The study by Wilfredo Quezada Pulido and Carlo Apablaza Ávila offers a comprehensive reconstruction of Rolando Chuaqui Kettlun’s career, articulating his intellectual biography, philosophical work, and interdisciplinary contributions. It highlights his role in the development of logic and mathematics in Chile and clarifies his formative and collaborative ties with Alfred Tarski and Newton da Costa.
In contrast to Kant’s metatheoretical (“transcendental”) approach grounded in the operations of an abstract logical subject, Herbart and Bolzano aimed to avoid subject-dependent foundations of logic. They instead proposed formal logics founded, respectively, on the form of conceptual content (Herbart) and on the extensions of logical forms (Bolzano). Among the traditions originating from Herbart and Bolzano, we consider the role of Bolzano’s informal disciple Robert Zimmermann, who subsequently adopted Herbartianism, and Zimmermann’s own disciple, the mathematical logician and philosopher Albino Nagy. Special attention is given to Nagy’s analysis of expressive completeness and his treatment of the Jevons-Clifford problem: how many and which types and representatives of types there are of Boolean propositional functions. We transform this latter problem into the question of types of sequent refutations and examine the corresponding admissibility of the cut rule.
This paper aims to re-examine the so-called Ja & sacute;kowski's criterion. This guiding principle was initially proposed as a methodological instrument for delineating formal systems as either satisfactory or unsatisfactory paraconsistent logics. It is a historical fact that Ja & sacute;kowski's criterion has been, and remains, frequently overlooked in the relevant literature. In order to provide a detailed explanation of this fact and the argument for it being a regrettable state, there are three main points elaborated upon. Firstly, the philosophical motivations of Ja & sacute;kowski's work. Secondly, an overview of some early attempts of meeting requirements of Ja & sacute;kowski's criterion. Thirdly, its relevance to contemporary research in the field of philosophical logic, particularly in relation to recent discussions on the notion of 'philosophical interpretation'. From the above, philosophical conclusions are drawn, and avenues for further research are identified.
We present a new axiomatization for a truth-functionally complete version of & Lstrok;3, a three-valued propositional logic, using two propositional constants and the & Lstrok;ukasiewicz implication as primitive symbols. We develop a corresponding proof system that incorporates & Lstrok;ukasiewicz's axioms, a variant of S & lstrok;upecki's postulates for a 0-ary connective, and some properties of the Baaz delta operation. Using a weakened Deduction Theorem and adapting Henkin's method to this three-valued setting, we establish a Completeness Theorem for the system.
Glidel's ontological proof for the existence of God - as an inference from the level of rational, logical structure to the level of being was strongly influenced by Leibniz's monadology, which is based on the Ur-Monade - God - conceived as absolutely infinite. Glidel defines God as the maximum of positive properties and he characterizes this maximum (of being) with his postulates implicitly as an ultrafilter structure. Using maximality as criterion we have compared the set of positive properties in Scott's version of Glidel's theistic theory with modified variants of it to see whether a maximum is still achieved, or not, and what effect this has on modal collapse. The motivation for these, typically weakened, variants of Glidel's theistic theory has been to avoid modal collapse and the limited determinism associated with it, and to show that the necessary existence of God can still be proved with less strong axioms. A weakening of the maximality criterion of the ultrafilter structure imposed by Glidel's (and Scott's) postulates have thus been applied to the set of positive properties in order to allow for alternative models in which modal collapse does not hold. In these alternative models, however, the maximality criterion for the basic quantity of positive properties is abandoned, so that in them God as a "maximum" is not reached. We argue that this is contrary to Glidel's intention, and that for his conception of God, maximality and absolute infinity are non-negotiable. What Glidel had in mind, as we argue here, was a holistic - absolutely infinite - view of the world, i.e. with modal collapse, but without limited determinism.
Lakatos's (1978) methodology of research programs is renowned for describing the evolution of scientific theories through a combination of heuristic mechanisms that guide their development. Various attempts to apply Lakatos's framework to research programs in logic and mathematics are found in (Priest, 1989; Aberdein and Read, 2009; Hallett, 1979a). The paper draws on Lakatosian assessments of progressive and degenerating mathematical theories found in (Hallett, 1979a) to advance a heuristic proposal for studying the progress and development of logical theories. The upshot is a characterization of Lakatos's framework as applied to logical theories, along with a defense of a holistic approach that acknowledges the iterative relationship between empirical input and logical refinement during the process of development of progressive logical theories.
This paper investigates the relationship between two semantics for Edward Zalta's Elementary Object Theory (OT): one proposed by Dana Scott and the other by Peter Aczel. We present some philosophical motivations underlying OT, characterize its second-order monadic fragment (MOT), and prove some of its theses. We define Scott and Aczel structures and establish a soundness theorem for MOT with respect to the latter. We indicate a class of Aczel structures in which a given formula is true iff it is true in all Scott structures. We also investigate two formulas: one concerning the extensionality of the identity of properties and another related to the overloading of extensions containing abstracta, meaning that if one abstract object exemplifies a property, then all abstract objects do.
In contrast to Kant's metatheoretical ("transcendental") approach grounded in the operations of an abstract logical subject, Herbart and Bolzano aimed to avoid subject-dependent foundations of logic. They instead proposed formal logics founded, respectively, on the form of conceptual content (Herbart) and on the extensions of logical forms (Bolzano). Among the traditions originating from Herbart and Bolzano, we consider the role of Bolzano's informal disciple Robert Zimmermann, who subsequently adopted Herbartianism, and Zimmermann's own disciple, the mathematical logician and philosopher Albino Nagy. Special attention is given to Nagy's analysis of expressive completeness and his treatment of the Jevons-Clifford problem: how many and which types and representatives of types there are of Boolean propositional functions. We transform this latter problem into the question of types of sequent refutations and examine the corresponding admissibility of the cut rule.
Higher-order likes and desires sometimes lead agents to have ungrounded or paradoxical preferences. This situation is particularly problematic in the context of games. If payoffs are interdependent, the overall assessment of particular courses of action becomes ungrounded; in such cases, the game's matrix is radically underdetermined. Paradigmatic examples of this phenomenon occur when players are 'perfect lovers' or 'perfect haters', in a sense to be explained. In this paper, I use a dynamic doxastic logic to mimic the search for a suitable matrix. Upgrades are triggered by conjectures about other players' utilities, which can, in turn, be based on behavioural or verbal cues. We can prove that, under certain conditions, pairs of agents with paradoxical preferences eventually come to believe they cannot interact in a game. As a result, I hope to provide a better understanding of game-theoretic ungroundedness and, more generally, of the structure of higher-order preferences and desires.
In this paper, I discuss the legitimacy of the cluster of logics that are considered to be relating logics. I argue that even if the cluster rRelating logics,1 satisfies the basic theoretical criteria for legitimate clusters, logicians and philosophers of logic have failed at providing the corresponding pragmatic backing. In response to this, I propose to provide the pragmatic justification of the cluster by showing a domain of application for which rRelating logics, fits more adequately the evidence than any of their rivals do; such a domain is the phenomenon of scientific understanding. Finally, I argue that characterizing scientific understanding as a relating phenomenon provides a more accurate description than those offered by rival views.
What does a Cretan mean when he says that all Cretans are liars? What is his intention? While formal logic only relates to the truth values of the Liar paradox, we relate to its normative and social aspects. We argue that such utterances are used to imply that certain behaviors, even if despicable, constitute local norms. One may posit such claims either to point out that he has transcended the local culture, to socialize others into local customs, or to deflect from being caught lying. This paradox exemplifies group self-deprecation, a communicative practice intended to get us to disagree, rather than agree, with the disparaging claim and blunt the negative consequences of poor behavior. Its rhetoric relies upon tu quoque, secundum quid and naturalistic fallacies.
In this paper we show how to extend the standard cut-elimination procedure from first-order intuitionistic stable logic to a class of intuitionistic stable theories. Building on previous works by Negri and von Plato, we aptly modify the underlying calculus for first-order intuitionistic logic so as to preserve the admissibility of all the structural rules, including cut, in the presence of a restricted version of the rule of classical reductio ad absurdum and of a special case of universal rules.
The Doxastic Arrogance Paradox, DAP, states that the belief that a proposition is an item of knowledge implies that the proposition holds/is true. Some possible sources of DAP, different from the negative introspection principle for knowledge, are pointed out. The emplacement of DAP among related paradoxes of epistemic-doxastic logics is characterized. Finally, some profits of identification of sources of DAP for the philosophical analysis of knowledge and belief are pointed out.
We present a construction of nondeterministic semantics for some deontic logics based on the class of paraconsistent logics known as Logics of Formal Inconsistency (LFIs), for the first time combining swap structures and Kripke models through the novel notion of swap Kripe models. We start by making use of Nmatrices to characterize systems based on LFIs that do not satisfy axiom (cl), while turning to RNmatrices when the latter is considered in the underlying LFIs. This paper also presents, for the first time, a full axiomatization and a semantics for the C^D_n hierarchy, by use of the aforementioned mixed semantics with RNmatrices. This includes the historical system C^D_1 of da Costa-Carnielli (1986), the first deontic paraconsistent system proposed in the literature.
This paper investigates how the use of different rules for making inferences affects our understanding of what certain Extensible Markup Language (XML) documents do not represent. The aim is to show that we can infer different, contrasting things from the same XML documents, thereby weakening the communication that XML is supposed to support. There are three main reasons why the paper focuses on XML. First, XML, as a metalanguage, has no inherent rules for making inferences, but it also has no constraints on the technologies, systems, or theories that support or define the rules that can be used in conjunction with it. Second, XML is still widely used, and there are many other markup languages based on XML. This means that the critical analysis of these pages is, in principle, extendable to contexts where XML is involved and/or the rules for making inferences are not inherently supported. Third, since XML is explicitly intended to support communication between people, between software applications, and between people and software applications, this analysis may also shed new light on some of the theoretical assumptions behind such communication.
The point of this paper is to show that deductive arguments may not be executed in deductive inferences, and reciprocally, deductive inferences, if they existed, may not instantiate deductive arguments. Three reasons for making this distinction between instances and executions of logical arguments are offered. For example, the pragmatic expectation of the consequent may instantiate a Modus Ponens argument form, but it does not execute a logical deduction. The proposed distinction clarifies the current state of the art on factual logically deductive inferences in two opposed directions: first, exemplifying a deductive argument is not enough for an inference to be deductive, and second, the same energy and time-consuming process may both execute and instantiate a deductive argument. Future experiments may show or discard the existence of recursive and semi-recursive processes being deductive inferences, that is, being at the same time instance and execution of a deductive argument form.
A translation of Priest’s paraconsistent logic daC into many-sorted logic is presented. Besides, following the project of (Manzano, 1996), the representation theorem, the main theorem and the calculi equivalence are proved. So, it is demonstrated that the formulated translation preserves the set of valid formulas, the consequence relation, and the derivation relation of daC. Furthermore, the compactness and Löwenheim-Skolem theorems are proved for this logic. Alternative proofs for the soundness and completeness theorems for daC based on the translation are also presented.
The paper "Tautology elimination, cut elimination, and S5" published in this journal presents a novel method for establishing by proof analysis the admissibility of the rule of tautology elimination for certain sequent calculi. Since tautology elimination will typically imply the admissibility of cut, the method promises a new path to show the admissibility of cut for cut-free calculi on which the standard techniques within structural proof theory seem inapplicable. This paper shows that the method as presented involves an error.
In this paper, we analyze Fitch's paradox of knowability in the framework of fusions of epistemic and alethic modal logics. The paradox arises from accepting the knowability principle, which states that all truths are knowable. However, this leads to the unacceptable conclusion that all truths are known. We introduce a logical system that incorporates all assumptions used by Fitch in his original reasoning, including the knowability principle. We present a natural semantics for this logic, proving the soundness and completeness theorem. Additionally, we present a new semantic proof of the knowability paradox, demonstrating that the problematic conclusion can be derived independently of Fitch's original proof and showing that the knowability principle itself is the source of the paradox. Using the formal tools introduced, we conduct a semantic analysis of the paradox, which allows us to identify the root cause of its occurrence. Finally, we propose a weakened version of the knowability principle that avoids paradoxical conclusions.
An expression occurs essentially in a formula (or sentence) when it occurs in every formula equivalent to the given formula, taking equivalence as logical equivalence relative to the logic in play in the discussion. Setting aside various niceties, this amounts to provable equivalence if that logic is presented via some proof system, and to valid equivalence if the salient characterization is couched in semantic terms. This notion of essential occurrence, or an informal analog thereof, has found its way into numerous philosophical discussions over the past seventy or more years, and here we tease out some issues of specifically logical interest it presents, stretching that description somewhat so as to subsume under it the frequently mooted connection between the essential occurrence of a singular term in a sentence and that sentence’s being genuinely about what the term denotes. This connection, stressed originally by Nelson Goodman, is touched on in several sections in the main body of the paper, but especially in §4, where it is contrasted with an alternative suggestion due to R. Demolombe and L. Fariñas del Cerro. Some issues raised by this and other parts of the discussion are also treated in several longer notes (referred to by means of letters A, B, . . . , K) which are postponed to an Appendix (§5) of roughly the same length as the main body of the paper. This enables readers with a special interest in one or more topics to consult them selectively, while allowing those with no such interest to avoid involvement with the further details supplied in the associated longer note(s).
