Grzegorczyk’s Non-Fregean Logics and Their Formal Properties

The paper discusses Grzegorczyk’s logic LD of descriptive equivalence, some of its extensions (logics LDD and LDT), and its more recent modifications, which are the logic of equimeaning LE and the logic of descriptions with Suszko’s axioms LDS. We present an improved semantics for LD and prove a corresponding extended soundness and completeness theorem. We also show that LD is paraconsistent. These results generalize to LDD, LDT, and LDS as well. We briefly study the properties of LE. Furthermore, we compare the strengths of the logics and prove, in particular, that LD is uncomparable with LE and LDS, and the logic LDD—the extension of LD with the so called Delusion Axiom—is the strongest among the logics in question. Next we show that descriptive equivalence can be defined in terms of descriptive implication in LDT but not in LD. We prove also that if we identify the descriptive equivalence with the implication of the other logics, then LD, LDD, and LDT are different from intuitionistic logic and relevance logics T, E, R, EM, RM. Moreover, descriptive equivalence cannot be identified with necessary equivalence in any class of Kripke frames. Finally, we study different ways to formulate the idea of extensionality, presenting three different extensionality principles and exploring which logics satisfy each of them.