1 Introduction

In human communication process, we often face situations where decisions have to be made, regardless of silence of one of the interlocutors, as often occurs in a dialogue. That is, we have to decide from incomplete information, guessing the knowledge or intentions of the silent person. This behavior has been studied by several disciplines but barely touched in logic or artificial intelligence. We have not found an approach to formalize the use of intentional silence in terms of logic, the closest attempt was that of [⁹] with an ”informal” logic.

According to Kurzon [¹⁰], there are two types of silence, intentional and unintentional.

Intentional silence is a deliberate action not to cooperate with the other party and unintentional silence is psychological in nature.

The interpretation of silence must be contextual. For example, in a normal conversation, silence is interpreted to the detriment of the person who is silent. The immediate reaction is that she hides something. For Kurzon [¹⁰], the silence is defined by language, and points to three types of silence:

The intentional silence is also a sign of group loyalty. To interpret intentional silence, first we have to discard the modal ”can” that expresses unintentional silence, as in ”I can not speak” [¹⁰]. Then, intentional silence can be interpreted with four manners:

Where manners 1 and 2 are intentional external silences ”by order”. And manners 3 and 4 are intentional internal silences ”by will”.

Bohnet and Frey [¹] state that the variants for the interpretation of silence can be: anonymous and not anonymous, where the latter can be with identification and face-to-face. They also studied silence in the communication process, specifically in the context of prisoner’s dilemma.

From the point of view of semiotics, silence is a sign. In a communication scheme that includes the interpretation of silence in its basic form, the speaker has to interpret the silence that the listener sends with a certain intention. The sender becomes the receiver of silence [³].

The categories proposed by Grice [⁸] to have a good communication are:

Category 4 is related to the way it is said and includes the supermaxim ’Be perspicuous’ and various maxims more. A participant in a talk exchange may fail to fulfill with ”Avoid obscure expressions”; he may say, indicate, or allow it to become plain that he is unwilling to cooperate in the way the maxim requires. He may say, for example, ’I cannot say more; my lips are sealed’. Nevertheless, this type of silence is intentional, details the context in which is presented and allows decisions to be made.

This paper offers a first logical approach to the study of intentional silence in a particular context, focusing on a puzzle formally expressed and previously solved, to analyze the implications of two different interpretations of silence, expressed in terms of the predicate Says.

The first intentional silence is Defensive Silence that is proactive, involving awareness and consideration of alternatives, followed by a conscious decision to withhold ideas, information, and opinions as the best personal strategy at the moment [²]. The second intentional silence, on the other hand, suggests disengaged behaviour that is more passive than active. We call it Acquiescent Silence, in this case, who remains silent agrees implicitly with what others say. Several conclusions are derived from both readings. After analyzing the case study, we propose a general strategy for the analysis of problems involving testimonies and silence.

This paper is organized as follows: Section 2 shows preliminaries concepts. Section 3 describes a case study for silence, includes the two interpretations and their consequences. Section 4 includes a formulation of a general strategy for analyzing problems involving testimonies. We conclude in Section 5, discussing in addition work in progress.

2 Preliminaries

A clause is a formula of the form H←B where H and B are arbitrary formulas in principle, called head and body of the clause respectively. There are several types of clauses. H={} the clause is called a constraint and we can write that clause as ←B. Analogously, if B={} then the clause is called a fact and can be written as H←. An augmented clause is a clause where H and B are some conjunction, disjunction or denial [⁶]. Hp and Bp contain positive atoms. Hn and Bn contain negative atoms.

A logical program is then a finite set of clauses. If all the clauses in a program are of a certain type, we say that the program is also of that type. For example, a set of augmented clauses specifies an augmented program, a set of free clauses is a free program and so is in the case of the disjunctive and definite programs [⁶]. Figure 1 shows the classification of types of clauses that can lead to the corresponding logical programs.

Fig. 1 Types of clauses 

A set consisting of literals X, satisfies a basic formula F (symbolically, X⊨F) recursively, as follows [¹¹]:

Let Π be a basic program. A consistent set of literals X is closed under Π if, for each rule F←G in Π, X⊨F when X⊨G.

We say that X is a answer set for the basic program Π if X is the minimum amount of the consistent set of literals closed under Π. For example, consider the program: q←p∨−p. The closure under this program is characterized by the following condition: if p∈X or −p∈X then q∈X. It is clear that the answer set for that program is empty. If we add the rule p (that is, p←⊤) to this program then {p,q} would be the answer set.

The reduction of a formula, rule or program Π, relative to a consistent X set of literals is defined recursively, as follows [¹¹]:

For example, let P be such that:

P:a←¬¬a,¬b←c∨b.

If we choose X={a} then the reduction is

PX:a←⊤,⊤←c∨b.

One can easily verify that {a} is closed under this reduction and the empty set or 0 is not, this is the minimum set with such property. So, it follows that {a} is a P answer set. However, this shows that the empty set or 0 is also a set of P , but produces a different closed reduction.

A consistent X set of literals is an answer set for a program Π if this is an answer set for the reduction ΠX.

Conversational Implicature is a potential inference that is not a logical implication and is connected with the meaning of the word ”say”.

Cooperative Principle consists of the participants making their conversational contribution as required in the scenario in which this occurs for the accepted purpose or direction of the speech exchange in which they were engaged [⁸]. In this sense, we employ the predicate ”Says” that has emerged previously in logics for access control [⁴]. Such predicate can be restricted by varied axioms, but we are using it here simply to express that “somebody” is asserting “something”, and considering that also intentional silence ”says” something when is interpreted in its context for decision making.

We define the predicate ”says” in the sense of Grice, as: Says(X, Y), it expresses that the agent X says Y (predicate). A predicates set ”says” can have a SaysGraph associated. A SaysGraph= <V,Ap>, where V is a set of agents and Ap is a set of Predicate Arcs. A SaysGraph represents the relations of the subset X (Says) of a program P, defined later.

Likewise, we give the following definition:

A Predicate Arc is a directed arc. The origin of the arc, labeled with a predicate, corresponds to the agent asserting something and the destination or destinations correspond to the agent or agents referred to.

If the origin of the arc is a black dot, the predicate includes, as an argument, the same agent that asserts.

Next, we define informally Defensive and Acquiescent Silence, as described by Dyne, Ang, and Botero [²], and then formalize such types of silence for further application and analysis.

Defensive Silence consists of withholding relevant ideas, information, or opinions as a form of self-protection, based on fear [²].

Let P be a logical program or a knowledge base (KB); A1,A2,…,An are agents; p1,p2,…,pm are predicates; XA1={Says(A1,p1),…,Says(A1,pl)}={Says(A1,∗)}; …; XAk={Says(Ak,∗)}; X=XA1UXA2U…UXAn; X⊂P; (1≤k≤n); (1≤l≤m).

PAi is Total Defensive Silence (TDS) for Ai, (1≤i≤n); where PAi=P−XAi.

PAij is Partial Defensive Silence (PDS) for Ai, (1≤i≤n), (1≤j≤m); where PAij=P−{Says(Ai,pj)}.

Acquiescent Silence expresses witholding relevant ideas, information, or opinions, based on resignation [²].

P′Ai is Acquiescent Silence (AS) for Ai; where P′Ai=PAiU{Says(Aj,∗)}∘λ, PAi is TDS for Ai, λ={Aj/Ai}, (1≤i,j≤n) and (j≠i).

Here the operator ∘ with λ sustitution denotes the replacement of Aj for Ai on Says subset.

We employ the Answer-Set Programming (ASP) paradigm to explore the implications of silence in our case study, given that is closely related to intuitionist logic, i.e. both are based on the concept of proof rather than truth (previously shown in [⁷] for intuitionist logic). This is a logical programming branch that computes stable models for difficult problems [⁶], where a stable model is a belief system that holds for a rational agent. This approach:

Clingo is an implementation of ASP that allows to find, if there exists, the answer set or stable model of a logical program [⁵]. Clingo is used to generate answer sets for the problem of a case study, and explore the implications of the silence interpretations formulated. Python is used to update the KB or logical program [¹⁴].

3 A Case Study for Silence

There is a puzzle, previously modeled and solved in [⁶], that includes testimonies of different people, and allows to model and explore our two interpretations of of silence. In this puzzle, a mystery related to the murder of a person is raised, where one can assume that a judge requests and records the testimony of three suspects:

Vinny has been murdered, and Andy, Ben, and Cole are suspects.

Andy says he did not do it. He says that Ben was the victim’s friend but that Cole hated the victim. Ben says he was out of town the day of the murder, and besides he didn’t even know the guy. Cole says he is innocent and he saw Andy and Ben with the victim just before the murder.

Figure 2 presents the testimony of the suspects through the predicate Says() as a SaysGraph. We must assume that all the people involved tell the truth except, possibly, the murderer. The story and testimony of these three people is formulated in the program Mystery.lp for Clingo (see Appendix A).

Fig. 2 SaysGraph for original puzzle 

The program for the puzzle produces as a result: murderer(ben). This means that according to the testimonies and the rules of common sense knowledge provided, the murderer is Ben.

3.1 Interpreting Silence

Based on the formulation of the puzzle previously described, we proceed to explore two interpretations of intentional silence linked to such context.

The first interpretation is a Defensive Silence, i.e. an agent intentionally simply remains silent, mainly by fear. While the second corresponds to Acquiescent Silence, understood as asserting with silence what others have said, commonly by resignation. We explore in both cases, the consequences of the interpretation.

3.1.1 Defensive Silence

Defensive silence, previously defined, is intentional and proactive behavior that is intended to protect the user from external threats.

If an agent investigating a case faces this kind of silence of one or more of those involved, he can not count on their testimonies. So, we have to remove the declaration of those people, as a rule.

So expressing this kind of silence in the context of our case study (puzzle); what would happen if silence with common sense is presented as a possibility? What conclusions the interrogator or judge can reach if some of the suspects decide to intentionally remain quiet?

Applying this first rule to each person giving his testimony and executing the Python program:

t_def_silence(Mystery.lp,andy)

t_def_silence(Mystery.lp,ben)

t_def_silence(Mystery.lp,cole)

The possible outcomes (guilty) when a one or more suspects decide intentionally to omit their testimonies are presented in Table 1. In this, we can notice that the culprit can be anyone depending on who decides to stay silent. For the possibilities, we can comment:

1. {} corresponds to the original scheme where nobody is silent, i.e. every testimony is taken into account. The only model for this case is Ben, as before.

2. When Andy is silent, the offender turns out to be either Ben or Cole. Each answer corresponds to a model, as shown below:

   • Answer: 1

   • murderer(cole).

   • Answer: 2

   • murderer(ben).

   • SATISFIABLE

   • The last line indicates that there are no more models.

3. When Ben is silent, any of the three suspects may be guilty. Intuitively we can think that Ben’s silence has more decision capability since anyone involved can turn out as guilty.

4. Cole’s silence can turn Andy or Ben guilty.

5. With the remaining possibilities, related to more than one person, any of the three involved can be guilty.

Table 1 Defensive Silence model for agent 

Silent agent(s)	Presumable culprit
{}	{ben}
{andy}	{cole, ben}
{ben}	{cole, ben, andy}
{cole}	{ben, andy}
{andy, ben}	{cole, ben, andy}
{ben, cole}	{cole, andy, ben}
{andy, cole}	{cole, ben, andy}
{andy, ben, cole}	{cole, ben, andy}

About the problem in general, we can add that Andy’s silence can be in his own benefit, in the same way as Cole.

For instance, in the case of Andy, in terms of logic programming, the rules that have to be deleted are: {Says(andy,∗)}. The existing relations are depicted in Figure 3.

Fig. 3 SaysGraph for Andy’s Defensive Silence 

We can further detail the analysis by considering partial silence. We can now wonder: What part of the testimony could be convenient to silence in the case of suspects? Who would be the culprit in the event that some person decide to remain partially silent? What possibilities would each of the suspects have if, before giving their allegation, they have access to the testimony of others?

Reflection for each suspect using the programs: p def silence(kb,agent,predicate) and Clingo.

1. In the case of Andy, with the silence of his first or second statement, the culprit can be Ben, with the silence of the third one, Cole also appears as presumable guilty. Table 2 shows the possibilities of Andy.

2. Ben is the most affected with his silence, either total (Table 1) or partial since he comes out in every model sharing the suspicion with somebody else, as Table 3 shows.

3. Cole can also decide, without incriminating himself, whom to reveal as guilty. Table 4 shows the different answers obtained.

Table 2 Partial Defensive Silence models for Andy 

Silenced testimony (predicate)	Presumable culprit
{says(andy, innocent(andy))}	{ben}
{says(andy, hated(cole,vinny))}	{ben}
{says(andy, friends(ben,vinny))}	{ben, cole}

Table 3 Partial Defensive Silence models for Ben 

Silenced testimony (predicate)	Presumable culprit
{says(ben,out-of-town(ben))}	{andy, ben}
{says(ben,know(ben,vinny))}	{ben, cole}

Table 4 Partial Defensive Silence models for Cole 

Silenced testimony (predicate)	Presumable culprit
{says(cole,innocent(cole))}	{ben}
{says(cole,together(andy,vinny))}	{ben}
{says(cole,together(ben,vinny))}	{andy, ben}

3.1.2 Acquiescent Silence

The second interpretation of silence is related with the old saying ”silence is consent”, expressing a passive disengaged attitude, previously defined.

In this interpretation, we operationalize it by omitting the whole person’s testimony and inserting new rules related with what he is implicitly assuming with his silence. For example, in the case of Ben, we execute the program acq_silence(Mystery.pl,ben), which leads to:

1. Ignore the following assertions, since he is not declaring anything:

   • Says(ben,pi), (1≤i≤2).

2. Add the following assertions, to model his consent on what others say:

   • Says(ben,q), where {Says(A,q)}⊂P and A≠ben.

Figure 4 presents the Acquiescent Silence for Ben, where he turns out to be guilty and is obtained by the execution of the program clingo 0 Mystery-as-ben.pl. The answer was similar to model of the original problem.

Fig. 4 SaysGraph of Ben’s Acquiescent Silence 

Table 5 shows the solutions reached for the puzzle when one or several persons are silenced under the interpretation of Acquiescent Silence. Again, the first line corresponds to the original situation where everybody has declared, leading to Ben as the murderer.

Table 5 Acquiescent Silence for agent 

Silent agent(s)	Presumable culprit
{}	{ben}
{andy}	UNSATISFIABLE
{ben}	{ben}
{cole}	UNSATISFIABLE
{andy, ben}	{ben, andy}
{ben, cole}	{ben, cole}
{andy, cole}	{cole, andy}
{andy, ben, cole}	{cole, ben, andy}

Notice that there is no model (solution) in cases 2 and 4, where UNSATISFIABLE is obtained. These situations can be interpreted that there is no evidence to blame any of the suspects, possibly leading to a mistrial. So, under this scheme, Andy and Cole are those who could benefit from remaining silent.

In cases 5, 6 and 7, the person who speaks is out of suspicion. In the latter case, as expected from common sense, when everybody is silent (no one has revealed any information), anyone can be the culprit.

4 A Strategy for Analysis

We now formulate a strategy for bringing intentional silence in the analysis of problems involving testimonies. Assuming that testimonies of different people involved are already available, the strategy is formulated as follows:

A key step in the strategy is 4. Here, the obvious case is when one of those involved recurs to his right to remain silent. We can then proceed to consider, one at a time, the two types of silence for such person.

However, other situations can emerge, for instance when two declarants A and B separately coincide in statements p and q, but let say A in addition declares r. We can then hypothesize an acquiescent silence of B, or even a partial defensive silence, since he is omiting r, and proceed accordingly to represent and analyze the problem.

A second puzzle is analyzed to illustrate the application of the strategy . This is formulated as follows [¹³]:

The Island of Knights and Knaves has two types of inhabitants: knights, who always tell the truth, and knaves, who always lie. One day, three inhabitants (A, B, and C) of the island met a foreign tourist and gave the following information about themselves:

What types are A, B, and C?.

5 Conclusions

Silence expresses valuable information that can be employed for decision making. In particular, when the intentional silent is interpreted according to its context, we achieve implicatures.

Understanding and modeling the implications of silence can be useful in agent interaction, either human or virtual. For instance, we foresee a useful analysis of different scenario in legal cases involving testimonies of varied people (witnesses) and different kinds of silence. The strategy sketched can serve as a basis for a system supporting judges or prosecutors for decision making.

Each model generated with a logical program in the ASP paradigm can have a formal proof in intuitionist logic, according to [¹²]. Thus, each of the applications of the interpretations of silence and their possible combinations, intuitively, represent a formal proof within the belief system constructed from the assertions (statements) of the rational agents involved.

As future work, we plan to extend the interpretations to incorporate prosocial silence, i.e. retaining work-related information or opinions with the goal of benefiting other people or an organization.

It remains to bring the interpretations of silence to a more general framework for agent interaction, beyond testimonies and puzzles. Also in this direction, we are exploring to consider payoffs of agents involved in the interaction, as well as to the predicates to know who or what has more gains with silence, as an instrument in making decisions.

The silence can be active, conscious, intentional, strategic, and purposeful.