<p>The no-no paradox and its generalizations (called no-no type) are self-referential statements whose paradoxicality arises from the impossibility of symmetric truth-value assignments imposed by syntactical symmetry. Their syntactic symmetry can be characterized by permutation groups. This paper addresses whether any permutation group can be represented by a no-no type paradox, in the sense that the symmetry of the paradox is precisely characterized by the group. By introducing the notion of invariance preorder for binary sequences, we extend the algebraic techniques from research on the representability of the permutation groups by Boolean functions. We prove that for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, the alternating group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and the symmetric group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> have the same invariance preorder. Consequently, we establish the main result of this paper: for any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, the alternating group <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> cannot be represented by any Boolean system of self-referential statements — and <i>a fortiori</i>, by any no-no type paradox. With the information of the invariance preorder for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, we give a systematic construction of both Boolean paradoxes and no-no type paradoxes with maximal symmetry. Our results reveal a profound connection between algebraic symmetry and semantic pathology.</p>

Symmetry Groups of Boolean Self-referential Systems

期刊论文
发表日期：

2026年5月4日

查看全文

Hangjie Cao,
Ming Hsiung

摘要

The no-no paradox and its generalizations (called no-no type) are self-referential statements whose paradoxicality arises from the impossibility of symmetric truth-value assignments imposed by syntactical symmetry. Their syntactic symmetry can be characterized by permutation groups. This paper addresses whether any permutation group can be represented by a no-no type paradox, in the sense that the symmetry of the paradox is precisely characterized by the group. By introducing the notion of invariance preorder for binary sequences, we extend the algebraic techniques from research on the representability of the permutation groups by Boolean functions. We prove that for $n\ge 4$ $n \geq 4$ , the alternating group $A_n$ $A_{n}$ and the symmetric group $S_n$ $S_{n}$ have the same invariance preorder. Consequently, we establish the main result of this paper: for any $n\ge 4$ $n \geq 4$ , the alternating group $A_n$ $A_{n}$ cannot be represented by any Boolean system of self-referential statements — and a fortiori, by any no-no type paradox. With the information of the invariance preorder for $S_n$ $S_{n}$ , we give a systematic construction of both Boolean paradoxes and no-no type paradoxes with maximal symmetry. Our results reveal a profound connection between algebraic symmetry and semantic pathology.