Logics

Principia Mathematica, 100 Years after. An Expository Essay. The paper is an expository essay in which I discuss the main goals, concepts, problems, and results of Whitehead’s and Russell’s Principia Mathematica (1910–1913) at its centennial anniversary. The great work by Whitehead and Russell may not be the birth certificate of modern logic, but it is nevertheless the legitimacy and consecration certificate of modern mathematical logic and of the logicist program into contemporary philosophy of mathematics. The emphasis is laid upon the (...) presentation of sentential and predicate logic, identity, singular definite descriptions, set-theoretic and semantic paradoxes, and type theory. (shrink)

The paper discusses an original and ingenious puzzle about time and thought due to Saul Kripke. Its connections with related issues and puzzles such as Russell’s paradox and Kaplan’s puzzle are addressed and explored. The problems raised by the puzzle are dealt with briefly, emphasizing the comparative worth of Kripke’s findings in terms of simplicity and modesty of its assumptions. Then the attempted solutions are analyzed and explored, the most important and promising avenue for future work being the extension of (...) the apparatus developed by Kripke in his groundbreaking paper on truth to intensional logic. (shrink)

This work examines the structure of paradoxes, phenomena characterized by the apparent derivation of contradictions or counter-intuitive conclusions from acceptable premises using acceptable reasoning. Motivated by the recurrence of certain features across diverse paradoxes, a specific structural hypothesis regarding their generation is proposed and analyzed. The hypothesis posits that paradoxes frequently arise from the interaction of three components: (1) concepts that establish or interact with boundaries (such as those pertaining to scope, classification, truth, state, or definition); (2) the reflexive application (...) of these concepts or related rules (involving self-reference, self-inclusion, recursion, or causal/definitional circularity); and (3) the involvement of specific operational triggers – negation, limit (including infinity, extremes, or boundary values), or transformation (change of state, identity, position, or logical status) – within the reflexive structure. The study first provides a detailed conceptual analysis of these components – boundaries, reflexivity, and the triggers – outlining their definitions and potential roles based on discussions in logic, philosophy, and related fields. Subsequently, the hypothesis is examined against a wide range of paradoxes drawn from logic, set theory, semantics, physics, metaphysics, epistemology, and decision theory. This examination involves analyzing the structure of each paradox to identify the presence and interaction of the hypothesized components. The analysis indicated a recurrent correspondence between the structures of the examined paradoxes and the proposed pattern involving boundaries, reflexivity, and triggers. Further analysis explored the representation of this structural pattern using elements of logical and mathematical formalism, sketching how the interaction of boundary predicates/sets, reflexive constructions, and operators corresponding to negation, limits, or transformations can lead to derivations of inconsistency (e.g., P ↔ ¬P or P ∧ ¬P) within systems adhering to standard logical principles. The discussion also addressed the context and limitations surrounding the hypothesis, including the meaning of "often arise," the relationship between the identified structure and strategies for paradox resolution (which often involve modifying the system to restrict or manage components of the pattern), the distinction between the structure and its outcome in different systems, and the acknowledgment that the pattern may not be the sole source of all paradoxes and that its identification can involve interpretation. Finally, potential implications arising from the identification of this recurring structure were considered. These include the possibility of providing a unifying framework for paradox analysis across disciplines, offering insights relevant to the design of consistent conceptual and formal systems by highlighting potential points of instability, suggesting connections between the structure and characteristics of natural language or thought involving self-reference and boundary definition, and noting structural resonances with limitative results in mathematical logic, such as Gödel's incompleteness theorems. Potential directions for future work, including further categorization, analysis of non-paradoxical structures, cross-disciplinary applications, and exploration of cognitive aspects, were also outlined. (shrink)

Este trabalho apresenta, em linhas gerais, uma visão esquemática da arquitetura filosófica de um novo Sistema fundada sobre três pilares principais: (1) a formulação de uma condição lógico-metafísica absolutamente necessária — S(Ⓣ(φ)) — que estabelece o Ser como condição universal e intranscendível de toda proposição e fato verdadeiros; (2) a demonstração metafísica, por via de um argumento alético-modal, da impossibilidade do nada absoluto e, por conseguinte, da necessidade de um Ser absoluto, identificado como Deus em sentido estritamente filosófico; (3) a (...) construção de uma ontologia estrutural estratificada unida a uma teoria da verdade, em que Deus é concebido como ultraestrutura da Verdade Transcendente (Verdade Absoluta ⇄ Vida Absoluta ⇄ Vontade Absoluta), fundamento superior da hiperestrutura axial imanente (Verdadeiridade ⇄ Belezidade ⇄ Bonidade), por sua vez fundamento da superestrutura fático-proposicional (T(p) ⇄ F). A tridimensão é unificada por um princípio dialético-ontoeinailógico radical — o da identidade autocontraditada —, pelo qual o sistema propõe uma filosofia do Ser como Verdade, em que o pensamento reencontra a possibilidade de afirmar o absoluto por meio da razão. Trata-se de uma proposta original que reabilita a ambição sistemática e restitui à filosofia a tarefa fundamental de pensar, de forma rigorosa e afirmativa, aquilo que “necessariamente é”, o objeto próprio da einailogia. (shrink)

This work presents, in general terms, a schematic view of the philosophical architecture of a new System based on three main pillars: (1) the formulation of an absolutely necessary logical-metaphysical condition — S((T)(φ)) — which establishes Being as a universal and untranscendable condition of every true proposition and fact; (2) the metaphysical demonstration, by means of an alethical-modal argument, of the impossibility of absolute nothingness and, consequently, of the need for an absolute Being, identified as God in a strictly philosophical (...) sense; (3) the construction of a stratified structural ontology united to a theory of truth, in which God is conceived as the ultrastructure of Transcendent Truth (Absolute Truth ⇄ Absolute Life ⇄ Absolute Will), the superior foundation of the immanent axial hyperstructure (Truthfulness ⇄ Beauty ⇄ Goodness), in turn the foundation of the factual-propositional superstructure (T(p) ⇄ F). The three-dimension is unified by a radical dialectical-ontoeinailogical principle — that of self-contradicted identity — by which the system proposes a philosophy of Being as Truth, in which thought rediscovers the possibility of affirming the absolute by means of reason. It is an original proposal that rehabilitates systematic ambition and restores to philosophy the fundamental task of thinking, in a rigorous and affirmative way, what "necessarily is", the proper object of einailogy. (shrink)

The propositional connectives – and, or, not, if–then, etc. – are fundamental building blocks in formal as well as natural languages. In the Western tradition, they were first studied as such by the Stoics, and Propositional Logic is the fundament of practically all current systems of logic; every beginning logic course starts with it. Still, the proof theory and semantics of systems of propositional logic are far from trivial, and have been studied intensely by logicians in the last one and (...) a half centuries, not least in recent decades. It is actually a vast area of research, as witnessed by Lloyd Humberstone’s 1500 page tome The Connectives (2011), which overviews much of that research. Perhaps the most familiar recent work in this area concerns conditionals in formal and natural languages. The 4th Tsinghua Interdisciplinary Workshop on Logic, Language, and Meaning (TLLM 2024) also focused on the apparently simpler connectives expressing (various versions of) conjunction, disjunction, and negation. -/- Researchers working from a cross-linguistic perspective also study how the connectives are encoded in different languages, and ask whether classical logic is capable of capturing the variations and universals exhibited. Even in well-studied languages such as English, there are intricate phenomena that remain challenging for classical logic, including free choice disjunction, non-Boolean conjunction, and metalinguistic negation, to name just a few. There is also growing interest in the acquisition and processing of natural language connectives. In the context of the hotly discussed Large Language Models (LLMs), understanding connectives presents novel challenges that deserve in-depth exploration. (shrink)

Este artigo apresenta o argumento alético-modal como uma demonstração filosófico-conceitual rigorosa da necessidade do Ser Absoluto. Argumenta-se que, se tudo fosse contingente, o nada absoluto seria possível; no entanto, tal hipótese revela-se logicamente e metafisicamente incoerente, pois contradiz as condições de inteligibilidade e de verdade. Conclui-se, portanto, que uma necessidade absoluta deve ser afirmada. O argumento defende que a necessidade, enquanto modalidade alética, pressupõe um fundamento no Ser e não pode ser reduzida à pura formalidade lógica. Assim, a necessidade afirmada (...) não é meramente epistêmica ou formal, mas metafísica - o que justifica a identificação de Deus como o Ser Absoluto e Necessário no passo final decisivo. (shrink)

This paper presents the alethic-modal argument as a rigorous philosophicalconceptual demonstration of the necessity of the Absolute Being. It is argued that if everything were contingent, absolute nothingness would be possible; however, such a hypothesis proves to be logically and metaphysically incoherent, as it contradicts the conditions of intelligibility and truth. It is concluded, therefore, that an absolute necessity must be affirmed. The argument maintains that necessity, as an alethic modality, presupposes a foundation in Being and cannot be reduced to (...) mere logical formality. Thus, the affirmed necessity is not merely epistemic or formal, but metaphysicalwhich justifies identifying God as the Absolute and Necessary Being in the final decisive step. (shrink)

This paper clarifies and defends the global approach to defining logical validity for meta- and higher-level inferences. This is contrary to an emerging consensus in favour of local validity. Prevalent recent arguments claim that global validity is either superfluous in virtue of collapsing into local, or else untenable because it overgenerates validities, compromises the formality of logic, or breaks symmetry with regular validity. Accordingly, the literature on higher inferential logic has come to focus almost exclusively on local validity. Many key (...) philosophical takeaways that have since been drawn from the field are effectively based on local monism: the view that local is the One True Validity Criterion. The present paper argues that this is a mistake, resulting largely from an imprecise understanding of global validity. I untangle some common confusions about how global generalizes beyond the metalevel, and submit that its proper formulation is schematic. These clarifications help dispel each of the aforementioned arguments against global, leaving local monism without any justification. Finally, I outline how much of the philosophical literature around higher-level inferences has been built on local monism, and sketch out the sweeping consequences of rejecting it. (shrink)

After introducing the debate between substance philosophy and process philosophy, and clarifying the relevance of the category of ‘substance’ in Peirce’s thought, the present paper reconstructs the role of ‘substance’ and ‘being’ from Peirce’s early works to his theory of the proposition, provided after his studies on the logic of relatives. If those two categories apparently disappear in Peirce’s writings from the mid-1890s onwards, the account of ‘subject’ and ‘copula’ in Peirce’s analysis of the proposition allows one to grasp the (...) reasons why Peirce omits ‘substance’ and ‘being’ in favor of his three categories (Firstness, Secondness, Thirdness), and to understand why his philosophy cannot be considered as a substance philosophy. (shrink)

The so-called Principle of Plenitude was ascribed to Leibniz by A. O. Lovejoy in The Great Chain of Being: A Study of the History of an Idea (1936). Its temporal version states that what holds always, holds necessarily (or that no genuine possibility can remain unfulfilled). This temporal formulation is the subject of the current paper. Lovejoy’s idea was criticised by Hintikka. The latter supported his criticisms by referring to specific Leibnizian notions of absolute and hypothetical necessities interpreted in a (...) possible-worlds semantics. In the paper, Hintikka’s interpretative suggestions are developed and enriched with a temporal component that is present in the characteristics of the real world given by Leibniz. We use in our approach the Leibnizian idea that change is primary to time and the idea that there are possible laws that characterize worlds other than the real one. We formulate a modal propositional logic with three primitive operators for change, temporal constancy, and possible lawlikeness. We give its axiomatics and show that our logic is complete with respect to the given semantics of possible worlds. Finally, we show that the counterparts of the considered versions of the Principle of Plenitude are falsified in this semantics and the same applies to the counterpart of Leibnizian necessarianism. (shrink)

In this paper, we present a logic MMLS5n which is a combination of multilattice logic and modal logic S5. MMLS5n is an extension of Kamide and Shramko’s modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MMLS5n in the spirit of Restall’s one for S5 and develop a Kripke semantics for MMLS5n, following Kamide and Shramko’s approach. Moreover, we prove theorems for embedding MMLS5n into S5 and vice versa. As a result, we (...) obtain completeness, cut elimination, decidability, and interpolation theorems for MMLS5n. Besides, we show the duality principle for MMLS5n. Additionally, we introduce a modification of Kamide and Shramko’s sequent calculus for their multilattice version of S4 which (in contrast to Kamide and Shramko’s original one) proves the interdefinability of necessity and possibility operators. Last, but not least, we present Hilbert-style calculi for all the logics in question as well as for a larger class of modal multilattice logics. (shrink)

The paper proposes a first approach to systems whose language includes two primitives (>+ and >-) as symbols for factual and counterfactual conditionals which are explicit, i.e. that are stated jointly with the truth or falsity of the antecedent clause. In systems based on this language, here called 2-conditional, the standard corner operator may be defined by (Def>) A > B := (A >+ B)∨(A >- B), while in classical conditional systems one could introduce the two symbols for explicit conditionals (...) by the definitions (Def>+) A >+ B := A∧(A>B) and (Def>-) A>- B := ¬A∧(A>B). Two 2-conditional systems, V² and VW², are axiomatized and proved to be definitionally equivalent to the monoconditional systems V and VW. A third system VWTr² is characterized by an axiom stating the transitivity of factual conditionals and is shown to be distinct from V², from VW² and from the 2-conditional version of Lewis’ well-known system VC, here named VC². The same may be said for a fourth system VW²◊ ±, based on an axiom inderivable in VC²: ◊(A >+ B) ⊃ ◊( ¬A >- ¬B). VC² contains what is here called a “semi-collapse” of the operator >+ and it is argued that it is inadequate as a logic for both factual and counterfactual conditionals. The last section shows that several different definitions of the corner operators in terms of >+ and >- may be introduced as an alternative to (Def>). (shrink)

In this introduction, we offer an overview of main systems developed in the growing literature on connexive logic, and also point to a few topics that seem to be collecting attention of many of those interested in connexive logic. We will also make clear the context to which the papers in this special issue belong and contribute.

The goal of this paper is to analyse Implicational Relevance Logic from the point of view of refutability. We also correct an inaccuracy in our paper “The RM paraconsistent refutation system”.

The paper introduces a variant of connexive logic in which connexivity is extended from the interaction of negation with implication to the interaction of negation also with conjunction and disjunction. The logic is presented by two deductively equivalent methods: an axiomatic one and a natural-deduction one. Both are shown to be complete for a four-valued model theory.

This paper discusses the relation between the natural deduction rules of deduction in sequent format and the provability valuation starting from Garson’s Local Expression Theorem, which is meant to establish that the natural deduction rules of inference enforce exactly the classical meanings of the propositional connectives if these rules are taken to be locally valid, i.e. if they are taken to preserve sequent satisfaction. I argue that the natural deduction rules for disjunction are in no better position than the axiomatic (...) calculi in uniquely determining the intended meaning of disjunction when the local models are used, if a satisfied sequent embeds, as a logical inferentialist should require, a formal derivability relation. This happens because, when governed by these rules and without additional semantic assumptions, the disjunction sign still expresses a non-extensional connective, i.e. a connective that, properly understood, has no unique logical characteristic. However, this is not a dead end for the logical inferentialists since both a multiple conclusions formalization of the disjunction operator and a bilateralist one do succeed in restoring the standard meaning of disjunction. (shrink)

A general mathematical framework, based on countable partitions of Natural Numbers [1], is presented, that allows to provide a Semantics to propositional languages. It has the particularity of allowing both the valuations and the interpretation Sets for the connectives to discriminate complexity of the formulas. This allows different adequacy criteria to be used to assess formulas associated with the same connective, but that differ in their complexity. The presented method can be adapted potentially infinite number of connectives and truth values, (...) therefore, it can be considered a general framework to provide semantics to several of the known logic systems (eg, LC, L3 LP, FDE). The presented semantics allow to converge to different standard semantics if the separation complexity procedure is annulled. Therefore, it can be understood as a framework that allows greater precision (in complexity terms) with respect to formula satisfaction. Naturally, because of how it is built, it can be incorporated into non-deterministic semantics. The presented procedure also allows generating valuations that grant a different truth value to each formula of propositional language. As a positive side effect, our method allows a constructive proof of the equipotence between N and N^n for all Natural n. (shrink)

A speculative exploration of the distinction between a relational formal ontology and a classical formal ontology for modelling phenomena in nature that exhibit relationally-mediated wholism, such as phenomena from quantum physics and biosemiotics. Whereas a classical formal ontology is based on mathematical objects and classes, a relational formal ontology is based on mathematical signs and categories. A relational formal ontology involves nodal networks (systems of constrained iterative processes) that are dynamically sustained through signalling. The nodal networks are hierarchically ordered and (...) exhibit characteristics of deep learning. Clarifying the distinction between classical and relational formal ontologies may help to clarify the role of interpretative context in physics (eg. the role of the observer in quantum theory) and the role of hierarchical nodal networks in computational simulations of learning in artificial intelligence. (shrink)

Neil Tennant was the first to propose a proof-theoretic criterion for paradoxicality, a framework in which a paradox, formalized through natural deduction, is derived from an unacceptable conclusion that employs a certain form of id est inferences and generates an infinite reduction sequence. Tennant hypothesized that any derivation in natural deduction that formalizes a genuine paradox would meet this criterion, and he argued that while the liar paradox is genuine, Russell's paradox is not. -/- The present paper delves into Tennant's (...) conjecture for genuine paradoxes and suggests that to validate the conjecture, one of two issues must be addressed. The first issue is the need for a philosophical consensus on the identification of a genuine paradox in an informal sense. The second issue is the requirement for a uniform approach to formalize paradoxes in natural deduction. If either of these issues is addressed, the conjecture could be validated, or at the very least, it could hold philosophical importance in delineating the proof-theoretic features of paradoxicality. (shrink)

This book develops a new approach to plural arbitrary reference and examines mereology, including considering four theses on the alleged innocence of mereology. The authors have advanced the notion of plural arbitrary reference in terms of idealized plural acts of choice, performed by a suitable team of agents. In the first part of the book, readers will discover a revision of Boolosʼ interpretation of second order logic in terms of plural quantification and a sketched structuralist reconstruction of second-order arithmetic based (...) on the axiom of infinite, a la Dedekind, as the unique non-logical axiom. The work goes on to analyse the pros and cons of the new interpretation, also with respect to Linneboʼs objections to the thesis that second order logic is genuine logic. A theory of concepts that can be labelled as a theory of logical concepts is expounded. In the second part of the book, the authors consider grounding megethology on plural arbitrary reference and argue that the arguments for the ontological innocence of mereology are not conclusive and that – for a certain use of mereology – a thesis of innocence, similar to that of plural arbitrary reference, is defensible. The work proposes a virtual theory of mereology in which the role of individuals is played by plural choices of atoms. This considered work will appeal to scholars from branches of analytic philosophy, logic and the philosophy of mathematics in particular. (shrink)

Gödel’s ontological proof is by now well known based on the 1970 version, written in Gödel’s own hand, and Scott’s version of the proof. In this article new manuscript sources found in Gödel’s Nachlass are presented. Three versions of Gödel’s ontological proof have been transcribed, and completed from context as true to Gödel’s notes as possible. The discussion in this article is based on these new sources and reveals Gödel’s early intentions of a liberal comprehension principle for the higher order (...) modal logic, an explicit use of second-order Barcan schemas, as well as seemingly defining a rigidity condition for the system. None of these aspects occurs explicitly in the later 1970 version, and therefore they have long been in focus of the debate on Gödel’s ontological proof. (shrink)

This paper argues for the significance of Kaplan's logic LD in two ways: first, by looking at how logic got along before we had LD, and second, by using it to bring out the similarity between David Hume's thesis that one cannot deduce claims about the future on the basis of premises only about the past, and the so‐called "essentiality" of the indexical.

Aristotle counts as the founder of formal logic. The logic he develops dominated until Frege and others introduced a new logic. This new logic is taken to be more powerful and better capable of capturing inference patterns. The new logic differs from Aristotelian logic in significant respects. It has been argued by Fred Sommers and Hanoch Ben-Yami that the new logic is not well equipped as a logic of natural language, and that a logic closer to Aristotle's is better suited (...) for this task. Each of them developed their own formalism - Sommers in form of term logic, Ben-Yami in form of his Quantified Argument Calculus (QUARC). I discuss Aristotle's logic - a term logic - and attempt a comparison between Aristotelian logic and (i) the new logic, (ii) Sommers' term logic, and (iii) Ben-Yami's QUARC. I consider differences between the systems, and show how they are related to and diverge from the new logic. (shrink)

This paper rigorously examines the primitive foundations of economic reasoning through an original framework based on symbolic logic. Extending previous work, it formalizes economic conceptions (\(\mathbb{C}\)), symbols (\(s_i\)), and introduces a structured language (\(\mathcal{L_{\mathbb{C}}}\)) to define their formation and interpretation. Organized as a continuous chain of declarations and illustrations, the paper offers a concise, systematic approach to understanding the philosophy of economic reasoning through formal representations.

This paper takes a stand on Hamblin semantics and its relation to the semantics-to-pragmatics interface. Biased questions, where the speaker finds one of the options more likely and expects the confirmation that p is true, raise a concern about the limits of Hamblin semantics. I argue that biased questions have modified Hamblin semantics, while unbiased questions have unconstrained Hamblin semantics. The optional bias feature explains compositionally. It is triggered by likelihood presuppositions ranging Hamblin sets and highlighting the preferred alternative(s). Biased (...) effects are exceptionally possible in alternative and wh-questions, but only in polar questions, they can be encoded lexically and grammatically. I argue that Hamblin semantics covers all core types of questions, while some of its applications for non-questions are problematic. (shrink)

Circumstantialists already have a logical semantics for impossibilities. They expand their logical space of possible worlds by adding impossible worlds. These are impossible circumstances serving as indices of evaluation, at which impossibilities are true. A variant of circumstantialism, namely modal Meinongianism (noneism), adds impossible objects as well. These are so-called incomplete objects that are necessarily non-existent. The opposite of circumstantialism, namely structuralism, has some catching-up to do. What might a structuralist logical semantics for impossibilities without impossibilia look like? This paper (...) makes a structuralist counterproposal. We present a semantics based on a procedural interpretation of the typed λ-calculus. The fundamental idea is that talk about impossibilities should be construed in terms of procedures: some yield as their product a condition that could not possibly have a satisfier, while the rest fail to yield a product altogether. Dispensing with a ‘bottom’ of impossibilia requires instead a ‘top’ consisting of structured hyperintensions, intensions, intensions defining other intensions, a typed universe, and dual (de dicto and de re) predication. We explain how the theory works by going through several examples. (shrink)

It is astonishing that we humans are able to have, act on and express moral beliefs. This dissertation aims to provide a better philosophical understanding of why and how this is possible especially when we assume metaethical expressivism. Metaethical expressivism is the combination of expressivism and noncognitivism. Expressivism is the view that the meaning of a sentence is explained by the mental state it is conventionally used to express. Noncognitivism is the view that the mental state expressed by a moral (...) sentence is a noncognitive, desire-like state. The central question of this dissertation is this: "How can metaethical expressivists explain that we are able to express moral beliefs?" The shortest answer this dissertation will give is the following: "by becoming `dispositional expressivists´." Dispositional expressivists hold that moral beliefs are dispositions to be in certain mental states that resemble desires. More precisely, a central thesis of this study is that to hold a moral belief is to be disposed to be in a desire-like, noncognitive state where this state has some specific structure. This dissertation arrives at and justifies this thesis by making a number of steps. It starts with preparing the ground by developing a general theory of what it is for a sentence to express a mental state. Roughly, this theory states that for a sentence to express a mental state is for that sentence to be conventionally used to perform a speech act which counts as sincere only if the user or speaker is in that mental state. This theory of expressing satisfies all the constraints that must be satisfied for it to be compatible with the purposes of metaethical expressivism. The next step is to provide a solution to the most notorious semantic problem confronting metaethical expressivism, the problem of explaining how sentences containing moral words can have compositional meaning and stand in logical relations also known as the Frege-Geach or embedding problem. Building on the research of Mark Schroeder, a new expressivist semantics, called `Attitude Semantics´, will be developed for a first-order language. The motivating idea underlying attitude semantics is that expressivists can avoid the problematic feature of so-called `biforcation´ if they treat the structures of the mental states expressed by moral sentences as well as that expressed by non-moral sentences as being isomorphic. In order to give those structures a philosophically interesting interpretation, it will be suggested that we treat sentences as expressing dispositional states. As a toy example we will treat the sentence `murder is wrong´ as expressing the disposition to be for blaming for murder. After dealing with the Frege-Geach problem, I turn to the metaethical expressivists' project of quasi-realism. Two major goals of this project are to explain how moral sentence can be truth-apt and how the states expressed by them can be regarded as moral beliefs. In order to establish those goals I critically examine a number of theories of truth-aptness by providing counterexamples to them. I then present my own theory which says that a sentence is truth-apt iff and because there is a conventional way to use it to express a belief. What is new about this theory is that it treats expressing a belief as being explanatorily more basic than being truth-apt. Belief, according to this theory, explains truth-aptness, not vice versa. This has the important consequence that it shows that what I call the `orthodox strategy´ of quasi-realists is mistaken and needs to be inverted: expressivists must `earn the right´ to moral belief before they can establish that moral sentences are truth-apt, and not the other way round. The second part of this dissertation is then concerned with the question of what it is to hold a moral belief and how they can figure in rational reasoning. In short: it is concerned with earning the right to moral belief. After discussing the major problems confronting a noncognitivist theory of belief, most importantly what I call the `tightrope problem´, I argue for a dispositional form of noncognitivism about moral belief that combines elements from Eric Schwitzgebel's dispositionalism about belief and Sebastian Köhler's `conceptual role expressivism´. This dispositional noncognitivism perfectly matches and justifies the dispositionalist interpretation of the expressivist semantics developed before. In order to fully earn the right to moral belief, however, metaethical expressivists must also deal with several epistemological objections concerning the rationality of moral belief. The first objection I deal with comes from Derek Baker and is directed against Mark Schroeder's specific development of expressivism. I deal with this objection because my own theory bears close resemblance to Schroeder's theory and so it might be worried that it confronts the same issues. Fortunately, we will see that Baker's objection rest on a number of mistaken assumptions and so neither applies to Schroeder's expressivism nor to the version which I defend in this study. The other epistemological objection to metaethical expressivism I deal with is Cian Dorr's famous wishful-thinking objection. In contrast to previous replies to this objection I attack it by directly questioning the central assumption on which it rests, namely that reasoners can never be justified in basing a factual belief on a desire-like state. I argue that this is false, and that under certain conditions reasoning from desire-like mental states to factual beliefs is justified. My argument shows that even if holding a moral belief is being in a desire-like noncognitive state, basing factual belief on a moral belief can give one an epistemic reason for the factual belief. Hence, even if moral thoughts are wishes, moral thinking is not wishful thinking. I take my arguments to earn the dispositional expressivist the right to moral belief. By the previously defended theory of truth-aptness in terms of belief, this allows the dispositional expressivists to conclude that moral sentences are also truth-apt. This finishes the quasi-realist project of this dissertation. In the last part of this dissertation I switch gears and present two unnoted problems for metaethical expressivists having to do with explaining moral motivation. Both arguments are surprisingly simple. The first one is this: if motivation requires a belief as well as a desire (Humeanism), and moral judgments motivate by themselves (internalism), then moral judgments cannot be desires only. If this argument is sound, it shows that the famous `motivation argument´ not only entails the falsity of cognitivism, but also the falsity of noncognitivism. The second argument is the following: moral judgments when combined with suitable external desires motivate to act (externalism), but if motivation requires belief as well as desire (Humeanism), then moral judgments cannot be desires only. I discuss several ways in which noncognitivists might want to deal with those problems, but leave it as an open question whether those solutions will ultimately be viable. Overall, this dissertation makes serious contributions to the research program of metaethical expressivism. It makes proposals for how to solve a number of notorious problems, such as the Frege-Geach problem in semantics and the wishful-thinking problem in epistemology. Besides this, however, it also raises two new problems for metaethical expressivism in the philosophy of action. For those reasons this dissertation will be of interest for anyone who wants to defend or criticize expressivism in metaethics, and more generally, anyone who wants to better understand our astonishing ability to express moral beliefs. (shrink)

A focussed issue of The Reasoner on the topic of 'Infinitary Reasoning'. Owen Griffiths and A.C. Paseau were the guest editors.

This short article discusses the fact that the word ‘ancestor’ features in certain arguments that a) are apparently logically valid, b) contain infinitely many premises, and c) are such that none of their finite sub-arguments are logically valid. The article's aim is to motivate, within its brief compass, the study of infinitary logics.

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules for (...) the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

In Metaphysics Z 8 Aristotle offers an infinite regress argument to deny that forms come to be. Briefly put, the argument states that, if we assume that every time an x composed of matter (m1) and form (f1) comes to be, f1 also comes to be, then there would be infinitely many xs coming to be – for f1 would itself be a compound, if it comes to be, and the same reasoning would in turn apply to it. This argument (...) has great significance in the history of philosophy, for some later thinkers take Aristotle to allow or make room for the creation ex nihilo of forms. However, the only direct Greek engagement with this argument in its proper context is provided by Asclepius of Tralles’ commentary on Book Z. A close study of Asclepius’ interpretation shows that he endorsed the argument on the tacit condition that it refers to the forms αὐτὰ καθ’ αὑτά as different from both Platonic Ideas and enmattered forms. At the same time, however, its elaboration includes alongside forms αὐτὰ καθ’ αὑτά also eternal separate forms and generated forms. His reconstruction also displays a sophisticated toning-down of the potential disagreements between Plato and Aristotle on the conception of pre-existing form which is assumed in the argument. Asclepius’ interpretation offers yet further proof of the commentators’ camouflaged and stratified way of tackling philosophical issues of paramount importance. (shrink)

Almost all existing genetic programming systems deal with fitness evaluation solely by testing. In this paper, by contrast, we present an original approach that combines genetic programming with Hoare logic with the aid of model checking and finite state automata, henceby proposing a brand new verification-focused formal genetic programming system that makes it possible to evolve reliable programs with mathematicallyverified properties.

The formal analysis of normative systems has traditionally focused on their deontic dimension rather than on their potestative dimension; yet, a growing amount of works aims at shedding light on the notion of power, its norm changing potential and its general interactions with deontic concepts. The present article contributes to this line of inquiry by adopting the following perspective: a normative system can be metaphorically seen as an agent that allocates abilities (powers) in order to promote the fulfillment of certain (...) desires (deontic directives), and in doing so regulates its behavioural domain. Our analysis emphasizes the instrumental nature of power, while clarifying the distinction between ‘being allowed’ and ‘being enabled’ and unveiling new patterns of interaction between deontic and potestative concepts. Operationally, we formulate this framework in terms of conditional rules, and provide a corresponding logic programming (ASP) implementation. (shrink)

Las máquinas aceleradas de Turing (ATMs) son dispositivos capaces de ejecutar súper-tareas. Sin embargo, el simple ejercicio de definirlas ha generado varias paradojas. En el presente artículo se definirán las nociones de súper-tarea y ATM de manera exhaustiva y se aclarará qué debe entenderse en un contexto lógico-formal cuando se pregunta por la existencia de un objeto. A partir de la distinción entre posibilidades lógicas y físicas se disolverán las paradojas y se concluirá que las ATMs son posibles y existen (...) como objetos abstractos. (shrink)

The purpose of this paper is to analyse and compare two concepts which tend to be treated as synonymous, and to show the difference between them: these are critical thinking and logical culture. Firstly, we try to show that these cannot be considered identical or strictly equivalent: i.e. that the concept of logical culture includes more than just critical thinking skills. Secondly, we try to show that Christian philosophers, when arguing about philosophical matters and teaching philosophy to students, should not (...) focus only on critical thinking skills, but rather also consider logical culture. This, as we argue, may help to improve debate both within and outside of Christian philosophy. (shrink)

This chapter introduces inferential role semantics (IRS) and some of the challenges it faces. It also introduces inferentialism and places it into the wider context of contemporary philosophy of language. The chapter focuses on what is standardly considered both the most important test case for and the most natural application of IRS: logical inferentialism, the view that the meanings of the logical expressions are fully determined by the basic rules for their correct use, and that to understand a logical expression (...) is to use it in accordance with the appropriate rules. It discusses some of the benefits of logical inferentialism, chiefly with regard to the epistemology of logic, and considers a number of objections. The chapter critically examines Robert Brandom's inferentialism about linguistic and conceptual content in general. Finally, it considers a number of general objections to IRS and possible responses on the inferentialist's behalf. (shrink)