Abstract
First-order logic augments the expressive power of propositional logic as it links the logical assertions to properties of objects of some non-empty universe: the domain of discourse. This is achieved by allowing the propositional symbols to take arguments that range over elements of the domain of discourse. These are now called predicate symbols and are interpreted as relations on the domain. Elements of the domain of discourse are denoted by terms built up from variables, constants, and functions applied to other terms. First-order logic also expands the lexicon of propositional logic with the quantifiers “for all” and “there exists” that are interpreted consistently with their natural language meaning.
This chapter is devoted to classical first-order logic. Our presentation is similar to the one conducted for propositional logic. We first define the syntax of first-order logic, followed by its semantics. Next we define a proof system for it and present the fundamental theoretical results of soundness and completeness. We also discuss the decision problems related to this logic. The remaining sections of the chapter cover variations and extensions of first-order logic, as well as first-order theories.
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Notes
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This argument assumes a fixed interpretation of equality, but can easily be rephrased without that assumption.
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This is an immediate corollary of Gödel’s incompleteness theorem, see Sect. 4.7.2. Note however that there exists an alternative semantics of second-order logic, Henkin models, that support complete proof systems and results such as compactness. Thisgeneral semantics, as opposed to thestandard semantics that we have been considering, essentially reduce second-order logic to many-sorted first-order logic [9].
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Note that in fact reflexivity is redundant as it follows from symmetry and transitivity.
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Almeida, J.B., Frade, M.J., Pinto, J.S., Melo de Sousa, S. (2011). First-Order Logic. In: Rigorous Software Development. Undergraduate Topics in Computer Science. Springer, London. https://doi.org/10.1007/978-0-85729-018-2_4
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