## 6 Cognitive Semantics
For any object \(x\in X_{t}\) and category \(c\), we assume that epistemic satisfaction relations \(SP(t)\vDash_{G}\text{Bel}(x\
ot\in c)\) and \(SP(t)\vDash_{G}\text{Pos}(x\in c)\) hold if and only if  
\[\left(\tau_{c}^{+}<f(o,o_{c}^{\star})<\tau_{c}^{-}\right)\wedge\left(\lambda_ {c}(EN_{c}(o,\varepsilon))<\lambda_{minBel}\right) \tag{22}\]  
where object \(x\) is observed realization of a mental object \(o\) at time point \(t\) and \(o_{c}^{\star}\) is a prototype of \(c\). The following examples illustrate the way in which the above definitions of cognitive semantics can shape language production:  
Example 5: Let us suppose that for the model of category \(c\), the thresholds are equal to \(\tau_{c}^{+}=5\) and \(\tau_{c}^{-}=8\). In \(Episode(t_{3})\), two objects \(x_{3,8}\) and \(x_{3,9}\) appeared in the agent's range of perception. Suppose that in the agent's cognition process the object \(x_{3,8}\) from working memory corresponds to the object \(o_{8}\) in embodied ontology, and the object \(x_{3,9}\) from working memory corresponds to the object \(o_{9}\) in embodied ontology. The distances between the objects and the prototype are \(f(o_{8},o_{c}^{\star})=3\) and \(f(o_{9},o_{c}^{\star})=10\). The above situation is presented in Figure 7.  
Since \(f(o_{8},o_{c}^{\star})=3\leq\tau_{c}^{+}=5\) according to Definition 8, an epistemic satisfaction relation holds for formula \(Know(x_{8}\in c)\) and such a formula could be generated by the agent. The intuitive meaning of the formula can be expressed as \"I know that object \(x_{3,8}\) belongs to category \(c\)\". Since \(f(o_{9},o_{c}^{\star})=10\geq\tau_{c}^{-}=8\) according to Definition 9, an epistemic satisfaction relation holds for formula \(Know(x_{9}\
ot\in c)\) and such a formula could be generated by an agent. The intuitive meaning of the formula can be expressed as \"I know that object \(x_{3,9}\) does not belong to category \(c\)\". **Example 6**. : _A more complicated case is when an object is included in the boundary of the category. In order to establish the right statement, the agent must then compare the considered object not only with the prototype but also with other objects in embodied ontology. Let us make similar assumptions as in the previous example, except that this time the distances between the objects and the prototype are \(f(o_{8},o_{c}^{\star})=6\) and \(f(o_{9},o_{c}^{\star})=6\). The above situation is presented in Figure 8._  
_Since \(\tau_{c}^{+}=5<f(o_{8},o_{c}^{\star})=6<\tau_{c}^{-}=8\) and \(\tau_{c}^{+}=5<f(o_{9},o_{c}^{\star})=6<\tau_{c}^{-}=8\), the relative grounding strength must be determined for both objects. The first step is to establish the radius of the epistemic neighborhood \(\varepsilon\). As already mentioned, we apply the formula for the linear dependence of the radius on the width of the boundary. Assuming \(\alpha=0.8\), we obtain \(\varepsilon=ER(\tau_{c}^{-},\tau_{c}^{+})=\alpha(\tau_{c}^{-}-\tau_{c}^{+})=0.8\cdot(8-5)=2.4\)_  
_According to Definition 10, we determine the epistemic neighborhood of objects, i.e., \(EN_{c}(o_{8},\varepsilon)\) and \(EN_{c}(o_{9},\varepsilon)\). For this purpose, it is necessary to calculate the distance from the objects \(o_{8}\) and \(o_{9}\) to the objects in embodied ontology in a model of \(c\). Let us assume that the above distances are given as in Table 6 and that furthermore:_  
* \(E^{+}(Exp_{c})=\{o_{1},o_{2},o_{3},o_{5}\}\)__
* \(E^{-}(Exp_{c})=\{o_{4},o_{6},o_{7}\}\)_._  
Figure 8: Grounding of objects in boundary of category model. Figure 7: Grounding of objects in core and outer region of category model. _Based on the above data, we determine \(EN_{c}(o_{8},\varepsilon)=EN_{c}(o_{8},2.4)=\{o_{2},o_{4},o_{5}\}\) and \(EN_{c}(o_{9},\varepsilon)=EN_{c}(o_{9},2.4)=\{o_{7}\}\).