Parenthesis Notation

The formal language of a logical system usually contains several types of symbols. In infix notation, two different kinds of symbols are used to construct compound formula and to indicate the order of combination. The logical constants such as Ø, Ú are used to construct compound formula, and auxiliary symbols such as ( ) are used to indicate the order of a combination. In Polish notation, there is no need for auxiliary symbols such as ( ), and only one class of symbols, N, C, K, etc., is used as a conjunction to make it function as a parenthesis. Contrary to Polish notation, a new parenthesis notation is put forward in this paper. Parenthesis notation uses only parentheses, and empowers them the function of connectives. More importantly, it is proved in this paper that we can define logical constants such as propositional connectives, quantifiers, modal operators and temporal operators in the same formula by using only parenthesis, which can greatly simplify the initial connectives needed to construct the formal system.