2,190 research outputs found
Constructive Provability Logic
We present constructive provability logic, an intuitionstic modal logic that
validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting
logical reflection over provability. Two distinct variants of this logic, CPL
and CPL*, are presented in natural deduction and sequent calculus forms which
are then shown to be equivalent. In addition, we discuss the use of
constructive provability logic to justify stratified negation in logic
programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl
From Quantum Metalanguage to the Logic of Qubits
The main aim of this thesis is to look for a logical deductive calculus (we
will adopt sequent calculus, originally introduced in Gentzen, 1935), which
could describe quantum information and its properties. More precisely, we
intended to describe in logical terms the formation of the qubit (the unit of
quantum information) which is a particular linear superposition of the two
classical bits 0 and 1. To do so, we had to introduce the new connective
"quantum superposition", in the logic of one qubit, Lq, as the classical
conjunction cannot describe this quantum link.Comment: 138 pages, PhD thesis in Mathematic
Cut Elimination for a Logic with Induction and Co-induction
Proof search has been used to specify a wide range of computation systems. In
order to build a framework for reasoning about such specifications, we make use
of a sequent calculus involving induction and co-induction. These proof
principles are based on a proof theoretic (rather than set-theoretic) notion of
definition. Definitions are akin to logic programs, where the left and right
rules for defined atoms allow one to view theories as "closed" or defining
fixed points. The use of definitions and free equality makes it possible to
reason intentionally about syntax. We add in a consistent way rules for pre and
post fixed points, thus allowing the user to reason inductively and
co-inductively about properties of computational system making full use of
higher-order abstract syntax. Consistency is guaranteed via cut-elimination,
where we give the first, to our knowledge, cut-elimination procedure in the
presence of general inductive and co-inductive definitions.Comment: 42 pages, submitted to the Journal of Applied Logi
Tonk Strikes Backâ
What is a logical constant? In which terms should we characterize the meaning of logical words like âandâ, âorâ, âimpliesâ? An attractive answer is: in terms of their inferential roles, i.e. in terms of the role they play in building inferences. More precisely, we favor an approach, going back to Dosen and Sambin, in which the inferential role of a logical constant is captured by a double line rule which introduces it as reflecting structural links (for example, multiplicative conjunction reflects comma on the right of the turnstyle). Rule-based characterizations of logical constants are subject to the well known objection of Priorâs fake connective, tonk. We show that some double line rules also give rise to such pseudo logical constants. But then, we are able to find a property of a double line rules which guarantee that it defines a genuine logical constant. Thus we provide an alternative answer to Belnapâs requirement of conservatity in terms of a local requirement on double line rules
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