Nonclassical Logics

We investigate an approach for drawing logical inference from inconsistent premisses. The main idea in this approach is that the inconsistencies in the premisses should be interpreted as uncertainty of the information. We propose a mechanism, based on Kinght’s [14] study of inconsistency, for revising an inconsistent set of premisses to a minimally uncertain, probabilistically consistent one. We will then generalise the probabilistic entailment relation introduced in [15] for propositional languages to the first order case to draw logical inference from (...) a probabilistic set of premisses. We will show how this combination can allow us to limit the effect of uncertainty introduced by inconsistent premisses to only the reasoning on the part of the premise set that is relevant to the inconsistency. (shrink)

An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion of (...) entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results. (shrink)

The Quantified Argument Calculus (Quarc) is a formal logic system, first developed by Hanoch Ben-Yami in (Ben-Yami 2014), and since then extended and applied by several authors. The aim of this paper is to further these contributions by, first, providing a philosophical motivation for the truth-valuational, substitutional approach of (Ben-Yami 2014) and defending it against a common objection, a topic also of interest beyond its specific application to Quarc. Second, we fill the formal lacunae left in the original presentation, which (...) did not incorporate identity systematically into Quarc, and although it proved the soundness of the system did not prove its completeness. (shrink)

Classical theists hold that God is omnipotent. But now suppose a critical atheologian were to ask: Can God create a stone so heavy that even he cannot lift it? This is the dilemma of the stone paradox. God either can or cannot create such a stone. Suppose that God can create it. Then there's something he cannot do – namely, lift the stone. Suppose that God cannot create the stone. Then, again, there's something he cannot do – namely, create it. (...) Either way, God cannot be omnipotent. Among the variety of known theological paradoxes, the paradox of the stone is especially troubling because of its logical purity. It purports to show that one cannot believe in both God and the laws of logic. In the face of the stone paradox, how should the contemporary analytic theist respond? Ought they to revise their belief in theology or their belief in logic? Ought they to lose their religion or lose their mind? (shrink)

Can God create an unliftable stone? Beall & Cotnoir have offered a new answer, suggesting that 'God can create an unliftable stone' is a truth-value gap--neither true nor false. I argue that their solution is susceptible to a revenge paradox, concerning whether God has the power to render the sentence non-gappy. More exactly, the revenge paradox concerns whether God has the power to realize the attribute "having the power to create an unliftable stone or not." Assuming a being with all (...) powers can realize that "dilemma attribute," then in brief, it still follows that there is some possible task which God cannot perform. In reply, B&C might suggest that the revenge paradox merely shows another truth-value gap--effectively, God neither has nor lacks the power to realize the dilemma attribute. But since I have the power to realize the attribute, God must as well. Beall & Cotnoir might respond that God does not have such a power since, per Aquinas, some powers are contrary to God's nature. Yet such an Aquinian suggestion is sufficient to block the original paradox, in which case, there is little motivation to proffer truth-value gaps in addition. I conclude that Beall & Cotnoir do not provide a well-motivated solution to the paradox. (shrink)

In natural language, conditionals are frequently used for giving explanations. Thus the antecedent of a conditional is typically understood as being connected to, being relevant for, or providing evidential support for the conditional's consequent. This aspect has not been adequately mirrored by the logics that are usually offered for the reasoning with conditionals: neither in the logic of the material conditional or the strict conditional, nor in the plethora of logics for suppositional conditionals that have been produced over the past (...) 50 years. In this paper I survey some recent attempts to come to terms with the problem of encoding evidential support or relevance in the logic of conditionals. I present models in a qualitative-modal and in a quantitative-probabilistic setting. Focussing on some particular examples, I show that no perfect match between the two kinds of settings has been achieved yet. (shrink)

We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to analyse certain family of affine logics with inclusion and convexity as primitives interpreted over real spaces of increasing dimensionality. In this article we show that logics of different dimensionalities must have different theories, thus justifying further work on different dimensions. (...) We then focus on the three-dimensional case, exploring the expressiveness of this logic and consequently showing that it is possible to construct formulas describing a three-dimensional coordinate frame. The final result, making use of the high expressive power of this logic, is that every region satisfies an affine complete formula, meaning that all regions satisfying it are affine equivalent. (shrink)

In the book, the decisive, foundational role of transcendental apperception for logic and transcendental philosophy in Kant is corroborated. The book contains many implicit connections with modern logic that could help a logician with a philosophical interest to gain a deeper insight into the origins and foundations of concepts such as object, truth, analyticity, identity, contradiction, judgment, existence, reference, quantification and others, as well as into the foundations and possible general features of logic. In distinction to the book, this review (...) emphasizes the objective validity of analytic judgements and the foundational role of hypthetical and disjunctive forms of a judgment in Kant. (shrink)

We introduce Arbitrary Public Announcement Logic with Memory (APALM), obtained by adding to the models a ‘memory’ of the initial states, representing the information before any communication took place (“the prior”), and adding to the syntax operators that can access this memory. We show that APALM is recursively axiomatizable (in contrast to the original Arbitrary Public Announcement Logic, for which the corresponding question is still open). We present a complete recursive axiomatization, that includes a natural finitary rule, and study this (...) logic’s expressivity and the appropriate notion of bisimulation. We then examine Group Announcement Logic with Memory (GALM), the extension of APALM obtained by adding to its syntax group announcement operators, and provide a complete finitary axiomatization (again in contrast to the original Group Announcement Logic, for which the only known axiomatization is infinitary). We also show that, in the memory-enhanced context, there is a natural reduction of the so-called coalition announcement modality to group announcements (in contrast to the memory-free case, where this natural translation was shown to be invalid). (shrink)

In this paper we study the interaction between symmetric logic and probability. In particular, we axiomatize the convex hull of the set of evaluations of symmetric logic, yielding the notion of probability in symmetric logic. This answers an open problem of Williams ( 2016 ) and Paris ( 2001 ).

Some think that logic concerns the “laws of truth”; others that logic concerns the “laws of thought.” This paper presents a way to reconcile both views by building a bridge between truth-maker theory, à la Fine, and normative bilateralism, à la Restall and Ripley. The paper suggests a novel way of understanding consequence in truth-maker theory and shows that this allows us to identify a common structure shared by truth-maker theory and normative bilateralism. We can thus transfer ideas from normative (...) bilateralism to truth-maker theory, such as non-transitive solutions to paradox, and vice versa, such as notions of factual equivalence and containment. (shrink)

The logical formalism of abductive reasoning is still an open discussion and various theories have been presented about it. Abduction is a type of non-monotonic and defeasible reasonings, and the logic containing such a reasoning is one of the types of non-nonmonotonic and defeasible logics, such as inductive logic. Abduction is a kind of natural reasoning and it is a solution to the problems having this form "the phenomenon of φ cannot be explained by the theory of Θ" and we (...) need to search for (or to adduce) an explanation for φ. Taking α as the explanation and <Θ,φ> as the abductive problem, Θ∪{α}⊨φ is the logical format of a strong abductive reasoning. In this article, the proof theory for first-order abductive calculus is proposed based on Gentzen-type first-order proof theory. Considering its dual, i.e., the semantic tableaux, and without the need to translate the formulas, this calculation is sound and complete. After examining the structural rules of this calculus, some metalogical considerations and open problems such as consistency, completeness and undecidability of this account have been addressed. (shrink)

By introducing and extending description logic (DLs) and growing up their application in knowledge representation and especially in OWLs and semantic web scope, many shortcomings and bugs were identified that weren’t resolvable in classical DLs and so logicians and computer scientists intended to non-classical and non-monotonic reasoning tools. In this paper, I discuses about abduction problem solvers, and by introducing A-Box abduction in description logics (DLs), such as ALC, discuss about decidability and complexity in different introduced algorithms, and report shortly (...) our team’s advantages in developing and implementation of ΑBox abduction in DLs. (shrink)

One of the main problems in t-norm fuzzy logic's meta-theorems is that despite the strong completeness of BL’s extensions such as Łukasiewicz (Ł), Gödel (G) and Product (Π) logics (i.e., Multi-valued, Gödel and Product standard algebras on [0,1] interval) in the propositional approach, in the first-order approach, given their standard chains and corresponding algebras, they aren't complete and strongly complete. One solution to this problem is that the first-order approaches of different fuzzy logics are complete and even strongly complete with (...) respect to non-standard single chains. But despite the success of this method in proving the strong completeness of many fuzzy logics such as TM, NM, BL, SBL, Ł and their first-order extensions, G and its first-order extensions, Π and its first-order extensions, the n-contraction logics SBLn, and Every finite valued extension of BL (such as finite valued Łukasiewicz (Łn) and finite valued Gödel (Gn)), there are three open problems: (1) Are MTL, IMTL, PMTL, WNM and their first-order extensions, (strongly) complete w.r.t. a single chain? (2) Although SMTL and SBL are strongly chain complete, Are SMTL∀ and SBL∀ also strongly completeness w.r.t. a single chain? And, (3) does chain completeness entail strong chain completeness or not? Answering these open problems and proving the completeness or incompleteness of these logics is the main purpose of this study, which will be achieved by some algebraic and meta-logical strategies. (shrink)

Spiritual Intelligence (S-Intelligence) introduced to achieve higher levels of knowledge and applying hidden knowledge. This intelligence is various in different people. When people work together in a team or set, these diverse influence on their performance. Facilitate the application of spiritual capacities by using abilities of S- intelligence is important to increase productivity teamwork. For this purpose, we provided a model of communication for individuals with different S-intelligence in performing team tasks. First, with the basics S-intelligence analysis and interviews with (...) experts, S-intelligence was divided in five categories. Each person according to their S-Intelligence and the role that play in teamwork (individualist, thinkist and pragmatist) from this aspect that he/she can by what kind of S-intelligences and what kind of role, has an effective team relationship, has been measured. With the 690 questionnaires … . (shrink)

Since McCall (1966), the heterodox principle of propositional logic that it is impossible for a proposition to be entailed by its own negation—in symbols, ¬(¬φ→φ)—has gone by the name of Aristotle’s thesis, since Aristotle apparently endorses it in Prior Analytics 2.4, 57b3–14. Scholars have contested whether Aristotle did endorse his eponymous thesis, whether he could do so consistently, and for what purpose he endorsed it if he did. In this article, I reconstruct Aristotle’s argument from this passage and show that (...) he accepts this thesis. Further, I show that the argument he gives is, making plausible assumptions, a correct proof in a consistent fragmentary nonclassical metalogic for a metatheorem he previously states concerning his assertoric syllogistic. In this way, Aristotle’s argument emerges as a fascinating case study in the use of a nonclassical metalogic to prove a result about a nonclassical object system. (shrink)

La introducción del grado de dependencia (y en consecuencia el grado de independencia) entre los componentes del conjunto difuso, y también entre los componentes del conjunto neutrosófico, se introduce por primera vez en la quinta edición del libro de Neutrosofía en el año 2006, basado en los elementos descritos en dicha edición del libro, se comienza a conocer conceptos de conjuntos neutrosóficos de los componentes borrosos así como los grados de dependencia e independencia, Por tal motivo el objetivo del presente (...) trabajo es extender el conjunto neutrosófico refinado, teniendo en cuenta la grado de dependencia o independencia de los subcomponentes que integran los conjuntos borrosos y neutrosóficos. (shrink)

We have introduced for the first time the degree of dependence (and consequently the degree of independence) between the components of the fuzzy set, and also between the components of the neutrosophic set in our 2006 book’s fifth edition [1]. Now we extend it for the first time to the refined neutrosophic set considering the degree of dependence or independence of subcomponets.

In this first books of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others dead or waiting, referring to neutrosophy – email messages to research colleagues, or replies, notes about authors, articles or books, so on.

In this second book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others dead or waiting, referring to many topics (see Topics) in different fields of research – email messages to research colleagues, or replies, notes about authors, articles, or books, so on – in an eager pursuit (consectatio) for meanings, reasons, and purports of (scientific) things (res).

In this third book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas!

In this fourth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring mostly to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas!

In this fifth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, mostly referring to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas!

In this sixth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, and so on. Feel free to budge in or just use the scilogs as open source for your own ideas!

In this seventh book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring to different scientific topics – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. -/- Exchanging ideas with Akeem Adesina A. Agboola, Muhammad Akram, Octavian Blaga, Said Broumi, Kajal Chatterjee, Vic Christianto, Octavian Cira, Mihaela Colhon, B. Davvaz, Luu Quoc Dat, R. Dhavaseelan, Jean Dezert, Hoda Esmail, Reza Farhadian, Ervin Goldfain, Muhammad Gulistan, (...) Yanhui Guo, Keli Hur, Saeid Jafari, W. B. Vasantha Kandasamy, J. Kim, J.G. Lee, Xinde Li, P. K. Lim, R. K. Mohanty, Mumtaz Ali, To Santanu Kumar Patro, Xu Peng, Dmitri Rabounski, Nouran Radwan, A.A. Salama, Musavarah Sarwar, Ganeshsree Selvachandran, Le Hoang Son, Selçuk Topal, Andrușa Vătuiu, Ștefan Vlăduțescu, Xiaohong Zhang.‬‬‬. (shrink)

In the fifth version of our response-paper [26] to Imamura’s criticism, we recall that NonStandard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century old neutrosophic operators (1995-1998) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the NonStandard Analysis hyperreals, monads and binads to tiny intervals with the desired accuracy – otherwise they (...) would be inapplicable. We point out several errors and false statements by Imamura [21] with respect to the inf/sup of nonstandard subsets, also Imamura’s “rigorous definition of neutrosophic logic” is wrong and the same for his definition of nonstandard unit interval, and we prove that there is not a total order on the set of hyperreals (because of the newly introduced Neutrosophic Hyperreals that are indeterminate), whence the Transfer Principle from R to R* is questionable. After his criticism, several response publications on theoretical nonstandard neutrosophics followed in the period 2018-2022. As such, I extended the NonStandard Analysis by adding the left monad closed to the right, right monad closed to the left, pierced binad (we introduced in 1998), and unpierced binad - all these in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of NonStandard Unit Interval and NonStandard Neutrosophic Logic, together with NonStandard Neutrosophic Operators are presented. (shrink)

In the fifth version of our reply article [26] to Imamura's critique, we recall that Neutrosophic Non-Standard Logic was never used by the neutrosophic community in any application, that the quarter-century old (1995-1998) neutrosophic operators criticized by Imamura were never used as they were improved soon after, but omits to talk about their development, and that in real-world applications we need to convert/approximate the hyperreals, monads and bi-nads of Non-Standard Analysis to tiny intervals with the desired precision; otherwise they would (...) be inapplicable. We pointed out several errors and false statements by Imamura [21] regarding the inf/sup of nonstandard subsets, also Imamura's "rigorous definition of neutrosophic logic" is incorrect, as is his definition of nonstandard unit interval, and we showed that there is no total order in Neutrosophic Computing and Machine Learning , Vol. 23, 2022 Florentin Smarandache, Definición mejorada de la lógica neutrosófica no estándar e introducción a los hiperreales neutrosóficos (Quinta versión) 2 the set of hyperreals (due to the recently introduced Neutrosophic Hyperreals which are indeterminate), so the Transfer Principle from R to R* is questionable. After his critique, several reply posts on non-standard theoretical neutrosophy followed in 2018-2022. As such, I extended the Nonstandard Analysis by adding the right-closed left monad, the left-closed right monad, the punctured binad (which we introduced in 1998), and the nonpunctured binad - all in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of the Nonstandard Unitary Interval and Nonstandard Neutrosophic Logic are presented, along with Nonstandard Neutrosophic Operators. (shrink)

One of the main controversies of the Logic Schools of the 12th century centered on the question: What follows from the impossible? In this paper arguments for two diametrically opposed positions are examined. The author of the ‘Avranches Text’ who probably belonged to the school of the Parvipontani defended the view that from an impossible proposition everything follows (‘Ex impossibili quodlibet’). In particular he developed a proof to show that by means of so-called ‘disjunctive syllogism’ any arbitrary proposition B can (...) be logically derived from a pair of contradictory propositions A and Not-A. The author of the Ars Meliduna instead argued that nothing follows from an impossible proposition (‘ex falso nihil sequitur’). This view is supported by various counterexamples which aimed to show that the admission of impossible premises would give rise to inconsistent conclusions. Upon closer analysis these inconsistencies do not, however, have the formal structure of a real contradiction like A and Not-A, but rather the structure of two rivalling conditionals like ‘If B then A’ and ‘If B then Not-A’. Hence these counterexamples rather have to be considered as refutations of the basic principles of ‘connexive logic’. (shrink)

The aim of this article is to generalize logics of formal inconsistency (LFIs) to systems dealing with the concept of incompatibility, expressed by means of a binary connective. The basic idea is that having two incompatible formulas to hold trivializes a deduction, and as a special case, a formula becomes consistent (in the sense of LFIs) when it is incompatible with its own negation. We show how this notion extends that of consistency in a non-trivial way, presenting conservative translations for (...) many simple LFIs into some of the most basic logics of incompatibility, thereby evidencing in a precise way how the notion of incompatibility generalizes that of consistency. We provide semantics for the new logics, as well as decision procedures, based on restricted non-deterministic matrices. The use of non-deterministic semantics with restrictions is justified by the fact that, as proved here, these systems are not algebraizable according to Blok-Pigozzi nor are they characterizable by finite Nmatrices. Finally, we briefly compare our logics to other systems focused on treating incompatibility, specially those pioneered by Brandom and further developed by Peregrin. (shrink)

This paper motivates the logic PCON, an extension of connexivity to conjunction and disjunction, called poly-connexivity. The motivation arises from differences in intonational stress patterns due to focus, where PCON turns out to be a logic of intentionally stressed connectives in focus.

We analyze the causal-observational languages that were introduced in Barbero and Sandu (2018), which allow discussing interventionist counterfactuals and functional dependencies in a unified framework. In particular, we systematically investigate the expressive power of these languages in causal team semantics, and we provide complete natural deduction calculi for each language. Furthermore, we introduce a generalized semantics which allows representing uncertainty about the causal laws, and we analyze the expressive power and proof theory of the causal-observational languages over this enriched semantics.

This paper starts the systematic study of inconsistency measures for propositional logics enriched with operators involving time. We use Prior’s operators for tense logic: H, G, P, and F; however, we apply different semantics to them. We define two logics. The first one, ATPL, allows formulas with the application of any of the four operators any number of times to propositional logic formulas. The semantics is given in terms of TPL structures. We then show how to measure the inconsistency of (...) a set of ATPL formulas using appropriately modified versions of standard inconsistency measures for propositional logic. Additionally, these new measures are checked for the satisfaction of various properties. We also investigate a new concept of inconsistency that cannot be defined for propositional logic. Then, ATPL is extended to CTPL by applying the propositional connectives to ATPL formulas and measuring inconsistency is extended to sets of CTPL formulas. (shrink)

The study of the theory of operators over modal pseudocomplemented De Morgan algebras was begun in papers [20] and [21]. In this paper, we introduce and study the class of modal pseudocomplemented De Morgan algebras enriched by a k-periodic automorphism -algebras). We denote by \ the automorphism where k is a positive integer. For \, the class coincides with the one studied in [20] where the automorphism works as a new unary operator which can be considered as a negation. In (...) the first place, we develop an algebraic study of the class of \-algebras; as consequence, we prove the class \-algebras is a semisimple variety and determine the generating algebras. After doing the algebraic study and using these properties, we built two families of sentential logics that we denote with \ and \ for every k. \ is a 1-assertional logic and \ is the degree-preserving logic both associated with the class of \-algebras. Working over these logics, we prove that \ is paraconsistent with respect to the de Morgan negation \, which is protoalgebraic and finitely equivalential but not algebraizable. In contrast, we prove that \ is algebraizable, sharing the same theorems with \, but not paraconsistent with respect to \. Furthermore, we show that \ and \ are paracomplete logics with respect to \ and \ and paraconsistent logics with respecto to \, for every k. (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 also 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)

Intuitionistic propositional logic with Galois negations ( \(\mathsf {IGN}\) ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf {IGN}\). We give the bi-tense (...) logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic _L_ is embedded into its tense companion \(\tau (L)\). (shrink)

In contemporary works on conditionals, the Ramsey test is a procedure for the evaluation of conditional sentences. There are several versions of the test, all inspired by a footnote by the British philosopher and mathematician Frank Ramsey, in his General Propositions and Causality. However, no study on Ramsey's own account of conditionals has been put forth so far. Furthermore, the footnote seems to cover indicative conditionals only, and this has led to the belief that no account of counterfactuals can be (...) found in Ramsey's work. In this paper, I recover Ramsey's account of counterfactuals and show that it is sketched in the footnote too. The result is a well-developed account of counterfactuals that resembles many contemporary ones. But Ramsey uses the same approach also for other types of conditionals, and this casts doubts on the current criteria for the classification of this type of sentences. (shrink)

The language of linear temporal logic can be interpreted on the class of dynamic topological systems, giving rise to the intuitionistic temporal logic ${\sf ITL}^{\sf c}_{\Diamond \forall }$, recently shown to be decidable by Fernández-Duque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.

We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic, it is also a logic of terms that admits a (sound) extensional semantics involving possible states-of-affairs in a given world. Applied to the analysis of Aristotle's modal syllogistic as found in the Prior Analytics A8-22, it sheds light on various fine-grained distinctions which when made allow us to clarify some ambiguities and obtain (...) a completely consistent system and prove all of the modal syllogisms considered valid by Aristotle. This logic allows us also to make a connection with the axioms of modern propositional modal logic and to perceive to what extent these are implicit in Aristotle's reasoning. Further work will involve addressing the question of the completeness of this logic (or variants thereof) together with an extension of the logic to include a calculus of relations (for instance the relational syllogistic treated in Galen's Introduction to Logic) which Slomkowsky has argued is already found in the Topics. (shrink)

Standpoint logic is a recently proposed formalism in the context of knowledge integration, which advocates a multi-perspective approach permitting reasoning with a selection of diverse and possibly conflicting standpoints rather than forcing their unification. In this paper, we introduce nested sequent calculi for propositional standpoint logics---proof systems that manipulate trees whose nodes are multisets of formulae---and show how to automate standpoint reasoning by means of non-deterministic proof-search algorithms. To obtain worst-case complexity-optimal proof-search, we introduce a novel technique in the context (...) of nested sequents, referred to as "coloring," which consists of taking a formula as input, guessing a certain coloring of its subformulae, and then running proof-search in a nested sequent calculus on the colored input. Our technique lets us decide the validity of standpoint formulae in CoNP since proof-search only produces a partial proof relative to each permitted coloring of the input. We show how all partial proofs can be fused together to construct a complete proof when the input is valid, and how certain partial proofs can be transformed into a counter-model when the input is invalid. These "certificates" (i.e. proofs and counter-models) serve as explanations of the (in)validity of the input. (shrink)

We generalize intuitionistic tense logics to the multi-modal case by placing grammar logics on an intuitionistic footing. We provide axiomatizations for a class of base intuitionistic grammar logics as well as provide axiomatizations for extensions with combinations of seriality axioms and what we call "intuitionistic path axioms". We show that each axiomatization is sound and complete with completeness being shown via a typical canonical model construction.

An extension of the propositional probability logic \ given in Ognjanović et al. that allows mixing of propositional formulas and probabilistic formulas is introduced. We describe the corresponding class of models, and we show that the problem of deciding satisfiability is in NP. We provide infinitary axiomatization for the logic and we prove that the axiomatization is sound and strongly complete.

An extension of the propositional probability logic \ given in Ognjanović et al. that allows mixing of propositional formulas and probabilistic formulas is introduced. We describe the corresponding class of models, and we show that the problem of deciding satisfiability is in NP. We provide infinitary axiomatization for the logic and we prove that the axiomatization is sound and strongly complete.