48,412 research outputs found
Metalogic and the Overgeneration Argument
A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic
Basic Logic and Quantum Entanglement
As it is well known, quantum entanglement is one of the most important
features of quantum computing, as it leads to massive quantum parallelism,
hence to exponential computational speed-up. In a sense, quantum entanglement
is considered as an implicit property of quantum computation itself. But...can
it be made explicit? In other words, is it possible to find the connective
"entanglement" in a logical sequent calculus for the machine language? And
also, is it possible to "teach" the quantum computer to "mimic" the EPR
"paradox"? The answer is in the affirmative, if the logical sequent calculus is
that of the weakest possible logic, namely Basic logic. A weak logic has few
structural rules. But in logic, a weak structure leaves more room for
connectives (for example the connective "entanglement"). Furthermore, the
absence in Basic logic of the two structural rules of contraction and weakening
corresponds to the validity of the no-cloning and no-erase theorems,
respectively, in quantum computing.Comment: 10 pages, 1 figure,LaTeX. Shorter version for proceedings
requirements. Contributed paper at DICE2006, Piombino, Ital
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