For example, a ring is a structure of the language $\{+,-,\times,0,1\}$, and a ring is not a structure of the language $\{+,-,\times,\cdot^{-1},0,1\}$ because the domain of the operation $\cdot^{-1}$ of the multiplicative inverse is not the whole. In general, it is not officially possible to introduce a function symbol into a partial function. In this paper, we consider ``a structure in a broad sense'' that allows a partial function as the interpretation of a function symbol, we give its semantics and a Hilbert-style formal deductive system, and we prove the completeness theorem. Regarding sequent calculus and natural deduction, it may not be difficult, but it is an unsolved problem. (This is the Japanese version.)

Japanese version