18 research outputs found
Interpolation and the Exchange Rule
It was proved by Maksimova in 1977 that exactly eight varieties of Heyting
algebras have the amalgamation property, and hence exactly eight axiomatic
extensions of intuitionistic propositional logic have the deductive
interpolation property. The prevalence of the deductive interpolation property
for axiomatic extensions of substructural logics and the amalgamation property
for varieties of pointed residuated lattices, their equivalent algebraic
semantics, is far less well understood, however. Taking as our starting point a
formulation of intuitionistic propositional logic as the full Lambek calculus
with exchange, weakening, and contraction, we investigate the role of the
exchange rule--algebraically, the commutativity law--in determining the scope
of these properties. First, we show that there are continuum-many varieties of
idempotent semilinear residuated lattices that have the amalgamation property
and contain non-commutative members, and hence continuum-many axiomatic
extensions of the corresponding logic that have the deductive interpolation
property in which exchange is not derivable. We then show that, in contrast,
exactly sixty varieties of commutative idempotent semilinear residuated
lattices have the amalgamation property, and hence exactly sixty axiomatic
extensions of the corresponding logic with exchange have the deductive
interpolation property. From this latter result, it follows also that there are
exactly sixty varieties of commutative idempotent semilinear residuated
lattices whose first-order theories have a model completion
Uniform interpolation and coherence
A variety V is said to be coherent if any finitely generated subalgebra of a
finitely presented member of V is finitely presented. It is shown here that V
is coherent if and only if it satisfies a restricted form of uniform deductive
interpolation: that is, any compact congruence on a finitely generated free
algebra of V restricted to a free algebra over a subset of the generators is
again compact. A general criterion is obtained for establishing failures of
coherence, and hence also of uniform deductive interpolation. This criterion is
then used in conjunction with properties of canonical extensions to prove that
coherence and uniform deductive interpolation fail for certain varieties of
Boolean algebras with operators (in particular, algebras of modal logic K and
its standard non-transitive extensions), double-Heyting algebras, residuated
lattices, and lattices
One-variable fragments of first-order logics
The one-variable fragment of a first-order logic may be viewed as an
"S5-like" modal logic, where the universal and existential quantifiers are
replaced by box and diamond modalities, respectively. Axiomatizations of these
modal logics have been obtained for special cases -- notably, the modal
counterparts S5 and MIPC of the one-variable fragments of first-order classical
logic and intuitionistic logic -- but a general approach, extending beyond
first-order intermediate logics, has been lacking. To this end, a sufficient
criterion is given in this paper for the one-variable fragment of a
semantically-defined first-order logic -- spanning families of intermediate,
substructural, many-valued, and modal logics -- to admit a natural
axiomatization. More precisely, such an axiomatization is obtained for the
one-variable fragment of any first-order logic based on a variety of algebraic
structures with a lattice reduct that has the superamalgamation property,
building on a generalized version of a functional representation theorem for
monadic Heyting algebras due to Bezhanishvili and Harding. An alternative
proof-theoretic strategy for obtaining such axiomatization results is also
developed for first-order substructural logics that have a cut-free sequent
calculus and admit a certain interpolation property.Comment: arXiv admin note: text overlap with arXiv:2209.0856
Universal Proof Theory: Semi-analytic Rules and Craig Interpolation
In [6], Iemhoff introduced the notion of a focused axiom and a focused rule
as the building blocks for a certain form of sequent calculus which she calls a
focused proof system. She then showed how the existence of a terminating
focused system implies the uniform interpolation property for the logic that
the calculus captures. In this paper we first generalize her focused rules to
semi-analytic rules, a dramatically powerful generalization, and then we will
show how the semi-analytic calculi consisting of these rules together with our
generalization of her focused axioms, lead to the feasible Craig interpolation
property. Using this relationship, we first present a uniform method to prove
interpolation for different logics from sub-structural logics ,
, and to their appropriate
classical and modal extensions, including the intuitionistic and classical
linear logics. Then we will use our theorem negatively, first to show that so
many sub-structural logics including \L_n, , , and and
almost all super-intutionistic logics (except at most seven of them) do not
have a semi-analytic calculus. To investigate the case that the logic actually
has the Craig interpolation property, we will first define a certain specific
type of semi-analytic calculus which we call PPF systems and we will then
present a sound and complete PPF calculus for classical logic. However, we will
show that all such PPF calculi are exponentially slower than the classical
Hilbert-style proof system (or equivalently ). We will then
present a similar exponential lower bound for a certain form of complete PPF
calculi, this time for any super-intuitionistic logic.Comment: 45 page
Theorems of Alternatives for Substructural Logics
A theorem of alternatives provides a reduction of validity in a substructural
logic to validity in its multiplicative fragment. Notable examples include a
theorem of Arnon Avron that reduces the validity of a disjunction of
multiplicative formulas in the R-mingle logic RM to the validity of a linear
combination of these formulas, and Gordan's theorem for solutions of linear
systems over the real numbers, that yields an analogous reduction for validity
in Abelian logic A. In this paper, general conditions are provided for
axiomatic extensions of involutive uninorm logic without additive constants to
admit a theorem of alternatives. It is also shown that a theorem of
alternatives for a logic can be used to establish (uniform) deductive
interpolation and completeness with respect to a class of dense totally ordered
residuated lattices
Algebraic semantics for one-variable lattice-valued logics
The one-variable fragment of any first-order logic may be considered as a
modal logic, where the universal and existential quantifiers are replaced by a
box and diamond modality, respectively. In several cases, axiomatizations of
algebraic semantics for these logics have been obtained: most notably, for the
modal counterparts S5 and MIPC of the one-variable fragments of first-order
classical logic and intuitionistic logic, respectively. Outside the setting of
first-order intermediate logics, however, a general approach is lacking. This
paper provides the basis for such an approach in the setting of first-order
lattice-valued logics, where formulas are interpreted in algebraic structures
with a lattice reduct. In particular, axiomatizations are obtained for modal
counterparts of one-variable fragments of a broad family of these logics by
generalizing a functional representation theorem of Bezhanishvili and Harding
for monadic Heyting algebras. An alternative proof-theoretic proof is also
provided for one-variable fragments of first-order substructural logics that
have a cut-free sequent calculus and admit a certain bounded interpolation
property
Idempotent residuated structures : some category equivalences and their applications
This paper concerns residuated lattice-ordered idempotent commutative
monoids that are subdirect products of chains. An algebra of this
kind is a generalized Sugihara monoid (GSM) if it is generated by the lower
bounds of the monoid identity; it is a Sugihara monoid if it has a compatible
involution :. Our main theorem establishes a category equivalence
between GSMs and relative Stone algebras with a nucleus (i.e., a closure
operator preserving the lattice operations). An analogous result is obtained
for Sugihara monoids. Among other applications, it is shown that Sugihara
monoids are strongly amalgamable, and that the relevance logic RMt has
the projective Beth de nability property for deduction.http://www.ams.org//journals/tran/hb201
Structure Theorems for Idempotent Residuated Lattices
In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has the amalgamation property. Finally, we address an open problem in the literature by giving an example of a noncommutative variety of idempotent residuated lattices that has the amalgamation property