Current Research

Proof-Theoretic Research of Coexistence of Different Logics Focusing on the Concepts of Negations

I study a combinations of two logics from proof-theoretic aspects. A logic is called a combination of two logics if its syntax has two types of logical operators and it is a conservative extension of both logics. Especially, I currently focus on a combination of intuitionistic and classical logic in terms of the concepts of negation. The motivation of this research is to properly formulate the discussion between advocates of classical logic and those of intuitionistic logic, the discussion in the field of philosophy of logic, philosophy of mathematics, and philosophy of language.  Moreover, I take into consideration a generalization of intuitionistic logic to subintuitionistic logics and that of classical negation to De Morgan negation.

Related papers: 1, 2, 3, 4, 5, 6 of Journal Papers and Full Papers. 

A Proof-theoretic Analysis of the Meaning of Logical Operators and Formulas 

I study a proof-theoretic analysis of the meaning of logical operators and formulas in intuitionsitic and classical logics and a combination of them. The meaning of logical operators and formulas is analyzed in terms of truth values, but there is an attempt to analyze them by using a proof theory. Such an attempt is called a proof-theoretic analysis of the meaning of logical operators and formulas. Such an analysis can be seen as an actualization of ``meaning as use,'' offered by Ludwig Wittgenstein, in the field of a logical language. A representative of proof-theoretic analyses of the meaning is proof-theoretic semantics, which was proposed mainly by Dag Prawitz and Michael Dummett. It is known that proof-theoretic semantics fits intuitionistic logic. Thus, the discussion between advocates of intuitionistic logic and those of  classical logic partially occurs as a confrontation about the explanation of the meaning. However, there have been studies trying to provide a proof-theoretic analysis of the meaning of classical logical operators and formulas recently. Based on this situation, I study what kind of explanation a proof-theoretic analysis of the meaning should be and what kind of proof-theoretic analysis of the meaning is possible for a combination of intuitionistic and classical logic. Moreover, by analyzing a combination of intuitionistic and classical logic, it is expected that we understand more deeply and properly the discussion between advocates of intuitionistic and classical logics about the explanation of meaning. 

Related papers: 3, 6, 7 of Journal Papers and Full Papers. 

Subintuitionistic Logics and an Addition of Various Logical Operators

I study the addition of various logical operators, especially special negations, to subintuitionistic logics. In the Kripke semantics for intuitionistic logic, the properties called Reflexivity, Transitivity, and Heredity are presupposed. A subintuitionistic logic is obtained by dropping one or some of these semantic properties. Such a logic was first proposed by Giovanna Corsi. There are two motivations to study subintuitionistic logics. The first one lies on various arguments claiming that it is plausible to drop one or some of the semantic properties in order to capture some phenomena. The second one is to provide an umbrella frame work, as in normal modal logics. In normal modal logics, an umbrella framework from K to S5 is provided. By considering subintuitionistic logics, it becomes possible to provide an umbrella framework of logics whose language is the same as intuitionistic logic. Moreover, I add various logical operators to intuitionistic logic in order to realize a combination of intuitionistic and classical logic, as already mentioned. However, it is clearly not ensured that the addition of a logical operator to a subintuitionistic logic has the consequence similar to that to intuitionistic logic. Taking this situation into consideration, I study consequences of the addition of various logical operators, especially special negations, to subintuitionistic logics. 

*These researches are partially supported by Grant-in-Aid for JSPS Fellows. If you would like to know more about this, see Grants, CV, and Others, .