29,171 research outputs found
From Many-Valued Consequence to Many-Valued Connectives
Given a consequence relation in many-valued logic, what connectives can be
defined? For instance, does there always exist a conditional operator
internalizing the consequence relation, and which form should it take? In this
paper, we pose this question in a multi-premise multi-conclusion setting for
the class of so-called intersective mixed consequence relations, which extends
the class of Tarskian relations. Using computer-aided methods, we answer
extensively for 3-valued and 4-valued logics, focusing not only on conditional
operators, but on what we call Gentzen-regular connectives (including negation,
conjunction, and disjunction). For arbitrary N-valued logics, we state
necessary and sufficient conditions for the existence of such connectives in a
multi-premise multi-conclusion setting. The results show that mixed consequence
relations admit all classical connectives, and among them pure consequence
relations are those that admit no other Gentzen-regular connectives.
Conditionals can also be found for a broader class of intersective mixed
consequence relations, but with the exclusion of order-theoretic consequence
relations.Comment: Updated version [corrections of an incorrect claim in first version;
two bib entries added
Formalizing Kant’s Rules
This paper formalizes part of the cognitive architecture that Kant develops in the Critique of Pure Reason. The central Kantian notion that we formalize is the rule. As we interpret Kant, a rule is not a declarative conditional stating what would be true if such and such conditions hold. Rather, a Kantian rule is a general procedure, represented by a conditional imperative or permissive, indicating which acts must or may be performed, given certain acts that are already being performed. These acts are not propositions; they do not have truth-values. Our formalization is related to the input/ output logics, a family of logics designed to capture relations between elements that need not have truth-values. In this paper, we introduce KL3 as a formalization of Kant’s conception of rules as conditional imperatives and permissives. We explain how it differs from standard input/output logics, geometric logic, and first-order logic, as well as how it translates natural language sentences not well captured by first-order logic. Finally, we show how the various distinctions in Kant’s much-maligned Table of Judgements emerge as the most natural way of dividing up the various types and sub-types of rule in KL3. Our analysis sheds new light on the way in which normative notions play a fundamental role in the conception of logic at the heart of Kant’s theoretical philosophy
Relevant Logics Obeying Component Homogeneity
This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues
Non normal logics: semantic analysis and proof theory
We introduce proper display calculi for basic monotonic modal logic,the
conditional logic CK and a number of their axiomatic extensions. These calculi
are sound, complete, conservative and enjoy cut elimination and subformula
property. Our proposal applies the multi-type methodology in the design of
display calculi, starting from a semantic analysis based on the translation
from monotonic modal logic to normal bi-modal logic
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