Digraphs provide an alternative syntax for propositional logic, with digraph kernels correspondin... more Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies of relevance.
Kernels of digraphs provide an alternative presentation of the classical propositional semantics,... more Kernels of digraphs provide an alternative presentation of the classical propositional semantics, [3, 2, 9]. Semikernels generalize kernels and, with some additional restrictions, provide semantics also for inconsistent theories, reducing to the classical kernels whenever the theory is consistent. The used semikernels are kernels of a maximal consistent subtheory (defined in the paper), so that semantics of any (also inconsistent) theory is the classical semantics of such a subtheory. We show that a classical resolution system, introduced in [9], is sound and complete (modulo weakening) for this new semantics, when used for direct, instead of refutational, reasoning. Natural interpretation of the Graph Normal Form, underlying the presentation, gives immediate applications to the analysis of paradoxes, which are used in the examples. §
Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discours... more Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discourses, with the semantics—equivalent to the classical one—defined by kernels of digraphs. The paper presents infinitary, resolution-based reasoning with GNF theories, which is refutationally complete for the classical semantics. Used for direct (not refutational) deduction it is not explosive and allows to identify in an inconsistent discourse, a maximal consistent subdiscourse with its classical consequences. Semikernels, generalizing kernels, provide the semantic interpretation.
History of Logic Basic Set Theory and Induction Turing Machines and Computability Propositional L... more History of Logic Basic Set Theory and Induction Turing Machines and Computability Propositional Logic: Hilbert's and Gentzen's Proof Systems Boolean and Set-Valued Semantics DNF and CNF Soundness and Completeness First-Order Logic: Hilbert's and Gentzen's Proof Systems Tarski's Semantics Prenex NF Term Models and Prolog Soundness and Completeness Exercises to Each Topic.
Multialgebras generalise algebraic semantics to handle nondeterminism. They model relational stru... more Multialgebras generalise algebraic semantics to handle nondeterminism. They model relational structures, representing relations as multivalued functions by selecting one argument as the “result”. This leads to strong algebraic properties missing in the case of relational structures. However, such strong properties can be obtained only by first choosing appropriate notion of homomorphism. We summarize earlier results on the possible notions of compositional homomorphisms of multialgebras and investigate in detail one of them, the outer-tight homomorphisms which yield rich structural properties not offered by other alternatives. The outer-tight homomorphisms are different from those obtained when relations are modeled as coalgebras and the associated congruence is the converse bisimulation equivalence. The category is cocomplete but initial objects are of little interest (essentially empty). On the other hand, the category does not, in general, possess final objects for the usual cardinality reasons. The main objective of the paper is to show that Aczel's construction of final coalgebras for set-based functors can be modified and applied to multialgebras. We therefore extend the category admitting also structures over proper classes and show the existence of final objects in this category.
We discuss briefly the duality (or rather, complementarity) of system descriptions based on actio... more We discuss briefly the duality (or rather, complementarity) of system descriptions based on actions and transitions, on the one hand, and states and their changes, on the other. We settle for the latter and present a simple language, for describing state changes, which is parameterized by an arbitrary language for describing properties of the states. The language can be viewed as a simple fragment of step logic, admitting however various extensions by appropriate choices of the underlying logic. Alternatively, it can be seen as a very specific fragment of temporal logic (with a variant of 'until' or 'chop' operator), and is interpreted over dense (possibly continuous) linear time. The reasoning system presented here is sound, as well as strongly complete and decidable (provided that so is the parameter logic for reasoning about a single state). We give the main idea of the completeness proof and suggest a wide range of possible applications (action based descript...
Digraphs provide an alternative syntax for propositional logic, with digraph kernels correspondin... more Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies of relevance.
Kernels of digraphs provide an alternative presentation of the classical propositional semantics,... more Kernels of digraphs provide an alternative presentation of the classical propositional semantics, [3, 2, 9]. Semikernels generalize kernels and, with some additional restrictions, provide semantics also for inconsistent theories, reducing to the classical kernels whenever the theory is consistent. The used semikernels are kernels of a maximal consistent subtheory (defined in the paper), so that semantics of any (also inconsistent) theory is the classical semantics of such a subtheory. We show that a classical resolution system, introduced in [9], is sound and complete (modulo weakening) for this new semantics, when used for direct, instead of refutational, reasoning. Natural interpretation of the Graph Normal Form, underlying the presentation, gives immediate applications to the analysis of paradoxes, which are used in the examples. §
Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discours... more Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discourses, with the semantics—equivalent to the classical one—defined by kernels of digraphs. The paper presents infinitary, resolution-based reasoning with GNF theories, which is refutationally complete for the classical semantics. Used for direct (not refutational) deduction it is not explosive and allows to identify in an inconsistent discourse, a maximal consistent subdiscourse with its classical consequences. Semikernels, generalizing kernels, provide the semantic interpretation.
History of Logic Basic Set Theory and Induction Turing Machines and Computability Propositional L... more History of Logic Basic Set Theory and Induction Turing Machines and Computability Propositional Logic: Hilbert's and Gentzen's Proof Systems Boolean and Set-Valued Semantics DNF and CNF Soundness and Completeness First-Order Logic: Hilbert's and Gentzen's Proof Systems Tarski's Semantics Prenex NF Term Models and Prolog Soundness and Completeness Exercises to Each Topic.
Multialgebras generalise algebraic semantics to handle nondeterminism. They model relational stru... more Multialgebras generalise algebraic semantics to handle nondeterminism. They model relational structures, representing relations as multivalued functions by selecting one argument as the “result”. This leads to strong algebraic properties missing in the case of relational structures. However, such strong properties can be obtained only by first choosing appropriate notion of homomorphism. We summarize earlier results on the possible notions of compositional homomorphisms of multialgebras and investigate in detail one of them, the outer-tight homomorphisms which yield rich structural properties not offered by other alternatives. The outer-tight homomorphisms are different from those obtained when relations are modeled as coalgebras and the associated congruence is the converse bisimulation equivalence. The category is cocomplete but initial objects are of little interest (essentially empty). On the other hand, the category does not, in general, possess final objects for the usual cardinality reasons. The main objective of the paper is to show that Aczel's construction of final coalgebras for set-based functors can be modified and applied to multialgebras. We therefore extend the category admitting also structures over proper classes and show the existence of final objects in this category.
We discuss briefly the duality (or rather, complementarity) of system descriptions based on actio... more We discuss briefly the duality (or rather, complementarity) of system descriptions based on actions and transitions, on the one hand, and states and their changes, on the other. We settle for the latter and present a simple language, for describing state changes, which is parameterized by an arbitrary language for describing properties of the states. The language can be viewed as a simple fragment of step logic, admitting however various extensions by appropriate choices of the underlying logic. Alternatively, it can be seen as a very specific fragment of temporal logic (with a variant of 'until' or 'chop' operator), and is interpreted over dense (possibly continuous) linear time. The reasoning system presented here is sound, as well as strongly complete and decidable (provided that so is the parameter logic for reasoning about a single state). We give the main idea of the completeness proof and suggest a wide range of possible applications (action based descript...
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Papers by Michal Walicki