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First order languages: further syntax and semantics. (English) Zbl 1276.03032
Summary: Third of a series of articles laying down the bases for classical first-order model theory. Interpretation of a language in a universe set. Evaluation of a term in a universe. Truth evaluation of an atomic formula. Reassigning the value of a symbol in a given interpretation. Syntax and semantics of a non-atomic formula are then defined concurrently. As a consequence, the evaluation of any w.f.f. string and the relation of logical implication are introduced. Depth of a formula. Definition of satisfaction and entailment (aka entailment or logical implication) relations.

##### MSC:
 03C07 Basic properties of first-order languages and structures 03B35 Mechanization of proofs and logical operations 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010)
Mizar
Full Text:
##### References:
 [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990. [2] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537-541, 1990. [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. · Zbl 1364.68157 [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. [5] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213-225, 1992. [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. [7] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990. [8] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990. [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. [11] 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. [12] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. [13] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. [14] 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 [15] Marco B. Caminati. Definition of first order language with arbitrary alphabet. Syntax of terms, atomic formulas and their subterms. Formalized Mathematics, 19(3):169-178, 2011, doi: 10.2478/v10037-011-0026-1. · Zbl 1276.03031 [16] M. B. Caminati. Basic first-order model theory in Mizar. Journal of Formalized Reasoning, 3(1):49-77, 2010. · Zbl 1211.03024 [17] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. [18] H. D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical logic. Springer, 1994. [19] Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992. [20] Jarosław Kotowicz and Yuji Sakai. Properties of partial functions from a domain to the set of real numbers. Formalized Mathematics, 3(2):279-288, 1992. [21] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990. [22] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990. [23] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990. [24] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990. [25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. [26] Edmund Woronowicz. Many-argument relations. Formalized Mathematics, 1(4):733-737, 1990. [27] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. [28] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
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