логика - Живые и мёртвые

Пусть $%P$% - множество людей, $%Hs$% - множество организмов вида Homo sapiens, $%G$% - множество богов, $%A$% - множество деятелей, $%T$% - время, $%So$% - множество душ.

Вопрос 1: Верны ли высказывания:

$% A = P \cup G \rightarrow \forall x (x \in A \rightarrow x \ is \ alive. \ \oplus \ x \ is \ dead.),$%

$% A = P \cup G \rightarrow \forall x (x \in A \rightarrow (x \ is \ immortal. \rightarrow x \ is \ alive.))$%

$% \begin {cases} A = P \cup G \wedge \varnothing \notin \{G, \ P \cap Hs, \ P \setminus Hs\} \\ \forall x (x \in A \rightarrow x \ is \ alive. \ \oplus \ x \ is \ dead.) \\ \forall x (x \in A \wedge x \ is \ immortal. \rightarrow x \ is \ alive.) \\ \forall x (x \in P \cap Hs \rightarrow \ x \ is \ mortal.) \\ \forall x (x \in G \cup (P \setminus Hs) \rightarrow x \ is \ immortal.) \end {cases} \rightarrow \begin {cases} \forall x (x \in P \cap Hs \rightarrow x \ is \ alive. \oplus \ x \ is \ dead.) \\ \forall x (x \in G \cup (P \setminus Hs) \rightarrow x \ is \ alive.) \end {cases}?$%

Вопрос 2: Верно ли, что:

$%(2.1.1) \ \ A = P \cup G \rightarrow \forall x (x \in A \rightarrow x \ is \ alive \ sometimes.),$%

$%(2.1.2) \ \ A = P \cup G \rightarrow \forall x (x \in A \rightarrow (x \ is \ immortal. \rightarrow x \ is \ alive \ always.)),$%

$%(2.1.3) \ \ A = P \cup G \rightarrow \forall x (x \in A \rightarrow (x \ is \ mortal. \leftrightarrow x \ is \ dead \ sometimes.)),$%

$%(2.2.1) \ \ x \ is \ alive \ sometimes. \leftrightarrow \exists \tau (\tau \in \wp (T) \setminus \{\varnothing\} \wedge x \ is \ alive \ during \ \tau.),$%

$%(2.2.2) \ \ x \ is \ alive \ always. \leftrightarrow \forall \tau (\tau \in \wp (T) \setminus \{\varnothing\} \rightarrow x \ is \ alive \ during \ \tau.),$%

$%(2.2.3) \ \ x \ is \ dead \ sometimes. \leftrightarrow \exists \tau (\tau \in \wp (T) \setminus \{\varnothing\} \wedge x \ is \ dead \ during \ \tau.),$%

$%(2.3.1) \ \ x \ is \ alive \ during \ \tau. \leftrightarrow \forall t (t \in \tau \rightarrow x \ and \ x's \ soul \ are \ connected \ at \ t.),$%

$%(2.3.2) \ \ x \ is \ dead \ during \ \tau. \leftrightarrow \exists t (t \in \tau \wedge x \ and \ x's \ soul \ are \ not \ connected \ at \ t.)?$%

Примечание

1. $%\tau \in \wp(T) \setminus \{\varnothing\} \ \Leftrightarrow \ \tau \subseteq T \wedge \tau \neq \varnothing \ \Leftrightarrow \ \tau \neq \varnothing \wedge \forall t (t \in \tau \rightarrow t \in T)$%
2. "x and x's soul are connected at t." $%\Leftrightarrow$% "x и x-ова душа связаны [между собой] в [момент] t."
3. "x and x's soul are not connected at t." $%\Leftrightarrow$% "x и x-ова душа не связаны [между собой] в [момент] t."