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.
We present a uniform method of density elimination for several semilinear substructural logics. Especially, the density elimination for the involutive uninorm logic IUL is proved. Then the standard completeness of IUL follows as a lemma by virtue of previous work by Metcalfe and Montagna.
In this paper, three semilinear substructural logics ULw, IULw and HpsUL*w are constructed. Then the completeness of ULw and IULw with respect to classes of finite UL and IUL-algebras, respectively, is proved. Algebraically, non-integral ULw and IULw-algebras have the finite embeddability property, which gives a characterization for finite UL and IUL-algebras. Furthermore, the standard completeness of ULw, IULw and HpsUL*w is proved, which shows that they are substructural fuzzy logics.
In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalizing the well-known Gödel-McKinsey-Tarski translation of intuitionistic logic into {\bf S4}. Došen's translation worked well for (variants of) {\bf BCI} and stronger systems ({\bf BCW}, {\bf BCK}), but not for systems below {\bf BCI}. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (…
We briefly examine the modal formulae that can be derived in Multiplicative Additive Linear Logic (MALL) and some extensions by using Tarksi's extensional modal operators. We also breifly compare this with a substructural form of the modal logic K.
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 completene…
We introduce a sequent calculus FL' for non-commutative substructural logic. It has at most one formula on the right side of sequent, and excludes three structural inference rules, i.e. contraction, weakening and exchange. (FL' is based on our investigations of the Gentzen-style natural deduction for non-commutative substructural logics.) FL' has the same proof strength as the standard sequent calculus FL (Full Lambek), which is the basic sequent calculus for all other substructural logics. For the standard FL, we use Ono's formulation. Although FL' and the standard FL are equivalent, there is…
In this paper, we investigate the proof complexity of a wide range of substructural systems. For any proof system $\mathbf{P}$ at least as strong as Full Lambek calculus, $\mathbf{FL}$, and polynomially simulated by the extended Frege system for some infinite branching super-intuitionistic logic, we present an exponential lower bound on the proof lengths. More precisely, we will provide a sequence of $\mathbf{P}$-provable formulas $\{A_n\}_{n=1}^{\infty}$ such that the length of the shortest $\mathbf{P}$-proof for $A_n$ is exponential in the length of $A_n$. The lower bound also extends to the…
Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics
School of Science, Xi’an Petroleum University（西安石油大学理学院）；School of Mathematics and Data Science, Shaanxi University of Science and Technology（陕西科技大学数学与数据科学学院）；School of Science, Xi Hang University（西航大学理学院）
We investigate the non-elementary computational complexity of a family of substructural logics without contraction. With the aid of the technique pioneered by Lazić and Schmitz (2015), we show that the deducibility problem for full Lambek calculus with exchange and weakening ($\mathbf{FL}_{\mathbf{ew}}$) is not in Elementary (i.e., the class of decision problems that can be decided in time bounded by an elementary recursive function), but is in PR (i.e., the class of decision problems that can be decided in time bounded by a primitive recursive function). More precisely, we show that this prob…
In this paper we study some fragments without implications of the (Hilbert) full Lambek logic $\mathbf{HFL}$ and also some fragments without implications of some of the substructural extensions of that logic. To do this, we perform an algebraic analysis of the Gentzen systems defined by the substructural calculi $\FL_σ$. Such systems are extensions of the full Lambek calculus $\FL$ with the rules codified by a subsequence, $σ$, of the sequence $e w_l w_r c$; where $e$ stands for \emph{exchange}, $w_l$ for \emph{left weakening}, $w_r$ for \emph{right weakening}, and $c$ for \emph{contraction}. …
A categorical equivalence for monadic algebras of first-order substructural logics motivated by Kalman's construction
School of Science, Xi’an Shiyou University（西安石油大学理学院）；School of Science, Xi Hang University（西安航天大学理学院）；CONICET and Departamento de Matematica, Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires（阿根廷布宜诺斯艾利斯省中部国立大学精确科学学院数学系与阿根廷国家科学研究委员会）
In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interp…
Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Actually, they provide a uniform and semantic way to axiomatise all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper we export the machi…
We formulate a general, signature-independent form of the law of the excluded middle and prove that a logic is semisimple if and only if it enjoys this law, provided that it satisfies a weak form of the so-called inconsistency lemma of Raftery. We then show that this equivalence can be used to provide simple syntactic proofs of the theorems of Kowalski and Kracht characterizing the semisimple varieties of FLew-algebras and Boolean algebras with operators, and to extend them to FLe-algebras and Heyting algebras with operators. Moreover, under stronger assumptions this correspondence works at th…
University of Strathclyde（斯特拉斯克莱德大学）；Independent Researcher（独立研究者）；Department of Computer Science, University of Sheffield（谢菲尔德大学计算机科学系）；Heriot-Watt and Southampton Universities（赫瑞-瓦特大学和南安普顿大学）；University of Western Australia（西澳大学）
Moisil logic, having as algebraic counterpart Łukasiewicz-Moisil algebras, provide an alternative way to reason about vague information based on the following principle: a many-valued event is characterized by a family of Boolean events. However, using the original definition of Łukasiewicz-Moisil algebra, the principle does not apply for subalgebras. In this paper we identify an alternative and equivalent definition for the $n$-valued Łukasiewicz-Moisil algebras, in which the determination principle is also saved for arbitrary subalgebras, which are characterized by a Boolean algebra and a fa…
Using the model theory of metric structures, I give an alternative proof of the following result by Thomas: If the Continuum Hypothesis fails then there are power of the continuum many universal sofic groups up to isomorphism. This method is also applicable to universal hyperlinear groups, giving a positive answer to a question posed by Thomas.
