11,175 research outputs found
A Meta-Logic of Inference Rules: Syntax
This work was intended to be an attempt to introduce the meta-language for
working with multiple-conclusion inference rules that admit asserted
propositions along with the rejected propositions. The presence of rejected
propositions, and especially the presence of the rule of reverse substitution,
requires certain change the definition of structurality
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
Suszko's Problem: Mixed Consequence and Compositionality
Suszko's problem is the problem of finding the minimal number of truth values
needed to semantically characterize a syntactic consequence relation. Suszko
proved that every Tarskian consequence relation can be characterized using only
two truth values. Malinowski showed that this number can equal three if some of
Tarski's structural constraints are relaxed. By so doing, Malinowski introduced
a case of so-called mixed consequence, allowing the notion of a designated
value to vary between the premises and the conclusions of an argument. In this
paper we give a more systematic perspective on Suszko's problem and on mixed
consequence. First, we prove general representation theorems relating
structural properties of a consequence relation to their semantic
interpretation, uncovering the semantic counterpart of substitution-invariance,
and establishing that (intersective) mixed consequence is fundamentally the
semantic counterpart of the structural property of monotonicity. We use those
to derive maximum-rank results proved recently in a different setting by French
and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various
structural properties (reflexivity, transitivity, none, or both). We strengthen
these results into exact rank results for non-permeable logics (roughly, those
which distinguish the role of premises and conclusions). We discuss the
underlying notion of rank, and the associated reduction proposed independently
by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve
compositionality in general, meaning that the resulting semantics is no longer
truth-functional. We propose a modification of that notion of reduction,
allowing us to prove that over compact logics with what we call regular
connectives, rank results are maintained even if we request the preservation of
truth-functionality and additional semantic properties.Comment: Keywords: Suszko's thesis; truth value; logical consequence; mixed
consequence; compositionality; truth-functionality; many-valued logic;
algebraic logic; substructural logics; regular connective
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
Professor Grzegorz Malinowski in Honorem
Zadanie „ Wdrożenie platformy Open Journal System dla czasopisma „ Bulletin of the Section of Logic” finansowane w ramach umowy 948/P-DUN/2016 ze środków Ministra Nauki i Szkolnictwa Wyższego przeznaczonych na działalność upowszechniającą naukę
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