Quantifier Elimination for a Class of Intuitionistic Theories

On elimination of quantifiers in some non‐classical mathematical theories

“…The additional ternary relation restrictions necessary for the logic R-mingle, or RM, are as follows (the axioms enforced by the principles are included): 6 • For any…”

“…3 Our toy theory will be the theory of dense linear orders without endpoints (henceforth, DLO), which is the simplest around. 4 This paper is (fundamentally) a contribution to the study of paraconsistent mathematical theories (i.e., theories where from a sentence and its negation not everything follows). Our work is motivated by the need of understanding just how different the behaviour of these theories is from that of their classical counterparts.…”

“…Quantifier elimination has been recently studied in for the also non‐classical context of intuitionistic logic.…”

“…Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back. 1 Quantifier elimination has been recently studied in [4] for the also non-classical context of intuitionistic logic. 2 Indeed, as can be extracted from § 5, if = is taken as a non-logical symbol then the pure theory of equality will certainly fail to have quantifier elimination.…”

“…[16][17][18][19]. 4 Details on the classical model theoretic properties of DLO, and other mathematical theories considered here can be found in [10], or any standard model theory textbook. Our axioms for RCF in § 3.2 are found in [22].…”