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In this paper we study the complexity of deontic logics grounded on norm-based semantics and apply norm-based deontic logic to access control. Four principal norm-based deontic logics have been proposed so far: imperative logic, input/output logic, deontic default logic and deontic defeasible logic. We present the readers that imperative logic is complete for the 2ed level of the polynomial hierarchy and deontic default logic is located in the 3ed level of the polynomial hierarchy. We then show how it is possible to impose restrictions to imperative logic such that the complexity goes down to be tractable, allowing the logic to be used in practical applications. We focus on a specific application: access control.

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language (using truth functional connectives and quantifiers). This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given.

Strong equivalence is an important concept in the theory of answer set programming. Informally speaking, two sets of rules are strongly equivalent if they have the same meaning in any context. Equilibrium logic was used to prove that sets of rules expressed as propositional formulas are strongly equivalent if and only if they are equivalent in the logic of here-and-there. We extend this line of work to formulas with infinitely long conjunctions and disjunctions, show that the infinitary logic of here-and-there characterizes strong equivalence of infinitary formulas, and give an axiomatization of that logic. This is useful because of the relationship between infinitary formulas and logic programs with local variables.

Stable model semantics had been recently generalized to non-Herbrand structures by several works, which provides a unified framework and solid logical foundations for answer set programming. This article focuses on the expressiveness of normal and disjunctive logic programs under general stable model semantics. A translation from disjunctive logic programs to normal logic programs is proposed for infinite structures. Over finite structures, some disjunctive logic programs are proved to be intranslatable to normal logic programs if the arities of auxiliary predicates and functions are bounded in a certain way. The equivalence of the expressiveness of normal logic programs and disjunctive logic programs over arbitrary structures is also shown to coincide with that over finite structures and coincide with whether the complexity class NP is closed under complement. Moreover, to obtain a more explicit picture of the expressiveness, some intertranslatability results between logic program classes, and fragments of second-order logic are established.

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is $\Pi _1^1$-complete. But third, it is only granted the assumption of countability that the class of tautologies is ${\Sigma _1}$-definable in set theory.Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects.

In this paper we introduce arbitrary arrow update logic (AAUL). The logic AAUL takes arrow update logic, a dynamic epistemic logic where the accessibility relations of agents are updated rather than the set of possible worlds, and adds a quantifier over such arrow updates.We investigate the relative expressivity of AAUL compared to other logics, most notably arbitrary public announcement logic (APAL). Additionally, we show that the model checking problem for AAUL is PSPACE-complete. Finally, we introduce a proof system for AAUL, and prove it to be sound and complete. (C) 2016 Elsevier B.V. All rights reserved.

We prove that the problem of tabularity over Johansson’s minimal logic J is decidable. Describing all pretabular extensions of the minimal logic, we find that there are seven of them and show that they are all recognizable over J. We find axiomatizations and semantic characterizations of all seven pretabular logics.

This paper deals with Husserl’s idea of pure logic as it is coined in the Logical Investigations (1900/1901). First, it exposes the formation of pure logic around a conception of completeness (Sect. 2); then, it presents intentionality as the keystone of such a structuring (Sect. 3); and finally, it provides a systematic reconstruction of pure logic from the semiotic standpoint of intentionality (Sect. 4). In this way, it establishes Husserlian pure logic as a phenomenological epistemology of mathematical logic.

MTL is a Monoidal t-norm based logic introduced by Esteva and Godo by omitting divisibility axiom from H´ajek's Basic logic (BL). Many logics can be obtained by adding axioms to MTL logic. Logic system WBL is obtained by adding weak divisibility axiom to logic system MTL. Logic system WMV is obtained by adding involution axiom to logic system WBL. WBL-algebra corresponding to logic system WBL and WMV-algebra to logic system WMV are defined respectively. It is proved that the both of logic system Luk and logic system Nilpotent minimum (NM) are the schematic extensions of logic system WMV. Weak Wajsberg algebra and the simplified form of logic system WMV are obtained.

The present author developed a calculus for the logic of ‘because’. In a recent paper in this journal, it has been claimed that the central inference rules for the logic are invalid and that the intuition upon which the rules are based is not accounted for. This note criticises these arguments and presents an independent argument in favour of the rules used in the logic.

This paper is devoted to study of the logic corresponding to intervals of the real line, where the modality is interpreted as "after." Since this logic is finitely axiomatizable, the proof of the finite model property given in the paper implies its decidability. Also, a description of the class of finite rooted Kripke frames corresponding to this logic is provided.

Absolute arithmetical realizability of predicate formulas is introduced. It is proved that the intuitionistic logic is not correct with this semantics, but the basic logic is correct.

Ordinary quantum logic has well known pathologies rendering it useless for the purposes of computation. However, loosely related logics, based upon variants of Girard's Linear Logic, have been found useful in the context of quantum computation. In one sense, the use of such computational schemes affords a meta level view of the possible provenance of certain expressions not otherwise apparent. Since such logics are presumed to encapsulate the essential behavior of quantum "resources" we may entertain the question as to whether this logical or computational approach could have any bearing upon quantum physics itself. In this article we address the question of the genesis of certain fundamental Lagrangians, namely those occurring in the standard model. If a certain set of sentences in a logic are added to the set of axioms of the logic the resulting structure is generally called a theory by logicians. In this paper we shall introduce a version of such a logic and deduce some of its physical ramifications. Namely, we will show that there is a single type of sequent that, when added to the logical calculus at hand as an axiom, generates in the theory so defined, series whose leading terms match exactly the Yang-Mills Lagrangian density (including a gauge fixing term) and also the Einstein-Hilbert Lagrangian density, most of the remaining terms being negligible at low intensities in both cases. By expanding the logic somewhat, in the manner of second quantization, we are able also to give an account of interaction terms in the Yang-Mills case. This shows that there is a common form ancestral to all the Lagrangians of the standard model in the ensemble of "evolutionary" trees provided by deductions in a certain clearly specified logic, and reveals the differences between the Yang-Mills and gravitational kinetic terms. Thus we acquire a new paradigm for "unification" of the fundamental forces at the level of the underlying logic.

We introduce a theorem proving strategy for Partial Classical Logic (PCL) thatis based on geometric logic. The strategy first translates PCL theories into sets of Kleene formulas. After that, the Kleene formulas are translated into 3-valued geometric logic. The resulting formulas can be refuted by an adaptation ofgeometric resolution.The translation to Kleene logic does not only open the way to theorem proving, butit also sheds light on the relation between PCL, Kleene Logic, and classical logic.

It is well known that the strongest t-norm, that is the largest with respect to the pointwise order, is the minimum. In 2001, the logic MTL was introduced as the base of a framework of many-valued logics, and in 2002 it was shown that it is the logic of all left-continuous t-norms and their residua. Within this family of logics, the many-valued logic associated with the minimum t-norm is the Godel one, whilst there is no logic associated to the drastic product t-norm. Indeed the drastic product is not left-continuous, and hence it does not have a residuum. However, in a recent paper the logic DP has been studied, by showing that the monoidal operation of every DP-chain is like the drastic product t-norm. In this paper we present the logic EMTL, whose algebraic variety is the smallest to contain the ones of Godel- and DP-algebras. We show that the chains in this algebraic variety are exactly all the Godel- and DP-chains, we classify and axiomatize all the subvarieties, and we show some limitative results concerning the amalgamation property.

We offer a new logic, a multimodal epistemic ?ukasiewicz logic, which is an extension of the infinitely valued ?ukasiewicz logic, the language of the logic is an extended by unary connectives that are interpreted as modal operators (knowledge operators). We propose the use such a logic in studying immune system. A relational system is developed as a semantic of this logic. The relational systems represent the immune system which in its turn is a part of relational biology.

A consideration of non-classical logic in terms of classical one allows us to show a role of designated truth values. In this way we show that our version of non-classical many-valued logic can be based on the structure of genetic code.

Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic (FOL). However, there are good reasons to avoid using full second-order logic (SOL) for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the logic which is obtained from FOL by augmenting it with the transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory (which is much more relevant for the task of mechanizing mathematics). Two natural Gentzen-style proof systems for ancestral logic are presented: one for the reflexive transitive closure, and one for the non-reflexive one. We show that these systems are sound for ancestral logic and provide evidence that they indeed encompass all forms of reasoning for this logic that are used in practice. The two systems are shown to be equivalent by providing translation algorithms between them. We end with an investigation of two main proof-theoretical properties: cut elimination and constructive consistency proof.

Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably construed, in accordance with an abundant or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational.

This article will show how René Girard’s hypothesis of the scapegoat victim is the indirect basis of the discipline that unfolds the necessary consequences of the distinction between two possible states; that is, formal logic. To support this unusual view on the foundation of logic, two systems of formal logic will be analyzed: the calculus of indications created by George Spencer-Brown in the 1960s and the system of existential graphs created by Charles S. Peirce in the late nineteenth century. The first system shows that logic is based on elemental difference, and the second displays that logic is based on an original act of expulsion, the expulsion of the opposite of logic, which will be called the illogical. It will then be argued that Girard’s theory allows the full understanding of the expulsion of the illogical; that is, the emergence of the elemental difference upon which logic is based. It is not argued that logic, as a symbolic system of inferences, is directly based on Girard’s theory. It is rather argued that, as shown by the work of Spencer-Brown, formal logic is based on elemental distinction and negation from where they had their roots in the scapegoat mechanism. Finally, concerning a real and specific pattern of inference, a link is suggested between apagoge and the founding murder.

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermore, we present a logic for reasoning about graded if-then rules which is obtained as particular case of the general result. A truth-functional logic for reasoning with graded inequalities is proposed.Its semantics is defined using algebras with fuzzy orders.Completeness in the Pavelka style is established for all complete residuated lattices.A complete abstract logic of graded attributes is shown as an application.

A proposition is noncontingent, if it is necessarily true or it is necessarily false. In an epistemic context, ‘a proposition is noncontingent’ means that you know whether the proposition is true. In this paper, we study contingency logic with the noncontingency operator Δ but without the necessity operator □. This logic is not a normal modal logic, because Δ(φ → ψ) → (Δφ → Δψ) is not valid. Contingency logic cannot define many usual frame properties, and its expressive power is weaker than that of basic modal logic over classes of models without reflexivity. These features make axiomatizing contingency logics nontrivial, especially for the axiomatization over symmetric frames. In this paper, we axiomatize contingency logics over various frame classes using a novel method other than the methods provided in the literature, based on the ‘almost-definability’ schema AD proposed in our previous work. We also present extensions of contingency logic with dynamic operators. Finally, we compare our work to the related work in the fields of contingency logic and ignorance logic, where the two research communities have similar results but are apparently unaware of each other’s work. One goal of our paper is to bridge this gap.

Based on the direct product of Boolean algebra and Lukasiewicz algebra, six lattice-valued logic is put forward in this paper. The algebraic structure and properties of the lattice are analyzed profoundly and the tautologies of six-valued logic system L6P(X) are discussed deeply. The researches of this paper can be used in lattice-valued logic systems and can be helpful to automated reasoning systems.

How should logic be understood? I outline and defend the view that I call “realism” about logic. This is a thesis with two elements: first, that certain logical constants are features or constituents of the world; and, second, that logical constants concepts and words have the function of referring to these features or constituents of the world.

Probability and logic are two branches of mathematics that have important philosophical applications. This article discusses several areas of intersection between them. Several involve the role for probability in giving semantics for logic or the role of logic in governing assignments of probability. Some involve probability over non-classical logic or self-referential sentences.

We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity.

This article introduces a deontic logic which aims to model the Canadian legal discourse. Category theory is assumed as a foundational framework for logic. A deontic deductive system DDS is defined as two fibrations: the logic for unconditional obligations OL is defined within a Cartesian closed category on the grounds of an intuitionistic propositional action logic PAL and an action logic AL , while a logic for conditional normative reasoning CNR is defined as a symmetric closed monoidal category. A typed syntax and typed arrows are used to define properly DDS . We show how it can solve the paradoxes of deontic logic and we provide some examples of application to legal reasoning.

The solution of the problem of the future random events truth is considered in Vasil’ev’s logic. N. A. Vasil’ev graded the logic according to two levels—the level of facts, i.e. time fixed events, and the level of notions or rules, governing these facts. The mathematical construction previously suggested for imaginary Vasil’ev’s logic, extends to the early variant of his logic—a logic of notions. In the paper, we investigate the meaning of problematic and uncertain assertions introduced by Vasil’ev. As a result, we developed a model of Vasil’ev’s logic of facts that resolves also the truth problem of future random events. The imaginary logic has also been extended to the level of notions, and the law of the excluded eighth is gotten in it. The correspondence between Vasil’ev’s terms “some” and “all” and modern quantifiers is discussed.

Its purpose is to prove information law by logic, mathematics and translation. The method involves: the generalized bilingual logic established on both Aristotle’s formal logic and Frege’s mathematical logic, the linkage function established on both Turing’ strong artificial intelligence using numbers and Searle’s weak artificial intelligence using characters, the ontological knowledge established on both Saussure’s general linguistics and Chomsky’s formal linguistics. The result is that the basic law can be proved by digital and textual twin matrices. Its significance lies in that the global positioning system should be regarded as a special case of the generalized bilingual system.

This paper presents a logic of questions developed as an extension of (S5) epistemic logic. We discuss many features that are important for erotetic logic (formalization and semantics of questions, answerhood conditions, and inferential structures with questions). The aim is to introduce an erotetic system which corresponds well with epistemic terms and can form an appropriate background for dynamic approaches in epistemic logic.

Predicate intuitionistic logic is a well established fragment of dependent types. According to the Curry-Howard isomorphism proof construction in the logic corresponds well to synthesis of a program the type of which is a given formula. We present a model of automata that can handle proof construction in full intuitionistic first-order logic. The automata are constructed in such a way that any successful run corresponds directly to a normal proof in the logic. This makes it possible to discuss formal languages of proofs or programs, the closure properties of the automata and their connections with the traditional logical connectives.

In the paper we make an attempt to reconstruct Nikolai Vasiliev’s logical systems by means of modern symbolic logic. We put forward the formalization of Vasiliev’s assertoric syllogistic with three initial types of categorical propositions and the Law of excluded fourth. We set up the axiomatic calculus and the adequate semantics for it in order to reconstruct the main version of Vasiliev’s imaginary logic. Metatheoretic relations between Vasiliev’s logic of n dimensions and n-valued predicate logic are established. We also make clear that Vasiliev has developed a sketch of original system of intensional logic (as an alternative version of imaginary logic). We offer the calculus and semantics for the explication of this idea. (The part of this paper was published earlier. In: Markin V.I. What Trends in Non-Classical Logic Were Anticipated by Nikolai Vasiliev? Logical Invest 19:122–135, 2013.)

We study a predicate extension of an unbounded real valued propositional logic that has been recently introduced. The latter, in turn, can be regarded as an extension of both the abelian logic and of the propositional continuous logic. Among other results, we prove that our predicate extension satisfies the property of weak completeness (the equivalence between satisfiability and consistency) and, under an additional assumption on the set of premisses, the property of strong completeness (the equivalence between logical consequence and provability). Eventually we discuss some topological properties of the space of types in our logic.

This paper is devoted to systematic studies of some extensions of first-order G\"odel logic. The first extension is the first-order rational G\"odel logic which is an extension of first-order G\"odel logic, enriched by countably many nullary logical connectives. By introducing some suitable semantics and proof theory, it is shown that the first-order rational G\"odel logic has the completeness property, that is any (strongly) consistent theory is satisfiable. Furthermore, two notions of entailment and strong entailment are defined and their relations with the corresponding notion of proof is studied. In particular, an approximate entailment-compactness is shown. Next, by adding a binary predicate symbol $d$ to the first-order rational G\"odel logic, the ultrametric logic is introduced. This serves as a suitable framework for analyzing structures which carry an ultrametric function $d$ together with some functions and predicates which are uniformly continuous with respect to the ultrametric $d$. Some model theory is developed and to justify the relevance of this model theory, the Robinson joint consistency theorem is proven.

Open logic is an attractive logic theory that can describe the growth and evolution of knowledge. However, related studies show that open logic is undecidable in first-order logic and thus is hard to be programmed. This paper proposes that open logic should be implemented with constraints, and provides a set of syntax constraints under which open logic is decidable. Furthermore, it is shown that the constraints are necessary and sufficient for the decidable formulas of open logic. A Domino problem-based algorithm R-CP that implements the R-calculus of the constrained open logic is presented and its reachability is proved.

A universally free logic is a system of quantification theory, with or without identity, whose theses remain logically true if (a) the domain of quantification is empty and (b) some of the singular terms present in the language do not denote existing objects. In the West, (inclusive) logics satisfying (a) and (free) ones satisfying (b) were developed starting in the 1950s. But Stanislaw Jaśkowski preceded all this work by some twenty years: his paper "On the Rules of Supposition in Formal Logic" of 1934 can be regarded as containing, at least implicitly, the first universally free logic. The system proposed there is an inclusive logic as it is, and a straightforward extension of it is also free.

The goal of this paper is to provide a detailed reading of John Venn's Logic of Chance (1866) as a work of logic or, more specifically, as a specific portion of the general system of so-called ‘material’ logic developed in his Principles of Empirical or Inductive Logic (1889) and to discuss it against the background of his Boolean-inspired views on the connection between logic and mathematics. It is by means of this situating of Venn 1866 [The Logic of Chance. An Essay on the Foundations and Province of the Theory of Probability. With Especial Reference to Its Application to Moral and Social Science, London: Macmillan] within the entirety of his oeuvre that it becomes both possible to revisit and necessary to re-articulate its place in the history of the frequency interpretation of probability. For it is clear that if Venn's approach to logic not only allowed him to establish its foundations on the basis of a process of idealization and define it as consisting of so-called hypothetical infinite series, bu...

After recalling the distinction between logic as reasoning and logic as theory of reasoning, we first examine the question of relativity of logic arguing that the theory of reasoning as any other science is relative. In a second part we discuss the emergence of universal logic as a general theory of logical systems, making comparison with universal algebra and the project of mathesis universalis. In a third part we critically present three lines of research connected to universal logic: logical pluralism, non-classical logics and cognitive science.

This note motivates a logic for a theory that can express its own notion of logical consequence—a ‘syntactically closed’ theory of naive validity. The main issue for such a logic is Curry’s paradox, which is averted by the failure of contraction. The logic features two related, but different, implication connectives. A Hilbert system is proposed that is complete and non-trivial.

This paper examines Hegel's accounts of the nature of judgements and inferences in the ‘subjective logic’ of the Science of Logic, and does so in light of the history of the tradition of formal logic to his time. It is argued that, contrary to the attitude often displayed by interpreters of Hegel's logic, it is important to understand the positive role played by formal logic, ‘logic commonly so called’, in Hegel's own conception of logic. It is argued that Hegel's own scientific presentation [Darstellung] of logic relies on a dialectic working through the tradition of formal logic from Aristotle to Leibniz. The positions within the dialectic are most easily brought into focus in terms of the distinction between Aristotelian and Stoic logic, but they can also be seen as internal to Aristotelian logic. The logical tradition can be regarded as presenting a type of reductio ad absurdum, and a science of logic must examine what it was about Aristotle's original project that brought it to this fate.

The logic of proofs is a refinement of modal logic introduced by Artemov in 1995 in which the modality ◻A is revisited as ⟦t⟧A where t is an expression that bears witness to the validity of A. It enjoys arithmetical soundness and completeness and is capable of reflecting its own proofs (⊦A implies ⊦ ⟦t⟧A, for some t). We develop the Hypothetical Logic of Proofs, a reformulation of LP based on judgemental reasoning Fil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y Tecnología; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Instituto Tecnológico de Buenos Aires; Argentina Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina

We describe an infinitary logic for metric structures which is analogous to L ω 1 , ω . We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.

We present a general method for inserting proofs in Frege systems for classical logic that produces systems that can internalize their own proofs.

The Halpern--Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We complete this classification by showing ---what we believe is quite an unexpected result---that the logic of subintervals, the fragment of the Halpern--Shoham logic where only the operator “during”, or D, is allowed, is undecidable over discrete structures. This is surprising as this, apparently very simple, logic is decidable over dense orders and its reflexive variant is known to be decidable over discrete structures. Our result subsumes a lot of previous undecidability results of fragments that include D.

Classical logic is typically concerned with abstract analysis. The problem for a synthetic logic is to transcend and unify available data to reconstruct the object as a totality. Three rules are proposed to pass from classic logic to synthetic logic. We present the category logic of qualitative opposition using examples from various sciences. This logic has been defined to include the neuter as part of qualitative opposition. The application of these rules to qualitative opposition, and, in particular, its neuter, demonstrated that a synthetic logic allows the truth of some contradictions. This synthetic logic is dialectical with a multi-valued logic, which gives every proposition a truth value in the interval [0,1] that is the square of the modulus of a complex number. In this dialectical logic, contradictions of the neuter of an opposition may be true.