# Are "(",")","," indispensable components for terms of first order language?

They are called auxiliary symbols in some books, which seems to say they are not necessary, at least for terms. But without them, how can we determine the categories of symbols in a formula such as:

$$abc\equiv def$$

All we know is that a,d are function symbols, but we don't know their arities. If they are 2-ary function, then b,c, e,f are all constants or variables. If a is a 1-ary function, then b is 1-ary function too, and c is a constant or variable.

You may say by convention, a,b,c,d,e are constant symbols and f is used as a function symbol, and we use v,w.. as variables. But since the constant/variable/function symbols are unlimited and only given by examples in the definition of FOL vocabulary, we cannot expect to use our intuition to determine the categories of each symbol.

If "(",")","," are indispensable symbols in FOL vocabulary, things would be simpler.

in formula $$a(b,c)\equiv d(e(f))$$

a would be an 2-ary function symbol; b,c are constants or variables(still indifferentiable); d,e are 1-ary function symbol; f is a constant or a variable(still indifferentiable).

• No. See Polish notation: no parentheses. Apr 11 at 15:34