Minghui Ma
Sun Yat-Sen University, Institute of Logic and Cognition, Faculty Member
- Sun Yat-Sen University, Department of Philosophy, Faculty Memberadd
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Peirce introduced the Alpha part of the logic of Existential Graphs (egs) as a diagrammatic syntax and graphical system corresponding to classical propositional logic. The logic of quasi-Boolean algebras (De Morgan algebras) is a... more
Peirce introduced the Alpha part of the logic of Existential Graphs (egs) as a diagrammatic syntax and graphical system corresponding to classical propositional logic. The logic of quasi-Boolean algebras (De Morgan algebras) is a weakening of classical propositional logic. We develop a graphical system of weak Alpha graphs for quasi-Boolean algebras, and show its soundness and completeness with respect to this algebra. Weak logical graphs arise with only minor modifications to the transformation rules of the original theory of egs. Implications of these modifications to the meaning of the sheet of assertion are then also examined.
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Some classes of topological quasi-Boolean algebras, including algebraic structures related with rough sets, are enriched with residuated and adjoint pairs. The strong finite model property for these classes of algebraic structures is... more
Some classes of topological quasi-Boolean algebras, including algebraic structures related with rough sets, are enriched with residuated and adjoint pairs. The strong finite model property for these classes of algebraic structures is established. The decidability of equational theories of these classes of algebras is derived from the finite model property.
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The finite model property of quasi-transitive modal logic $\mathsf{K}_2^3=\mathsf{K}\oplus \Box\Box p\rightarrow \Box\Box\Box p$ is established. This modal logic is conservatively extended to the tense logic $\mathsf{Kt}_2^3$. We present... more
The finite model property of quasi-transitive modal logic $\mathsf{K}_2^3=\mathsf{K}\oplus \Box\Box p\rightarrow \Box\Box\Box p$ is established. This modal logic is conservatively extended to the tense logic $\mathsf{Kt}_2^3$. We present a Gentzen sequent calculus $\mathsf{G}$ for $\mathsf{Kt}_2^3$. The sequent calculus $\mathsf{G}$ has the finite algebra property by a finite syntactic construction. It follows that $\mathsf{Kt}_2^3$ and $\mathsf{K}_2^3$ have the finite model property.
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A sequent calculus \(\mathbf {wG5}\) is introduced for the variety of partition topological quasi-Boolean algebras. The sequent calculus \(\mathbf {wG5}\) has the cut elimination property, i.e., every sequent derivable in \(\mathbf... more
A sequent calculus \(\mathbf {wG5}\) is introduced for the variety of partition topological quasi-Boolean algebras. The sequent calculus \(\mathbf {wG5}\) has the cut elimination property, i.e., every sequent derivable in \(\mathbf {wG5}\) has a cut-free derivation. Furthermore, a sequent calculus \(\mathbf {wG4}_t\) is introduced for the variety of topological quasi-Boolean algebras with tense operators, and it is a conservative extension of a sequent calculus \(\mathbf {wG4}\) for the variety of topological quasi-Boolean algebras.
The language of Belnap–Dunn modal logic $${\mathscr {L}}_0$$ expands the language of Belnap–Dunn four-valued logic (having constant symbols for the values 0 and 1) with the modal operator $$\Box $$ . We introduce the polarity semantics... more
The language of Belnap–Dunn modal logic $${\mathscr {L}}_0$$ expands the language of Belnap–Dunn four-valued logic (having constant symbols for the values 0 and 1) with the modal operator $$\Box $$ . We introduce the polarity semantics for $${\mathscr {L}}_0$$ and its two expansions $${\mathscr {L}}_1$$ and $${\mathscr {L}}_2$$ with value operators. The local finitary consequence relation $$\models _4^k$$ in the language $${\mathscr {L}}_k$$ with respect to the class of all frames is axiomatized by a sequent system $$\mathsf {S}_k$$ where $$k=0, 1, 2$$ . We prove by using translations between sequents and formulas that these languages under the polarity semantics have the same expressive power on the level of frames with the language $${\mathscr {L}}_0$$ under the relational semantics for classical modal logic.
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This article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus,... more
This article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted by PC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to present PC as a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, in PC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection between PC and the alpha system.
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Graded epistemic logic is a logic for reasoning about uncertainties. Graded epistemic logic is interpreted on graded models. These models are generalizations of Kripke models. We obtain completeness of some graded epistemic logics. We... more
Graded epistemic logic is a logic for reasoning about uncertainties. Graded epistemic logic is interpreted on graded models. These models are generalizations of Kripke models. We obtain completeness of some graded epistemic logics. We further develop dynamic extensions of graded epistemic logics, along the framework of dynamic epistemic logic. We give an extension with public announcements, i.e., public events, and an extension with graded event models, a generalization also including nonpublic events. We present complete axiomatizations for both logics.
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The covering-based rough sets in the system C2 and bounded N -morphisms form a category CFN. This category is dually equivalent to the category KFS4 in which objects are Kripke S4-frames and arrows are bounded morphisms. Morphisms over... more
The covering-based rough sets in the system C2 and bounded N -morphisms form a category CFN. This category is dually equivalent to the category KFS4 in which objects are Kripke S4-frames and arrows are bounded morphisms. Morphisms over covering structures are defined in order to characterize the modal C2-definability of classes of finite covering frames.
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Charles Peirce’s alpha system $$\mathfrak {S}_\alpha $$Sα is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof... more
Charles Peirce’s alpha system $$\mathfrak {S}_\alpha $$Sα is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof analysis of $$\mathfrak {S}_\alpha $$Sα is given in terms of two embedding theorems: the system $$\mathfrak {S}_\alpha $$Sα and Brünnler’s deep inference system for classical propositional logic can be embedded into each other; and the system $$\mathfrak {S}_\alpha $$Sα and Gentzen sequent calculus $$\mathbf {G3c}^*$$G3c∗ can be embedded into each other.
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We study the residuated basic logic ($\mathsf{RBL}$) of residuated basic algebra in which the basic implication of Visser's basic propositional logic ($\mathsf{BPL}$) is interpreted as the right residual of a non-associative binary... more
We study the residuated basic logic ($\mathsf{RBL}$) of residuated basic algebra in which the basic implication of Visser's basic propositional logic ($\mathsf{BPL}$) is interpreted as the right residual of a non-associative binary operator $\cdot$ (product). We develop an algebraic system $\mathsf{S_{RBL}}$ of residuated basic algebra by which we show that $\mathsf{RBL}$ is a conservative extension of $\mathsf{BPL}$. We present the sequent formalization $\mathsf{L_{RBL}}$ of $\mathsf{S_{RBL}}$ which is an extension of distributive full non-associative Lambek calculus ($\mathsf{DFNL}$), and show that the cut elimination and subformula property hold for it.
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Visser's basic propositional logic $$\mathbf {BPL}$$ is the subintuitionistic logic determined by the class of all transitive Kripke frames, and his formal provability logic $$\mathbf {FPL}$$, an extension of $$\mathbf {BPL}$$, is... more
Visser's basic propositional logic $$\mathbf {BPL}$$ is the subintuitionistic logic determined by the class of all transitive Kripke frames, and his formal provability logic $$\mathbf {FPL}$$, an extension of $$\mathbf {BPL}$$, is determined by the class of all irreflexive and transitive finite Kripke frames. While Visser showed that $$\mathbf {FPL}$$ is embeddable into the modal logic $$\mathbf {GL}$$, we first show that $$\mathbf {BPL}$$ is embeddable into the modal logic $$\mathbf {wK4}$$, which is determined by the class of all weakly transitive Kripke frames, and we also show that $$\mathbf {BPL}$$ is characterized by the same frame class. Second, we introduce the proper successor semantics under which we prove that $$\mathbf {BPL}$$ is characterized by the class of weakly transitive frames, transitive frames, pre-ordered frames, and partially ordered frames. Third, we introduce topological semantics by interpreting implication in terms of the co-derived set operator and prove that $$\mathbf {BPL}$$ is characterized by the class of all topological spaces, $$T_0$$-spaces and $$T_d$$-spaces. Finally, we establish the topological completeness of $$\mathbf {FPL}$$ with respect to the class of scattered spaces.