On Negation for Non-classical Set Theories
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On Negation for Non-classical Set Theories S. Jockwich Martinez1 · G. Venturi1 Received: 27 October 2019 / Accepted: 13 September 2020 / © Springer Nature B.V. 2020
Abstract We present a case study for the debate between the American and the Australian plans, analyzing a crucial aspect of negation: expressivity within a theory. We discuss the case of non-classical set theories, presenting three different negations and testing their expressivity within algebra-valued structures for ZF-like set theories. We end by proposing a minimal definitional account of negation, inspired by the algebraic framework discussed. Keywords Negation · American plan · Australian plan · Algebra-valued structures · ZF · Non-classical set theory · Paraconsistency Mathematics Subject Classification (2010) 03E40 · 03E70 · 03B53
1 Introduction In the last decade there has been an intense debate on the possibility to ground the concept of negation on more primitive notions. Two main views have been put forward, with the names of, respectively, American and Australian Plan. The main point of disagreement is found on where to discharge the complexity that negation provides to language. While the American Plan proposes to approach this complexity from a purely semantic side, allowing more than two truth values, the Australian Plan has a more modal take, viewing this complexity as arising from the interaction between the local and the global properties of a two-valued evaluation relation on Kripke frames. Among the supporters of the Australian Plan we find authors such as Berto, Meyer, Martin and Restall, [2, 3, 11, 14], who present the basic tenants of this viewpoint as S. Jockwich Martinez
[email protected] G. Venturi [email protected] 1
Department of Philosophy, University of Campinas (UNICAMP), Bar˜ao Geraldo, Campinas, SP, Brazil
S. Jockwich Martinez, G. Venturi
(1) negation is a device meant to capture a notion of exclusion or ruling out, and (2) negation has a modal character grounded on the concept of (in)compatibility. These points are meant to justify a two-valued approach to semantics and to offer a way to express the complexity of the evaluation process by means of a Kripke style or an algebraic semantics [4]. For example, in [2] we find a characterization of negation by generalized Kripke semantics, where a sentence ∼ϕ is said to be true at a world w if all worlds v that are compatible with w (here compatibility is understood in terms of the accessibility relation) do not validate ϕ. On the other hand, advocates of the American Plan, like Wansing, Omori, and De [6, 7, 17] give a non-modal account of negation. They sustain an intuitive interpretation of negation in terms of a switch operator between truth and falsity, where no restraints are made on the exclusiveness or exhaustivity of these truth-values. Concretely, one can say that a logical connective ∼ is a negation if and only if ∼ is a contradictory-forming operator: i.e., for any formula ϕ we have that (1) ϕ is true iff ∼ϕ is false and (2) ϕ is false iff ∼ϕ is tr
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