Definition of first-order language with arbitrary alphabet. Syntax of terms, atomic formulas and their subterms.

*(English)*Zbl 1276.03031Summary: Second of a series of articles laying down the bases for classical first-order model theory. A language is defined basically as a tuple made of an integer-valued function (adicity), a symbol of equality and a symbol for the NOR logical connective. The only requests for this tuple to be a language is that the value of the adicity in = is \(-2\) and that its preimage (i.e. the variables set) in 0 is infinite. Existential quantification will be rendered by mere prefixing a formula with a letter. Then the hierarchy among symbols according to their adicity is introduced, taking advantage of attributes and clusters.

The strings of symbols of a language are depth-recursively classified as terms using the standard approach; technically, this is done here by deploying the ‘-multiCat’ functor and the ‘unambiguous’ attribute previously introduced, and the set of atomic formulas is introduced. The set of all terms is shown to be unambiguous with respect to concatenation; we say that it is a prefix set. This fact is exploited to uniquely define the subterms both of a term and of an atomic formula without resorting to a parse tree.

The strings of symbols of a language are depth-recursively classified as terms using the standard approach; technically, this is done here by deploying the ‘-multiCat’ functor and the ‘unambiguous’ attribute previously introduced, and the set of atomic formulas is introduced. The set of all terms is shown to be unambiguous with respect to concatenation; we say that it is a prefix set. This fact is exploited to uniquely define the subterms both of a term and of an atomic formula without resorting to a parse tree.

##### MSC:

03C07 | Basic properties of first-order languages and structures |

03B35 | Mechanization of proofs and logical operations |

68T15 | Theorem proving (deduction, resolution, etc.) (MSC2010) |

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\textit{M. Caminati}, Formaliz. Math. 19, No. 3, 169--178 (2011; Zbl 1276.03031)

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