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Definition of first-order language with arbitrary alphabet. Syntax of terms, atomic formulas and their subterms. (English) Zbl 1276.03031
Summary: 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.

##### 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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##### References:
 [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. · Zbl 1364.68157 [3] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589-593, 1990. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. [6] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990. [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. [8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. [9] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990. [10] Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155-167, 2011, doi: 10.2478/v10037-011-0025-2. · Zbl 1276.03030 [11] Marco B. Caminati. First order languages: Further syntax and semantics. Formalized Mathematics, 19(3):179-192, 2011, doi: 10.2478/v10037-011-0027-0. · Zbl 1276.03032 [12] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. [13] Katarzyna Jankowska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5):711-717, 1991. [14] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990. [15] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990. [16] W. Pohlers and T. Glaß. An introduction to mathematical logic. Vorlesungsskriptum, WS, 93, 1992. [17] Marta Pruszyńska and Marek Dudzicz. On the isomorphism between finite chains. Formalized Mathematics, 9(2):429-430, 2001. [18] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990. [19] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. [20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. [21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. [22] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
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