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.

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)

A collection of all but two of the author's philosophical essays and lectures originally published or presented before August 1976.

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.

In this paper, a family of paraconsistent propositional logics with constructive negation, constructive implication, and constructive co-implication is introduced. Although some fragments of these logics are known from the literature and although these logics emerge quite naturally, it seems that none of them has been considered so far. A relational possible worlds semantics as well as sound and complete display sequent calculi for the logics under consideration are presented.

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 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)

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.

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)

A critical view of the alleged significance of Belnap four-valued logic for reasoning under inconsistent and incomplete information is provided. The difficulty lies in the confusion between truth-values and information states, when reasoning about Boolean propositions. So our critique is along the lines of previous debates on the relevance of many-valued logics and especially of the extension of the Boolean truth-tables to more than two values as a tool for reasoning about uncertainty. The critique also questions the significance of partial (...) logic. (shrink)

In the first chapter, we discuss Dummett’s idea that the notion of truth arises from the one of the correctness of an assertion. We argue that, in a first-order language, the need of defining truth in terms of the notion of satisfaction, which is yielded by the presence of quantifiers, is structurally analogous to the need of a notion of truth as distinct from the one of correctness of an assertion. In the light of the analogy between predicates in Frege (...) and open formulas in Tarksi, we concentrate on the semantic status of predicates. We register a dual attitude of Dummett towards Frege’s ascription of reference to predicates. On the one it is needed to endow quantifiers with their appropriate meaning. On the other hand, the introduction of concepts, as semantic correlate of predicates, smuggles a realist flavor in the overall semantic picture. In concluding the excursus (and with it the chapter), we stress Dummett’s will of developing a semantic picture free from this realist trait. In the second chapter, we present the idea of a proof-theoretic semantics. As for Frege true sentences denotes the truth-value True, so here closed' (i.e. categorical) valid argumentations denotes proofs. If a language contains implication-like operators, in order to characterize the condition of validity of a closed argumentation one has to introduce a notion of validity applying to ‘open’ (i.e. hypothetical) argumentations. We argue that this problem is analogous to the one posed by quantifiers in Frege-Tarksi’s style semantics. That is, implication forces one to introduce notion of validity for argumentations, which is more substantial than the correctness of the assertion of their conclusions. As we saw, the corresponding claim in the truth-based approach—that quantifiers require one to introduce a notion of truth, which is more substantial than the one of an assertion being correct—was the source of realism. Hence, Dummett proposes to reduce the semantic contribution of open argumentations to the one of their ‘closed instances’. In the truth-based perspective, this would correspond to the denial of the need of introducing concepts as the semantic correlates of predicates. We argue that Dummett’s fear, that an irreducible notion of function (represented by the need of ascribing validity to open argumentations) would lead to realism, turns out to be ill-founded. In the third chapter, we discuss the role played by the notion of truth in the anti-realist account. The notion of truth is what the anti-realist needs to cope with the so-called paradox of deduction. The analysis of the paradox yields to distinguishing between the truth of a sentence and the truth of a sentence being recognized. In terms of these conceptual couple, we reconsider the relationship between truth and assertion in an anti-realist perspective. Grounds are provided for a thesis (which was already advanced in chapter two), according to which the notion of the assertion of a sentence being correct is primarily connected only with the canonical means of establishing a sentence. The possibility of establishing a sentence by indirect means is conceptually dependent on the practice of establishing logical relationship of dependence among sentences. That is, the notion of a closed valid non-canonical argumentation is of any theoretical relevance only in presence of a notion of validity applying to open argumentations. In the fourth chapter, we discuss the possibility of characterizing in the proof-theoretic-semantics a notion of refutation. We develop an original characterization of refutations starting from an informal inductive specification of the condition of refutations of logically complex sentences. A sub-structural logic, called dual-intuitionistic logic, stands to this notion in the same relationship in which intuitionistic logic stands to the notion of proof so far consdered. All notions developed in chapter 2 have their corresponding (dual) ones in the framework developed. In particular, the distinctions canonical/non-canonical and closed/open argumentations. In the refutation based perspective, elimination rules have priority over introductions and the (only) assumption over the (many) conclusions. In the conclusions, we indicate the ingredients that an anti-realist approach to meaning should incorporate, in order to avoid the difficulties we registered. The core of an alternative view is a different conception of the relationship between categorical and hypothetical notions, in which the validity of open argumentations is not reduced to that of their instances, but rather it is directly defined. As a limit case, one would get a notion of validity applying to closed argumentations. (shrink)

One approach to the paradoxes of self-referential languages is to allow some sentences to lack a truth value (or to have more than one). Then assigning truth values where possible becomes a fixpoint construction and, following Kripke, this is usually carried out over a partially ordered family of three-valued truth-value assignments. Some years ago Matt Ginsberg introduced the notion of bilattice, with applications to artificial intelligence in mind. Bilattices generalize the structure Kripke used in a very natural way, while making (...) the mathematical machinery simpler and more perspicuous. In addition, work such as that of Yablo fits naturally into the bilattice setting. What I do here is present the general background of bilattices, discuss why they are natural, and show how fixpoint approaches to truth in languages that allow self-reference can be applied. This is not new work, but rather is a summary of research I have done over many years. (shrink)