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Epistemic Sentiment Modeling

1. knowledge

1.1 sentiment annotators

we introduce three symbols: plus + for positive associations, circle o for neutral associations, and dash/minus - for negative associations. we call these sentiment annotators

when a sentiment annotators between two people are denoted over an arrow indicating directionality (mono-directional or bidirectional)

$A \xrightarrow{+} B$ reads “A likes B”

$A \xleftrightarrow{+} B$ reads “A and B like eachother”

1.2 epistemic logic

it is possible that two people like eachother but don’t know the other likes them, orr iso knows allo likes them, but allo doesn’t know iso likes them. this becomes important in the construction of self-reinforcing social structures (“granules” as they are called later in the text). we bring in concepts from epistemic logic.

Knowledge and belief are represented via the modal operators K and B, often with a subscript indicating the agent that holds the attitude. Formulas Kaφ and Baφ are then read “agent a knows that phi” and “agent a believes that phi”, respectively.

: Epistemic Logic, Stanford Encyclopedia of Philosophy

$K_A (B \xrightarrow{+} A)$ reads “A knows B likes A”

$E_{\{A,B,C\}} \phi$ reads “everyone in A, B, and C knows phi”

$K_X (Y \xrightarrow{+} X) : X, Y \in (A, B)$ reads “a and b like eachother, and a and b knows that the other likes them”

here we introduce the hetero-diallel function

1.3 forming a stable bidirectional relationship

let’s take this from a third point observer z. (not from the point of view of iso or allo, because one of them could not know things that would influence the other’s behavior). iso and allo have standins a and b, respectively. past a certain point, knowledge modeling becomes recursive and there is no payoff to modeling it, but before the point of behavior being affected, it is worth modeling it.

the sentiment table

number	symbolic representation	relationship in english	how this might affect behavior
a.1	$A \xrightarrow{+} B$	iso likes allo	iso could either conceal the fact but still act on it, by doing favors from afar, or could be up front about it, by doing things in the open
a.2	$K_B (A\xrightarrow{+}B$)	allo knows iso likes allo	this does nothing to affect iso’s behavior, but it can affect allo’s. allo could continue acting the same, which doesn’t reveal their epistemic position to iso, so iso doesn’t change their behavior
a.3	$K_A$ a.2	iso knows allo knows iso likes allo	now any action iso takes towards allo will be done with the fact that iso knows allo knows iso likes them
b.1	$B \xrightarrow{+} A$	allo likes iso	allo could either conceal the fact but still act on it, by doing favors from afar, or could be up front about it, by doing things in the open
b.2	$K_A(B\xrightarrow{+}A)$	iso knows allo likes iso	this does nothing to affect allo’s behavior, but it can affect iso’s. iso could continue acting the same, which doesn’t reveal their epistemic position to allo, so allo doesn’t change their behavior
b.3	$K_B$ b.2	allo knows iso knows allo likes iso	now any action allo takes towards iso will be done with the fact that allo knows iso knows allo likes them.

where:

$$(a.1 \prec a.2 \prec a.3) \parallel (b.1 \prec b.2 \prec b.3)$$

at this point you have reciprocity and a stable relationship and no further need for derivations because otherwise by necessity it will be reflexive. after this point, any more reflexive knowledge does not affect behavior. if all six of these conditions are satisfied, than we have a stable bidirectional relationship, where the stable bidirectional relationship is represented by:

$$a.3 \land b.3$$

and with the arrow notation, we use the Kleene star to indicate that all the states of the relationship have been satisfied. (without the star, it just means that A and B like eachother, without any of the epistemic steps)

$$A \xleftrightarrow{+*} B$$ a heuristic to quickly determine weather A and B are in a stable bidirectional relationship or not, this question can be asked:

will observer Z be able to tell that A likes B and B likes A?

if this is the case, than it is probably obvious enough that a.1 that a.2 and a.3, enough so that a non-party will be able to tell, and the same for b - satisfying the above requirement. if a non-party can tell, then it is very likely that the participant parties can tell eachother’s sentiment. (of course, a keen observer will be able to tell at the stage one or two states, but this is usually only if someone is actively paying attention)

2. gestalts and granules

when everyone likes eachother but does not know that everyone likes eachother, that is called a gestalt. it takes its name from the German word for “shape”. it is denoted with lowercase letter $\gamma$

Example:

$$\gamma_{ABC} := x_1 \xrightarrow{+} x_2: x \in☒\{A,B,C\}$$

a granule is when there are three or more people who all like eachother and everyone knows everyone else likes eachother. these units are indispensable for understanding social interactions. it is denoted with capital letter $\Gamma$

Example:

$$\Gamma_{ABC} := E_{\{A,B,C\}}(x_1 \xrightarrow{+*} x_2) : x \in☒\{A,B,C\}$$

does a granule have an immune system?

2.1 tertiary granule

joining

step	formula	comment
1.a	$$B\xleftrightarrow{+*}C, A \xleftrightarrow{+*} B, A \xleftrightarrow{+*}C$$	B and C are in a stable bidirectional relationship. A and B and A and c are in a stable bidirectional relationship
1.b	$$\neg K_{d_1} (d_2 \xleftrightarrow{+*} A) : d \in ☒\{B,C\}$$	neither B nor C knows the other is in a stable bidirectional relationship with A
1.c	$$\gamma_{ABC}$$	A, B, and C all like eachother but do not all know it. a gestalt
1.d	$$K_A (B\xleftrightarrow{+*}C)$$	A knows B and C like eachother
2	$$K_{d_1} (d_2 \xleftrightarrow{+*} A) : d \in ☒\{B,C\}$$	B and C both know the other likes A. this is the catalyst and it has to independently of A’s direct involvement - either through B and C finding out through eachother, or one of B or C having a notable interaction with A that the other observes. an inversion of 1.b
3	$$K_A 2$$	A knows that B and C both know the other likes A. this has to happen as well. there can be a scenario where A does something defensive that would undo the above step but that is not very common
4	$$\Gamma_{ABC}$$	A, B, and C all like eachother and all know they like eachother. the final granule

the funny part is, this table is not complete. this is annoying.

synthesizing

step	formula	comment
1	$$A \xleftrightarrow{+*} B, B \xleftrightarrow{+*}, A \xleftrightarrow{+*} C$$	A and B like eachother, B and C like eachother, A and C like eachother
2	$$K_{d_1} (A \xleftrightarrow{+} d_2) : d \in ☒(\{B,C\})$$	B and C both know A likes the other. the gestalt symbol isn’t used here because it satisfies B+C, which isn’t necessary information here. (if A introduces B and C, than this is obvious, but it is not if B and C meet by any other means - by another mutual, or by chance)
3	$$K_A (B \xleftrightarrow{+} C)$$	A knows B and C like eachother. this is not obvious, such as in the case that a introduces b and c but b and c might not like eachother, and a doesn’t know for sure or not
4	$$\Gamma_{ABC}$$	A, B, and C all like eachother and all know they like eachother. the final granule

now the amount of connections that need to be broken are 3 instead of one for group dissolution. this is a terniary-granule

quaternary granule

these ones currently do not model recursive knowledge.

existing granule + infiltrator

step	formula	comment
1	$$\Gamma_{BCD}$$	pre-existing granule of B, C, and D
2	$$A \xleftrightarrow{+} B, A \xleftrightarrow{+} C, A \xleftrightarrow{+} D$$	A and each member of the granule like eachother
3	$$K_A (B \xleftrightarrow{+} C, C \xleftrightarrow{+} D, B \xleftrightarrow{+} D)$$	A knows the internal structure of the granule
4	$$E_{\{B,C,D\}} (B \xleftrightarrow{+} A, C \xleftrightarrow{+} A, D \xleftrightarrow{+} A)$$	everyone in B, C, and D know that each of them and A like eachother
5	$$\Gamma_{ABCD}$$	Complete quaternary granule formed

two pairs merge

step	formula	comment
1	$$A \xleftrightarrow{+} B, C \xleftrightarrow{+} D$$	Two initial pairs exist independently
2	$$A \xleftrightarrow{+} C, B \xleftrightarrow{+} D$$	Cross connections form
3	$$E_{\{A,B\}} (C \xleftrightarrow{+} D)$$	both A and B know that C and D like each other
4	$$E_{\{C,D\}} (A \xleftrightarrow{+} B))$$	both C and D know that A and B like each other
5	$$B \xleftrightarrow{+} C, A \xleftrightarrow{+} D$$	Final connections form (catalyst event)
6	$$\Gamma_{ABCD}$$	Complete quaternary granule formed

• (this epistemic knowledge could also be used for secret sharing (a knows b knows cs secret, b knows a knows c’s secret, now the secret becomes a topic of conversation between a and b)

double triangle fusion

step	formula	comment
1	$$\Gamma_{ABC}, \Gamma_{ACD}$$	two tertiary granules with A as bridge node
2	$$\neg K_B (D \xrightarrow{+} A)$$	B initially doesn’t know D likes A
3	$$\neg K_D (B \xrightarrow{+} A)$$	D initially doesn’t know B likes A
4	$$B \xleftrightarrow{+} D$$	B and D form connection (catalyst moment)
5	$$K_B (D \xleftrightarrow{+} A), K_D (B \xleftrightarrow{+} A), K_B (D \xleftrightarrow{+} C), K_D (B \xleftrightarrow{+} C)$$	B and D learn about each other’s connection to A and C
6	$$\Gamma_{ABD}, \Gamma_{BCD}$$	two more tertiary granules
7	$$E_{\{A,C\}} (B \xleftrightarrow{+} D), E_{\{B,D\}} (A \xleftrightarrow{+} C)$$	A and C know B and D like eachother, and B and D know A and C like eachother
8	$$\Gamma_{ABCD}$$	Complete quaternary granule formed

3. Formation of Groups

to be added later