Quasi-set theory by S. French and D. Krause (2006) has been so far the most promising attempt of a formal theory of non-individuals. However, due to its sharp bivalent truth valuations, maximally fine-grained binary relations are readily found, in which members of equivalence classes are substitutable for each other in formulas salva veritate. Hence its mentioning and non-mentioning of individuals differs from existing set theory with defined identity merely by the range of nominal definitions. On a semantic level, quasi-set theory does not provide an interpretation with explanatory power of its language terms as non-individuals, and it is not easy to see how such an interpretation can be set up.

We think that the formation and development of the logic of language in China can be divided into three periods: (1) Initial Period (the fifties - the eighties of the 20th century). Zhou Liquan is the initiator and founder of the logic of language research in China; under his advocation and influence, some scholars begin to introduce the logical thought of foreign languages systematically, discussing the objects of research and contents of the logic of language. The representative work of this period is Wang Weixian, Li Xiankun and Chen Zongming’s “The Introduction to the Logic of Language”. (2) Formative Period (1990s - the end of the 20th century). The representative achievements in this period are as follows, Zhou Liquan’s “Logic-A Theory of Accurate Thought and Effective Communication”, Zou Chongli’s “A Part of Lexical System that Utilizes Montague Grammar and General Quantifiers Theory to Analyse the Quantified Phrases of Chinese”, Cai Shushan’s “Speech Acts and the Logic of Language”. These three achievements indicate The Logic of Language has already formed in China. (3) Developing Period (the beginning of 21st century). It starts with Zou Chongli’s “Logic, Language and Communication” that was published in 2002. From the view of research approach, the studies of logic of language by Zhou Liquan, Li Xiankun and Chen Zongming belong to the Descriptive Logic of Language, and those by Cai Shushan and Zou Chongli belong to the Formal Logic of Language.

In 1934 Jaśkowski and Gentzen independently published their groundbreaking works on Natural Deduction. The aim of this paper is to provide some criteria for division of the diversity of existing systems on some natural subcategories and to show that despite the differences all these systems are descendants of original systems of Jaśkowski and Gentzen. Three criteria are discussed:The kind of items which are building-blocks of the proof.The format of proof.The kind of rules.The first leads to the division of ND into two main classes: F-systems working on (occurences of) formulas and S-systems working on sequents (but not to be confused with sequent calculi). The second distinguishes between T-systems with tree-proofs and L-systems with linear proofs. Finally, the third leads to several minor divisions in the main categories.

The paper develops an axiomatic theory of truth and falsehood operators, including the non-classical case. Their domain is a set of sentences, which then is extended to the set of symbol expressions of the language. In general, sentences do not necessarily have to be two-valued. In case of statements about the truth or falsity of sentences, classical logic is applied. We restrict ourselves to the set of sentences for which the truth and falsehood operators are well-defined. This proposed theory differs from Kripke’s theory of truth. Iterations of truth and falsehood operators are allowed. Thus, the pro-posed theory differs from Tarski’s semantic theory of truth. Note that the use of truth and falsehood operators instead of the corresponding predicates allows avoiding the liar paradox. Non-truthfulness in general does not necessarily mean falsehood. Therefore, truth and falsehood operators will be regarded as logically independent. On the basis of the above considerations, the truth and falsehood operator theory is constructed and formulated and it is further ex-tended to the universe of symbolic expressions.

Routley-Meyer ternary relational semantics (RM-semantics, for short) was introduced in the early seventies of the past century. RM-semantics was intended to model classical relevant logics such as the logic of the relevant conditional R and the logic of Entailment E. But, ever since Routley and Meyer’s first papers on the topic, this essentially malleable semantics has been used for characterizing more general relevant logics or even non-relevant logics. The aim of this paper is to provide an RM-semantics with respect to which Łukasiewicz 3-valued logic Ł3 is sound and complete. Ł3 is understood as (any axiomatization of) the set of all valid formulas in Łukasiewicz 3-valued matrices MŁ3. In this sense, leaning on previous work by us, Ł3 is axiomatized as an extension of Routley and Meyer’s basic positive logic B+, labelled Ł3(B+). And the RM-semantics for Ł3 is actually defined for this particular axiomatization of Ł3(B+). The result presented in the paper is interesting from the Universal Logic perspective, in the sense that it connects Łukasiewicz many-valued logics and similar systems to relevant logics from the point of view of the latter, the 3-termed relational point of view, in particular.

As it is well known, in the forties of the past century, Curry proved that in any logic S closed under Modus Ponens, uniform substitution of propositional variables and the Contraction Law (W), the naïve Comprehension axiom (CA) trivializes S in the sense that all propositions are derivable in S plus CA. Not less known is the fact that, ever since Curry published his proof, theses and rules weaker than W have been shown to cause the same effect as W causes. Among these, the Contraction rule (RW) or the Modus Ponens axiom (MPa), for example, are to be noted. But, moreover, as Brady has proved, even the Generalized Modus Ponens axiom (gMPa) or the Generalized Contraction rule (gRW) give rise to “Curry’s Paradox” under the same circumstances as W does. In some previous work by us, “weak relevant model structures” (wr-ms, for short) are defined on “weak relevant matrices” by generalizing Brady’s model structure MCL built upon Meyer’s Crystal matrix CL. We have proved that wr-ms only verify logics with the “depth relevance condition” (drc). The aim of this paper is to show how to falsify gMPa and gRW (and so, W, RW and MPa) in certain wr-ms. In particular, it will be shown that gMPa is falisfied in any wr-ms and gRW in any wr-ms verifying Routley and Meyer’s basic positive logic B+.