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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).

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    $\begingroup$ No. See Polish notation: no parentheses. $\endgroup$ Apr 11 at 15:34

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All we know is that a,d are function symbols, but we don't know their unitary.

That is not true. When we specify a first-order language, the arities of the function symbols are part of the information we must specify.

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    $\begingroup$ Quite so, but you have to admit that we often cheat and let function symbols like the minus sign be unary and binary (but parsers can easily sort that potential source of amibiguity out). $\endgroup$
    – Rob Arthan
    Apr 11 at 19:59
  • $\begingroup$ @Ted When we write a constant symbol, do we need to also specify its ary, i.e., a 0-ary function, so that it can be recognized as a constant instead of a variable? $\endgroup$
    – William
    Apr 12 at 6:29
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    $\begingroup$ @William Yes. For each symbol, you need to specify exactly what kind of symbol it is. That is part of the specification of the language. $\endgroup$
    – Ted
    Apr 12 at 12:53
  • $\begingroup$ @Ted but I find we only need two pieces of information when writing symbols in a formula: the symbol itself and the ary of the symbol(if it has). We don't even need to know the kind of the symbol; we can decide its kind based on the two pieces of information of all the symbols in the formula. $\endgroup$
    – William
    Apr 12 at 13:21

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