## 7 Verification of the Model
: _If relation \(SP(t)\vDash_{G}Bel(x\
otin c)\) holds, then_  
* _relation_ \(SP(t)\vDash_{G}Bel(x\in c)\) _does not hold,_
* _relation_ \(SP(t)\vDash_{G}Pos(x\
otin c)\) _does not hold._  
Proof. : The epistemic satisfaction relation \(SP(t)\vDash_{G}Bel(x\
otin c)\) holds (Definition 13) if and only if  
\[\left(\tau_{c}^{+}<f(o,o_{c}^{*})<\tau_{c}^{-}\right)\wedge\left(\lambda_{c}( EN_{c}(o,\varepsilon))<\lambda_{minBel}\right) \tag{30}\]  
where \(x\) is the observed realization of mental object \(o\) at time point \(t\) and \(EN_{c}(o,\varepsilon)\) is the epistemic neighborhood of the object \(o\) with radius \(\varepsilon\). Thus, the condition \(\lambda_{c}(EN_{c}(o,\varepsilon))\geq\lambda_{minBel}\) required for epistemic satisfaction relations \(SP(t)\vDash_{G}Bel(x\in c)\) and \(SP(t)\vDash_{G}Pos(x\
otin c)\) is not fulfilled (Definition 12). Analogically to the above, if the agent generates a statement indicating that it is possible that an object belongs (or does not belong) to category \(c\), then it should not produce at the same time a statement with stronger confidence--Theorems 10 and 11. **Theorem 10**. : _If relation \(SP(t)\vDash_{G}Pos(x\in c)\) holds, then relation \(SP(t)\vDash_{G}Bel(x\in c)\) does not hold._  
Proof. : This theorem is the contraposition of the second part of Theorem 8. **Theorem 11**. : _If relation \(SP(t)\vDash_{G}Pos(x\
otin c)\) holds, then relation \(SP(t)\vDash_{G}Bel(x\
otin c)\) does not hold._  
Proof. : This theorem is the contraposition of the second part of Theorem 9. If the agent expresses the belief that an object belongs to some category \(c\), it is rational that the agent accepts the possibility that this object does not belong to category \(c\). It should therefore be possible for the agent to express both of the above opinions at the same state of knowledge. Such pairs of statements could potentially be connected with an additional language connector (not defined formally in this work), e.g., \"I believe that object \(x\) belongs to category \(c\), however, it is possible that object \(x\) does not belong to category \(c\)\". Similarly, if the agent expresses the belief that an object does not belong to some category \(c\), it is rational that the agent accepts the possibility that this object does belong to category \(c\). The next two theorems are proven for the above situations. **Theorem 12**. : _Relations \(SP(t)\vDash_{G}Bel(x\in c)\) and \(SP(t)\vDash_{G}Pos(x\
otin c)\) hold in the same state of knowledge._  
Proof. : It follows directly from Definition 12, where the conditions of the epistemic satisfaction relation are the same for both formulas. The general conclusion resulting from the list of theorems formulated above, supplemented by the theorems discussed in work [18], is that the design and implementation of an artificial cognitive agent as is described by the proposed theory (including specific definitions of cognitive semantics) makes it possible to realize an intelligent interactive system generating linguistic behavior consistent with human linguistic behavior, at least in terms of processing autoepistemic membership statements communicating beliefs about the belonging of observed objects to a category with a prototype. After taking into account the results presented in works [19; 20; 21; 22; 23; 24; 25; 26; 27] (some of which also presented simulation studies),the given conclusion can be extended to the case of autoepistemic extensions of more complex membership statements.