Interpreting Common Appearances

In this chapter we will look at the common appearances of two atomic conditionals and try to find out from the CA-tables whether the two conditionals hold a balanced sharing. We gave the chapter a more general name, as we assume that the CA-tables might reveal even more information about the two related conditionals.

We have already seen in section 3.3 that an imbalanced usage of ground atoms cannot be determined from a given CS and that therefore the function $tr_{CS}$ is unlikely to exist. Due to that all we restrict ourselves in this chapter to conditionals which are balanced with respect to the usage of ground atoms.

We will try to interpret the common appearances of two atomic conditionals based on the patterns within their CA-table. From those we will see that it seems possible to detect from these patterns whether the two conditionals hold a balanced sharing. These patterns look differently based on the relation of the two predicate symbols in the conclusion and the antecedence of the two atomic conditionals and also based on the (non-local) instantiation restrictions used. In order to make these cases easier to differentiate, we assign specific case-names to them.

We will not fully cover all cases. The goal of this chapter is to show a possible approach on how CA-Tables can be interpreted. A full study would have gone beyond what is achievable within this thesis.

The findings of this chapter will not be described as formal propositions but as assumptions. We will introduce a large number of example cases for which we will only partially try to explain why certain patterns occur in the CA-tables.

10.1 Definitions

The atomic conditionals in this chapter will all make use of the same set of predicates, variables and sorts which we define upfront.

Example 57 (Predicates, Variables and Sorts Used in Chapter 10)

Let $C_1, C_2, A_1, A_2$ be unary predicates.

Let $V_1, V_2, W_1, W_3$ be variables ranging over sort $s=\{~a,~b,~c~\}$ .

Note that in this chapter we only use unary predicates, i.e. non of them makes use of exactly one variable. Nevertheless, the findings of this chapter also hold for predicates which make use of more than one variable.

Definition 48 (Common Appearance Block)

Let $\R_1$ and $\R_2$ be two atomic conditionals.

Let $[mos_1]$ be a MOS within the conditional impact of $\R_1$ and let $[mos_2]$ be a MOS within the conditional impact of $\R_2$ .

Then the common appearance block (CA-block) of $mos_1$ and $mos_2$ within the common appearance table of $\R_1$ and $\R_2$ is the section of the table where the lines and columns of $[mos_1]$ and $[mos_2]$ intersect. We will denote such CA-blocks by ( $[mos_1]$ / $[mos_2]$ ).

A full CA-block is a CA-block in which all cells indicate a "*", we say that such CA-blocks are full.

A full CA-table is a CA-table in which all CA-blocks indicate at least a "*", we say that such CA-tables are full.

A complete CA-table is a CA-table in which all CA-blocks are full, we say that such CA-tables are complete.

A CA-block-line is either vertical, horizontal or diagonal single line of "*" symbols within a CA-block.

A CA-table-line is either a vertical, horizontal or diagonal single line of full CA-blocks within a CA-table.

We will see examples for all these definitions in the upcoming chapters.

Our goal will be to show, based on examples that two atomic conditionals are balanced with respect to sharing of ground atoms if either their CA-table is complete or there exists at least one CA-block-line or a CA-table-line within their CA-table.

10.2 Complete CA-Tables

We start by showing that a complete CA-Table, i.e. a CA-table in which all CA-blocks are full, is always balanced with respect to sharing.

The first case we look at we call the $C_1$ - $C_2$ case, which represents the CA of two c-conditionals $\R_1^c$ and $\R_2^c$ which both hold no instantiation restrictions and for which the conclusion arguments are different, i.e. it is $\R_1^c = \acond{C_1(V_1)}{\top}{\emptyset}$ and $\R_2^c = \acond{C_1(V_2)}{\top}{\emptyset}$

Example 58 ( $C_1$ - $C_2$ Case)

Continuing from example 57.

Let $\Rc_1 = \acond{C_1(V_1)}{\top}{\emptyset}$ and $\Rc_2 = \acond{C_2(V_1)}{\top}{\emptyset}$ .

Table 10.1 shows the vf truth table and table 10.2 the CA-table of $\Rc_1$ and $\Rc_2$ .

Table 10.1
$C_1(a)$	$C_1(b)$	$C_2(a)$	$C_2(b)$	$\gamma(\R_1^c)$	$\gamma(\R_2^c)$
$0$	$0$	$0$	$0$	$(0,~2)$	$(0,~2)$
$0$	$0$	$0$	$1$	$(0,~2)$	$(1,~1)$
$0$	$0$	$1$	$0$	$(0,~2)$	$(1,~1)$
$0$	$0$	$1$	$1$	$(0,~2)$	$(2,~0)$

$0$	$1$	$0$	$0$	$(1,~1)$	$(0,~2)$
$0$	$1$	$0$	$1$	$(1,~1)$	$(1,~1)$
$0$	$1$	$1$	$0$	$(1,~1)$	$(1,~1)$
$0$	$1$	$1$	$1$	$(1,~1)$	$(2,~0)$

$1$	$0$	$0$	$0$	$(1,~1)$	$(0,~2)$
$1$	$0$	$0$	$1$	$(1,~1)$	$(1,~1)$
$1$	$0$	$1$	$0$	$(1,~1)$	$(1,~1)$
$1$	$0$	$1$	$1$	$(1,~1)$	$(2,~0)$

$1$	$1$	$0$	$0$	$(2,~1)$	$(0,~2)$
$1$	$1$	$0$	$1$	$(2,~1)$	$(1,~1)$
$1$	$1$	$1$	$0$	$(2,~1)$	$(1,~1)$
$1$	$1$	$1$	$1$	$(2,~0)$	$(2,~0)$

Table 10.1: vf Truth Table for the

C_1

C_2

Case of Example 58

Table 10.2
		$[2]$
$CA(\R_1^c,\R_2^c)$		$( 0, 2)$	$( 1, 1)$	$( 2, 0)$
$[2]$	$( 0, 2)$	$*$	$*$	$*$
	$( 1, 1)$	$*$	$*$	$*$
	$( 2, 0)$	$*$	$*$	$*$

Table 10.2: CA-table for the

C_1

C_2

Case of Example 58

Based on definition 48 we can see that the CA-table only includes one CA-block, as both conditional contributions include only a single MOS. Based on the same definition we see that the CA-table for this case is complete, as all CA-blocks are full.

Example 58 shows two c-conditionals which hold different predicates in their conclusion part. We see from the vf truth table that every vf-pair contributed by $\R_1^c$ has a common appearance with every vf-pair contributed by $\R_2^c$ . This is obviously the case due to the independence of the ground atoms, i.e. as it holds that $\ovl(\R_1^c,\R_2^c)=\emptyset$ . This is why every cell of the CA-table indicates a "*". We therefore formulate this result as an assumption.

Assumption 4 (Balanced Sharing and Complete CA-Tables).

Let $\R_1$ and $\R_2$ be two conditionals.

If the CA-table of $\Ra_1$ and $\Ra_2$ is a complete CA-table then it holds that $\herbrand(\{\R_1\})\cap\herbrand(\{\R_2\})=\emptyset$ .

Then two conditionals of a complete CA-table are balanced with respect to sharing of ground atoms.

Note that the assumption applies to any type of conditional, i.e. it is not restricted to atomic conditionals only and also does not limit the use of instantiation restrictions. Also note that there is no equivalence between the statement that the CA-table is complete and the ground atoms of both conditionals are mutually exclusive.

The following example shows that assumption 4 also holds for two atomic conditionals, where one includes a local instantiation restrictions (called the $C_1$ - $C_2$ -localIR case). As for most of the following examples we will not show the vf truth table or how we derive the vf-pairs. For the examples given this is regarded as a straight forward task.

Example 59 ( $C_1$ - $C_2$ -localIR Case)

Continuing from example 57.

Let $\R_3^c = \acond{C1(V1)}{\top}{V1 \neq x}$ and $\R_4^c = \acond{C2(V2)}{\top}{\emptyset}$ .

Table 10.3 shows the CA-table of $\R_3^c$ and $\R_4^c$ .

Table 10.3
		$[3]$
$CA(\Rc_3,\Rc_4)$		$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$
$[2]$	$( 0, 2)$	$*$	$*$	$*$	$*$
	$( 1, 1)$	$*$	$*$	$*$	$*$
	$( 2, 0)$	$*$	$*$	$*$	$*$

Table 10.3: CA-table for the

C_1

C_2

-localIR case of Example 59

Finally we look at a more sophisticated example of two ca-conditionals which make use of four different predicate symbols and where one holds a local instantiation restriction of type $V=W$ (called the $C_1A_1$ - $C_2A_2$ -localIR case) .

Example 60 ( $C_1A_1$ - $C_2A_2$ -localIR Case)

Continuing from example 57.

Let $\Rca_1 = \acond{C_1(V_1)}{A_1(W_1)}{V_1\neq W_1}$ and $\Rca_2 = \acond{C_2(V_2)}{A_2(W2)}{\emptyset}$

Table 10.4 shows the CA-table of $\Rca_1$ and $\Rca_2$ .

Table 10.4
		$[0]$	$[3]$				$2[3]$				$3[3]$
$CA(\Rca_1,\Rca_2)$		$( 0, 0)$	$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$	$( 0, 6)$	$( 2, 4)$	$( 4, 2)$	$( 6, 0)$	$( 0, 9)$	$( 3, 6)$	$( 6, 3)$	$( 9, 0)$
$[0]$	$( 0, 0)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$[2]$	$( 0, 2)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 1, 1)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 2, 0)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$[4]$	$( 0, 4)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 1, 3)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 2, 2)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 3, 1)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 4, 0)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$2[3]$	$( 0, 6)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 2, 4)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 4, 2)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 6, 0)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$

Table 10.4: CA-table for the

C_1A_1

C_2A_2

-localIR case of Example 60

We finally get to see some CA-blocks in this table. For example the cells which are in the crossing of those lines where the conditional contribution of $\Rca_1$ is $[4]$ (i.e. the lines with vf-pairs $(0,4),~(1,3),~(2,2),~(3,1)$ and $(4,0)$ ) and those columns where the conditional contribution of $\Rca_2$ is $2[3]$ (i.e. the columns with vf-pairs $(0,6),~(2,4),~(4,2)$ and $(6,0)$ )), form the CA-block $([4]/2[3])$ . As we can see that all these cells indicate a "*" this is a full CA-block, based on definition 48.

Within the CA-tables we will always depict CA-blocks as surrounded by solid lines, so that they can be easily identified.

We also see for example the CA-block $([0]/[0])$ , which consists of a single set cell, which therefore is also a full CA-block.

For the two conditionals we again have $\herbrand(\{\Rca_1\})\cap\herbrand(\{\Rca_2 \}) = \emptyset$ and therefore it holds that they are balanced with respect to sharing of ground atoms. We again see a complete CA-table, which is in-line with assumption 4.

Example 60 shows a full CA-table for the same reasons as in example 58, i.e. due to the fact that $\herbrand(\Rca_1)\cap\herbrand(\Rca_2)=\emptyset$ .

10.3 Single Lines

In this section we will show that as long as there is a CA-Block-line within at least a single CA-block of a CA-table, the two related conditionals are balanced with respect to sharing of ground atoms.

We first look into c-conditionals which share common ground atoms and start with a very simple example, the $C_1$ - $C_1$ case where both c-conditionals use the same predicate and there is no instantiation restriction in any of the two conditionals.

Example 61 ( $C_1$ - $C_1$ Case)

Continuing from example 57.

Let $\Rc_5 = \acond{C_1(V_1)}{\top}{\emptyset}$ and $\Rc_6 = \acond{C_1(V_2)}{\top}{\emptyset}$ .

Table 10.5 shows the vf truth table and table 10.6 the CA-table of $\Rc_5$ and $\Rc_6$ .

Table 10.5
$C_1(a)$	$C_1(b)$	$C_1(c)$	$\gamma(\Ra_1)$	$\gamma(\Ra_2)$
$0$	$0$	$0$	$(0,~3)$	$(0,~3)$
$0$	$0$	$1$	$(1,~2)$	$(1,~2)$
$0$	$1$	$0$	$(1,~2)$	$(1,~2)$
$0$	$1$	$1$	$(2,~1)$	$(2,~1)$
$1$	$0$	$0$	$(1,~2)$	$(1,~2)$
$1$	$0$	$1$	$(2,~1)$	$(2,~1)$
$1$	$1$	$0$	$(2,~1)$	$(2,~1)$
$1$	$1$	$1$	$(3,~0)$	$(3,~0)$

Table 10.5: vf Truth Table of

R^a_1

and

R^a_2

of Example 61

Table 10.6
		$[3]$
$CA(\Rc_5,\Rc_6)$		$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$
$[3]$	$( 0, 3)$	$*$
	$( 1, 2)$		$*$
	$( 2, 1)$			$*$
	$( 3, 0)$				$*$

Table 10.6: CA-table for the

C_1

C_1

Case of Example 61

Based on definition 48 we see that CA-table consists of a single CA-block which includes a diagonal CA-block-line.

As both atomic conditionals use exactly the same conclusion predicate and have no antecedence predicate they generate the same vf-pairs for the same possible worlds. Therefore within their CS only identical vf-pairs of $\Ra_1$ and $\Ra_2$ have a common appearance. Therefore it is clear that both conditionals (which are in fact identical) are balanced with respect to sharing of ground atoms.

The next example looks at two ca-conditionals which share the same conclusion predicate. Both ca-conditionals are free of instantiation restrictions. We call this the $C_1A_1$ - $C_1A_2$ case.

Example 62 ( $C_1A_1$ - $C_1A_2$ Case)

Continuing from example 57.

Let $\Rca_3 = \acond{C_1(V_1)}{A_1(W_1)}{\emptyset}$ and $\Rca_4 = \acond{C_1(V_1)}{A_1(W_1)}{\emptyset}$ .

Table 10.7 shows the CA-table of $\Rca_3$ and $\Rca_4$ .

Table 10.7
		$[0]$	$[3]$				$2[3]$
$CA(\Rca_3,\Rca_4)$		$( 0, 0)$	$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$	$( 0, 6)$	$( 2, 4)$	$( 4, 2)$	$( 6, 0)$
$[0]$	$( 0, 0)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$[3]$	$( 0, 3)$	$*$	$*$				$*$
	$( 1, 2)$	$*$		$*$				$*$
	$( 2, 1)$	$*$			$*$				$*$
	$( 3, 0)$	$*$				$*$				$*$
$2[3]$	$( 0, 6)$	$*$	$*$				$*$
	$( 2, 4)$	$*$		$*$				$*$
	$( 4, 2)$	$*$			$*$				$*$
	$( 6, 0)$	$*$				$*$				$*$

Table 10.7: CA-table for the

C_1A_1

C_1A_2

case of Example 62

The CA-table indicates diagonal CA-block-lines in the CA-blocks $([3]/[3])$ , $([3]/2[3])$ , $(2[3]/[3])$ and $(2[3]/2[3])$ . The CA-blocks $([0]/[3])$ , $([0]/2[3])$ , $([3]/[0])$ , $(2[3]/[0])$ and $([0]/[0])$ are full CA-blocks.

As said already, our goal is to show that the occurrence of single CA-lines within a CA-table implies that the two atomic conditionals hold a balanced sharing. We will outline now a basic proof that this holds for $C_1A_1$ - $C_1A_2$ cases.

We know that the ground atoms of the two conditionals overlap, whereby $\ovl(\Rca_3,\Rca_4) = \cat(\Rca_3) = \cat(\Rca_4)$ and that it holds that ${\aat(\Rca_3)\cap\aat(\Rca_4)=\emptyset}$ .

It follows that all a-segments $\omega^{a_3}\in\OmegaA(\Rca_3)$ and all a-segments $\omega^{a_4}\in\OmegaA(\Rca_4)$ operate exactly on the same c-atoms when generating their respective vf-pairs. Therefore any two a-segments $\omega^{a_3}$ and $\omega^{a_4}$ will either both count or both not count such c-atom, depending whether it is verified or falsified, i.e. it will never occur that $\omega^{a_3}$ counts a specific c-atom as verified and $\omega^{a_4}$ counts the same c-atom as falsified. With this it follows that

the counting function for CA-blocks related to any a-segment $\omega^{a_3}_{()}$ (e.g. $([0]/[0])$ or $([0]/2[3])$ in example 62) never count any c-atom, but any $\omega^{a_4}$ contributes its respective vf-pairs. As the vf truth table needs to reflect all combinations of $\aat(\Rca_3)$ and $\aat(\Rca_4)$ it follows that also for the case when $\omega^{a_3}_{()}$ is not counting (i.e. all a-atoms of $\Rca_3$ set to $false$ ) the a-atoms of $\Rca_4$ run through all possible combinations of truth values and therefore each $\omega{a_4}$ contributes its individual vf-pair. It follows that $(0,0)$ has a common appearance with all vf-pairs contributed by any vf-pair of $\omega^{a_4}$ . With this it follows that the related CA-blocks are full.

The same can be shown for $\omega^{a_4}_{()}$ .
the counting functions in CA-blocks generated by canonical a-segments $\omega^{a_3}_{(i)}$ and $\omega^{a_4}_{(j)}$ (e.g. $([3]/[3])$ in example 62) count exactly the same c-atoms and therefore the same truth values, thereby generating identical vf-pairs and from this it follows that there has to be a diagonal line in this CA-block.
for the counting functions in CA-blocks which are generated by one or two composed a-segments (e.g. $(2[3]/[3])$ or $(2[3]/2[3])$ in example 62) it holds again that a composed a-segment counts a multiple of the verified (or falsified) c-atoms whilst the other a-segment can only count the same amount (if canonical) or multiple of that amount (if composed) of c-atoms. Again we get a diagonal in the related CA-block.

Obviously the above is just a very rough outline of a possible proof. For further formalization of the proof we would need more specific terminology. The outline also does not show in full clarity the connection between the single lines in the CA-table and the fact that the two ca-conditionals are balanced with respect to the sharing of ground atoms. We will see later on that it is easier to show that there are no single lines in CA-tables where that balance does not exist.

With this outline we have shown a way how we can deduct from combined vf truth tables some basic patterns within CA-tables. We will not show such proof outlines for all possible cases. Nevertheless the mentioned relationship between balanced sharing and single lines in CA-tables seems to hold for all cases of c-conditionals. In order to underline this we will show some more examples which show different kinds of CA-lines.

The following example might come as a surprise regarding the patterns in CA-tables. It shows the $C_1A_1$ - $C_1A_2$ -2nolocalIR case, i.e. the same case as in example 62, but with both of the conditionals including a non-local instantiation restriction.

Example 63 ( $C_1A_1$ - $C_1A_2$ -2nolocalIR Case)

Continuing from example 57.

Let $\Rca_5 = \acond{C_1(V_1)}{A_1(W_1)}{V_1\neq W_1}$
and $\Rca_6 = \acond{C_1(V_2)}{A_2(W_2)}{V_2\neq W_2}$ .

Table 10.8 shows the CA-table of $\Rca_5$ and $\Rca_6$ .

Table 10.8
		$[0]$	$[2]$			$[4]$					$2[3]$
$CA(\Rca_5,\Rca_6)$		$( 0, 0)$	$( 0, 2)$	$( 1, 1)$	$( 2, 0)$	$( 0, 4)$	$( 1, 3)$	$( 2, 2)$	$( 3, 1)$	$( 4, 0)$	$( 0, 6)$	$( 2, 4)$	$( 4, 2)$	$( 6, 0)$
$[0]$	$( 0, 0)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$[2]$	$( 0, 2)$	$*$	$*$	$*$		$*$	$*$	$*$			$*$	$*$
	$( 1, 1)$	$*$	$*$	$*$	$*$		$*$	$*$	$*$			$*$	$*$
	$( 2, 0)$	$*$		$*$	$*$			$*$	$*$	$*$			$*$	$*$
$[4]$	$( 0, 4)$	$*$	$*$			$*$					$*$
	$( 1, 3)$	$*$	$*$	$*$			$*$	$*$				$*$
	$( 2, 2)$	$*$	$*$	$*$	$*$		$*$	$*$	$*$			$*$	$*$
	$( 3, 1)$	$*$		$*$	$*$			$*$	$*$				$*$
	$( 4, 0)$	$*$			$*$					$*$				$*$
$2[3]$	$( 0, 6)$	$*$	$*$			$*$					$*$
	$( 2, 4)$	$*$	$*$	$*$			$*$	$*$				$*$
	$( 4, 2)$	$*$		$*$	$*$			$*$	$*$				$*$
	$( 6, 0)$	$*$			$*$					$*$				$*$

Table 10.8: CA-table for the

C_1A_1

C_1A_2

-2nolocalIR Case of Example 63

We can see that only the CA-block $(2[3]/2[3])$ in this CA-table includes a single CA-bock-line, as there are two lines e.g. in CA-block $([2]/[3])$ and two partial lines in e.g. CA-block $([4]/2[3])$ . Nevertheless we know that $\isBS(\Rca_5,\Rca_6)$ as $\herbrand(\{\Rca_5\})\cap\herbrand(\{\Rca_5\})=\emptyset$ .

What the above example shows are two conditionals which include non-local instantiation restrictions but which don’t hold a imbalanced sharing (as they do not include any local instantiation restriction). We see that the CA-table still includes a block with a single CA-block-line for CA-block (2[3]/2[3]). This is again not too complicated to understand and we try to outline the related reasoning again in a colloquial manner.

As both conditionals operate the same c-atoms and have different a-atoms it follows that at least those composed a-segments which verify the complete set of a-atoms (i.e. those a-segments which contribute the vf-pairs with the highest values, as they are added up from all available canonical a-segments) count the same amount of verified and falsified c-atoms. Therefore the related vf-pairs of both conditionals result in the same values and therefore we see at least one CA-block-line in the resulting table.

The following two examples show the $C_1A_1$ - $A_1A_2$ case (example 64)) and the $C_1A_1$ - $C_2C_1$ case (example 66) each for two conditionals where both share a common predicate, one in the conclusion and one in the antecedence. Both examples show single CA-block-lines (either vertical or horizontal) in several of the CA-blocks. We will not further discuss why these lines indicate that the two conditionals are balanced with respect to sharing.

Example 64 ( $C_1A_1$ - $A_1A_2$ Case)

Continuing from example 57.

Let $\Rca_7 = \acond{C_1(V_1)}{A_1(W_1)}{\emptyset}$ and $\Rca_8 = \acond{A_1(V_2)}{A_2(W_2)}{\emptyset}$ .

Table 10.9 shows the CA-table of $\Rca_7$ and $\Rca_8$ .

Table 10.9
		$[0]$	$[2]$			$2[2]$
$CA(\Rca_7,\Rca_8)$		$( 0, 0)$	$( 0, 2)$	$( 1, 1)$	$( 2, 0)$	$( 0, 4)$	$( 2, 2)$	$( 4, 0)$
$[0]$	$( 0, 0)$	$*$	$*$			$*$
$[3]$	$( 0, 3)$	$*$		$*$			$*$
	$( 1, 2)$	$*$		$*$			$*$
	$( 2, 1)$	$*$		$*$			$*$
	$( 3, 0)$	$*$		$*$			$*$
$2[3]$	$( 0, 6)$	$*$			$*$			$*$
	$( 2, 4)$	$*$			$*$			$*$
	$( 4, 2)$	$*$			$*$			$*$
	$( 6, 0)$	$*$			$*$			$*$

Table 10.9: CA-table for the

C_1A_1

A_1A_2

case of Example 64

Example 65 ( $C_1A_1$ - $C_2C_1$ Case)

Continuing from example 57.

Let $\Rca_9 = \acond{C_1(V_1)}{A_1(W_1)}{\emptyset}$ and $\Rca_{10} = \acond{C_2(V_2)}{C_1(W_2)}{\emptyset}$ .

Table 10.10 shows the CA-table of $\Rca_9$ and $\Rca_{10}$ .

Table 10.10
		$[0]$	$[3]$				$2[3]$				$3[3]$
$CA(\Rca_{9},\Rca_{10})$		$( 0, 0)$	$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$	$( 0, 6)$	$( 2, 4)$	$( 4, 2)$	$( 6, 0)$	$( 0, 9)$	$( 3, 6)$	$( 6, 3)$	$( 9, 0)$
$[0]$	$( 0, 0)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$[3]$	$( 0, 3)$	$*$
	$( 1, 2)$		$*$	$*$	$*$	$*$
	$( 2, 1)$						$*$	$*$	$*$	$*$
	$( 3, 0)$										$*$	$*$	$*$	$*$
$2[3]$	$( 0, 6)$	$*$
	$( 2, 4)$		$*$	$*$	$*$	$*$
	$( 4, 2)$						$*$	$*$	$*$	$*$
	$( 6, 0)$										$*$	$*$	$*$	$*$

Table 10.10: CA-table for the

C_1A_1

C_2C_1

case of Example 65

We finally show a rather special example of a single line within a CA-table which occurs for the $C_1A_1$ - $C_2A_1$ case, where both conditionals share the same antecedence.

Example 66 ( $C_1A_1$ - $C_2A_1$ Case)

Continuing from example 57.

Let $\Rca_{11} = \acond{C_1(V_1)}{A_1(W_1)}{\emptyset}$ and $\Rca_{12} = \acond{C_2(V_2)}{A_1(W_2)}{\emptyset}$ .

Table 10.11 shows the CA-table of $\Rca_{11}$ and $\Rca_{12}$ .

Table 10.11
		$[0]$	$[3]$				$2[3]$
$CA(\Rca_{11},\Rca_{12})$		$( 0, 0)$	$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$	$( 0, 6)$	$( 2, 4)$	$( 4, 2)$	$( 6, 0)$
$[0]$	$( 0, 0)$	$*$
$[3]$	$( 0, 3)$		$*$	$*$	$*$	$*$
	$( 1, 2)$		$*$	$*$	$*$	$*$
	$( 2, 1)$		$*$	$*$	$*$	$*$
	$( 3, 0)$		$*$	$*$	$*$	$*$
$2[3]$	$( 0, 6)$						$*$	$*$	$*$	$*$
	$( 2, 4)$						$*$	$*$	$*$	$*$
	$( 4, 2)$						$*$	$*$	$*$	$*$
	$( 6, 0)$						$*$	$*$	$*$	$*$

Table 10.11: CA-table for the

C_1A_1

C_2A_1

case of Example 66

The resulting CA-table shows a single CA-table line.

We outline why a balanced sharing between the two conditionals can be read from table 10.11. Both conditionals share exactly the same a-atoms and therefore also the same a-segments. Therefore, if for a given possible world $\omega$ the same a-atoms of $\Rca_{11}$ and and of $\Rca_{12}$ are verified then their respective a-segments are always identical. It follows that if within $\omega$ the a-atoms with indexes $a,b$ and $c$ are set then both a-segments $\omega^{a_{11}}_{(a,b,c)}$ and $\omega^{a_{12}}_{(a,b,c)}$ contribute their vf-pairs to the conditional contributions of $\Rca_{11}$ and $\Rca_{12}$ . In the same way, if one or more of the a-atoms $a,b$ or $c$ is not set within $\omega$ , then the two a-segments both do not contribute a vf-pair to the conditional contribution of their respective conditional. In short: if a vf-pair is generated for one of the two conditionals then there is also a vf-pair generated for the other.

What we have seen in this section is obviously only a small set of examples, but the related explanations given for several of the tables give hope that the cases shown can be generalized. We therefore state the following assumption.

Assumption 5 (Balanced Sharing and Single Lines).

Let $\R_1$ and $\R_2$ be two conditionals.

If the CA-table of $\Ra_1$ and $\Ra_2$ either shows at least one single CA-block-line or a CA-table-line then it holds that $\herbrand(\{\R_1\})\cap\herbrand(\{\R_2\})=\emptyset$ .

Then the two conditionals of a complete CA-table are balanced with respect to sharing of ground atoms.

10.4 Imbalanced Sharing and CA-Tables

In this chapter we will show three cases of conditionals which hold a imbalanced sharing and for which the related CA-tables are neither complete nor show a single line. We will not explain how the related appear. The examples are just meant to show how CA-tables for conditionals look like when there is an imbalanced sharing.

The first example shows the $C_1$ - $C_1$ -IRc1c case, where two c-conditionals which both use the same predicate are investigated, whereby one holds a local instantiation restriction.

Example 67 ( $C_1$ - $C_1$ -IRc1c Case)

Continuing from example 57.

Let $\Rc_7 = \big<\big(~C_1(V_1)~|~\top~\big), V_1\neq c~\big>$ and $\Rc_8 = \big<\big(~C_1(V_2)~|~\top~\big)~\big>$ .

Table 10.12 shows the vf truth table and table 10.13 the CA-table of $\Rc_8$ and $\Rc_9$ .

Table 10.12
$C_1(a)$	$C_1(b)$	$C_1(c)$	$\gamma(\Ra_1)$	$\gamma(\Ra_2)$
$0$	$0$	$0$	$(0,~2)$	$(0,~3)$
$0$	$0$	$1$	$(0,~2)$	$(1,~2)$
$0$	$1$	$0$	$(1,~1)$	$(1,~2)$
$0$	$1$	$1$	$(1,~1)$	$(2,~1)$
$1$	$0$	$0$	$(1,~1)$	$(1,~2)$
$1$	$0$	$1$	$(1,~1)$	$(2,~1)$
$1$	$1$	$0$	$(2,~0)$	$(2,~1)$
$1$	$1$	$1$	$(2,~0)$	$(3,~0)$

Table 10.12: vf Truth Table of Example 67

Table 10.13
		$[3]$
$CA(\Rc_7,\Rc_8)$		$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$
$[2]$	$( 0, 2)$	$*$	$*$
	$( 1, 1)$		$*$	$*$
	$( 2, 0)$			$*$	$*$

Table 10.13: CA-table for the

C_1

C_1

-IRc1c Case of Example 67

Here we see two parallel diagonals within the CA-table. There is no single line in the CA-table, nor is the CA-table complete.

The next examples shows the $C_1A_1$ - $C_1A_2$ -IRc1c case, where to ca-conditionals share the same conclusion predicate and one of them holds a local instantiation restriction against that predicate.

Example 68 ( $C_1A_1$ - $C_1A_2$ -IRc1c Case)

Continuing from example 57.

Let $\Rca_{13} = \big<~\big(~C_1(V_1)~|~A_1(W_1)~\big), V1 \neq x~\big>$ and $\Rca_{14} = \big<~\big(~C_1(V_2)~|~A_2(W_2)~\big)~\big>$

Table 10.14 shows the CA-table of $\Rca_{13}$ and $\Rca_{14}$ .

Table 10.14
		$[0]$	$[3]$				$2[3]$
$CA(\Rca_{13},\Rca_{14})$		$( 0, 0)$	$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$	$( 0, 6)$	$( 2, 4)$	$( 4, 2)$	$( 6, 0)$
$[0]$	$( 0, 0)$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$[2]$	$( 0, 2)$	$*$	$*$	$*$			$*$	$*$
	$( 1, 1)$	$*$		$*$	$*$			$*$	$*$
	$( 2, 0)$	$*$			$*$	$*$			$*$	$*$
$2[2]$	$( 0, 4)$	$*$	$*$	$*$			$*$	$*$
	$( 2, 2)$	$*$		$*$	$*$			$*$	$*$
	$( 4, 0)$	$*$			$*$	$*$			$*$	$*$

Table 10.14: CA-table for the

C_1A_1

C_1A_2

-IRc1c Case of Example 68

In the above example we see the lines within the CA-blocks further distorting. There is again no single line in any block nor for the whole table.

The last example shows the $C_1A_1$ - $C_2A_1$ -IRa1c case, in which the two conditionals share the antecedence atom and one of them holds an instantiation restriction against that predicate.

Example 69 ( $C_1A_1$ - $C_2A_1$ -IRa1c Case)

Continuing from example 57.

Let $\Rca_{15} = \big<~\big(~C_1(V_1)~|~A_1(W_1)~\big), W1 \neq a~\big>$ and $\Rca_{16} = \big<~\big(~C_2(V_2)~|~A_1(W_2)~\big)~\big>$

Table 10.15 shows the CA-table of $\Rca_{15}$ and $\Rca_{16}$ .

Table 10.15
		$[0]$	$[3]$				$2[3]$				$3[3]$
$CA(\Rca_{15},\Rca_{16})$		$( 0, 0)$	$( 0, 3)$	$( 1, 2)$	$( 2, 1)$	$( 3, 0)$	$( 0, 6)$	$( 2, 4)$	$( 4, 2)$	$( 6, 0)$	$( 0, 9)$	$( 3, 6)$	$( 6, 3)$	$( 9, 0)$
$[0]$	$( 0, 0)$	$*$	$*$	$*$	$*$	$*$
$[3]$	$( 0, 3)$		$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 1, 2)$		$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 2, 1)$		$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 3, 0)$		$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$2[3]$	$( 0, 6)$						$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 2, 4)$						$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 4, 2)$						$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 6, 0)$						$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$

Table 10.15: CA-table for the

C_1A_1

C_2A_1

-IRa1c Case of Example 69

Also this table does not show a single line in a block or for the whole table.

The examples in this section were chosen to be representative for a large number of cases. We have checked a great deal more such examples and they all support assumptions 4 and 5.

10.5 Determining Imbalanced Sharing

The examples and assumptions in this chapter have shown that we have reason to believe that an imbalanced sharing between two conditionals can be detected from their CA-table.

Assumption 6 (Imbalanced Sharing Can be Determined from CS).

Let $\R_1,R_2$ be atomic conditionals.

From the CS $\gamma(\{~\R_1,~R_2~\})$ it can be determined whether $\isBS(\{~\R_1,~R_2~\}) = \true$ or $\isBS(\{~\R_1,~R_2~\}) = \false$ .

This is interesting when compared to proposition 2 in section 3.3, as there it is stated that an imbalanced usage cannot be read from a given CS.

We were unfortunately not able to proof this assumption in a strict formal way. But with the given examples we hopefully have shown some good starting points for further investigations.

10.6 Summary and Discussion

The results of this chapter can only be regarded as a collection of different findings and some attempts to generalize the found patterns in CA-tables in order to interpret the relationships between the two involved atomic conditionals. We only looked a some specific scenarios and did not investigate e.g. CA-tables of cc-conditionals or of mixed conditionals, e.g. of a c-conditional and a ca-conditional.

Still, the findings are interesting and we showed that at least for some of them we found assumptions and proof outlines which give hope that further generalization is possible.

CA-Tables therefore are not just a tool for visualizing the common appearances of two conditionals but also reveal certain additional aspects of the relationship between two conditionals.

Finally, the result shown in section 10.5 gives hope that at least imbalanced sharings can be determined on CS-level.