A Language for Ontological Nihilism

According to ontological nihilism there are, fundamentally, no individuals. Both natural languages and standard predicate logic, however, appear to be committed to a picture of the world as containing individual objects. This leads to what I call the \emph{expressibility challenge} for ontological nihilism: what language can the ontological nihilist use to express her account of how matters fundamentally stand? One promising suggestion is for the nihilist to use a form of \emph{predicate functorese}, a language developed by Quine. This proposal faces a difficult objection, according to which any theory in predicate functorese will be a notational variant of the corresponding theory stated in standard predicate logic. Jason Turner (2011) has provided the most detailed and convincing version of this objection. In the present paper, I argue that Turner's case for the notational variance thesis relies on a faulty metasemantic principle and, consequently, that an objection long thought devastating is in fact misguided

An intermediate term functor logic

Neste artigo, tentamos fazer algo bastante simples: conhecer os avanços de Sommers e Englebretsen (a saber, uma álgebra mais-menos para silogística) juntamente com os desenvolvimentos de Peterson e Thompson (ou seja, uma extensão da silogística com “a maioria”, “Muitos” e “poucos”). O resultado é uma silogística intermediária que lida com uma ampla gama de padrões lógicos, mas com as virtudes de uma abordagem algébrica

Complexity Classifications via Algebraic Logic

Complexity and decidability of logics is an active research area involving a wide range of different logical systems. We introduce an algebraic approach to complexity classifications of computational logics. Our base system GRA, or general relation algebra, is equiexpressive with first-order logic FO. It resembles cylindric algebra but employs a finite signature with only seven different operators, thus also giving a very succinct characterization of the expressive capacities of first-order logic. We provide a comprehensive classification of the decidability and complexity of the systems obtained by limiting the allowed sets of operators of GRA. We also discuss variants and extensions of GRA, and we provide algebraic characterizations of a range of well-known decidable logics

A Logic for Natural Language

This paper describes a language called £N whose structure mirrors that of natural language. £N is characterized by absence of variables and individual constants. Singular predicates assume the role of both individual constants and free variables. The role of bound variables is played by predicate functors called selection operators. Like natural languages, £N is implicitly many-sorted. £N does not have an identity relation. Its expressive power lies between the predicate calculus without identity and the predicate calculus with identity. The loss in expressiveness relative to the predicate calculus with identity however is not significant. Deduction in £N is intended to parallel reasoning in natural language, and therefore is termed surface reasoning. In contrast to deduction in a disparate underlying logic such as clausal form, each step of a proof in £N has a direct counterpart in the surface language. A sound and complete axiomatization is given. Derived rules, corresponding to monotonicity and conservativity of quantifiers and to unification and resolution in conventional logic, are presented. Several problems are worked to illustrate reasoning in £N

Algebraic classifications for fragments of first-order logic and beyond

Complexity and decidability of logics is a major research area involving a huge range of different logical systems. This calls for a unified and systematic approach for the field. We introduce a research program based on an algebraic approach to complexity classifications of fragments of first-order logic (FO) and beyond. Our base system GRA, or general relation algebra, is equiexpressive with FO. It resembles cylindric algebra but employs a finite signature with only seven different operators. We provide a comprehensive classification of the decidability and complexity of the systems obtained by limiting the allowed sets of operators. We also give algebraic characterizations of the best known decidable fragments of FO. Furthermore, to move beyond FO, we introduce the notion of a generalized operator and briefly study related systems.Comment: Significantly updates the first version. The principal set of operations change

Diagrammatic Algebra of First Order Logic

We introduce the calculus of neo-Peircean relations, a string diagrammatic extension of the calculus of binary relations that has the same expressivity as first order logic and comes with a complete axiomatisation. The axioms are obtained by combining two well known categorical structures: cartesian and linear bicategories

Foundations of Generalism: Symmetries, Non-individuals and Ontological Nihilism

The topic of this thesis is the metaphysical theory of generalism: the view that the world is constituted by purely general facts. Whilst the connection may not be immediately obvious, generalism is also touted as a qualitative metaphysics: a theory that seeks to elevate, in some important metaphysical sense, the notion of qualities (i.e. properties and relations) over that of objects. As such, generalism is just as well individuated by its categorial commitments—its commitment to the fundamentality of certain metaphysical categories—as it is by its construal of fundamental facts. My aim in this thesis is to make explicit these connections, providing a proper explication of the generalist position, as well as its motivations and its apparent consequences. Beyond this, the thesis can also be read as an extended argument in favour of individualism: the view that holds, contrary to generalism, that the category of individual, or object, is at least as fundamental as that of property and relation. The subtitle of this thesis, ‘symmetries, non-individuals and ontological nihilism’, alludes to the topic addressed by each of the three chapters. In chapter 1 I explicate and critique the generalist’s primary argument against individualism, one based on the notion of a symmetry. In chapter 2 I investigate the tenability of a position dubbed ‘quantifier generalism’, a position that, I argue, can be further explicated through the notion of a non-individual. And in chapter 3 I turn to the most widely-discussed form of generalism found in the literature: algebraic generalism, a (purported) form of ontological nihlism

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic