Nonrepresentability of the Alternating Group A4 on Boolean Systems
The no-no paradox is a typical type of symmetrical semantic paradox with its sym-metry,which can be characterized by permutation groups.When a permutation group characterizes the symmetry of a paradox,we call that the group is represented by this paradox.Hisung(2022)proposed the conjecture that any permutation group can be rep-resented by a no-no type of paradox.In this paper,by using concepts such as group map,we analyze the orbits and stabilizers of group actions on the set of truth value sequences,and prove that the alternating group A4 cannot be represented by any Boolean system.This indicates that if no-no type paradoxes are restricted within the scope of Boolean paradoxes,not every permutation group can be represented by a no-no type of paradox.This paper also provides examples of permutation groups that cannot be represented by Boolean systems through the construction of direct sum of permutation groups.Our re-search shows that permutation groups that cannot be represented by Boolean systems are not isolated,but have a certain basis for its existence.
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