One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
WoLLIC '08 Proceedings of the 15th international workshop on Logic, Language, Information and Computation
Proceedings of the Fourth International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice
Electronic Notes in Theoretical Computer Science (ENTCS)
Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
Information and Computation
Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming
Presenting functors on many-sorted varieties and applications
Information and Computation
Permissive-nominal logic: First-order logic over nominal terms and sets
ACM Transactions on Computational Logic (TOCL)
PNL to HOL: From the logic of nominal sets to the logic of higher-order functions
Theoretical Computer Science
An algebraic presentation of predicate logic
FOSSACS'13 Proceedings of the 16th international conference on Foundations of Software Science and Computation Structures
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The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions about freshness of variables with respect to meta-variables. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within it. We discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation for future implementation.