pp. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. First-order logic, second-order logic, and completeness. We take it upon ourselves in this paper to compare the two approaches, second order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. Pure second-order logic is second-order logic without functional or first-order variables. In Vincent Hendricks, Fabian Neuhaus, Stig Andur Pedersen, Uwe Scheffler & Heinrich Wansing (eds. What I do realize is that first-order logic was adopted at a time before its own deficiencies were fully understood. SOL (second-order logic) has two main flavours, one being Henkin semantics and the other being full semantics. 194 14. Marcus Rossberg. Second of all, in order to even ask if the completeness theorem holds for second-order logic, we need to define what a "proof" is in second-order logic. that has two components—first, it incorporates the axioms of a Σ0 1-complete system of arithmetic (in fact, a version of PRA), second, it involves the ω-rule. Both second order logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. Goldrei's Propositional and Predicate Calculus states (in my words; any mistake is mine) that first-order logic is complete, i.e. Second order logic and completeness A; Thread starter jordi; Start date May 15, 2019; May 15, 2019 #1 jordi. Logos. SOL has a computable deductive system that is complete for Henkin semantics (since it can be reduced to FOL), but does not have any computable deductive system for full semantics. As a matter of fact we can show that there is no reasonable notion of proof for second-order logic, at all! ), First-Order Logic Revisited. Second-order logic differs from the usual first-order predicate calculus in that it has variables and quantifiers not only for individuals but also for subsets of the universe (sometimes variables for n-ary relations as well, but this is not important in this context). In propositional logic, the truth tree test will give you a "yes" (all branches closed) for all inferences and a "no" (at least one branch open) for all non-inferences: Propositional logic is decidable. But the completeness of first-order logic was only proved by Kurt Gödel at a time when first-order logic had already displaced second-order logic. 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