Proof-Theoretic Treatment of Various Negations Based on Natural Deduction

The differences and relations of various negations have attracted interest in logical investigation, and most of the studies have utilized a model-theoretic approach (cf. Dunn 1999, Shramko 2005, Ripley 2009). This study will propose a general framework to treat various negations from the perspective of proof-theoretic semantics (PTS) based on natural deduction. It is well known that when we define logical connectives by inferential rules, it is effectual to focus on the meta-linguistic structures that correspond to the connectives, particularly in the case of sequent calculus (cf. Dos̆en 1989, Sambin et al., 2000). The core meaning of logical connectives is defined by the operational rules that “reflect” the meta-linguistic structures, and the differences among various logics are specified by the structural rules that regulate these structures. According to this idea, some prominent studies of various logics and their negations have been recently offered based on display calculus (cf. Wansing 2010, Onishi 2015). Rummfitt’s system R is a natural deduction system that has meta-linguistic labels (+ and −) expressing assertion and denial (as speech acts), and their coordination rules that regulate the relation between + and − (cf. Rummfitt 2000). In this system R, the labels are introduced mainly to give an adequate definition of negation from the perspective of PTS, and negation is a counterpart of denial in the object language. With the meta-linguistic labels, we can to some extent consider negation as a “reflection” as well as display calculus. While Rummfitt’s motivation is to justify classical logic (and its negation), we propose an ordering, or “kite” of various negations in terms of PTS based on natural deduction, focusing on the relation between the meta-linguistic labels and, as we will see later, some types of proof. A recent study by Suzuki (2015) offers an alternative system, E, to modify the problem of R indicated by Ferreira (2008). Besides + and −, the system E has two additional labels, ! and ?, that express criticism of assertion and denial, respectively. The important point for our purpose is that Suzuki’s four-labeled system seems to be more suitable for treating various negations (and logics) in the same light. Indeed, Suzuki indicates that if we take some types of PTS, first degree entailment, the logic of paradox, and strong Kleene logic, are justifiable as fragments of E. I will supplement this approach, and propose a “kite” of various negations based on E (and its expansion). This examination will intuitively clarify the correspondence between different proof-theoretic interpretations and each negation. In the first section, I will review E and its expansion with co-implication and constructive negations. Subsequently, I will show that in terms of an extended Brouwer-Heyting-Kolmogorov (BHK) interpretation, a proof tree whose conclusion is prefixed with one of the four labels, +, −, !, and ?, can be interpreted as proof, disproof, dual proof, or dual disproof, respectively (cf. Wansing 2010). In the second section, I will elucidate the relation between various negations and proof-theoretic interpretations based on the expansion of E. In the third section, I will examine how the above consideration relates to display calculus, multiple conclusions, and logical pluralism.