We show that there are continuum many different extensions of SCI (the basic theory of non-Fregean propositional logic) that lie below WF (the Fregean extension) and are closed under substitution. Moreover, continuum many of them are independent from WB (the Boolean extension), continuum many lie above WB and are independent from WH (the Boolean extension with only two values for the equality relation), and only countably many lie between WH and WF.

The development of the method of logical matrices at the turn of 19th Century made it possible to define the concept of many-valued logic. Since the first construction of the system of three-valued logic by ukasiewicz in 1918 several matrix based logics have been proposed, cf. [8]. The aim of the present paper is to touch upon some problems related to the topic, which would permit one to get a viewpoint upon the nature of many-valuedness. First, we show that the (...) multiplication of logical values is not a sufficient condition to obtain a non-two-valued logic. Second, we discuss an ingenious solution by R. Suszko [11] explaining through the sentential identity an ontological nature of non-classical logical values. Next, we present a kind of metalogical relation of inference, so-called q-consequence, being three-valued in its spirit. The last chapter will bring a concise description of two ukasiewicz “manyvalued” systems of modalities and an application of the paradigm of q-consequence to these systems. (shrink)