How to use arities in a sentence? Find typical usage patterns (collocations)/phrases/context for "arities" and check conjugation/comparative form for "arities". Mastering all the usages of "arities" from sentence examples published by news publications.
Higher arities lead to proper trees. For instance, in the term :f(a,g(a),f(a,b,c)), a,b,c are constants, g is unary, and f is ternary. Contrariwise, :f(a,f(a)) cannot be a valid term, as the symbol f appears once as binary, and once as unary, which is illicit, as Arity must be a function.
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The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions. However, it is also possible to admit generalized signatures in which function and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables. Because an infinite formula cannot be represented by a finite string, it is necessary to choose some other representation of formulas; the usual representation in this context is a tree. Thus formulas are, essentially, identified with their parse trees, rather than with the strings being parsed.
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A generalization of first order logic considers languages with more than one sort of variables. The idea is different sorts of variables represent different types of objects. Every sort of variable can be quantified; thus an interpretation for a many-sorted language has a separate domain for each of the sorts of variables to range over (there is an infinite collection of variables of each of the different sorts). Function and relation symbols, in addition to having arities, are specified so that each of their arguments must come from a certain sort.
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In the general case, the algebraic relations need not be associative, in which case the starting point is not the set of all words, but rather, strings punctuated with parentheses, which are used to indicate the non-associative groupings of letters. Such a string may equivalently be represented by a binary tree or a free magma; the leaves of the tree are the letters from the alphabet. The algebraic relations may then be general arities or finitary relations on the leaves of the tree. Rather than starting with the collection of all possible parenthesized strings, it can be more convenient to start with the Herbrand universe.
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