Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$-valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$. The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

基于幺半 t 范数的逻辑$\mathbf {MTL}$是最弱的基于t范数的残差模糊逻辑，它是一个$[0,1]$值的命题逻辑系统，具有 t 范数及其残差作为真值函数用于连接和暗示。一元模糊谓词逻辑$\mathbf {mMTL\forall }$由带有一元谓词和一个对象变量的公式组成，是模糊谓词逻辑$\mathbf {MTL\forall }$的一元片段，它确实是谓词基于幺半群 t 范数的逻辑$\mathbf {MTL}$的版本。本文的主要目的是给出一元模糊谓词逻辑$\mathbf {mMTL\forall }$的完备性定理及其一些公理扩展的代数证明。首先，我们考察了基于t范数的剩余模糊逻辑的一元代数公理系统，并对其中的一些进行了修正，从而表明这些一元代数的关系完全继承了相应代数的关系。随后，利用一元模糊谓词逻辑$\mathbf {mMTL\forall }$和类 S5 模糊模态逻辑$\mathbf {S5(MTL)}$之间的等价性，我们证明一元 MTL 代数的多样性实际上是逻辑$\mathbf {mMTL\forall }$的等效代数语义，通过函数一元 MTL 代数给出该逻辑的完整性定理的代数证明。最后，我们进一步得到了逻辑$\mathbf {mMTL\forall }$的一些公理扩展的完备性定理，从而给出了一个重要的应用，即证明基于内卷幺半群t的一元模糊谓词逻辑的强完备性定理-范数逻辑$\mathbf {mIMTL\forall }$通过有限次直接不可约一元 IMTL 代数的函数表示。