The logics of formalinconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the (...) literature. The case of QmbC, the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called QLFI1o is also studied, which is equivalent to the quantified version of da Costa and D'Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and QLFI1o with a standard equality predicate is also considered. (shrink)
In this paper we present a philosophical motivation for the logics of formalinconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non-contradiction. We defend the view according to which logics of formalinconsistency may be interpreted as theories of logical consequence of an epistemological character. We also (...) argue that in order to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we show that mbC, a logic of formalinconsistency based on classical logic, may be enhanced in order to express the basic ideas of an intuitive interpretation of contradictions as conflicting evidence. (shrink)
One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of FormalInconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)
In this paper we present a philosophical motivation for the logics of formalinconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non- contradiction. We defend the view according to which logics of formalinconsistency may be interpreted as theories of logical consequence of an epistemological character. We (...) also argue that in order to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we argue that an intuitive reading of the bivalued semantics for the logic mbC, a logic of formalinconsistency based on classical logic, fits in well with the basic ideas of an intuitive interpretation of contradictions. On this interpretation, the acceptance of a pair of propositions A and ¬A does not mean that A is simultaneously true and false, but rather that there is conflicting evidence about the truth value of A. (shrink)
We present a philosophical motivation for the logics of formalinconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language. We shall defend the view according to which logics of formalinconsistency are theories of logical consequence of normative and epistemic character. This approach not only allows us to make inferences in the presence of contradictions, but offers a philosophically (...) acceptable account of paraconsistency. (shrink)
This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics (...) of FormalInconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning. (shrink)
Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of FormalInconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap (...) structures are given. Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics. (shrink)
In this paper two systems of AGM-like Paraconsistent Belief Revision are overviewed, both defined over Logics of FormalInconsistency (LFIs) due to the possibility of defining a formal consistency operator within these logics. The AGM° system is strongly based on this operator and internalize the notion of formal consistency in the explicit constructions and postulates. Alternatively, the AGMp system uses the AGM-compliance of LFIs and thus assumes a wider notion of paraconsistency - not necessarily (...) related to the notion of formal consistency. (shrink)
The aim of this paper is to explore the peculiar case of infectious logics, a group of systems obtained generalizing the semantic behavior characteristic of the -fragment of the logics of nonsense, such as the ones due to Bochvar and Halldén, among others. Here, we extend these logics with classical negations, and we furthermore show that some of these extended systems can be properly regarded as logics of formalinconsistency and logics of (...) class='Hi'>formal undeterminedness. (shrink)
In 1988, J. Ivlev proposed some (non-normal) modal systems which are semantically characterized by four-valued non-deterministic matrices in the sense of A. Avron and I. Lev. Swap structures are multialgebras (a.k.a. hyperalgebras) of a special kind, which were introduced in 2016 by W. Carnielli and M. Coniglio in order to give a non-deterministic semantical account for several paraconsistent logics known as logics of formalinconsistency, which are not algebraizable by means of the standard techniques. Each swap (...) structure induces naturally a non-deterministic matrix. The aim of this paper is to obtain a swap structures semantics for some Ivlev-like modal systems proposed in 2015 by M. Coniglio, L. Fariñas del Cerro and N. Peron. Completeness results will be stated by means of the notion of Lindenbaum–Tarski swap structures, which constitute a natural generalization to multialgebras of the concept of Lindenbaum–Tarski algebras. (shrink)
In this paper, we approach the problem of classical recapture for LP and K3 by using normality operators. These generalize the consistency and determinedness operators from Logics of FormalInconsistency and Underterminedness, by expressing, in any many-valued logic, that a given formula has a classical truth value (0 or 1). In particular, in the rst part of the paper we introduce the logics LPe and Ke3 , which extends LP and K3 with normality operators, and we (...) establish a classical recapture result based on the two logics. In the second part of the paper, we compare the approach in terms of normality operators with an established approach to classical recapture, namely minimal inconsistency. Finally, we discuss technical issues connecting LPe and Ke3 to the tradition of Logics of FormalInconsistency and Underterminedness. (shrink)
There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of FormalInconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent (...) scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formalinconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices. (shrink)
Two systems of belief change based on paraconsistent logics are introduced in this article by means of AGM-like postulates. The first one, AGMp, is defined over any paraconsistent logic which extends classical logic such that the law of excluded middle holds w.r.t. the paraconsistent negation. The second one, AGMo , is specifically designed for paraconsistent logics known as Logics of FormalInconsistency (LFIs), which have a formal consistency operator that allows to recover all the (...) classical inferences. Besides the three usual operations over belief sets, namely expansion, contraction and revision (which is obtained from contraction by the Levi identity), the underlying paraconsistent logic allows us to define additional operations involving (non-explosive) contradictions. Thus, it is defined external revision (which is obtained from contraction by the reverse Levi identity), consolidation and semi-revision, all of them over belief sets. It is worth noting that the latter operations, introduced by S. Hansson, involve the temporary acceptance of contradictory beliefs, and so they were originally defined only for belief bases. Unlike to previous proposals in the literature, only defined for specific paraconsistent logics, the present approach can be applied to a general class of paraconsistent logics which are supraclassical, thus preserving the spirit of AGM. Moreover, representation theorems w.r.t. constructions based on selection functions are obtained for all the operations. (shrink)
In this paper, we look at applying the techniques from analyzing superintuitionistic logics to extensions of the cointuitionistic Priest-da Costa logic daC (introduced by Graham Priest as “da Costa logic”). The relationship between the superintuitionistic axioms- definable in daC- and extensions of Priest-da Costa logic (sdc-logics) is analyzed and applied to exploring the gap between the maximal si-logic SmL and classical logic in the class of sdc-logics. A sequence of strengthenings of Priest-da Costa logic is examined and (...) employed to pinpoint the maximal non-classical extension of both daC and Heyting-Brouwer logic HB . Finally, the relationship between daC and Logics of FormalInconsistency is examined. (shrink)
Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of (...) ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)
In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formalinconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a (...) new proof of S4 as being "the logic of topological spaces", but now under the perspective of paraconsistency. (shrink)
Jaakko Hintikka, in a series of talks in Brazil in 2008, defended that IF logic and paraconsistent logic are, in a sense, very similar. Having sketched the proposal of a new paraconsistent system, he maintains that several achievements of IF logic could be reproducible in paraconsistent logic. One of the major difficulties, left as a challenge, would be to formulate some truth conditions for this new paraconsistent first-order language in order to make IF logic and paraconsistent logic more inter-related. My (...) proposal is that this would demand an innovative game-theoretical semantic approach to paraconsistentism, but also that the syntax of the paraconsistent “Logics of FormalInconsistency” can model the internal logic of Socratic elenchi. I aim to discuss these, and other points posed by Hintikka, as challenges and opportunities for paraconsistentism, paraconsistent logics and IF logics, as well as to raise some criticisms on Hintikka’s view about paraconsistency. (shrink)
In this paper the class of Fidel-structures for the paraconsistent logic mbC is studied from the point of view of Model Theory and Category Theory. The basic point is that Fidel-structures for mbC (or mbC-structures) can be seen as first-order structures over the signature of Boolean algebras expanded by two binary predicate symbols N (for negation) and O (for the consistency connective) satisfying certain Horn sentences. This perspective allows us to consider notions and results from Model Theory in order to (...) analyze the class of mbC-structures. Thus, substructures, union of chains, direct products, direct limits, congruences and quotient structures can be analyzed under this perspective. In particular, a Birkhoff-like representation theorem for mbC-structures as subdirect poducts in terms of subdirectly irreducible mbC-structures is obtained by adapting a general result for first-order structures due to Caicedo. Moreover, a characterization of all the subdirectly irreducible mbC-structures is also given. An alternative decomposition theorem is obtained by using the notions of weak substructure and weak isomorphism considered by Fidel for Cn-structures. (shrink)
Formal relationism in the philosophy of mind is the thesis that folk psychological states should be individuated, at least partially, in terms of the purely formal inference-licensing relations between underlying mental representations. It's supposed to provide a Russellian alternative to a Fregean theory of propositional attitudes. I argue that there's an inconsistency between the motivation for formal relationism and the use to which it's put in defense of Russellian propositions. Furthermore, I argue that formal relationism (...) is committed to epiphenomenalism about singular mental content. (shrink)
We propose a typology of representational artifacts for health care and life sciences domains and associate this typology with different kinds of formal ontology and logic, drawing conclusions as to the strengths and limitations for ontology in a description logics framework. The four types of domain representation we consider are: (i) lexico-semantic representation, (ii) representation of types of entities, (iii) representations of background knowledge, and (iv) representation of individuals. We advocate a clear distinction of the four kinds of (...) representation in order to provide a more rational basis for using ontologies and related artifacts to advance integration of data and enhance interoperability of associated reasoning systems. We highlight the fact that only a minor portion of scientifically relevant facts in a domain such as biomedicine can be adequately represented by formal ontologies as long as the latter are conceived as representations of entity types. In particular, the attempt to encode default or probabilistic knowledge using ontologies so conceived is prone to produce unintended, erroneous models. (shrink)
Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging (...) and analyzing at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic. (shrink)
In a first part, I defend that formal semantics can be used as a guide to ontological commitment. Thus, if one endorses an ontological view \(O\) and wants to interpret a formal language \(L\) , a thorough understanding of the relation between semantics and ontology will help us to construct a semantics for \(L\) in such a way that its ontological commitment will be in perfect accordance with \(O\) . Basically, that is what I call constructing formal (...) semantics from an ontological perspective. In the rest of the paper, I develop rigorously and put into practice such a method, especially concerning the interpretation of second-order quantification. I will define the notion of ontological framework: it is a set-theoretical structure from which one can construct semantics whose ontological commitments correspond exactly to a given ontological view. I will define five ontological frameworks corresponding respectively to: (i) predicate nominalism, (ii) resemblance nominalism, (iii) armstrongian realism, (iv) platonic realism, and (v) tropism. From those different frameworks, I will construct different semantics for first-order and second-order languages. Notably I will present different kinds of nominalist semantics for second-order languages, showing thus that we can perfectly quantify over properties and relations while being ontologically committed only to individuals. I will show in what extent those semantics differ from each other; it will make clear how the disagreements between the ontological views extend from ontology to logic, and thus why endorsing an ontological view should have an impact on the kind of logic one should use. (shrink)
In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic (...) which individually, but not jointly, lack the problematic feature. (shrink)
2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory (...) to cognitive modeling, and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)
We survey main developments, results, and open problems on interval temporal logics and duration calculi. We present various formal systems studied in the literature and discuss their distinctive features, emphasizing on expressiveness, axiomatic systems, and (un)decidability results.
In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent (...) class='Hi'>logics once the right set-up is chosen. Our logic L4 has a semantics that also underlies Belnap’s [4] and is related to the logic of bilattices. L4 is in focus most of the time, but it is also shown how results obtained for L4 can be transferred to several variants. (shrink)
The general methodology of "algebraizing" logics is used here for combining different logics. The combination of logics is represented as taking the colimit of the constituent logics in the category of algebraizable logics. The cocompleteness of this category as well as its isomorphism to the corresponding category of certain first-order theories are proved.
The aim of this paper is to explore what insights relevant logics may provide for the understanding of literary fictional narrative. To date, hardly anyone has reflected on the intersection of relevant logics and narratology, and some could think that there is good reason for it. On the one hand, relevance has been a prominent issue in pragmatics, in the tradition of Grice, and Sperber and Wilson; thus framed, relevance is highly context-sensitive, so it seems unsuitable for (...) class='Hi'>formal analysis. On the other hand, the very idea of a logic of narrative has been criticized, arguing that logic brings to a stasis the time of human action (Ricœur, II: 29-60), or that its emphasis on rules misses the creative, unpredictable character of literature (De Man)... First, I will briefly introduce relevant logics, with an eye to showing their interest for narratological concerns, rather than to here providing a coherent (let alone comprehensive) survey. Secondly, lest I get drawn into purely abstract discussion, I will analyse several stories in order to give some instances of the kind of topics congenial to narratology that may be addressed with a relevantist toolkit. Thirdly (and lastly), I will expand in more theoretical fashion on certain issues raised in the second section and bring them into connection with pragmatic relevance theory. (shrink)
We here make preliminary investigations into the model theory of DeMorgan logics. We demonstrate that Łoś's Theorem holds with respect to these logics and make some remarks about standard model-theoretic properties in such contexts. More concretely, as a case study we examine the fate of Cantor's Theorem that the classical theory of dense linear orderings without endpoints is $\aleph_{0}$-categorical, and we show that the taking of ultraproducts commutes with respect to previously established methods of constructing nonclassical structures, namely, (...) Priest's Collapsing Lemma and Dunn's Theorem in 3-Valued Logic. (shrink)
Logics based on weak Kleene algebra (WKA) and related structures have been recently proposed as a tool for reasoning about flaws in computer programs. The key element of this proposal is the presence, in WKA and related structures, of a non-classical truth-value that is “contaminating” in the sense that whenever the value is assigned to a formula ϕ, any complex formula in which ϕ appears is assigned that value as well. Under such interpretations, the contaminating states represent occurrences of (...) a flaw. However, since different programs and machines can interact with (or be nested into) one another, we need to account for different kind of errors, and this calls for an evaluation of systems with multiple contaminating values. In this paper, we make steps toward these evaluation systems by considering two logics, HYB1 and HYB2, whose semantic interpretations account for two contaminating values beside classical values 0 and 1. In particular, we provide two main formal contributions. First, we give a characterization of their relations of (multiple-conclusion) logical consequence—that is, necessary and sufficient conditions for a set Δ of formulas to logically follow from a set Γ of formulas in HYB1 or HYB2 . Second, we provide sound and complete sequent calculi for the two logics. (shrink)
Formal ontologies are nowadays widely considered a standard tool for knowledge representation and reasoning in the Semantic Web. In this context, they are expected to play an important role in helping automated processes to access information. Namely: they are expected to provide a formal structure able to explicate the relationships between different concepts/terms, thus allowing intelligent agents to interpret, correctly, the semantics of the web resources improving the performances of the search technologies. Here we take into account a (...) problem regarding Knowledge Representation in general, and ontology based representations in particular; namely: the fact that knowledge modeling seems to be constrained between conflicting requirements, such as compositionality, on the one hand and the need to represent prototypical information on the other. In particular, most common sense concepts seem not to be captured by the stringent semantics expressed by such formalisms as, for example, Description Logics (which are the formalisms on which the ontology languages have been built). The aim of this work is to analyse this problem, suggesting a possible solution suitable for formal ontologies and semantic web representations. The questions guiding this research, in fact, have been: is it possible to provide a formal representational framework which, for the same concept, combines both the classical modelling view (accounting for compositional information) and defeasible, prototypical knowledge ? Is it possible to propose a modelling architecture able to provide different type of reasoning (e.g. classical deductive reasoning for the compositional component and a non monotonic reasoning for the prototypical one)? We suggest a possible answer to these questions proposing a modelling framework able to represent, within the semantic web languages, a multilevel representation of conceptual information, integrating both classical and non classical (typicality based) information. Within this framework we hypothesise, at least in principle, the coexistence of multiple reasoning processes involving the different levels of representation. (shrink)
In this paper we prove the completeness of three logical systems I LI, IL2 and IL3. IL1 deals solely with identities {a = b), and its deductions are the direct deductions constructed with the three traditional rules: (T) from a = b and b = c infer a = c, (S) from a = b infer b = a and (A) infer a = a(from anything). IL2 deals solely with identities and inidentities {a ± b) and its deductions include both (...) the direct and the indirect deductions constructed with the three traditional rules. IL3 is a hybrid of IL1 and IL2: its deductions are all direct as in IL1 but it deals with identities and inidentities as in IL2. IL1 and IL2 have a high degree of naturalness. Although the hybrid system IL3 was constructed as an artifact useful in the mathematical study of IL1 and IL2, it nevertheless has some intrinsically interesting aspects. The main motivation for describing and studying such simple systems is pedagogical. In teaching beginning logic one would like to present a system of logic which has the following properties. First, it exemplifies the main ideas of logic: implication, deduction, non-implication, counterargument(or countermodel), logical truth, self-contradiction, consistency,satisfiability, etc. Second, it exemplifies the usual general metaprinciples of logic: contraposition and transitivity of implication, cut laws, completeness,soundness, etc. Third, it is simple enough to be thoroughly grasped by beginners. Fourth, it is obvious enough so that its rules do not appear to be arbitrary or purely conventional. Fifth, it does not invite confusions which must be unlearned later. Sixth, it involves a minimum of presuppositions which are no longer accepted in mainstream contemporary logic. (shrink)
This special issue of the Logic Journal of the IGPL includes revised and updated versions of the best work presented at the fourth edition of the workshop Formal Ap- proaches to Multi-Agent Systems, FAMAS'09, which took place in Turin, Italy, from 7 to 11 September, 2009, under the umbrella of the Multi-Agent Logics, Languages, and Organisations Federated Workshops (MALLOW). -/- Just like its predecessor, research reported in this FAMAS 2009 special issue is very much inspired by practical concerns. (...) This time the authors of all the five selected papers are concerned with knowledge and beliefs in multi-agent settings: How to create a group belief in a fair way from individual plausibility orderings? How to close gaps and resolve ambiguities in a tractable way, when information comes from multiple sources? How to reason about a spatial environment? How to compare the strengths of an agent's beliefs in a principled way? How to decide as efficiently as possible whether a given formula concerning group beliefs is valid? These questions and their answers lead to a multi-faceted and at the same time coherent special issue. We concisely introduce the five articles. (shrink)
Over the last decade, multi-agent systems have come to form one of the key tech- nologies for software development. The Formal Approaches to Multi-Agent Systems (FAMAS) workshop series brings together researchers from the fields of logic, theoreti- cal computer science and multi-agent systems in order to discuss formal techniques for specifying and verifying multi-agent systems. FAMAS addresses the issues of logics for multi-agent systems, formal methods for verification, for example model check- ing, and formal approaches (...) to cooperation, multi-agent planning, communication, coordination, negotiation, games, and reasoning under uncertainty in a distributed environment. In 2007, the third FAMAS workshop, FAMAS'007, was one of the agent workshops gathered together under the umbrella of Multi-Agent Logics, Languages, and Organ- isations - Federated Workshops, MALLOW'007, taking place from 3 to 7 September 2007 in Durham. This current special issue of the Logic Journal of the IGPL gathers together the revised and updated versions of the five best FAMAS'007 contributions. (shrink)
Description Logics are nowadays widely accepted as formalisms which provide reasoning facilities which allow us to discover inconsistencies in ontologies in an automatic fashion. Where ontologies are developed in modular fashion, they allow changes in one module to propogated through the system of ontologies automatically in a way which helps to maintain consistency and stability. For this feature to be utilized effectively, however, requires that domain ontologies be represented in a normalized form.
Description logics and other formal devices are frequently used as means for preventing or detecting mistakes in ontologies. Some of these devices are also capable of inferring the existence of inter-concept relationships that have not been explicitly entered into an ontology. A prerequisite, however, is that this information can be derived from those formal definitions of concepts and relationships which are included within the ontology. In this paper, we present a novel algorithm that is able to suggest (...) relationships among existing concepts in a formal ontology that are not derivable from such formal definitions. The algorithm exploits cross-lingual information that is implicitly present in the collection of terms used in various languages to denote the concepts and relationships at issue. By using a specific experimental design, we are able to quantify the impact of cross-lingual information in coping with underspecification in formal ontologies. (shrink)
To eliminate incompleteness, undecidability and inconsistency from formal systems we only need to convert the formal proofs to theorem consequences of symbolic logic to conform to the sound deductive inference model. -/- Within the sound deductive inference model there is a (connected sequence of valid deductions from true premises to a true conclusion) thus unlike the formal proofs of symbolic logic provability cannot diverge from truth.
A general framework for translating various logical systems is presented, including a set of partial unary operators of affirmation and negation. Despite its usual reading, affirmation is not redundant in any domain of values and whenever it does not behave like a full mapping. After depicting the process of partial functions, a number of logics are translated through a variety of affirmations and a unique pair of negations. This relies upon two preconditions: a deconstruction of truth-values as ordered and (...) structured objects, unlike its mainstream presentation as a simple object; a redefinition of the Principle of Bivalence as a set of four independent properties, such that its definition does not equate with normality. (shrink)
ABSTRACT: This 1974 paper builds on our 1969 paper (Corcoran-Weaver [2]). Here we present three (modal, sentential) logics which may be thought of as partial systematizations of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of these three logics coincide with one another and with those of standard formalizations of Lewis's S5. These logics, when regarded as logistic systems (cf. Corcoran [1], p. 154), are seen (...) to be equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be as closely linked as previously thought. -/- This 1974 paper uses the linear notation for natural deduction presented in [2]: each two-dimensional deduction is represented by a unique one-dimensional string of characters. Thus obviating need for two-dimensional trees, tableaux, lists, and the like—thereby facilitating electronic communication of natural deductions. The 1969 paper presents a (modal, sentential) logic which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of this logic coincides those of standard formalizations of Lewis’s S4. Among the paper's innovations is its treatment of modal logic in the setting of natural deduction systems--as opposed to axiomatic systems. The author’s apologize for the now obsolete terminology. For example, these papers speak of “a proof of a sentence from a set of premises” where today “a deduction of a sentence from a set of premises” would be preferable. 1. Corcoran, John. 1969. Three Logical Theories, Philosophy of Science 36, 153–77. J P R -/- 2. Corcoran, John and George Weaver. 1969. Logical Consequence in Modal Logic: Natural Deduction in S5 Notre Dame Journal of Formal Logic 10, 370–84. MR0249278 (40 #2524). 3. Weaver, George and John Corcoran. 1974. Logical Consequence in Modal Logic: Some Semantic Systems for S4, Notre Dame Journal of Formal Logic 15, 370–78. MR0351765 (50 #4253). (shrink)
This paper presents two systems of natural deduction for the rejection of non-tautologies of classical propositional logic. The first system is sound and complete with respect to the body of all non-tautologies, the second system is sound and complete with respect to the body of all contradictions. The second system is a subsystem of the first. Starting with Jan Łukasiewicz's work, we describe the historical development of theories of rejection for classical propositional logic. Subsequently, we present the two systems of (...) natural deduction and prove them to be sound and complete. We conclude with a ‘Theorem of Inversion’. (shrink)
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued (...) first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)
We present two defeasible logics of norm-propositions (statements about norms) that (i) consistently allow for the possibility of normative gaps and normative conflicts, and (ii) map each premise set to a sufficiently rich consequence set. In order to meet (i), we define the logic LNP, a conflict- and gap-tolerant logic of norm-propositions capable of formalizing both normative conflicts and normative gaps within the object language. Next, we strengthen LNP within the adaptive logic framework for non-monotonic reasoning in order to (...) meet (ii). This results in the adaptive logics LNPr and LNPm, which interpret a given set of premises in such a way that normative conflicts and normative gaps are avoided ‘whenever possible’. LNPr and LNPm are equipped with a preferential semantics and a dynamic proof theory. (shrink)
Formal principles governing best practices in classification and definition have for too long been neglected in the construction of biomedical ontologies, in ways which have important negative consequences for data integration and ontology alignment. We argue that the use of such principles in ontology construction can serve as a valuable tool in error-detection and also in supporting reliable manual curation. We argue also that such principles are a prerequisite for the successful application of advanced data integration techniques such as (...) ontology-based multi-database querying, automated ontology alignment and ontology-based text-mining. These theses are illustrated by means of a case study of the Gene Ontology, a project of increasing importance within the field of biomedical data integration. (shrink)
The paper presents a new method for generating typologies of formal-ontological relations. The guiding idea is that formal relations are those sorts of relations which hold between entities which are constituents of distinct ontologies. We provide examples of ontologies (in the spirit of Zemach’s classic “Four Ontologies” of 1970), and show how these can be used to give a rich typology of formal relations in a way which also throws light on the opposition between threeand four-dimensionalism.
In the latter half of the twentieth century, philosophers of science have argued (implicitly and explicitly) that epistemically rational individuals might compose epistemically irrational groups and that, conversely, epistemically rational groups might be composed of epistemically irrational individuals. We call the conjunction of these two claims the Independence Thesis, as they together imply that methodological prescriptions for scientific communities and those for individual scientists might be logically independent of one another. We develop a formal model of scientific inquiry, define (...) four criteria for individual and group epistemic rationality, and then prove that the four definitions diverge, in the sense that individuals will be judged rational when groups are not and vice versa. We conclude by explaining implications of the inconsistency thesis for (i) descriptive history and sociology of science and (ii) normative prescriptions for scientific communities. (shrink)
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