In a very preliminary way, the pluralistic perspectives on logic are based on the idea that there is more than one correct, true, or adequate logic. This means that there is more than one way to say which arguments are logically valid and which logical principles hold. In this talk, I offer an overview of the current scenario on this topic. The first step is to motivate logical pluralism by giving its context and how it contributes to our present debate on philosophy of logic, for instance, what are the scope and limitations of classical logic, how to deal with paradoxes and weird operations like material implication and so on. The second step is to search for a definition of pluralism, which mostly requires a discussion of the notions of logical consequence and validity. Depending on how it is done, we will see that there are many ways one can be a pluralist about logic, be it by cases, contexts, domains, inferential practices, etc. Finally, we need to clarify how only some logics, and not all of them, get to be legit, that is, how can we avoid a trivial theory and how logical relativism fits here. In order to do so, I will present a brief characterization of these approaches.
SEMINAR - Finished
Tomando como ponto de partida o cálculo de sequentes para a lógica ecumênica (LEci) desenvolvido e exposto por Elaine Pimentel, Luiz Carlos Pereira e Valéria de Paiva em artigo de 2019, apresentarei uma proposta de tablô para a lógica ecumênica proposicional clássico-intuicionista. Ao longo do seminário, apresentarei algumas das motivações que perpassam o desenvolvimento dessas lógicas, bem como as regras de expansão da árvore e os aspectos centrais de sua formalização em andamento na linguagem do assistente de provas Coq.
My goal is to assess the evolution of logical formalism that saw, especially in the first half of the twentieth century, a shift from a logic-as-practice to a logic-as-object perspective, a process that I will call logical de-pragmatization. I will claim that this shift is the product of a long historical process of reform that saw the abandonment of Judgements and the systematization of some primitive notions with modern meta-theoretic perspective, such as Hilbert's formalization of Proof and Tarski's semantic conception of Truth. The nineteenth-century dispute over Psychologism also played a significant role in de-pragmatizing logic, as I will argue. Albeit a decisive moment in the history of logic, I will conclude provisionally that the de-pragmatization was never fully reached in most formalisms. Even in current standards, the adoption of performative components is an inescapable fact in logical and mathematical practice.
Cantor abstractionist account of cardinal numbers has been discredited as a psychological theory of numbers which leads to contradiction – see Frege (1884) and Dummett (1991). The talk intends to resist these objections by arguing for the coherence and plausibility of Cantor's proposal. The defence of Cantorian abstraction will be built upon the set theoretic framework of Bourbaki (1968) – called BK – which is formulated in First-order Logic extended with the ε-operator. I will first introduce the axiomatic setting of BK and the definition of cardinal numbers. Then, I will present Cantor (1915) abstractionist theory, stressing the legacy with his early work of Cantor (1883), which will highlight two assumptions concerning the definition of cardinal numbers. I will argue that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to the ones of Zermelo-von Neumann and Frege-Russell. Based on the similarities between the BK definition and Cantor (1915) abstractionist theory, I will characterize the BK abstractionist account of cardinal numbers by examining two objections originally made by Frege (1884) to Cantor (1915) proposal. A key ingredient in the defence of Cantorian abstraction will be played by reasoning about arbitrary sets, as denoted by the ε-operator in the definition of cardinal numbers.
Free logics reject the classical assumption that every term of a given first-order language refers to an individual in the domain of quantification. According to the Quinean interpretation of the quantifiers, this move amounts to the rejection of the idea that every term must always denote an existent object. On the other hand, paraconsistent and paracomplete logics differ from classical logic by rejecting the validity of the principles ofexplosion and excluded middle, respectively. Although paraconsistent and paracomplete logics are usually characterized as subsystems of classical logic, some paraconsistent logics (that may also be paracomplete), the so-calledLFIs (LFIUs), are sufficiently expressive to allow recovering every classical inferences in certain circumstances. In this talk, I present free versions of some LFIs and indicate some of its potential philosophical applications. Specifically, we shall discuss how the existence predicate E, that is often used in the formulation of free logics, may be combined with the LFIs’ consistency (classicality) operator to allow specifying the scope of classical logic in terms of the kinds of object one is reasoning about. As we shall see, in the systems to be presented, although every classical inference turns out to be valid with respect to objects that satisfy E(i.e., objects that really exist), violations of certain classical principles may still occur whenever objects that fail to do so are concerned.