2,071 research outputs found
Resolution in Linguistic Propositional Logic based on Linear Symmetrical Hedge Algebra
The paper introduces a propositional linguistic logic that serves as the
basis for automated uncertain reasoning with linguistic information. First, we
build a linguistic logic system with truth value domain based on a linear
symmetrical hedge algebra. Then, we consider G\"{o}del's t-norm and t-conorm to
define the logical connectives for our logic. Next, we present a resolution
inference rule, in which two clauses having contradictory linguistic truth
values can be resolved. We also give the concept of reliability in order to
capture the approximative nature of the resolution inference rule. Finally, we
propose a resolution procedure with the maximal reliability.Comment: KSE 2013 conferenc
Encoding many-valued logic in {\lambda}-calculus
We extend the well-known Church encoding of two-valued Boolean Logic in
-calculus to encodings of -valued propositional logic (for ) in well-chosen infinitary extensions in -calculus. In case
of three-valued logic we use the infinitary extension of the finite
-calculus in which all terms have their B\"ohm tree as their unique
normal form. We refine this construction for . These -valued
logics are all variants of McCarthy's left-sequential, three-valued
propositional calculus. The four- and five-valued logic have been given
complete axiomatisations by Bergstra and Van de Pol. The encodings of these
-valued logics are of interest because they can be used to calculate the
truth values of infinitary propositions. With a novel application of McCarthy's
three-valued logic we can now resolve Russell's paradox. Since B\"ohm trees are
always finite in Church's original -calculus, we believe
their construction to be within the technical means of Church. Arguably he
could have found this encoding of three-valued logic and used it to resolve
Russell's paradox.Comment: 15 page
First-Order Logical Duality
From a logical point of view, Stone duality for Boolean algebras relates
theories in classical propositional logic and their collections of models. The
theories can be seen as presentations of Boolean algebras, and the collections
of models can be topologized in such a way that the theory can be recovered
from its space of models. The situation can be cast as a formal duality
relating two categories of syntax and semantics, mediated by homming into a
common dualizing object, in this case 2. In the present work, we generalize the
entire arrangement from propositional to first-order logic. Boolean algebras
are replaced by Boolean categories presented by theories in first-order logic,
and spaces of models are replaced by topological groupoids of models and their
isomorphisms. A duality between the resulting categories of syntax and
semantics, expressed first in the form of a contravariant adjunction, is
established by homming into a common dualizing object, now \Sets, regarded
once as a boolean category, and once as a groupoid equipped with an intrinsic
topology. The overall framework of our investigation is provided by topos
theory. Direct proofs of the main results are given, but the specialist will
recognize toposophical ideas in the background. Indeed, the duality between
syntax and semantics is really a manifestation of that between algebra and
geometry in the two directions of the geometric morphisms that lurk behind our
formal theory. Along the way, we construct the classifying topos of a decidable
coherent theory out of its groupoid of models via a simplified covering theorem
for coherent toposes.Comment: Final pre-print version. 62 page
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