). \begin{align} [{\mathsf O}_{123}^{(1)};{\Xi_1}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [\omega] }) }{ \Longrightarrow} [{\mathsf O}_{123}^{(2)};{\Xi_2}] , \quad [{\mathsf O}_{123}^{(2)};{\Xi_2}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [\omega] }) }{ \Longleftarrow} [{\mathsf O}_{123}^{(3)};{\Xi_3}] \end{align} implies \begin{align} & \; \; \rm{Rep}_\omega^{\Xi_1 \times \Xi_3}[{}{\mathsf O}^{(13)}_{123}] = \left[\begin{array} [{}F^{(13)}_{123} (\Xi_1 \times \Xi_3)] (\omega) & [{}F^{(13)}_{123} (\Xi_1 \times \Xi_3^c)] (\omega) \\ {} [{}F^{(13)}_{123} (\Xi_1^c \times \Xi_3)] (\omega) & [{}F^{(13)}_{123} (\Xi_1^c \times \Xi_3^c)] (\omega) \end{array}\right] \\ = & \left[\begin{array} \alpha_{_{\Xi_1 \times \Xi_3}} (\omega) &\;\;\;\; [F^{(1)}_{123}(\Xi_1)](\omega)-\alpha_{_{\Xi_1 \times \Xi_3}} (\omega) \\ {} [F^{(3)}_{123}(\Xi_3)](\omega)-\alpha_{_{\Xi_1 \times \Xi_3}} (\omega) &\;\;\;\; 1-\alpha_{_{\Xi_1 \times \Xi_3}} (\omega) - [F^{(1)}_{123}(\Xi_1)](\omega) - [F^{(3)}_{123}(\Xi_3)](\omega) \end{array}\right] \end{align} where \begin{align} & \hspace{-1cm} \max \{ 0 , [F^{(1)}_{123}(\Xi_1)](\omega) + [F^{(3)}_{123}(\Xi_3)](\omega) - [F^{(2)}_{123}(\Xi_2)](\omega) \} \\ & \hspace{1cm} {{\; \leqq \;}} \alpha_{_{\Xi_1 \times \Xi_3}} (\omega) {{\; \leqq \;}} \min \{[F^{(1)}_{123}(\Xi_1)](\omega) , [F^{(3)}_{123}(\Xi_3)](\omega) \} \end{align}

Proof. $\;\;$ (i): By the condition, we see $\;\;$ \begin{align} & 0= [F^{(12)}_{123}(\Xi_1 \times \Xi_2^c )](\omega) = [F_{123}(\Xi_1 \times \Xi_2^c \times \Xi_3 )](\omega) + [F_{123}(\Xi_1 \times \Xi_2^c \times \Xi_3^c )](\omega) \\ & 0= [F^{(23)}_{123}(\Xi_2 \times \Xi_3^c )](\omega) = [F_{123}(\Xi_1 \times \Xi_2 \times \Xi^c_3 )](\omega) + [F_{123}(\Xi_1^c \times \Xi_2 \times \Xi_3^c )](\omega) \end{align} Therefore, \begin{align} & 0= [F_{123}(\Xi_1 \times \Xi_2^c \times \Xi_3 )](\omega) = [F_{123}(\Xi_1 \times \Xi_2^c \times \Xi_3^c )](\omega) \\ & 0= [F_{123}(\Xi_1 \times \Xi_2 \times \Xi^c_3 )](\omega) = [F_{123}(\Xi_1^c \times \Xi_2 \times \Xi_3^c )](\omega) \end{align} Hence, \begin{align} & [F^{(13)}_{123}(\Xi_1 \times \Xi_3^c )](\omega) = [F_{123}(\Xi_1 \times \Xi_2 \times \Xi_3^c )](\omega) + [F^{(13)}_{123}(\Xi_1 \times \Xi_2^c \times \Xi_3^c )](\omega) =0 \end{align} Thus, we get (8.12).

For the proof of (ii) and (iii), see my papers.

Example 8.13 [Continued from Example 8.5]

Let ${\mathsf O}_{{1}}$ ${{=}}$ ${\mathsf O}_{{{\scriptsize{\mbox{SW}}}}}$ ${{=}}$ $(X_{{\scriptsize{\mbox{SW}}}} ,$ $2^{ X_{{\scriptsize{\mbox{SW}}}} } ,$ $F_{{\scriptsize{\mbox{SW}}}})$ and ${\mathsf O}_{{3}}$ ${{=}}$ ${\mathsf O}_{{\scriptsize{\mbox{RD}}}}$ ${{=}}$ $(X_{{\scriptsize{\mbox{RD}}}} ,$ $2^{ X_{{\scriptsize{\mbox{RD}}}} } ,$ $F_{{\scriptsize{\mbox{RD}}}})$ be as in Example 8.5.

Putting $X_{{\scriptsize{\mbox{RP}}}} = \{ y_{{\scriptsize{\mbox{RP}}}} , n_{{\scriptsize{\mbox{RP}}}} \}$, consider the new observable ${\mathsf O}_{{2}}$ ${{=}}$ ${\mathsf O}_{{{\scriptsize{\mbox{RP}}}}}$ ${{=}}$ $(X_{{\scriptsize{\mbox{RP}}}} , 2^{ X_{{\scriptsize{\mbox{RP}}}} } , F_{{\scriptsize{\mbox{RP}}}})$. Here, "$y_{{\scriptsize{\mbox{RP}}}}$" and "$n_{{\scriptsize{\mbox{RP}}}}$" respectively means "ripe" and "not ripe". Put

\begin{align} \rm{Rep}[{}{\mathsf O}_1{}] & = \big[ [{}F_{{\scriptsize{\mbox{SW}}}} (\{ y_{{{\scriptsize{\mbox{SW}}}}} \}) {}] ({\omega_k}), [{}F_{{\scriptsize{\mbox{SW}}}} (\{ n_{{{\scriptsize{\mbox{SW}}}}} \}) {}] ({\omega_k}) \big] \\ \rm{Rep}[{}{\mathsf O}_2{}] & = \big[ [{}F_{{\scriptsize{\mbox{RP}}}} (\{ y_{{{\scriptsize{\mbox{RP}}}}} \}) {}] ({\omega_k}), [{}F_{{\scriptsize{\mbox{RP}}}} (\{ n_{{{\scriptsize{\mbox{RP}}}}} \}) {}] ({\omega_k}) \big] \\ \rm{Rep}[{}{\mathsf O}_3{}] & = \big[ [{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{{\scriptsize{\mbox{RD}}}}} \}) {}] ({\omega_k}), [{}F_{{\scriptsize{\mbox{RD}}}} (\{ n_{{{\scriptsize{\mbox{RD}}}}} \}) {}] ({\omega_k}) \big] \end{align}

Consider the following quasi-product observable:

\begin{align} & {\mathsf O}_{12} = (X_{{\scriptsize{\mbox{SW}}}} \times X_{{\scriptsize{\mbox{RP}}}} , 2^{ X_{{\scriptsize{\mbox{SW}}}} \times X_{{\scriptsize{\mbox{RP}}}} }, F_{12} {{=}} F_{{\scriptsize{\mbox{SW}}}} {\mathop{\overset{qp}{\times}}} F_{{\scriptsize{\mbox{RP}}}}) \\ & {\mathsf O}_{23} = (X_{{\scriptsize{\mbox{RP}}}} \times X_{{\scriptsize{\mbox{RD}}}} , 2^{ X_{{\scriptsize{\mbox{RP}}}} \times X_{{\scriptsize{\mbox{RD}}}} }, F_{23} {{=}} F_{{\scriptsize{\mbox{RP}}}} {\mathop{\overset{qp}{\times}}} F_{{\scriptsize{\mbox{RD}}}}) \end{align}

Let ${{\omega_k}}$ $\in \Omega$. And assume that

\begin{align} & [{\mathsf O}_{123}^{(1)};{\{y_{{\scriptsize{\mbox{SW}}}} \}}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [{\omega_k}] }) }{ \Longrightarrow} [{\mathsf O}_{123}^{(2)};{\{y_{{\scriptsize{\mbox{RP}}}} \}}] , \nonumber \\ & [{\mathsf O}_{123}^{(2)};{\{y_{{\scriptsize{\mbox{RP}}}} \}}] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{123} , S_{ [{\omega_k}] }) }{ \Longrightarrow} [{\mathsf O}_{123}^{(3)};{\{y_{{\scriptsize{\mbox{RD}}}} \}}] \tag{8.14} \end{align}

Then, by Theorem 8.12 (i), we get:

\begin{align} & \; \;\;\; \rm{Rep} [{}{\mathsf O}_{13}{}] = \left[\begin{array} [{}F_{13} ( \{ y_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ y_{{{\scriptsize{\mbox{RD}}}}} \}){}] ({{\omega_k}} ) & [{}F_{13} ( \{ y_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ n_{{{\scriptsize{\mbox{RD}}}}} \}){}] ({{\omega_k}} ) \\ {}[{}F_{13} ( \{ n_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ y_{{{\scriptsize{\mbox{RD}}}}} \}){}] ({{\omega_k}} ) & [{}F_{13} ( \{ n_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ n_{{{\scriptsize{\mbox{RD}}}}} \}){}] ({{\omega_k}} ) \\ \end{array}\right] \\ & = \left[\begin{array} [{}F_{{\scriptsize{\mbox{SW}}}} (\{ y_{{\scriptsize{\mbox{SW}}}} \}){}]({{\omega_k}} ) & 0 \\ {}[{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{\scriptsize{\mbox{RD}}}} \}){}]({{\omega_k}} ) - [{}F_{{\scriptsize{\mbox{SW}}}} (\{ y_{{\scriptsize{\mbox{SW}}}} \}){}] ({{\omega_k}} ) & 1 - [{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{\scriptsize{\mbox{RD}}}} \}){}]({{\omega_k}} ) \\ \end{array}\right] \end{align}

Therefore, when we know that the tomato ${{\omega_k}}$ is sweet by measurement ${\mathsf M}_{L^\infty (\Omega)} ( {\mathsf O}_{123} , S_{[{}{{{\omega_k}} }]})$, the probability that ${{\omega_k}}$ is red is given by

\begin{align} & \frac { [{}F_{13} ( \{ y_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ y_{{{\scriptsize{\mbox{RD}}}}} \}){}] ({{\omega_k}} ) } { [{}F_{13} ( \{ y_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ y_{{{\scriptsize{\mbox{RD}}}}} \}){}] ({{\omega_k}} ) + [{}F_{13} ( \{ y_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ n_{{{\scriptsize{\mbox{RD}}}}} \}){}] ({{\omega_k}} ) } = \frac{ [{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{\scriptsize{\mbox{RD}}}} \}){}] ({{\omega_k}}) } { [{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{\scriptsize{\mbox{RD}}}} \}){}] ({{\omega_k}}) } = 1 \\ & \tag{8.15} \end{align}

Of course, (8.14) means

\begin{align} \mbox{ "Sweet" $\Longrightarrow$ "Ripe" } \qquad \mbox{} \qquad \mbox{ "Ripe" $\Longrightarrow$ "Red" } \end{align}

Therefore, by (8.12), we get the following conclusion.

\begin{align} \mbox{ "Sweet" $\Longrightarrow$ "Red" } \end{align}

However, it is not useful in the market. What we want to know is such as

\begin{align} \mbox{ "Red" $\Longrightarrow$ "Sweet" } \end{align} This will be discussed in the following example.

Example 8.14 [Continued from Example 8.5] Instead of (8.14), assume that \begin{align} {\mathsf O}_1^{ \{y_1 \}} \underset{ {\mathsf M}_{L^\infty(\Omega) }({\mathsf O}_{12} ,S_{ [\delta_{\omega_n}{}] }) } { \Longleftarrow} {\mathsf O}_2^{ \{y_2 \}} , \qquad {\mathsf O}_2^{ \{y_2 \}} \underset{ {\mathsf M}_{L^\infty(\Omega) }({\mathsf O}_{23} ,S_{[{}\delta_{\omega_n}{}] }) } { \Longrightarrow} {\mathsf O}_3^{ \{y_3 \}} . \tag{8.16} \end{align}

When we observe that the tomato $\omega_n$ is "RED"$\!\!,\;$ we can infer, by the fuzzy inference ${\mathsf M}_{L^\infty(\Omega)}( {\mathsf O}_{13} ,$ $S_{ [\delta_{\omega_n}{}] })$, the probability that the tomato $\omega_n$ is "SWEET" is given by

\begin{align} Q = \frac { [{}F_{13} ( \{ y_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ y_{{{\scriptsize{\mbox{RD}}}}} \}){}] (\omega_n ) } { [{}F_{13} ( \{ y_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ y_{{{\scriptsize{\mbox{RD}}}}} \}){}] (\omega_n ) + [{}F_{13} ( \{ n_{{{\scriptsize{\mbox{SW}}}}} \}\times \{ y_{{{\scriptsize{\mbox{RD}}}}} \}){}] (\omega_n ) } \end{align} which is, by (8.3), estimated as follows: \begin{align} & \; \; \; \max \left\{ \frac{ [{}F_{{\scriptsize{\mbox{RP}}}} (\{ y_{{\scriptsize{\mbox{RP}}}} \})] (\omega_n)} { [{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{\scriptsize{\mbox{RD}}}} \})] (\omega_n)} , \frac{ [{}F_{{\scriptsize{\mbox{SW}}}} (\{ y_{{\scriptsize{\mbox{SW}}}} \})] + [{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{\scriptsize{\mbox{RD}}}} \})] -1 } { [{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{\scriptsize{\mbox{RD}}}} \})] (\omega_n)} \right\} \le Q \le \min \{ \frac{ [{}F_{{\scriptsize{\mbox{SW}}}} (\{ y_{{\scriptsize{\mbox{SW}}}} \})] (\omega_n) } {[{}F_{{\scriptsize{\mbox{RD}}}} (\{ y_{{\scriptsize{\mbox{RD}}}} \})] (\omega_n)} , \; 1 \} . \\ & \tag{8.17} \end{align}

Note that (8.16) implies (and is implied by)

\begin{align} \mbox{ "RIPE" $\Longrightarrow$ "SWEET" } \qquad \mbox{and} \qquad \mbox{ "RIPE" $\Longrightarrow$ "RED" } . \end{align}

And note that the conclusion (8.17) is somewhat like

\begin{align} \mbox{ "RED" $\Longrightarrow$ "SWEET" }. \end{align}

Therefore， the estimation (8.17) may be useful in marckets.

If some may think that the (8.17) is strange, see the following figure:
Since $$\frac{| \mbox{"Sweet"} \land \mbox{"Red"} |}{|\mbox{"Red"}|} \doteqdot 1$$ some may agree to the (8.17).

Supplement
 $\qquad \quad$Fig. 1.1: The history of world-descriptions

Of course, the most inportant is Theore 8.12 (i)[usual syllogizm], i.e.,
 $(\sharp_1):$ Since Socrates is a man and all men are mortal, it follows that Socrates is mortal.
However, unless the right figure is accepted in general, I am not accepted as the first prover of Theore 8.12 (i)[usual syllogizm],