134 research outputs found
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
A Short and Readable Proof of Cut Elimination for Two 1st Order Modal Logics
Since 1960s, logicians, philosophers, AI people have cast eyes on modal logic. Among various modal logic systems, propositional provability logic which was established by Godel modeling provability in axiomatic Peano Arithmetic (PA) was the most striking application for mathematicians. After Godel, researchers gradually explored the predicate case in provability logic. However, the most natural application QGL for predicate provability logic is not able to admit cut elimination. Recently, a potential candidate for the predicate provability logic ML3 and its precursors BM and M3 introduced by Toulakis,Kibedi, Schwartz dedicated that A is always closed. Although ML3, BM and M3 are cut free, the cut elimination proof with the unfriendly nested induction of high multiplicity is difficult to understand. In this thesis, I will show a cut elimination proof for all (Gentzenisations) of BM, M3 and ML3, with much more readable inductions of lower multiplicity
The Modal Logics of Kripke-Feferman Truth
We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results
The provability logic of all provability predicates
We prove that the provability logic of all provability predicates is exactly
Fitting, Marek, and Truszczy\'nski's pure logic of necessitation .
Moreover, we introduce three extensions , , and
of and investigate the arithmetical semantics of
these logics. In fact, we prove that , , and
are the provability logics of all provability predicates
satisfying the third condition of the derivabiity conditions, all
Rosser's provability predicates, and all Rosser's provability predicates
satisfying , respectively.Comment: 34 page
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