Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values

Abstract

A conditional is natural if it fulfills the three following conditions. (1) It coincides with the classical conditional when restricted to the classical values  and  ; (2) it satisfies the Modus Ponens; and (3) it is assigned a designated value whenever the value assigned to its antecedent is less than or equal to the value assigned to its consequent. The aim of this paper is to provide a "bivalent" Belnap-Dunn semantics for all natural implicative expansions of Kleene's strong 3-valued matrix with two designated elements. (We understand the notion "natural conditional" according to N. Tomova, "A lattice of implicative extensions of regular Kleene's logics",

Reduced Routley–Meyer semantics for the logics characterized by natural implicative expansions of Kleene’s strong 3-valued matrix

“…On the other hand, in Appendix 2, it is proved that the axiomatizations of Lt1, Lt2, Lt3, Lt4, Lt5 and Lt6 given in the present section are (deductively) equivalent (cf. Definition 2.3) to the original ones provided in [14] and [15].…”

“…By 9, 10 and 12, we obtain (13) ¬( ∨ ) ∨ ( ∨ ) ∈ . Now, (14) ¬( ∨ )  ∈  is derivable from 11 by Definition 7.3, whence (15)  ∨  ∈  follows by 13 and primeness of . But 5 and 8 contradict 15.…”

“…[18]). In [14] and [15], it is presented a "bivalent" Belnap-Dunn semantics (cf. [3], [4], [7] and [8]) for all natural implicative expansions of Kleene's strong 3-valued matrix (cf.…”

“…[14]) as well as with only one designated value (cf. [15]). Well then, the aim of this paper is to provide a Routley-Meyer semantics for the logics characterized by all natural implicative expansions of Kleene's strong 3-valued matrix (with both only one and also two designated values) susceptible to be interpreted in this type of semantics.…”

“…Then, Kleene's strong 3-valued matrix MK3 (our label) is defined. In §3, the notion of a "natural conditional" (according to [18]) is introduced and all natural implicative expansions of MK3 (with both one and also two designated values), defined in [14] and [15], are recalled (we remark that there are stricter notions of "natural" in the literature; cf., e.g., [2]). It develops that only six of these implicative expansions of MK3 verify all axioms and rules of Sylvan and Plumwood's minimal logic B M .…”