19 research outputs found
On the injectivity of the Leibniz operator
The class of weakly algebrizable logics is defined as the class of logics having
monotonic and injective Leibniz operator. We show that \monotonicity" can-
not be discarded on this definition, by presenting an example of a system with
injective and non monotonic Leibniz operator.
We also show that the non injectivity of the non protoalgebraic inf-sup
fragment of the Classic Propositional Calculus, CPC_{inf,sup}, holds only from the fact that the empty set is a CPC_{inf,sup}-filter.FCT via UIM
A unified relational semantics for intuitionistic logic, basic propositional logic and orthologic with strict implication
In this paper, by slightly generalizing the notion of 'proposition' in
'Propositional Logic and Modal Logic - A Connection via Relational Semantics'
by Shengyang Zhong, we propose a relational semantics of propositional language
with bottom, conjunction and imlication, which unifies the relational semantics
of intuitionistic logic, Visser's basic propositional logic and orthologic with
strict implication. We study the semantic and syntactic consequence relations
and prove the soundness and completeness theorems for eight propositional
logics
On subreducts of subresiduated lattices and logic
Subresiduated lattices were introduced during the decade of 1970 by Epstein
and Horn as an algebraic counterpart of some logics with strong implication
previously studied by Lewy and Hacking. These logics are examples of
subuintuitionistic logics, i.e., logics in the language of intuitionistic logic
that are defined semantically by using Kripke models, in the same way as
intuitionistic logic is defined, but without requiring of the models some of
the properties required in the intuitionistic case. Also in relation with the
study of subintuitionistic logics, Celani and Jansana get these algebras as the
elements of a subvariety of that of weak Heyting algebras.
Here, we study both the implicative and the implicative-infimum subreducts of
subresiduated lattices. Besides, we propose a calculus whose algebraic
semantics is given by these classes of algebras. Several expansions of this
calculi are also studied together to some interesting properties of them
The intensional side of algebraic-topological representation theorems
Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems.Peer reviewe
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