MoL201614: Colacito, Almudena (2016) Minimal and Subminimal Logic of Negation. [Report]

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Abstract
Starting from the original formulation of minimal propositional logic proposed by Johansson, this thesis aims to investigate some of its relevant subsystems. The main focus is on negation, defined as a primitive unary operator in the language. Each of the subsystems considered is defined by means of some ‘axioms of nega tion’: different axioms enrich the negation operator with different properties. The basic logic is the one in which the negation operator has no properties at all, except the property of being functional. A Kripke semantics is developed for these subsystems, and the clause for negation is completely determined by a function between upward closed sets. Soundness and completeness with respect to this se mantics are proved, both for Hilbertstyle proof systems and for defined sequent calculus systems. The latter are cutfree complete proof systems and are used to prove some standard results for the logics considered (e.g., disjunction property, Craig’s interpolation theorem). An algebraic semantics for the considered sys tems is presented, starting from the notion of Heyting algebras without a bottom element. An algebraic completeness result is proved. By defining a notion of descriptive frame and developing a duality theory, the algebraic completeness result is transferred into a framebased completeness result which has a more generalized form than the one with respect to Kripke semantics.
Item Type:  Report 

Report Nr:  MoL201614 
Series Name:  Master of Logic Thesis (MoL) Series 
Year:  2016 
Uncontrolled Keywords:  logic, computation 
Subjects:  Logic 
Date Deposited:  12 Oct 2016 14:39 
Last Modified:  12 Oct 2016 14:39 
URI:  https://eprints.illc.uva.nl/id/eprint/986 
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