Abstract
The method of synthetic tableaux is a cut-based tableau system with synthesizing rules introducing complex formulas. In this paper, we present the method of synthetic tableaux for Classical First-Order Logic, and we propose a strategy of extending the system to first-order theories axiomatized by universal axioms. The strategy was inspired by the works of Negri and von Plato. We illustrate the strategy with two examples: synthetic tableaux systems for identity and for partial order.
Highlights
The method of synthetic tableaux is a cut-based tableau system with synthesizing rules introducing complex formulas
This paper presents the system of synthetic tableaux for First-Order Logic, which is an extension of the method for Classical Propositional Logic presented by Urbański [8] and Urbański [9], but the inspiration for the first-order version comes from D’Agostino [10] and Mondadori [11]
If the number of distinct variables of A is k, the number of branches of the canonical synthetic tableau for A is 2k. This result shows that the ST-system for CPL is a standard proof system in the sense of D’Agostino and Mondadori, which is not at all surprising, as synthetic tableaux for CPL in the canonical version constitute a formal representation of the familiar truth-tables method
Summary
We present the synthetic tableaux system (ST-system, for short) for Classical Propositional. A linear rule may be applied in the construction of a synthetic tableau for formula A provided that each premise and conclusion of the rule belongs to the set Sub( A) ∪ ¬Sub( A). If the number of distinct variables of A is k, the number of branches of the canonical synthetic tableau for A is 2k In the margin, this result shows that the ST-system for CPL is a standard proof system in the sense of D’Agostino and Mondadori (see, e.g., [14]), which is not at all surprising, as synthetic tableaux for CPL in the canonical version constitute a formal representation of the familiar truth-tables method. On the level of a synthetic tableau, it means that the branching rule (our PB-rule) is never applied on the same branch more than once with respect to the same propositional variable This condition warrants consistency of each synthetic inference. The outcomes are presented in the research report [27]; the basic metalogical results—soundness and completeness—are missing
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