The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years \cite{aA75,nB77,aA89,gP89,sH96}. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle \emph{ex falso quodlibet}, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency.
Topoi are categories which have enough structure to interpret higher order logic. They admit two notions of morphism: logical morphisms which preserve all of the structure and therefore the interpretation of higher order logic, and geometric morphisms which only preserve only some of the structure and therefore only some of the interpretation of higher order logic. The question then arises: what kinds of higher order theories are preserved by geometric morphisms? It is known that certain first order theories called internal geometric theories are preserved by geometric morphisms, and these adm…
We consider a family U of finite universes. The second order quantifier Q_R, means for each u in U quantifying over a set of n(R)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Q_R, ever Q_R is interpretable by quantifying over subsets of u and one to one functions on u both of bounded order, or the logic L(Q_R) (first order logic plus the quantifier Q_R) is undecidable.
The Boolean logic of subsets, usually presented as `propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical." But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of negation and implication in that other mathematical logic of partitions.
In this paper we intend to study implications in their most general form, generalizing different classes of implications including the Heyting implication, sub-structural implications and weak strict implications. Following the topological interpretation of the intuitionistic logic, we will introduce non-commutative spacetimes to provide a more dynamic and subjective interpretation of an intuitionistic proposition. These combinations of space and time are natural sources for well-behaved implications and we will show that their spatio-temporal implications represent any other reasonable abstra…
Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. The question is how to introduce the connective implication in such a logic. We show that this is possible when the orthomodular poset in question is of finite height. The main point is that the corresponding algebra, called implication orthomodular poset, i.e. a poset equipped with a binary operator of implication, corresponds to the original orthomodular poset and this operator is everywhere defined. We present here the complete list of axioms for implication orthomodular posets. This enables us to derive …
The unified correspondence theory for distributive lattice expansion logics (DLE-logics) is specialized to strict implication logics. As a consequence of a general semantic consevativity result, a wide range of strict implication logics can be conservatively extended to Lambek Calculi over the bounded distributive full non-associative Lambek calculus (BDFNL). Many strict implication sequents can be transformed into analytic rules employing one of the main tools of unified correspondence theory, namely (a suitably modified version of) the Ackermann lemma based algorithm $\msf{ALBA}$. Gentzen-st…
It is well-known that intuitionistic logics can be formalized by means of Brouwerian semilattices, i.e. relatively pseudocomplemented semilattices. Then the logical connective implication is considered to be the relative pseudocomplement and conjunction is the semilattice operation meet. If the Brouwerian semilattice has a bottom element 0 then the relative pseudocomplement with respect to 0 is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the Ascending Chain Condition, whi…
In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g. (generalized) orthomodular lattices, and MV-algebras, which admit a natural notion of implication. In fact, it turns out that skew Hilbert algebras play a similar role for (strongly) sectionally pseudocomplemented posets as Hilbert algebras do for relatively pseudocomplemented ones. We will discuss basic properties of closed, dense, and weakly dense elements …
In this paper we investigate the fragment of intuitionistic logic which only uses conjunction (meet) and implication, using finite duality for distributive lattices and universal models. We give a description of the finitely generated universal models of this fragment and give a complete characterization of the up-sets of Kripke models of intuitionistic logic which can be defined by meet-implication-formulas. We use these results to derive a new version of subframe formulas for intuitionistic logic and to show that the uniform interpolants of meet-implication-formulas are not necessarily unifo…
Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras E we investigate a natural implication and prove that the implication reduct of E is term equivalent to E. Then we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then we study effect implication algebras and prove the correspondence between these a…
Both propositional dependence logic and inquisitive logic are expressively complete. As a consequence, every formula with intuitionistic disjunction or intuitionistic implication can be translated equivalently into a formula in the language of propositional dependence logic without these two connectives. We show that although such a (non-compositional) translation exists, neither intuitionistic disjunction nor intuitionistic implication is uniformly definable in propositional dependence logic.
Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serves as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is is equipped with a connective implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain "unsharp" residuation which is formulated on the bas…
The non-deterministic algorithmic procedure PEARL (an acronym for `Propositional variables Elimination Algorithm for Relevance Logic') has been recently developed for computing first-order equivalents of formulas of the language of relevance logics RL in terms of the standard Routley-Meyer relational semantics. It succeeds on a large class of axioms of relevance logics, including all so-called inductive formulas. In the present work we re-interpret PEARL from an algebraic perspective, with its rewrite rules seen as manipulating quasi-inequalities interpreted over Urquhart's relevant algebras, …
Gödel-McKinsey-Tarski and (not quite) Blok-Esakia for Heyting-Lewis Implication
School of Computing The Australian National University（澳大利亚国立大学计算机学院）；Chair For Theoretical Computer Science FAU Erlangen-Nuremberg（埃尔朗根-纽伦堡大学理论计算机科学教席）
Effect algebras were introduced in order to describe the structure of effects, i.e. events in quantum mechanics. They are partial algebras describing the logic behind the corresponding events. It is natural to ask how to introduce the logical connective implication in effect algebras. For lattice-ordered effect algebras this task was already solved by several authors, including the present ones. We concentrate on effect algebras that need not be lattice-ordered since these can better describe the events occurring in the quantum physical system. Although an effect algebra is only partial, we tr…
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}. …
Since orthomodular posets serve as an algebraic axiomatization of the logic of quantum mechanics, it is a natural question how the connective of implication can be defined in this logic. It should be introduced in such a way that it is related with conjunction, i.e. with the partial operation meet, by means of some kind of adjointness. We present here such an implication for which a so-called unsharp residuated poset can be constructed. Then this implication is connected with the operation meet by the so-called unsharp adjointness. We prove that also conversely, under some additional assumptio…
We give a novel approach to proving soundness and completeness for a logic (henceforth: the object-logic) that bypasses truth-in-a-model to work directly with validity. Instead of working with specific worlds in specific models, we reason with eigenworlds (i.e., generic representatives of worlds) in an arbitrary model. This reasoning is captured by a sequent calculus for a meta-logic (in this case, first-order classical logic) expressive enough to capture the semantics of the object-logic. Essentially, one has a calculus of validity for the object-logic. The method proceeds through the perspec…
The intuitionistic implication and hence the notion of function space in constructive disciplines is both non-geometric and impredicative. In this paper we try to solve both of these problems by first introducing weak exponential objects as a formalization for predicative function spaces and then by proposing modal spaces as a way to introduce a natural family of geometric predicative implications based on the interplay between the concepts of time and space. This combination then leads to a brand new family of modal propositional logics with predicative implications and then to topological se…
The logic of bunched implications (BI) is a substructural logic that forms the backbone of separation logic, the much studied logic for reasoning about heap-manipulating programs. Although the proof theory and metatheory of BI are mathematically involved, the formalization of important metatheoretical results is still incipient. In this paper we present a self-contained formalized, in the Coq proof assistant, proof of a central metatheoretical property of BI: cut elimination for its sequent calculus. The presented proof is *semantic*, in the sense that is obtained by interpreting sequents in a…
This paper deals with join-semilattices whose sections, i.e. principal filters, are pseudocomplemented lattices. The pseudocomplement of a\vee b in the section [b,1] is denoted by a\rightarrow b and can be considered as the connective implication in a certain kind of intuitionistic logic. Contrary to the case of Brouwerian semilattices, sections need not be distributive lattices. This essentially allows possible applications in non-classical logics. We present a connection of the semilattices mentioned in the beginning with so-called non-classical implication semilattices which can be converte…
Stellenbosch University（斯泰伦博斯大学）；National Institute for Theoretical and Computational Sciences（国家理论与计算科学研究所）；Université Paris Cité, CNRS, IRIF（巴黎西岱大学，法国国家科学研究中心，IRIF）；National Research Foundation of South Africa（南非国家研究基金会）
The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the "logical" content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system for s-logic, and explore in detail two fragmen…
Three classes of models of QHC, the joint logic of problems and propositions, are constructed, including a class of subset/sheaf-valued models that is related to solutions of some actual problems (such as solutions of algebraic equations). To test the models, we consider a number of principles and rules, which empirically appear to cover all "sufficiently simple" natural conjectures about the behaviour of the operators ! and ?, and include two hypotheses put forward by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these turns out to be either derivable in QHC or equival…
Probabilistic logic programming is a major part of statistical relational artificial intelligence, where approaches from logic and probability are brought together to reason about and learn from relational domains in a setting of uncertainty. However, the behaviour of statistical relational representations across variable domain sizes is complex, and scaling inference and learning to large domains remains a significant challenge. In recent years, connections have emerged between domain size dependence, lifted inference and learning from sampled subpopulations. The asymptotic behaviour of stati…
We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Vaananen, and their compositional semantics due to Hodges. We show how Hodges' semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives. Thi…
In this paper we introduce a new class of fuzzy implications called ($S$,$N$,$T$)-implications inspired in the logical equivalence $p\rightarrow q \equiv \neg(p\wedge\neg q)\vee\neg p$ and present a brief study of some of the main properties that characterize this class. We present methods of obtaining $t$-norms and $t$-conorms from an ($S$,$N$,$T$)-implication and a fuzzy negation.
We present an elementary introduction to a new logic for reasoning about behaviors that occur over time. This logic is based on temporal type theory. The syntax of the logic is similar to the usual first-order logic; what differs is the notion of truth value. Instead of reasoning about whether formulas are true or false, our logic reasons about temporal landscapes. A temporal landscape may be thought of as representing the set of durations over which a statement is true. To help understand the practical implications of this approach, we give a wide variety of examples where this logic is used …
