In most systems, there is an operation that is simply described by juxtaposition, with no special operator.
In Church's system this is the application of a function to its argument, in Russell's system it is conjunction and in algebra it is multiplication. In such systems, the short formulas are less restricted than the short formulas as defined here. It is also common to omit some points in the short formulas, introducing a point wherever it is necessary to fulfill the above conditions (a), (b), (c). [⇐ General Bracketing Theory]
The power of such points can be determined simultaneously with the other power conventions. There is one point that has remained doubtful in the introduction of these points. If a pair of parentheses is adjacent to operators at both ends, one of the parentheses must be "protected" by a dot before its operator, but only by one to satisfy (b); which parenthesis should that be? The following three rules are equivalent:
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TURING, A. M., 1942. The use of dots as brackets in Church’s system. Journal of Symbolic Logic. December 1942. Vol. 7, no. 4, p. 146–156. DOI 10.2307/2268111. [Accessed 10 December 2023].