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From http://en.wikipedia.org/wiki/First-order_logic#Non-logical_symbols

  1. Relations of valence 0 can be identified with propositional variables. For example, P, which can stand for any statement.

    Is a relation of valence 0 the empty set? Why can it be identified with a propositional variable?

  2. Function symbols of valence 0 are called constant symbols, and are often denoted by lowercase letters at the beginning of the alphabet a, b, c,... .

    What is a function of valence 0? WHy can it be a constant symbol?

Thanks.

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1 Answer 1

An $n$-ary (or $n$-place) function symbol is an expression which combines with $n$ terms to form another term. (If you like, think of the symbol coming with $n$ slots to be filled in; and when the slots are filled, we then get a complete term.)

For example, in arithmetic the function symbol '$+(\ldots,\ldots)$' combines with the two numerals '$2$' and '$4$' to give us the term '$+(2, 4)$' which denotes, in fact, the number 6.

So what is a $0$-ary (or $0$-place -- or in another jargon, a 0-valence) function symbol? It is an expression which combines with $0$ terms to form a term. That it is to say it is already a term. And since not a variable, it is a constant term.

Similarly, an $n$-ary (or $n$-place) predicate symbol is an expression which combines with $n$ terms to form a complete proposition. So what is a $0$-ary (or $0$-place -- or in another jargon, a $0$-valence) predicate symbol? It is an expression which combines with $0$ terms to form a proposition. That it is to say it is already a proposition.

That's the quick explanation of the Wikipedia quotes. But for more discussion of some interesting issues in the background here, see http://www.logicmatters.net/2011/12/and-who-is-for-nullary-functions/

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