Sums of vf–Pairs of CA-Conditionals

In this chapter we will show that every vf-pair within the conditional contribution of a ca-conditional can be calculated as the sum of those vf-pairs which are counted by the canonical a-segments. This result will serve us as a base for the upcoming chapters.

The results of this chapter only apply to ca-conditionals. In chapters 6 and 8 we will show how these results can also be applied to c- and ca-conditionals.

5.1 vf–Pairs of A-Segments in a CA-Conditional Without Non-Local Instantiation Restrictions

In this section we will show that for a ca-conditional which is free of non-local instantiation restrictions the vf-pairs contributed by the composed a-segments can be calculated by summing up the vf-pairs contributed by the canonical a-segments.

This result will be further generalized in the next section to ca-conditionals with all type of instantiation restrictions. We decided to start from this more restrictive case here to allow for a step-by-step introduction of the key concepts. The current section therefore can be regarded as a first introduction to the methods which allow us to create sums of vf-pairs. Also only in this section we will rely on the assumptions expressed in section 4.9 concerning the influence of instantiation restrictions on the counting functions.

The next definition introduces canonical and composed vf-pairs, which are those vf-pairs which are contributed by the respective a-segments, as defined by in definition 28.

Definition 36 (Canonical and Composed vf-Pairs)

Let $\Rca$ be a ca-conditional. Let $\omega^c \in \Omega^C(\Rca)$ be a c-segment of $\Rca$ .

A vf-pair $\vf\in\gamma(\omega^c)$ is called a canonical vf-pair iff it is contributed by a canonical a-segment of $\Rca$ .

A vf-pair $\vf\in\gamma{\omega^c}$ is called a composed vf-pair iff it contributed by a composed a-segment of $\Rca$ .

We will now show that the composed vf-pairs of a ca-conditional (which is free of non-local instantiation restrictions) can be calculated as the sums of canonical vf-pairs.

Proposition 9 (vf-Pairs of A-Segments in a CA-Conditional With Only Local Instantiation Restrictions)

Let $\Rca = \big<\big(~C(V_1,\cdots,V_v)~|~A(W_1,\cdots,W_w)~\big),~\xi~\big>$ be a ca-conditional whereby $\xi$ includes no non-local instantiation restrictions.

Then it holds for all $\omega^c_{(i_1,\cdots,i_k)}\in\cat(\Rca)$ and for all $\omega^a_{(j_1,\cdots,j_l)}\in\aat(\Rca)$ that

\vf_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_1,\cdots,j_l)}) = \sum\limits^{l}_{t=1} \vfa_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)})

Proof

Let $n = |\cat(\Rca)|$ be the number of c-atoms and let $m = |\aat(\Rca)|$ be the number of a-atoms of $\Rca$ .

We first show that the semantical combinations creates combinations of all c-segments with all a-segments.

As $\Rca$ is a ca-conditional it holds that $C\neq A$ and therefore it is $\cat(\Rca) \cap \aat(\Rca) = \emptyset$ and if follows that $\ovl(\Rca)=\emptyset$ . It therefore holds that the set of c-atoms and the set of a-atoms are independent of each other. The semantical combination function $\OmegaC(\Rca)\timesM\OmegaA(\Rca)$ will therefore create for every c-segment $\omega^c\in\Omega(\Rca)$ a combination with every a-segment $\omega^a\in\Omega(\Rca)$ , regardless of how the truth values are set. This means specifically that $\omega^c\timesM\omega^a \in \Omega(\Rca)$ .

We now show that for a given c-segment every canonical a-segment contributes the same vf-pair.

Let $a_t\in\aat(\Rca)$ be any a-atom of $\Rca$ and $j_t$ be the index of $a_t$ . Then it holds based on definition 27 that $\omega^a_{(j_t)} \models a_t$ . It follows that the atomic counting function $\vera_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)})$ counts all groundings of the form $\big<\big(c|a_t\big)\big>$ for which $\omega^c_{(i_1,\cdots,i_k)} \models c$ .

As there are no non-local instantiation restrictions the instantiation of groundings of $\Rca$ is the combination of all c-atoms with all a-atoms of $\Rca$ . It follows that for $a_t$ and for every c-atom $c$ there exists a grounding of the form $\big<\big(c|a_t\big)\big>$ . It follows for the case of the canonical a-segment $\omega^a_{(j_t)}$ that $\vera_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)})$ counts exactly the number of c-atoms which are verified by $\omega^c_{(i_1,\cdots,i_k)}$ , so that $\vera_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)}) = k$ .

In the same way it can be shown that $\fala_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)}) = n-k$ , so that $\vfa_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)}) = (k,~n-k)$ .

We now show that for a given c-segment every composed a-segment contributes the same vf-pair.

If more than one a-atom is set within an a-segment $\omega^a_{(j_1,\cdots,j_k)}$ , it holds for every $t$ with $1\leq t\leq l~$ that $~\omega^a_{(j_1,\cdots,j_l)} \models a_t$ . For each such verified a-atom $a_t$ the counting function $\vera_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)})$ counts all c-atoms $c\in\cat(\Rca)$ for which $\omega^c_{(i_1,\cdots,i_k)}\models c$ . This is the same result as shown for the canonical a-segment in the last paragraph, i.e. the verified a-atom $a_t$ contributes to the composed vf-pair ${\vfa_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_1,\cdots,j_l)})}$ the canonical vf-pair ${\vfa_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)}) = (k,~n-k)}$ .

As this holds for all verified a-atoms in $\omega^a_{(j_1,\cdots,j_l)}$ it is $\vfa_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_1,\cdots,j_l)}) = \sum\limits^{l}_{t=1}= \vf_{\Rca}(\omega^c_{(i_1,\cdots,i_k)},~\omega^a_{(j_t)})$ .□

Proposition 9 shows why we chose the terms "canonical" and "composed" in definition 28 and definition 36. The vf-pairs generated by the a-segments of ca-conditionals are very similar to the canonical vectors in a vector space, as each composed vf-pair can be expressed as a sum of the canonical vf-pairs contributed by canonical a-segments. With this we can regard any vf-pair as a sum of vf-pairs (a canonical vf-pair is are just the sum of itself with nothing else). Therefore it is sufficient to perform the counting functions only for the canonical a-segments, all the other vf-pairs can then just be summed up.

The next definition is just for wording purposes and was in fact already used in an informal way in the last proof. We will use it not only for ca-conditionals but for cac-conditionals in general.

Definition 37 (Contribution of a Canonical A-Segment)

Let $\Rcac$ be a cac-conditional. Let $a_t\in\OmegaA(\Rcac)$ be an a-atom of $\Rcac$ and let $j_t$ be the index of $a_t$ , so that $\omega^a_{(j_t)}\in\OmegaA(\Rcac)$ is a canonical a-segment of $\Rcac$ . Let $\omega^a_{(j_1,\cdots,j_t,\cdots,j_l)}$ be an a-segment and $\omega^c\in\OmegaC(\Rcac)$ a c-segment of $\Rcac$ .

We say that $\omega^a$ contributes $\vfa_{\Rcac}(\omega^c,\omega^a_{(j_t)})$ to $\vfa_{\Rcac}(\omega^c,\omega^a_{(j_1,\cdots,j_t,\cdots,j_l)})$ . We also say that $a_t$ contributes $\vfa_{\Rcac}(\omega^c,\omega^a_{(j_t)})$ to $\vfa_{\Rcac}(\omega^c,\omega^a_{(j_1,\cdots,j_t,\cdots,j_l)})$ .

It is $\vfa_{\Rcac}(\omega^c,\omega^a_{(j_t)})$ the contribution of $\omega^a$ (or $a_t$ respectively).

The following proposition states that in case of a ca-conditional with only local instantiation restrictions, every canonical a-segment contributes the same vf-pair to the composed vf-pairs within a conditional impact.

Proposition 10 (Contributions of A-Segments of a CA-Conditional Without Non-Local Instantiation Restrictions)

Let $\Rca$ be a ca-conditional with only local instantiation restrictions. Let $\omega^c \in \Omega^C(\Rca)$ be a c-segment of $\Rca$ .

Then it holds that for the conditional impact $\gamma(\omega^c)$ all canonical vf-pairs are identical.

Proof

This was already shown in the proof of proposition 9.□

Example 30 (vf-Pairs of A-Segments in a CA-Conditional without Instantiation Restrictions)

Let $s=\{~a,~b~\}$ be a sort, $V,~W$ be variables ranging over $s$ and ${\Rca = \big<\big(C(V)|A(W)\big)\big>}$ be a ca-conditional.

Table 5.1 shows the vf truth table of $\Rca$ . The lines in the table which indicate $\omega^a_{(1,2)}$ represent those possible worlds which verify both $A(a)$ and $A(b)$ and due to proposition 9 the related vf-pairs can be calculated from the vf-pairs of the respective two lines above, in which either only $A(a)$ or only $A(b)$ is verified (all for the same c-segment).

We then can say that e.g. in the fourth line ( $\omega^c_{()}\timesM\omega^a_{(1,2)}$ ) of table 5.1 the canonical a-segments $\omega^a_{(1)}$ and $\omega^a_{(2)}$ both contribute the canonical vf-pair $(0,~2)$ to the composed vf-pair $(0,~4)$ .

Table 5.1
$\OmegaC(\Rc) \timesM \OmegaA(\Rc)$	$C(a)$	$C(b)$	$A(a)$	$A(b)$	$vf$
$\omega^c_{()} \timesM\omega^a_{()}$	$0$	$0$	$0$	$0$	$(0,~0)$
$\omega^c_{()} \timesM\omega^a_{(1)}$	$0$	$0$	$1$	$0$	$(0,~2)$
$\omega^c_{()} \timesM\omega^a_{(2)}$	$0$	$0$	$0$	$1$	$(0,~2)$
$\omega^c_{()} \timesM\omega^a_{(1,2)}$	$0$	$0$	$1$	$1$	$(0,~4) = (0,~2) + (0,~2)$

$\omega^c_{(1)} \timesM\omega^a_{()}$	$1$	$0$	$0$	$0$	$(0,~0)$
$\omega^c_{(1)} \timesM\omega^a_{(1)}$	$1$	$0$	$1$	$0$	$(1,~1)$
$\omega^c_{(1)} \timesM\omega^a_{(2)}$	$1$	$0$	$0$	$1$	$(1,~1)$
$\omega^c_{(1)} \timesM\omega^a_{(1,2)}$	$1$	$0$	$1$	$1$	$(2,~2) = (1,~1) + (1,~1)$

$\omega^c_{(2)} \timesM\omega^a_{()}$	$0$	$1$	$0$	$0$	$(0,~0)$
$\omega^c_{(2)} \timesM\omega^a_{(1)}$	$0$	$1$	$1$	$0$	$(1,~2)$
$\omega^c_{(2)} \timesM\omega^a_{(2)}$	$0$	$1$	$0$	$1$	$(1,~2)$
$\omega^c_{(2)} \timesM\omega^a_{(1,2)}$	$0$	$1$	$1$	$1$	$(2,~2) = (1,~1) + (1,~1)$

$\omega^c_{(1,2)}\timesM\omega^a_{()}$	$1$	$1$	$0$	$0$	$(0,~0)$
$\omega^c_{(1,2)}\timesM\omega^a_{(1)}$	$1$	$1$	$1$	$0$	$(2,~0)$
$\omega^c_{(1,2)}\timesM\omega^a_{(2)}$	$1$	$1$	$0$	$1$	$(2,~0)$
$\omega^c_{(1,2)}\timesM\omega^a_{(1,2)}$	$1$	$1$	$1$	$1$	$(4,~0) = (2,~0) + (2,~0)$

Table 5.1: Extended vf Truth Table with vf-Pair Sums of Example 30

We now introduce a new style of vf truth tables, the reduced vf truth table which allows for better overview and readability. Note that we will further modify the reduced vf truth table in the upcoming chapters but the basic structure of the table will stay the same.

Example 31 (Reduced vf Truth Tables)

Continuing from example 30.

Table 5.2 shows a more compact version of the $vf$ truth table in table 5.1.

Table 5.2
		$C(a)$	$C(b)$		$A(a)$	$A(b)$	$\sum$
				$\omega^a_{()}$	$\omega^a_{(1)}$	$\omega^a_{(2)}$	$\omega^a_{(1,2)}$
	$\omega^c_{()}$	$0$	$0$	$(0,~0)$	$(0,~2)$	$(0,~2)$	$(0,~4)$
$C(a)$	$\omega^c_{(1)}$	$0$	$1$	$(0,~0)$	$(1,~1)$	$(1,~1)$	$(2,~2)$
$C(b)$	$\omega^c_{(2)}$	$1$	$0$	$(0,~0)$	$(1,~1)$	$(1,~1)$	$(2,~2)$
	$\omega^c_{(2,1)}$	$1$	$1$	$(0,~0)$	$(2,~0)$	$(2,~0)$	$(4,~0)$
				$[0]$	$[2]$	$[2]$	$2[2]$

Table 5.2: Reduced vf Truth Table of Example 31

On its left side the table lists all c-segments as rows and on the top all a-segments as columns. This is similar to the tables we have seen in example 26 for the unification function. For better readability the rows/columns of the canonical segments indicate the related ground atoms which are set in the possible world segment. The combination of truth values can be read from the indexes of the different possible world segments (e.g. $\omega^c_{(2)}$ indicates the c-segment where $C(a)=0$ and $C(b)=1$ ). We also show the c-atoms as separate columns and below them the truth values assigned to them by the c-segments.

The cells below the a-segments indicate the vf-pairs resulting from the semantical combination of the two possible world segments in the row (c-segment) and the column (a-segment). All vf-pairs to the right of the canonical a-segments can be calculated by simply adding the canonical vf-pairs.

Finally the last row shows the mos notation of the vf-pairs in the related column, which makes the generated conditional contribution easy to read, as $\gamma(\Rca) = \{~[0],~[2],~2[2]~\}$ .

5.2 vf-Pairs of A-Segments in a CA-Conditional With All Types Of Instantiation Restrictions

The results of section 5.2 only work for ca-conditionals which are free of non-local instantiation restrictions. Due to this restriction the contributions of the different canonical a-segments are identical within a c-segment. In this section we de-couple the canonical a-segments in a way which allows us to generate and accumulate their contributions individually. With this it is then possible to generate the vf-pairs of all ca-conditionals, i.e. their conditional contribution, regardless of the type of instantiation restriction used.

Definition 38 (Set of C-Atoms Counted by an A-Atom)

Let $\Rcac$ be a cac-conditional. Let $a\in\aat(\Rcac)$ be an a-atom of $\Rcac$ .

The set of c-atoms counted by $a$ is

\acountsc(\Rca,~a) := \{~c~|~\big<(c|a)\big>~\in\gnd(\Rcac)~\}.

Example 32 (Set of C-Atoms Counted by an A-Atom)

Let $s=\{~a,~b,~c~\}$ be a sort, $V,~W$ be variables ranging over $s$ and ${\Rca = \big<\big(C(V)|A(W)\big), V\neq W \big>}$ be a ca-conditional.

Then we get

$\acountsc(\Rca,~A(a))$	$= \{~C(b),~C(c)~\}$
$\acountsc(\Rca,~A(b))$	$= \{~C(a),~C(c)~\}$
$\acountsc(\Rca,~A(c))$	$= \{~C(a),~C(b)~\}$

Definition 38 assigns to every canonical a-segment the set of those c-atoms which the a-segment contributes to the relevant counting functions $vera$ and $fala$ , which then form the vf-pair. With these sets we have de-coupled the canonical a-segments from each other, so that they can contribute different values to the counting functions.

The following proposition is nearly identical to proposition 9 but allows any type of instantiation restrictions. Nevertheless, it is only valid for a ca-conditional, i.e. it requires that $C\neq A$ .

Proposition 11 (vf-Pairs of A-Segments in a CA-Conditional)

Let $\Rca = \big<\big(~C(V_1,\cdots,V_v)~|~A(W_1,\cdots,W_w)~\big),~\xi~\big>$ be a ca-conditional.

It holds for all $\omega^c\in\OmegaC(\Rca)$ and for all $\omega^a_{(j_1,\cdots,j_l)}\in\OmegaA(\Rca)$ that

\vf_{\Rca}(\omega^c,~\omega^a_{(j_1,\cdots,j_l)}) = \sum\limits^{l}_{t=1} \vfa_{\Rca}(\omega^c,~\omega^a_{(j_t)})

Proof

Let $a_t\in\aat(\Rca)$ be an a-atom of $\Rca$ and $j_t$ be the index of $a_t$ so that $\omega^a_{(j)}\in\OmegaA(\Rca)$ is the canonical a-segment of $\Rca$ related to $a_t$ . Let $\omega^c\in\OmegaC(\Rca)$ be a c-segment of $\Rca$ .

Let $v_t=\vera_{\Rca}(\omega^c,~\omega^a_{(j_t)})$ and $f_t=\fala_{\Rca}(\omega^c,~\omega^a_{(j_t)})$ be the results of the counting functions for $\omega^c$ and $\omega^a_{(j_t)}$

Based on the definition of a-segments it holds for every composed a-segment $\omega^a_{(j_1,\cdots,j_k)}$ and for every $t$ with $1\leq t\leq l$ that $\omega^a_{(j_1,\cdots,j_l)} \models a_t$ .

Let $\omega^a_{(j_1,\cdots,j_l)} \in\OmegaA(\Rca)$ be a composed a-segment of $\Rca$ with $\omega^a_{(j_1,\cdots,j_l)}\models a_t$ .

It then holds that $a_t$ counts those groundings $\big<\big(c|a_t\big)\big>\in\gnd(\Rca)$ of $\Rca$ in which $c$ is either verified or falsified, i.e. $a_t$ contributes the canonical vf-pair $(v_t,~f_t)$ to the composed vf-pair $\vfa_{\Rca}(\omega^c,~\omega^a_{(j_t)})$ . As this holds for all verified a-atoms in $\omega^a_{(j_1,\cdots,j_l)}$ it follows

\vfa_{\Rca}(\omega^c,~\omega^a_{(j_1,\cdots,j_l)}) = \sum\limits^{l}_{t=1}= \vf_{\Rca}(\omega^c,~\omega^a_{(j_t)})

□

Example 33 (vf-Pairs of A-Segments in a CA-Conditional)

Let $s=\{~a,~b,~c~\}$ be a sort, $V,~W$ be variables ranging over $s$ and $\Rca = \big<\big(C(V)|A(W,a)\big), V\neq c, V\neq W\big>$ be a ca-conditional.

It is

$\acountsc(\Rca,~A(a,a))$	$= \{~C(b)~\}$
$\acountsc(\Rca,~A(b,a))$	$= \{~C(a)~\}$
$\acountsc(\Rca,~A(c,a))$	$= \{~C(a),~C(b)~\}$

For better overview, 5.3 shows an (incomplete) old-fashioned vf truth table, of which a number of lines have been left out, to make the table more readable.

Table 5.3
$\OmegaC(\Rca) \timesM \OmegaA(\Rca)$	$C(a)$	$C(b)$	$A(a,a)$	$A(b,a)$	$A(c,a)$
$\omega^c_{()}\timesM \omega^a_{()}$	$0$	$0$	$0$	$0$	$0$	$(0,~0)$
$\omega^c_{()}\timesM \omega^a_{(1)}$	$0$	$0$	$1$	$0$	$0$	$(0,~1)$
$\omega^c_{()}\timesM \omega^a_{(2)}$	$0$	$0$	$0$	$1$	$0$	$(0,~1)$
$\omega^c_{()}\timesM \omega^a_{(3)}$	$0$	$0$	$0$	$0$	$1$	$(0,~2)$
$\omega^c_{()}\timesM \omega^a_{(1,2)}$	$0$	$0$	$1$	$1$	$0$	$(0,~2) = (0,~1) + (0,~1)$
$\omega^c_{()}\timesM \omega^a_{(1,3)}$	$0$	$0$	$1$	$0$	$1$	$(0,~3) = (0,~1) + (0,~2)$
$\omega^c_{()}\timesM \omega^a_{(2,3)}$	$0$	$0$	$0$	$1$	$1$	$(0,~3) = (0,~1) + (0,~2)$
$\omega^c_{()}\timesM \omega^a_{(1,2,3)}$	$0$	$0$	$1$	$1$	$2$	$(0,~4) = (0,~1) + (0,~1) + (0,~2)$

$\omega^c_{(1)}\timesM \omega^a_{()}$	$1$	$0$	$0$	$0$	$0$	$(0,~0)$
$\omega^c_{(1)}\timesM \omega^a_{(1)}$	$1$	$0$	$1$	$0$	$0$	$(0,~1)$
$\omega^c_{(1)}\timesM \omega^a_{(2)}$	$1$	$0$	$0$	$1$	$0$	$(1,~0)$
$\omega^c_{(1)}\timesM \omega^a_{(3)}$	$1$	$0$	$0$	$0$	$1$	$(1,~1)$
$\omega^c_{(1)}\timesM \omega^a_{(1,2)}$	$1$	$0$	$1$	$1$	$0$	$(1,~1) = (0,~1) + (0,~1)$
$\omega^c_{(1)}\timesM \omega^a_{(1,3)}$	$1$	$0$	$1$	$0$	$1$	$(1,~2) = (0,~1) + (1,~1)$
$\omega^c_{(1)}\timesM \omega^a_{(2,3)}$	$1$	$0$	$0$	$1$	$1$	$(2,~1) = (1,~0) + (1,~1)$
$\omega^c_{(1)}\timesM \omega^a_{(1,2,3)}$	$1$	$0$	$1$	$1$	$2$	$(2,~2) = (0,~1) + (1,~0) + (1,~1)$
$\vdots$
$\omega^c_{(1,2)}\timesM \omega^a_{()}$	$1$	$1$	$0$	$0$	$0$	$(0,~0)$
$\omega^c_{(1,2)}\timesM \omega^a_{(1)}$	$1$	$1$	$1$	$0$	$0$	$(1,~0)$
$\omega^c_{(1,2)}\timesM \omega^a_{(2)}$	$1$	$1$	$0$	$1$	$0$	$(1,~0)$
$\omega^c_{(1,2)}\timesM \omega^a_{(3)}$	$1$	$1$	$0$	$0$	$1$	$(2,~0)$
$\omega^c_{(1,2)}\timesM \omega^a_{(1,2)}$	$1$	$1$	$1$	$1$	$0$	$(2,~0) = (1,~0) + (1,~0)$
$\omega^c_{(1,2)}\timesM \omega^a_{(1,3)}$	$1$	$1$	$1$	$0$	$1$	$(3,~0) = (1,~0) + (2,~0)$
$\omega^c_{(1,2)}\timesM \omega^a_{(2,3)}$	$1$	$1$	$0$	$1$	$1$	$(3,~0) = (1,~0) + (2,~0)$
$\omega^c_{(1,2)}\timesM \omega^a_{(1,2,3)}$	$1$	$1$	$1$	$1$	$2$	$(4,~0) = (1,~0) + (1,~0) + (2,~0)$

Table 5.3: Extended vf Truth Table with vf-Pair Sums of Example 33

Table 5.4 shows the reduced vf truth table.

Table 5.4
		$C(a)$	$C(b)$	$A(a,a)$	$A(b,a)$	$A(c,a)$	$\sum$
				$\omega^a_{()}$	$\omega^a_{(1)}$	$\omega^a_{(2)}$	$\omega^a_{(3)}$	$\omega^a_{(1,2)}$	$\omega^a_{(1,3)}$	$\omega^a_{(2,3)}$	$\omega^a_{(1,2,3)}$
	$\omega^c_{()}$	$0$	$0$	$(0,~0)$	$(0,~1)$	$(0,~1)$	$(0,~2)$	$(0,~2)$	$(0,~3)$	$(0,~3)$	$(0,~4)$
$C(a)$	$\omega^c_{(1)}$	$0$	$1$	$(0,~0)$	$(0,~1)$	$(1,~0)$	$(1,~1)$	$(1,~1)$	$(1,~2)$	$(2,~1)$	$(2,~2)$
$C(b)$	$\omega^c_{(2)}$	$1$	$0$	$(0,~0)$	$(1,~0)$	$(0,~1)$	$(1,~1)$	$(1,~1)$	$(2,~1)$	$(1,~2)$	$(2,~2)$
$C(a), C(b)$	$\omega^c_{(1,2)}$	$1$	$1$	$(0,~0)$	$(1,~0)$	$(1,~0)$	$(2,~0)$	$(2,~0)$	$(3,~0)$	$(3,~0)$	$(4,~0)$
				$[0]$	$[1]$	$[1]$	$[2]$	$[2]$	$[3]$	$[3]$	$2[2]$

Table 5.4: Reduced vf Truth Table of Example 33

It is $\gamma(\Rca) = \{~[0],~[1],~[2],~[3],~2[2]~\}$ .

The following proposition shows that for a ca-conditional the contributions of a single a-segment over all c-segments is always a MOS. This finding is essential for the next chapter where we will look into the calculation of conditional contributions.

Proposition 12 (Canonical A-Segments of CA-Conditionals Generate Maximum Ordered Sum)

Let $\Rca$ be an ca-conditional with ${|\cat(\Rca)| = n}$ and ${\OmegaC(\Rca)=\{~\omega^c_1,\cdots,~\omega^c_{2^n}~\}}$ .

For any canonical a-segment $\omega^a_{(j)}$ it holds that the set of vf-pairs $\vera_{\Rca}(\omega^c_1,~\omega^a_{(j)}),~\cdots,~\vera_{\Rca}(\omega^c_{2^n},~\omega^a_{(j)})$ is the MOS $\big[|\acountsc(\Rca,~a_{(j)})|\big]$ .

Proof

We first show the sum of the atomic counting functions for any given c-atom and any given canonical a-atom $a_j$ is $|\acountsc(\Rca,~a_j)|$ .

Let $a_j$ be the a-atom of $\Rca$ with index $j$ .

It is $\OmegaC(\Rca)$ the set of all c-segments of $\Rca$ , whereby every c-segment assigns a different combination of truth values to the ground atoms within $\cat(\Rca)$ .

As $\omega^a_{(j)}$ is a canonical a-segment it holds for all $a_i\in\aat(\Rca)$ with $i\neq j$ that $\omega^a_{(j)} \not\models a_i$ .

It then holds that

\vera_{\Rca}(\omega^c,\omega^a_{(j)}) = \Big|\big\{~\big<(c|a_j)\big>~\in\gnd(\Rca)~| ~\omega^c \models c,~\omega^a_{(j)} \models a_j~\big\}\Big|

which is equivalent to

\vera_{\Rca}(\omega^c,\omega^a_{(j)}) = \Big|\big\{~\big<(c|a_j)\big>~\in\gnd(\Rca)~| ~\omega^c \models c,~c\in\acountsc(\Rca,~a_j)~\big\}\Big|.

In the same way it holds that

\fala_{\Rca}(\omega^c,\omega^a_{(j)}) = \Big|\big\{~\big<(c|a_j)\big>~\in\gnd(\Rca)~\big| ~\omega^c \models \neg c,~c\in\acountsc(\Rca,~a_j)~\big\}\Big|.

As $\omega^c$ either verifies or falsifies every c-atom it follows that the sum of the atomic counting functions $\vera_{\Rca}(\omega^c,\omega^a_{(j)})+\fal_{\Rca}(\omega^c,\omega^a_{(j)})$ is the number of c-atoms which occur together with a-atom $a_{(j)}$ in any grounding of $\Rca$ , which is $|\{~c~|~\big<(c|a)\big>~\in\gnd(\Rca)~\}| = |\acountsc(\Rca,~a_j)|$ . It follows that for all $\omega^c \in \OmegaC(\Rca)$ it holds that

\vera_{\Rca}(\omega^c,\omega^a_{(j)})+\fal_{\Rca}(\omega^c,\omega^a_{(j)})= |\acountsc(\Rca,~a_j)|

and therefore it holds that $0\leq \vera_{\Rca}(\omega^c,\omega^a_{(j)}),\fala_{\Rca}(\omega^c,\omega^a_{(j)})\leq |\acountsc(\Rca~a_j)|$ .

We now show that the set of all vf-pairs contributed by $\omega^a_{(j)}$ is $[~|\acountsc(\Rca,~a_j)|~]$ .

Let $k = |\acountsc(\Rca,~a_j)|$ and let

V := \bigcup_{\omega^c_i\in\OmegaC(\Rca)}\vera_{\Rca}(\omega^c_i,~\omega^a_{(j)})

and

F := \bigcup_{\omega^c_i\in\OmegaC(\Rca)} \fala_{\Rca}(\omega^c_i,\omega^a_{(j)}).

As every c-segment assigns a different combination of truth values to the c-atoms and as $\OmegaC(\Rca)$ is the set of all c-segments, i.e. all possible combinations of such truth values, it follows that $V = \{~0,~1,~\cdots,~k-1,~k~\}$ and in the same way that $F = \{~0,~1,~\cdots,~k-1,~k~\}$ . It follows that

	$\bigcup\limits_{\omega^c_i~\in~\OmegaC(\Rca)}(~\vera_{\Rca}(\omega^c_i,\omega^a_{(j)}),~\fala_{\Rca}(\omega^c_i,\omega^a_{(j)})~)$
$=$	$\{~ (0,~k),~(1,~k-1),~\cdots,~(k-1,~1),~(k,~0)~\}$
$=$	$\{~[k]~\}$
$=$	$\{~[~\|\acountsc(\Rca,~a_j)\|~]\}$

□

It is important to note that the above proposition only holds for ca-conditionals and, as we will show in section 6.1, for c-conditionals. It does not hold for cc-conditionals, as we will show in section 6.3.

Proposition 12 also makes it to check the completeness of reduced vf truth tables, as the columns related to a canonical a-segment within a reduced vf truth table always include a complete set of MOSs.

Example 34 (Canonical A-Segments of CA-Conditionals Generate Maximum Ordered Sum)

Examples for the MOSs generated by canonical a-segments can e.g. be found in table 5.4 in the reduced vf truth table in example 33.

From proposition 12 we can easily follow that composed a-segments of ca-conditionals contribute either a MOS or a multiple of a MOS.