545 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
Epistemic systems and Flagg and Friedman's translation
In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of
the faithfulness of G\"{o}del (or Rasiowa-Sikorski) translation
of Heyting arithmetic to Shapiro's epistemic arithmetic . In
\S 2, we shall prove the faithfulness of without using
stability, by introducing another translation from an epistemic system to
corresponding intuitionistic system which we shall call \it the modified
Rasiowa-Sikorski translation\rm . That is, this introduction of the new
translation simplifies the original Flagg and Friedman's proof. In \S 3, we
shall give some applications of the modified one for the disjunction property
() and the numerical existence property () of
Heyting arithmetic. In \S 4, we shall show that epistemic Markov's rule
in is proved via . So and are equivalent. In \S 5, we
shall give some relations among the translations treated in the previous
sections. In \S 6, we shall give an alternative proof of Glivenko's theorem. In
\S 7, we shall propose several(modal-)epistemic versions of Markov's rule for
Horsten's modal-epistemic arithmetic . And, as in \S 4, we shall study
some meta-implications among those versions of Markov's rules in and
one in . Friedman and Sheard gave a modal analogue (i.e.
Theorem in \cite{fs}) of Friedman's theorem (i.e. Theorem 1 in
\cite {friedman}): \it Any recursively enumerable extension of which
has also has \rm . In \S 8, we shall give a proof
of our \it Fundamental Conjecture \rm proposed in Inou\'{e}
\cite{ino90a} as follows: This is a new type of proofs. In \S 9, I
shall give discussions.Comment: 33 page
A Galois connection between classical and intuitionistic logics. I: Syntax
In a 1985 commentary to his collected works, Kolmogorov remarked that his
1932 paper "was written in hope that with time, the logic of solution of
problems [i.e., intuitionistic logic] will become a permanent part of a
[standard] course of logic. A unified logical apparatus was intended to be
created, which would deal with objects of two types - propositions and
problems." We construct such a formal system QHC, which is a conservative
extension of both the intuitionistic predicate calculus QH and the classical
predicate calculus QC.
The only new connectives ? and ! of QHC induce a Galois connection (i.e., a
pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying
posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation
translation of propositions into problems extends to a retraction of QHC onto
QH; whereas Goedel's provability translation of problems into modal
propositions extends to a retraction of QHC onto its QC+(?!) fragment,
identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic
modal logic, whose modality !? is a strict lax modality in the sense of Aczel -
and thus resembles the squash/bracket operation in intuitionistic type
theories.
The axioms of QHC attempt to give a fuller formalization (with respect to the
axioms of intuitionistic logic) to the two best known contentual
interpretations of intiuitionistic logic: Kolmogorov's problem interpretation
(incorporating standard refinements by Heyting and Kreisel) and the proof
interpretation by Orlov and Heyting (as clarified by G\"odel). While these two
interpretations are often conflated, from the viewpoint of the axioms of QHC
neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a
simplified version of Isabelle's meta-logic
Inferentialism
This article offers an overview of inferential role semantics. We aim
to provide a map of the terrain as well as challenging some of the inferentialist’s
standard commitments. We begin by introducing inferentialism and
placing it into the wider context of contemporary philosophy of language. §2
focuses on what is standardly considered both the most important test case
for and the most natural application of inferential role semantics: the case
of the logical constants. We discuss some of the (alleged) benefits of logical
inferentialism, chiefly with regards to the epistemology of logic, and consider
a number of objections. §3 introduces and critically examines the most influential
and most fully developed form of global inferentialism: Robert Brandom’s
inferentialism about linguistic and conceptual content in general. Finally, in
§4 we consider a number of general objections to IRS and consider possible
responses on the inferentialist’s behalf
Topic-Sensitive Epistemic 2D Truthmaker ZFC and Absolute Decidability
This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance an epistemic two-dimensional truthmaker semantics, if hyperintenisonal approaches are to be preferred to possible worlds semantics. I examine the relation between epistemic truthmakers and epistemic set theory
Stone-Type Dualities for Separation Logics
Stone-type duality theorems, which relate algebraic and
relational/topological models, are important tools in logic because -- in
addition to elegant abstraction -- they strengthen soundness and completeness
to a categorical equivalence, yielding a framework through which both algebraic
and topological methods can be brought to bear on a logic. We give a systematic
treatment of Stone-type duality for the structures that interpret bunched
logics, starting with the weakest systems, recovering the familiar BI and
Boolean BI (BBI), and extending to both classical and intuitionistic Separation
Logic. We demonstrate the uniformity and modularity of this analysis by
additionally capturing the bunched logics obtained by extending BI and BBI with
modalities and multiplicative connectives corresponding to disjunction,
negation and falsum. This includes the logic of separating modalities (LSM), De
Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics
extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as
corollaries soundness and completeness theorems for the specific Kripke-style
models of these logics as presented in the literature: for DMBI, the
sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene
BI (connecting our work to Concurrent Separation Logic), this is the first time
soundness and completeness theorems have been proved. We thus obtain a
comprehensive semantic account of the multiplicative variants of all standard
propositional connectives in the bunched logic setting. This approach
synthesises a variety of techniques from modal, substructural and categorical
logic and contextualizes the "resource semantics" interpretation underpinning
Separation Logic amongst them
Mathematical Logic: Proof theory, Constructive Mathematics
The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
A Substructural Epistemic Resource Logic: Theory and Modelling Applications
We present a substructural epistemic logic, based on Boolean BI, in which the
epistemic modalities are parametrized on agents' local resources. The new
modalities can be seen as generalizations of the usual epistemic modalities.
The logic combines Boolean BI's resource semantics --- we introduce BI and its
resource semantics at some length --- with epistemic agency. We illustrate the
use of the logic in systems modelling by discussing some examples about access
control, including semaphores, using resource tokens. We also give a labelled
tableaux calculus and establish soundness and completeness with respect to the
resource semantics
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