Logical Expressions

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)

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)

I distinguish between instantial sentences (_There are elephants that swim_), particular quantification, and predication of existence in natural language. I explore the logical relation between the first two, while the last one is shown independent of either. I continue to consider the incorporation of the three kinds of sentence in the Quantified Argument Calculus (Quarc). I provide formalisations that preserve the logical relations specified earlier. I also extend the analysis to quantified instantial sentences (_There are_ five _elephants that swim_) and (...) modal ones. A conclusion of this work is that quantification and instantial sentences provide no guide to ontology. (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)

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.

Two printed works on logic by Georgius Benignus / Juraj Dragišić are preserved: Dialectica nova (Florentiae 1488 [1489]) and Artis dialecticaes praecepta (Romae 1520 [1519]). The presentations of Dragišić's logic by Carl Prantl and Stjepan Zimmermann are assessed. Dragišić’s 1488 logic contains a very early exposition of nearly all modes of the fourth figure. He listed four direct modes of the fourth figure, and he may also have had in mind the fifth mode (Fresison), along with an additional mode, probably (...) the indirect Fimeno. Furthermore, Dragišić presented a specific version of the terminist doctrine of supposition of terms (suppositio terminorum), incorporating his initial distinction between the mode and subject of supposition. Dragišić’s operational doctrine of consequences (consequentiae) reflects his interest in systematizing and reducing the number of rules (as seen in the differences between his two logical works). His detailed argumentation for the rejection of the principles ex impossibile and ad necessarium makes his work particularly interesting from the perspective of modern paraconsistent logics. (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)

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.

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)

Qualifiers such as “insofar as” and “in itself” have always been important ingredients in key philosophical claims. Descartes, for instance, famously argues that insofar as he is a thinker, he is not made of matter, and Kant equally famously argues that we cannot know things in themselves. Neither of these claims is meant to be true without qualification. Descartes is not simply denying that humans consist of matter, and Kant is not simply denying that we know things. Therefore, we cannot (...) even begin to understand such claims without knowing how qualifiers work. Unlike the logic of quantification, however, the logic of qualification is rather underexplored. In this paper, I examine several instances of philosophical uses of qualifiers, taken from Aristotle, Avicenna, Descartes, Kant, and 20th c. action theory. In the light of these examples, I discuss several accounts of how such qualifiers work. (shrink)

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)

Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic `of nonsense' introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic K3W by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) (...) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules. (shrink)

We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inferenceis, orconsists in. We arrive at a proposed solution that places a surprisingly heavy load on the prospect of being able to understand and deal with specifications of rules that are essentiallyself-referring. That is, any rule$\rho $is to be understood via a specification that involves, embedded within it, reference to rule$\rho $itself. Just how we arrive at this position is explained by (...) reference to familiar rules as well as less familiar ones with unusual features. An inquiry of this kind is surprisingly absent from the foundations of inferentialism—the view that meanings of expressions (especially logical ones) are to be characterized by the rules of inference that govern them. (shrink)

Stephen Read presented harmonious inference rules for identity in classical predicate logic. I demonstrate here how this approach can be generalised to a setting where predicate logic has been extended with epistemic modals. In such a setting, identity has two uses. A rigid one, where the identity of two referents is preserved under epistemic possibility, and a non-rigid one where two identical referents may differ under epistemic modality. I give rules for both uses. Formally, I extend Quantified Epistemic Multilateral Logic (...) with two identity signs. I argue that a uniform meaning for identity tout court can be given by adopting Maria Aloni’s account of reference using conceptual covers. We obtain a harmonious set of rules for identity that is sound and complete for Aloni’s model theory. (shrink)

According to the interpretational theory of logical validity (IR), logical validity is preservation of truth in all interpretations compatible with the intended meaning of logical expressions. IR suffers from a seemingly defeating objection, the so-called cardinality problem: any instance of the statement ‘There are n things’ is true under all interpretations, since it can be written down using only logical expressions that are not to be reinterpreted; yet ‘There are n things’ is not logically true. I argue that the cardinality (...) problem is indeed a serious problem for IR, when understood in terms of ‘asymmetry of information’. I then argue that IR can be rehabilitated by making quantifiers context-sensitive: what we do not reinterpret is the Kaplanian character of a quantifier, rather than its content. ‘There are n things’ is false in a context where fewer than n things are relevant, so it is not logically true in IR. I finally discuss some objections and ramifications of my account: I discuss how to make space for the possibility of an explicitly absolutely general quantifier in my framework, how terms can be logical even though context-sensitive, and how to recapture classical logic within my framework. (shrink)

Statements containing epistemic modals (e.g., “by spring 2023 most European countries may have the Covid-19 pandemic under control”) are common expressions of epistemic uncertainty. In this paper, previous published findings (Knobe & Yalcin, 2014; Khoo & Phillips, 2018) on the opposition between Contextualism and Relativism for epistemic modals are re-examined. It is found that these findings contain a substantial degree of individual variation. To investigate whether participants differ in their interpretation of epistemic modals, an experiment with multiple phases and sessions (...) is used to classify participants according to the three semantic theories of Relativism, Contextualism, and Objectivism. Through this study, some of the first empirical evidence for the kind of truth-value shifts postulated by semantic Relativism is presented. It is furthermore found that participants’ disagreement judgments match their truth evaluations and that participants are capable of distinguishing between truth and justification. In a second experimental session, it is investigated whether participants thus classified follow the norm of retraction which Relativism uses to account for argumentation with epistemic modals. Here the results are less favorable for Relativism. In a second experiment, these results are replicated and the normative beliefs of participants concerning the norm of retraction are investigated following work on measuring norms by Bicchieri (2017). Again, it is found that on average participants show no strong preferences concerning the norm of retraction for epistemic modals. Yet, it was found that participants who had committed to Objectivism and had training in logics applied the norm of retraction to might-statements. These results present a substantial challenge to the account of argumentation with epistemic modals presented in MacFarlane (2014), as discussed. Keywords: Epistemic Modals, Relativism, Norm Conflicts, Retraction, Semantics, Truth Conditions, Argumentation. (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)

A metainference is usually understood as a pair consisting of a collection of inferences, called premises, and a single inference, called conclusion. In the last few years, much attention has been paid to the study of metainferences—and, in particular, to the question of what are the valid metainferences of a given logic. So far, however, this study has been done in quite a poor language. Our usual sequent calculi have no way to represent, e.g. negations, disjunctions or conjunctions of inferences. (...) In this paper we tackle this expressive issue. We assume some background sentential language as given and define what we call an inferential language, that is, a language whose atomic formulas are inferences. We provide a model-theoretic characterization of validity for this language—relative to some given characterization of validity for the background sentential language—and provide a proof-theoretic analysis of validity. We argue that our novel language has fruitful philosophical applications. Lastly, we generalize some of our definitions and results to arbitrary metainferential levels. (shrink)

The content for an advanced logic course is presented, which includes the properties of first-order logic language, soundness and completeness of the first-order logic deductive system, Peano arithmetic, Gödel's incompleteness theorems, higher-order logic and its properties. As a reminder, a brief description of first-order logic is included.

The book contains an introduction to basic logical concepts and methods. It covers traditional logic of categorical judgment and syllogism, modern propositional logic, and introductory elements of predicate logic with corresponding methods (truth tables, natural deduction, truth trees).

This book is an introduction to elementary logic (classical propositional and first-order logic), comprising brief summaries of the basics of elementary logic, with the emphasis on typical questions and procedure descriptions and with a large number of corresponding exercises and problems. Solutions are given for each problem and exercise, often with commentaries. The first part, Basics of Logic, deals with (a) formal language, models, Venn diagrams for sentences, and translation from natural into formal language and vice versa, (b) deduction and (...) refutation methods in the format of natural deduction (Fitch and Suppes-Lemmon styles), tableau system and Venn diagrams, and (c) basic logical properties and relations (consistency, validity, consequence, equivalence, various types of opposition, logical completeness and independency, relationship between syntax and semantics). The second part, Logic in Understanding and Analysis, introduces into (a) logical approach to problem solving in concrete situations and events as presented in brief tales and anecdotes, (b) discovering the meaning, and reconstruction of logical interconnections, in ordinary, scholarly and philosophical texts, (c) application of already mastered logical tools in examining and defining further theoretical (logical, scientific) concepts and in building formal languages. The shorter third part, From the Traditional Logic, is dedicated to the traditional logic of concepts, judgements and syllogisms, largely with the use of Venn diagrams. (shrink)

It is shown, first, that Georgius Benignus [Juraj Dragišić, ca 1445-1520] at the end of the 15th century almost fully disposed with the theory of the fourth syllogistic figure. Prantl's view on Benignus' logic is critically examined. Besides, Benignus' doctrines on the supposition of terms and logical consequences are examined with respect to his two versions of logic (1488/1489 and 1519) and to the main influences (e.g., Strodus, Paulus Venetus, Ferebrich, Paulus Pergolensis). A reduction of the rules of consequences of (...) Benignus' logic 1488/1489 to the rules of logic 1519 is reconstructed. Benignus rejects the reasoning ex impossible quodlibet and e quolibet necessarium in what he defines as material consequences. (shrink)

It is shown in what sense essential characteristics of formalism, which is normative for exact knowledge, can be found already in Aristotle. It is described how exactness and formalism are pre-conditioned by sensible intuition. The machine character (Turing machine) of a formalism is considered. The general concept of provability leads to an expanding and dynamic understanding of a formalism, with the final source of logical patterns in the "forms of life" (Wittgenstein).

According to an influential view, when it comes to representing reality, some words are better suited for the job than others. This is elitism. There is reason to believe that the set of the best, or elite, words should not be redundant or arbitrary. However, we are often forced to choose between these two theoretical vices, especially in cases involving theories that seem to be mere notational variants. This is the riddle of redundancy: both redundancy and arbitrariness are vicious, but (...) there are cases in which one must be picked. Logical realists admit that there are some logical constants among the elite words. This leads to awkward questions, such as which among conjunction and disjunction is elite. In this paper, I show how the riddle of redundancy arises for logical realists, and offer a solution. This approach requires us to change how we represent negation. Instead of using some particular symbol, we represent negation by flipping formulae over the horizontal axis. (shrink)

This paper is a short introduction to Carnap’s writings on semantics with an emphasis on the transition from the syntactic period to the semantic one. I claim that one of Carnap’s main aims was to investigate the possibility of the symmetry between the syntactic and the semantic methods of approaching philosophical problems, both in logic and in the philosophy of science. This ideal of methodological symmetry could be described as an attempt to obtain categorical logical systems, i.e., systems that allow (...) only the intended semantical interpretation. (shrink)

I try to identify elements of our mental capacities that separate us from animals. I focus on our command of logical concepts, demonstrable already in children in the second or third year of their life, which to date no animal has been shown to master. I draw various conclusions about the behavioural, intellectual, emotional, and moral capacities that depend on this mastery, and discuss recent empirical research that either supports or apparently disagrees with the claim that animals, even those we (...) consider intelligent, are limited in these respects. My discussion is built around observations extracted from Wittgenstein. (shrink)

Habilitation thesis in psychology. The book consists of a collection of reasoning studies. The experimental investigations will take us from people’s reasoning about probabilities, entailments, pragmatic factors, argumentation, and causality to morality. An overarching theme of the book is norm pluralism and individual differences in rationality research.

This paper attempts to demonstrate that the logical problematic of signification, has a very dangerous socio-political effect due to the ontological implication that is connected to the signification of terms in logic. It expounds the notion of signification in Formal Logic as exposed by William of Ockham. It thus, employs this notion of signification of terms, to discuss the term “Fulani”, to show the danger potent in distorting the signification of the term “Fulani” as in every conventional and connotative terms. (...) This work claims that as a result of the connection between logic and ontology, conceptualization and perception, names and reality, the distortion of terms, which are the building blocks of propositions in logic could lead to a dangerous cognition and perception of ontological realities. To this end, it posits a critique on the dangerous change in the signification of the term “Fulani”, in the conceptualization and perception of present day Nigeria people, of West Africa. A dangerous change, as this paper claims, if not speedily reconstructed and re-conceptualized, could lead to a disastrous situation in the socio-political reality of the Nigerian State. (shrink)

We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an$\omega $-rule, and prove that the derivability problem in this calculus is$\Pi _1^0$-hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with (...) unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007). (shrink)

In one of their papers, Michael De and Hitoshi Omori observed that the notion of classical negation is not uniquely determined in the context of so-called Belnap-Dunn logic, and in fact there are 16 unary operations that qualify to be called classical negation. These varieties are due to different falsity conditions one may assume for classical negation. The aim of this paper is to observe that there is an interesting way to make sense of classical negation independent of falsity conditions. (...) We discuss two equivalent semantics, and offer a Hilbert-style system that is sound and complete with respect to the semantics. (shrink)

In a recent article, “Logical Consequence and Natural Language”, Michael Glanzberg claims that there is no relation of logical consequence in natural language (2015). The present paper counters that claim. I shall discuss Glanzberg’s arguments and show why they don’t hold. I further show how Glanzberg’s claims may be used to rather support the existence of logical consequence in natural language.

This dissertation is about the structure of thought. Following Gottlob Frege, I define a thought as the sort of content relevant to determining whether an assertion is true or false. The historical component of the dissertation involves interpreting Frege’s actual views on the structure of thought. I argue that Frege did not think that a thought has a unique decomposition into its component senses, but rather the same thought can be decomposed into senses in a variety of distinct ways. I (...) extend Frege’s position and use it to develop an account of the hierarchy of senses, the senses expressed by indexicals and demonstratives, and the distinction between logical and non-logical structure. I also discuss various connections with the nature of meta-representation, our capacity for reflective judgment, some aspects of the structure of conscious experience, the way we perceive regions of space and durations of time, and our conscious awareness of our own perceptions and events of thinking. (shrink)

In his Formal Logic, Arthur N. Prior declared that Jan Łukasiewicz's logical notation is ‘unquestionably the best logical symbolism for most purposes’. Whether he had a substantive, and...

This paper aims to reassess a notion in the works of the later Husserl that is both historically important and philosophically insightful, but remains understudied, namely, that of type. In opposition to a standard reading which treats Husserl’s type presentations as pre-conceptual habits, this paper argues that these representations are a specific kind of concept. More precisely, it shows that Husserl’s account of type presentations is akin to the contemporary prototype theory of concepts. This is historically important, since the predecessor (...) of the prototype theory is usually said to be Wittgenstein. From a philosophical standpoint, the paper shows that Husserl has an innovative account of the connection between type concepts and their extension. Contrary to the standard view of extensions as sets and thus sharp entities, Husserl develops a correlationalist theory of concepts, according to which, for the specific characteristics in the structure of a concept, there are corresponding characteristics in the arrangement of its members, and vice versa. According to this theory, while sharp concepts lead to sharp extensions, vague concepts such as (proto)type concepts lead to vague extensions. The paper presents this understanding of Husserl in detail and explains its philosophical significance. (shrink)

We present a generalization of the algebra-valued models of \ where the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get intermediate values, in a set of designated values. Under this generalization there are many algebras which are neither Boolean, nor Heyting, but that still validate \.

An exclusive disjunction is true when exactly one of the disjuncts is true. In the case of the familiar binary exclusive disjunction, we have a formula occurring as the first disjunct and a formula occurring as the second disjunct, so, if what we have is two formula-tokens of the same formula-type—one formula occurring twice over, that is—the question arises as to whether, when that formula is true, to count the case as one in which exactly one of the disjuncts is (...) true, counting by type, or as a case in which two disjuncts are true, counting by token. The latter is the standard answer: counting by tokens. James McCawley once suggested that, when the exclusively disjunctive construction in natural language is at issue, the construction should be treated as involving a multigrade connective whose semantic treatment is sensitive to the set of disjuncts rather than the corresponding multiset. Without any commitment as to whether there actually is such a construction, and conceding that for obvious pragmatic reasons such ‘repeated disjunct’ cases would be at best highly marginal, we note that for the binary case, this requires a nonstandard answer—count by type rather than by token—to the earlier question, and thus, an idempotent exclusive disjunction connective. Section 2 explores that idea and Section 3, a further idempotent variant for which it is the propositions expressed by the disjuncts, rather than the disjuncts themselves, that get counted once only in the case of repetitions. Sections 1 and 4 respectively set the stage for these investigations and conclude the discussion. More detailed considerations of points arising from the discussion but otherwise in danger of interrupting the flow are deferred to a ‘Longer Notes’ appendix at the end. (shrink)

In this article, we present a modal logic that extends the basic modal logic with two dynamic operators: copy ( ), which replicates the current model, labelling each copy with a different propositional symbol and respecting accessibility relations even between distinct copies; and remove ( ), which deletes paths in the model that satisfy certain intermediate conditions. We call the resulting logic. We study its computational complexity, and its relative expressivity with respect to (static) modal logics and, and the dynamic (...) epistemic Action Model Logic,. (shrink)

There has been a recent interest in hierarchical generalizations of classic incompleteness results. This paper provides evidence that such generalizations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “$\mathrm {T}$is$\Sigma _n$-ill” (...) is a canonical example of a$\Sigma _{n+1}$formula that is$\Pi _{n+1}$-conservative over$\mathrm {T}$. (shrink)

In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same role as the compound modality ∀pA. Unlike the propositional quantifier by itself, the global quantificational modality can be straightforwardly interpreted in any Boolean Algebra Expansion (BAE). We present a logic GQM for this language and prove that it is complete with respect to (...) the intended algebraic semantics. This logic enables a conceptual shift, as what have traditionally been called different “modal logics” now become [∀p]-universal theories over the base logic GQM: instead of defining a new logic with an axiom schema such as □φ→□□φ, one reasons in GQM about what follows from the globally quantified formula [∀p](□p→□□p). (shrink)

The paper studies the containment companion of a logic \. This consists of the consequence relation \ which satisfies all the inferences of \, where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. In accordance with the work started in [10], we show that a different generalization of the Płonka sum construction, adapted from algebras to logical matrices, allows to provide a matrix-based semantics for containment logics. In particular, (...) we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization. (shrink)

L.L. Maksimova and L. Esakia, V. Meskhi showed that the modal logic \ has exactly 5 pretabular extensions PM1–PM5. In this paper, we study the problem of unification for all given logics. We showed that PM2 and PM3 have finitary, and PM1, PM4, PM5 have unitary types of unification. Complete sets of unifiers in logics are described.

The aim of this article is to study the notion of derivability and its semantic counterpart in the context of non-transitive and non-reflexive substructural logics. For this purpose we focus on the study cases of the logics _S__T_ and _T__S_. In this respect, we show that this notion doesn’t coincide, in general, with a nowadays broadly used semantic approach towards metainferential validity: the notion of local validity. Following this, and building on some previous work by Humberstone, we prove that in (...) these systems derivability can be characterized in terms of a notion we call absolute global validity. However, arriving at these results doesn’t lead us to disregard local validity. First, because we discuss the conditions under which local, and also global validity, can be expected to coincide with derivability. Secondly, because we show how taking into account certain families of valuations can be useful to describe derivability for different calculi used to present _S__T_ and _T__S_. (shrink)

We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal$M_1$, there is a model of$\mathsf {ZFC}$in which$\mbox {NS}_{\omega _{1}}$is$\aleph _2$-saturated and${\Delta }_{1}$-definable with$\omega _1$as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with$\mbox {NS}_{\omega _{1}}$saturated and${\Delta }_{1}$-definable with a ladder system$\vec {C}$and a full Suslin treeTas parameters. Both results rely on (...) a new coding technique whose presentation is the main goal of this article. (shrink)

According to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that (...) propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise. (shrink)

This sentence G ↔ ¬(F ⊢ G) and its negation G ↔ ~(F ⊢ ¬G) are shown to meet the conventional definition of incompleteness: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)). They meet conventional definition of incompleteness because neither the sentence nor its negation is provable in F (or any other formal system). -- .

Peano was one of the driving forces behind the development of the current mathematical formalism. In this paper, we study his particular approach to notational design and present some original features of his notations. To explain the motivations underlying Peano's approach, we first present his view of logic as a method of analysis and his desire for a rigorous and concise symbolism to represent mathematical ideas. On the basis of both his practice and his explicit reflections on notations, we discuss (...) the principles that guided Peano's introduction of new symbols, the choice of characters, and the layout of formulas. Finally, we take a closer look, from a systematic and historical perspective, at one of Peano's most striking innovations, his use of dots for the grouping of subformulas. (shrink)

This work will focus on Józef Maria Bocheński’s inclination towards seeing the world and its logical structure from the point of view of ontology. In section 2, we shall discuss the perception of the world deriving from Bocheński, while in the third section – issues of its logical structure will be dealt with. In section 4, we will present a formal framework of the structure of the world.