216 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
On Constructive Connectives and Systems
Canonical inference rules and canonical systems are defined in the framework
of non-strict single-conclusion sequent systems, in which the succeedents of
sequents can be empty. Important properties of this framework are investigated,
and a general non-deterministic Kripke-style semantics is provided. This
general semantics is then used to provide a constructive (and very natural),
sufficient and necessary coherence criterion for the validity of the strong
cut-elimination theorem in such a system. These results suggest new syntactic
and semantic characterizations of basic constructive connectives
Carnap's problem for intuitionistic propositional logic
We show that intuitionistic propositional logic is \emph{Carnap categorical}:
the only interpretation of the connectives consistent with the intuitionistic
consequence relation is the standard interpretation. This holds relative to the
most well-known semantics with respect to which intuitionistic logic is sound
and complete; among them Kripke semantics, Beth semantics, Dragalin semantics,
and topological semantics. It also holds for algebraic semantics, although
categoricity in that case is different in kind from categoricity relative to
possible worlds style semantics.Comment: Keywords: intuitionistic logic, Carnap's problem, nuclear semantics,
algebraic semantics, logical constants, consequence relations, categoricity.
Versions: 3rd version has minor additions, and correction of an error in 2nd
version (not in 1st version
- …