It is shown that the class of reduced matrices of a logic is a 1 st order -class provided the variety associated with has the finite replacement property in the sense of [7]. This applies in particular to all 2-valued logics. For 3-valued logics the class of reduced matrices need not be 1 st order.

The logic I is the sentential logic defined in the language with just implication → by the axiom of reflexivity or identity “φ→φ” and the rule of Modus Ponens “from φ and φ→ψ to infer ψ”. The theorems of this logic are exactly all formulas of the form φ→φ. We argue that this is the simplest protoalgebraic logic, and that in it every set of assumptions encodes in itself not only all its consequences but also their proofs. In this paper (...) we study this logic from the point of view of abstract algebraic logic, and in particular we use it as a relatively natural counterexample to settle some open problems in this theory. It appears that this logic has almost no properties: it is neither equivalential nor weakly algebraizable; it does not have an algebraic semantics; it does not satisfy any form of the Deduction Theorem, other than the most general parameterized and local one that all protoalgebraic logics satisfy; it is not filter‐distributive; and so on. It satisfies some forms of the interpolation property but in a rather trivial way. Very few things are known about its algebraic counterpart, save that its intrinsic variety is the class of all algebras of the similarity type. (shrink)

We give a characterization theorem for multiple-conclusion consequence relations with the conjunctive reading of conclusions. As in the case of disjunctive multiple-conclusion consequence relations, we define consequence relations in terms of sets of two-set partitions of formulae. We see that a binary relation between sets of formulae is a conjunctive multiple-conclusion consequence relation if it is closed under the properties of inclusion, transitivity and reducibility. To prove this result we use only the definition and some basic properties of conjunctive multiple-conclusion (...) consequence relations. (shrink)

The aim of this paper is to study the paraconsistent deductive systemP 1 within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse thatP 1 is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebraS being the unique quasivariety semantics forP 1. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated byS is not equivalent to any algebraizable (...) deductive system. We also show thatP 1 has no algebraic semantics in the sense of Czelakowski. Among other results, we study the variety generated by the algebraS. This enables us to prove in a purely algebraic way that the only proper non-trivial axiomatic extension ofP 1 is the classical deductive systemPC. Throughout the paper we also study those abstract logics which are in a way similar toP 1, and are called hereabstract Sette logics. We obtain for them results similar to those obtained for distributive abstract logics by Font, Verdú and the author. (shrink)

The topic of this thesis lies in the intersection between proof theory and algebraic logic. The main object of discussion, constructive reasoning, was introduced at the beginning of the twentieth century by Brouwer, who followed Kant’s explanation of human intuition of spacial forms and time points: these are constructed step by step in a finite process by certain rules, mimicking constructions with straightedge and compass and the construction of natural numbers, respectively.The aim of the present thesis is to show how (...) classical reasoning, which admits some forms of indirect reasoning, can be made more constructive. The central tool that we are using are induction principles, methods that capture infinite collections of objects by considering their process of generation instead of the whole class. We start by studying the interplay between certain structures that satisfy induction and the calculi for some non-classical logics. We then use inductive methods to prove a few conservation theorems, which contribute to answering the question of which parts of classical logic and mathematics can be made constructive.Abstract prepared by Giulio FellinE-mail: [email protected]: https://researchportal.helsinki.fi/en/publications/constructivisation-through-induction-and-conserva tion. (shrink)

We generalize the notion of consequence relation standard in abstract treatments of logic to accommodate intuitions of relevance. The guiding idea follows the use criterion, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each be used in some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining between multisets. We motivate and state (...) basic definitions of relevant consequence relations, both in single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as well as derivations and theories, guided by the use intuitions, and prove a number of results indicating that the definitions capture the desired results (at least in many cases). (shrink)

In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or natural implication) is strictly functionally complete. Further, finding axiomatizations (...) of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic. (shrink)

The present paper concerns a technical study of PRIEST'S logic of paradox [Pri 79], We prove that this logic has no proper paraconsistent strengthening. It is also proved that the mentioned logic is the largest paraconsistent one satisfaying TARSKI'S conditions for the classical conjunction and disjunction together with DE MORGAN'S laws for negation. Finally, we obtain for the logic of paradox an algebraic completeness result related to Kleene lattices.

In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system ⊢ is algebraized by a variety K, then ⊢ has an inconsistency lemma—in the abstract sense—iff every algebra in K has a dually pseudo‐complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1) ⊢ has a classical inconsistency lemma; (2) ⊢ has a greatest compact theory and K is filtral, i.e., (...) semisimple with EDPC; (3) the compact congruences of any algebra in K form a Boolean lattice; (4) the compact congruences of any A∈K constitute a Boolean sublattice of the full congruence lattice of A. These results extend to quasivarieties and relative congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction‐detachment theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The converses are false. (shrink)

This is the continuation of the paper :417–446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we consider extensions (...) of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property. (shrink)

The best known algebraizable logics with a conjunction and an implication have the property that the conjunction defines a meet semi-lattice in the algebras of their algebraic counterpart. This property makes it possible to associate with them a semi-lattice based deductive system as a companion. Moreover, the order of the semi-lattice is also definable using the implication. This makes that the connection between the properties of the logic and the properties of its semi-lattice based companion is strong. We introduce a (...) class of algebraizable deductive systems that includes those systems, and study some of their properties and of their semi-lattice based companions. We also study conditions which, when satisfied by a deductive system in the class, imply that it is strongly algebraizable. This brings some information on the open area of research of Abstract Algebraic Logic which consists in finding interesting characterizations of classes of algebraizable logics that are strongly algebraizable. (shrink)

We define the category QM $\mathcal {QM}$ script upper Q upper M of quantales and their modules and prove the existence of coproducts, and the existence of pushout and amalgamated coproducts under certain conditions. Then we define the non-full subcategory DS 0 $\mathcal {DS}_0$ script upper D upper S 0 of propositional deductive systems, show that it is equivalent to the one of “real” propositional logics whose morphisms are interpretations (modulo a language translation, when needed), and prove that the coproduct (...) in DS 0 $\mathcal {DS}_0$ script upper D upper S 0 is precisely the deductive system called “logical coproduct” in [17]. Last, we discuss amalgamation in DS 0 $\mathcal {DS}_0$ script upper D upper S 0. (shrink)

We continue the work of Blok and Jónsson by developing the theory of structural closure operators and introducing the notion of a representation between them. Similarities and equivalences of Blok-Jónsson turn out to be bijective representations and bijective structural representations, respectively. We obtain a characterization for representations induced by a transformer. In order to obtain a similar characterization for structural representations we introduce the notions of a graduation and a graded variable of an M-set. We show that several deductive systems, (...) Gentzen systems among them, are graded M-sets having graded variables, and describe the graded variables in each case. In the last section we show that, for a sentential logic, having an algebraic semantics is equivalent to being representable in an equational consequence. This motivates the extension of the notion of having an algebraic semantics for Gentzen systems, hypersequents systems, etc. We prove that if a closure operator is representable by a transformer, then every extension of it is also representable by the same transformer. As a consequence we obtain that if one of these systems has an algebraic semantics, then so does any of its extensions with the same defining equations. (shrink)

This paper provides a finer analysis of the well-known form of the Local Deduction Theorem in contraction-free logics. An infinite hierarchy of its natural strengthenings is introduced and studied. The main results are the separation of its initial four members and the subsequent collapse of the hierarchy.

Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the law of double negation and such that $\neg(a\wedge\neg a)$ is a theorem which can be algebraized by a technique similar to the Tarski-Lindenbaum technique.

This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in (...) terms of an extension property and a syntactical metarule. This is the final part of the investigation started and developed in the papers :417–446, 2010; Arch Math Logic 53:353–372, 2016). (shrink)

ABSTRACT In this paper we study consequence relations on the set of many sided sequents over a propositional language. We deal with the consequence relations axiomatized by the sequent calculi defined in [2] and associated with arbitrary finite algebras. These consequence relations are examples of what we call Gentzen systems. We define a semantics for these systems and prove a Strong Completeness Theorem, which is an extension of the Completeness Theorem for provable sequents stated in [2]. For the special case (...) of the finite linear MV-algebras, the Strong Completeness Theorem was proved in [10], as a consequence of McNaughton's Theorem. The main tool to prove this result for arbitrary algebras is the deduction-detachment theorem for Gentzen systems. (shrink)

This paper examines a series of works by Radzki, who addresses the problem of axiomatizing Łukasiewicz’s groundbreaking three-valued logic Ł3T (Ł3 with Słupecki’s operator T) and n-valued logic Łn for n⩾3. According to Radzki, the solution presented in Słupecki’s textbook proof for the case n = 3 is flawed. Furthermore, Radzki demonstrates that the textbook solutions provided by Rosser and Turquette, as well as by Grigolia, for the case n>3 are also inadequate. As a result of Radzki’s studies, the only (...) indisputable solution is attributed to Tuziak. However, Radzki highlights certain shortcomings in this solution: specifically, Tuziak’s completeness proof is indirect and relies on non-logical algebraic methods. This paper is inspired by the open question posed by Radzki, namely: ‘How exactly we can construct, e.g. the Kalmár-type completeness proof for the Tuziak axiom systems for Łn’? We adopt a proof-theoretic method known as correspondence analysis and present natural deduction systems for Łukasiewicz’s three-valued logic Ł3T and n-valued (n⩾3) logic Łn. The completeness of these systems is established in a unified manner using the one hundred-percent logical Henkin method. Additionally, we provide formal proofs for the formulas Radzki uses to demonstrate the failure of the aforementioned solutions. Finally, we address another issue raised by Radzki: while applying correspondence analysis, we propose the first proof system for Radzki’s logic of atomic transactions AtL, which he defines semantically using the interjunction connective introduced by Blamey. (shrink)

The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules (...) in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal. (shrink)

Neo-Fregeanism aims to provide a possible route to knowledge of arithmetic via Hume’s principle, but this is of only limited significance if it cannot account for how the vast majority of arithmetic knowledge, accrued by ordinary people, is obtained. I argue that Hume’s principle does not capture what is ordinarily meant by numerical identity, but that we can do much better by buttressing plural logic with plural versions of the ancestral operator, obtaining natural and plausible characterizations of various key arithmetic (...) concepts, including finiteness, equinumerosity and addition and multiplication of cardinality—revealing these to be logical concepts, and obtaining much of ordinary arithmetic knowledge as logical knowledge. Supplementing this with an abstraction principle and a simple axiom of infinity (known either empirically or modally) we obtain a full interpretation of arithmetic. (shrink)

In our previous paper Algebraic Logic for Classical Conjunction and Disjunction we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA and of all distributive (...) closure operators overA. In the present paper we study the lattice structure of this last set, give a description of its finite and infinite operations, and obtain a topological representation. We also apply the mentioned isomorphism and other results to obtain proofs with a logical flavour for several new or well-known lattice-theoretical properties, like Hashimoto's characterization of distributive lattices, and Priestley's topological representation of the congruence lattice of a bounded distributive lattice. (shrink)

We define when a ternary term m of an algebraic language \ is called a distributive nearlattice term -term) of a sentential logic \. Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a \-term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras associated with a selfextensional logic with a \-term is a variety, and we obtain (...) that the logic is in fact fully selfextensional. (shrink)

Say that a property is topological if and only if it is invariant under homeomorphism. Homeomorphism would be a successful criterion for the equivalence of logical systems only if every logically significant property of every logical system were topological. Alas, homeomorphisms are sometimes insensitive to distinctions that logicians value: properties such as functional completeness are not topological. So logics are not just devices for exploring closure topologies. One still wonders, though, how much of logic is topological. This essay examines some (...) logically significant properties that are topological (or are topological in some important class). In the process, we learn something about the conditions under which the meaning of a connective can be "given by the connective's role in inference.". (shrink)

The logic FBCI given by linearly ordered BCI-matrices is known not to be an axiomatic extension of the well-known BCI logic. In this paper we axiomatize FBCI by adding a recursively enumerable set of schemes of inference rules to BCI and show that there is no finite axiomatization for FBCI.