Sequent calculi for trilattice logics, including those that are determined by the truth entailment, the falsity entailment and their intersection, are given. This partly answers the problems in Shramko-Wansing.

The trilattice SIXTEEN3 introduced in Shramko & Wansing (2005) is a natural generalization of the famous bilattice FOUR2. Some Hilbert-style proof systems for trilattice logics related to SIXTEEN3 have recently been studied (Odintsov, 2009; Shramko & Wansing, 2005). In this paper, three sequent calculi GB, FB, and QB are presented for Odintsovs coordinate valuations associated with valuations in SIXTEEN3. The equivalence between GB, FB, and QB, the cut-elimination theorems for these calculi, and the decidability of B are proved. In addition, (...) it is shown how the sequent systems for B can be extended to cut-free sequent calculi for Odintsov’s LB, which is an extension of B by adding classical implication and negation connectives. (shrink)

In a recent paper we have defined an analytic tableau calculus PL_16 for a functionally complete extension of Shramko and Wansing's logic based on the trilattice SIXTEEN_3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic---such as the relations |=_t, |=_f, and |=_i that each correspond to a lattice order in SIXTEEN_3; and |=, the intersection of |=_t and |=_f,. -/- It turns out that our method of characterising these semantic relations---as (...) intersections of auxiliary relations that can be captured with the help of a single calculus---lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that |=, when restricted to L_{tf}, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano. (shrink)

The work of Arnon Avron and Ofer Arieli has shown a deep relationship between the theory of bilattices and the Belnap-Dunn logic \. This correspondence has been interpreted as evidence that \ is “the” logic of bilattices, a consideration reinforced by the work of Yaroslav Shramko and Heinrich Wansing in which \ is shown to be similarly entrenched with respect to the theories of trilattices and, more generally, multilattices. In this paper, we export Melvin Fitting’s “cut-down” connectives—propositional connectives that (...) “cut down” available evidence—to the case of multilattices and show that two related first degree systems—Harry Deutsch’s four-valued \ and Richard Angell’s \—emerge just as elegantly and are as intimately connected to the theories of bilattices and trilattices as the Belnap-Dunn logic. (shrink)

A sequent calculus for Odintsov’s Hilbert-style axiomatization of a logic related to the trilattice SIXTEEN3 of generalized truth values is introduced. The completeness theorem w.r.t. a simple semantics for is proved using Maehara’s decomposition method that simultaneously derives the cut-elimination theorem for . A first-order extension of and its semantics are also introduced. The completeness and cut-elimination theorems for are proved using Schütte’s method.

A short description of a toy theorem prover for 16-valued trilattice logics. Written for the occasion of my friend's Jan van Eijck's retirement. With a link to a swish interface to the prolog prover.

This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations, \ and \ , considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN 3 . The solution is based on the fact that a certain algebra isomorphic to SIXTEEN 3 generates the variety of commutative and distributive (...) bilattices with conflation. (shrink)

In Belnap's useful 4-valued logic, the set 2 = {T, F} of classical truth values is generalized to the set 4 = ������(2) = {Ø, {T}, {F}, {T, F}}. In the present paper, we argue in favor of extending this process to the set 16 = ᵍ (4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR₂ with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN₃ with an (...) information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN₃ coincide with the logic of FOUR₂, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arieli's and Avron's notion of a logical bilattice and state a number of open problems for future research. (shrink)

In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN 3 with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are a — incoming data is (...) asserted, d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT 4 . Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a –order (truth order) and d –order (falsity order) coincide with first-degree entailment. Finally logic with two kinds of operations ( a –connectives and d –connectives) and consequence relation defined via a –ordering is considered. An adequate axiomatization for this logic is proposed. (shrink)

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of “revenge Liar” arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of (...) higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's “useful four-valued logic”, one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a “bi-and-a-half” lattice which determines seven-valued logics different from Priest's Logic of Paradox. (shrink)

The trilattice SIXTEEN₃ is a natural generalization of the wellknown bilattice FOUR₂. Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃, are presented.

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN3 with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are a — incoming data is asserted, (...) d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT4. Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a–order and d–order coincide with first-degree entailment. Finally logic with two kinds of operations and consequence relation defined via a–ordering is considered. An adequate axiomatization for this logic is proposed. (shrink)

This work treats the problem of axiomatizing the truth and falsity consequence relations, $ \vDash _t $ and $ \vDash _f $, determined via truth and falsity orderings on the trilattice SIXTEEN₃. The approach is based on a representation of SIXTEEN₃ as a twist-structure over the two-element Boolean algebra.

This work treats the problem of axiomatizing the truth and falsity consequence relations, ⊨ t and ⊨ f, determined via truth and falsity orderings on the trilattice SIXTEEN 3 (Shramko and Wansing, 2005). The approach is based on a representation of SIXTEEN 3 as a twist-structure over the two-element Boolean algebra.