427 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
Slanted canonicity of analytic inductive inequalities
We prove an algebraic canonicity theorem for normal LE-logics of arbitrary
signature, in a generalized setting in which the non-lattice connectives are
interpreted as operations mapping tuples of elements of the given lattice to
closed or open elements of its canonical extension. Interestingly, the
syntactic shape of LE-inequalities which guarantees their canonicity in this
generalized setting turns out to coincide with the syntactic shape of analytic
inductive inequalities, which guarantees LE-inequalities to be equivalently
captured by analytic structural rules of a proper display calculus. We show
that this canonicity result connects and strengthens a number of recent
canonicity results in two different areas: subordination algebras, and transfer
results via G\"odel-McKinsey-Tarski translations.Comment: arXiv admin note: text overlap with arXiv:1603.08515,
arXiv:1603.0834
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
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