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There are three differing accounts of restricted quantification in relevant logics under the author’s name, singly or jointly. The first was Brady (Relevant logics and their rivals, vol 2. Ashgate, Aldershot, 2003), in which an axiomatization was given for the restricted quantifiers (∀xP) and (∃xP), where the respective quantifiers have sub-domains restricted by the predicate P which must satisfy a non-emptiness requirement. The second was in the group work, Beall, Brady, Hazen, Priest and Restall (J Philos Log 35:587–598, 2006), where criteria were set up for restricted quantifiers to satisfy in relevant logics with the appropriate restricted quantifiers being introduced accordingly. The third was in (Brady RT, Logic – the big picture. In: Beziau J-Y, Chakraborty M, Dutta S (eds) New directions in paraconsistent logic, Springer, pp 353–373, 2015), where the classical account, using ⊃ and & to respectively restrict the universal and existential quantifiers, was briefly given within a big-picture view of logic. We will rectify this situation by presenting a novel and simpler account of restricted quantification in accordance with its conceptual analysis. We introduce a new connective > which will replace both the classical ⊃ and &, but behave as a pure positive restrictor without the negative requirements on the restricting predicate which indeed separate the two classical restrictors. We axiomatize it with a basic axiom and rule and a number of general rules with existential premises. We then determine how this new connective fits into the context of the standard connectives and quantifiers of a range of relevant logics, including the logic MCQ of meaning containment. Using this, we then provide a simple account of the square of opposition. We also relate > to the rule ⟹, which is also interpreted positively with no negative requirements on its antecedent/premise. We then relate it as best as we can to the classical account. We finish by examining the extent of the application of >, focussing on the Curry Paradox of naive set theory.

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A Conceptual Approach to Restricted Quantification in Relevant Logics

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  • 发表日期：

    2025年

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  • Ross T. Brady

摘要

There are three differing accounts of restricted quantification in relevant logics under the author’s name, singly or jointly. The first was Brady (Relevant logics and their rivals, vol 2. Ashgate, Aldershot, 2003), in which an axiomatization was given for the restricted quantifiers (∀xP) and (∃xP), where the respective quantifiers have sub-domains restricted by the predicate P which must satisfy a non-emptiness requirement. The second was in the group work, Beall, Brady, Hazen, Priest and Restall (J Philos Log 35:587–598, 2006), where criteria were set up for restricted quantifiers to satisfy in relevant logics with the appropriate restricted quantifiers being introduced accordingly. The third was in (Brady RT, Logic – the big picture. In: Beziau J-Y, Chakraborty M, Dutta S (eds) New directions in paraconsistent logic, Springer, pp 353–373, 2015), where the classical account, using ⊃ and & to respectively restrict the universal and existential quantifiers, was briefly given within a big-picture view of logic. We will rectify this situation by presenting a novel and simpler account of restricted quantification in accordance with its conceptual analysis. We introduce a new connective > which will replace both the classical ⊃ and &, but behave as a pure positive restrictor without the negative requirements on the restricting predicate which indeed separate the two classical restrictors. We axiomatize it with a basic axiom and rule and a number of general rules with existential premises. We then determine how this new connective fits into the context of the standard connectives and quantifiers of a range of relevant logics, including the logic MCQ of meaning containment. Using this, we then provide a simple account of the square of opposition. We also relate > to the rule ⟹, which is also interpreted positively with no negative requirements on its antecedent/premise. We then relate it as best as we can to the classical account. We finish by examining the extent of the application of >, focussing on the Curry Paradox of naive set theory.