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\section{Valuations}
\label  {valuations}

Let $H_\odot$ and $J_\odot$ be a ZHA and a J-operator on it, and let
$v_\odot$ be a function from the set $\{P,Q\}$ to $H$. By an abuse of
language $v_\odot$ will also denote the triple $(H_\odot, J_\odot, v_\odot)$ --- and by a second abuse of language $v_\odot$ will also
denote the obvious extension of $v_\odot: \{P,Q\}→H$ to the set of all
valid expressions formed from $P$, $Q$, $·^*$, $⊤$, $⊥$, and the
connectives.

Let $i,j∈\{0,\ldots,7\}$. Then $(\oand_i,\oand_j)∈\SCube^*_\land$
means that $\oand_i ≤ \oand_j$ is a theorem, and so $v_\odot(\oand_i) ≤ v_\odot(\oand_j)$ holds; i.e.,
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$$\SCube^*_\land ⊆ \setofst {(\oand_i,\oand_j)} {i,j∈\{0,\ldots,7\}, \; v_\odot(\oand_i) ≤ v_\odot(\oand_j)}$$
%
and the same for:
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$$\begin{array}{c} \SCube^*_\lor ⊆ \setofst {(\oor_i,\oor_j)} {i,j∈\{0,\ldots,7\}, \; v_\odot(\oor_i) ≤ v_\odot(\oor_j)} \\ \SCube^*_\to ⊆ \setofst {(\oimp_i,\oimp_j)} {i,j∈\{0,\ldots,7\}, \; v_\odot(\oimp_i) ≤ v_\odot(\oimp_j)} \\ \end{array}$$

Some valuations that turn these $⊆$'s into $=$'. Let
%
%L mp = mpnew({def="orCube", scale="11pt"}, "12321L"):addcuts("c 21/0 0|12")
%L mp:put(v"10", "P"):put(v"20", "P*", "P^*")
%L mp:put(v"01", "Q"):put(v"02", "Q*", "Q^*")
%L mp:output()
%
%L mp = mpnew({def="andCube", scale="11pt"}, "12321"):addcuts("c 2/10 01|2")
%L mp:put(v"20", "P"):put(v"21", "P*", "P^*")
%L mp:put(v"02", "Q"):put(v"12", "Q*", "Q^*")
%L mp:output()
%
%L mp = mpnew({def="impCube", scale="11pt"}, "12R1L"):addcuts("c 2/10 01|2")
%L mp:put(v"10", "P")  -- :put(v"20", "P*", "P^*")
%L mp:put(v"01", "Q")  -- :put(v"02", "Q*", "Q^*")
%L mp:output()
%
\pu
%
$$\begin{array}{c} (H_∧, J_∧, v_∧) = \andCube \qquad (H_∨, J_∨, v_∨) = \orCube \\ (H_→, J_→, v_→) = \impCube \\ \end{array}$$
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then
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$$\begin{array}{c} \SCube^*_\land = \setofst {(\oand_i,\oand_j)} {i,j∈\{0,\ldots,7\}, \; v_∧(\oand_i) ≤ v_∧(\oand_j)} \\ \SCube^*_\lor = \setofst {(\oor_i,\oor_j)} {i,j∈\{0,\ldots,7\}, \; v_∨(\oor_i) ≤ v_∨(\oor_j)} \\ \SCube^*_\to = \setofst {(\oimp_i,\oimp_j)} {i,j∈\{0,\ldots,7\}, \; v_→(\oimp_i) ≤ v_→(\oimp_j)} \\ \end{array}$$
%
or, in more elementary terms:

\newpage

{\sl A very important fact.}
For any $i$ and $j$,
%
$$\pu \begin{array}{rcl} \oand_i≤\oand_j & \text{ is a theorem iff it is true in } & \andCube \;\; , \\ \\ \oor_i≤\oor_j & \text{ is a theorem iff it is true in } & \orCube \;\; , \\ \\ \oimp_i≤\oimp_j & \text{ is a theorem iff it is true in } & \impCube \;\; . \\ \end{array}$$

The very important fact, and the valuations $v_∧$, $v_∨$, $v_→$, give
us:

\begin{itemize}

\item a way to {\sl remember} which sentences of the forms
$\oand_i≤\oand_j$, $\oor_i≤\oor_j$, $\oimp_i≤\oimp_j$ are theorems;

\item countermodels for all the sentences of these forms not in
$\SCube_∧$, $\SCube_∨$, $\SCube_→$. For example, $\oor_7≤\oor_4$ is
not in $\SCube_∨$; and $v_∨(\oor_7)≤v_∨(\oor_4)$, which shows that
$\oor_7≤\oor_4$ can't be a theorem.

\end{itemize}

% (find-books "__cats/__cats.el" "bell")
% (find-books "__cats/__cats.el" "bell" "163")

{\sl An observation.} I arrived at the cubes $\ECube_∧^*$,
$\ECube_∨^*$, $\ECube_→^*$ by taking the material in the corollary 5.3
of chapter 5 in \cite{BellLST} and trying to make it fit into less
mental space (as discussed in \cite{OchsIDARCT}); after that I wanted
to be sure that each arrow that is not in the extended cubes has a
countermodel, and I found the countermodels one by one; then I
wondered if I could find a single countermodel for all non-theorems in
$\ECube_∧^*$ (and the same for $\ECube_∨^*$ and $\ECube_→^*$), and I
equivalence classes in $\ECube_∧^*$, and change it bit by bit, getting
valuations that distinguished more equivalence classes at every step.
Eventually I arrived at $v_∧$, $v_∨$ and at $v_→$, and at the ---
surprisingly nice --- very important fact'' above.

% (ph2p 20 "ZHA-vals-good")
% (ph2     "ZHA-vals-good")

Note that this valuation
%
%L mp = mpnew({def="orand", scale="11pt"}, "1234321L"):addcuts("c 432/10 01|23")
%L mp:put(v"20", "P"):put(v"31", "P*", "P^*")
%L mp:put(v"02", "Q"):put(v"13", "Q*", "Q^*")
%L mp:output()
%
$$(H_{∧∨},J_{∧∨},v_{∧∨}) \;\; = \;\; \pu\orand$$
%
distinguishes all equivalence classes in $\ECube^*_∧$ and in
$\ECube^*_∨$, but not in $\ECube^*_→$... it thinks'' that $P→Q$ and
$P^*→Q$ are equal.

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