Variants of classical linear logics are presented based on the modal version of new structural rule !?mingle instead of the known rules !weakening and ?weakening. The cut-elimination theorems, the completeness theorems and a characteristic property named the mix separation principle are proved for these logics.

We study interpolation for elementary fragments of classical linear logic. Unlike in intuitionistic logic (see [Renardel de Lavalette, 1989]) there are fragments in linear logic for which interpolation does not hold. We prove interpolation for a lot of fragments and refute it for the multiplicative fragment (→, +), using proof nets and quantum graphs. We give a separate proof for the fragment with implication and product, but without the structural rule of permutation. This is (...) nearly the Lambek calculus. There is an appendix explaining what quantum graphs are and how they relate to proof nets. (shrink)

In this paper, constructions of free algebras corresponding to multiplicative classical linear logic, its affine variant, and their extensions with -contraction () are given. As an application, the cardinality problem of some one-variable linear fragments with -contraction is solved.

In an unpublished note Reddy introduced an extended intuitionistic linear calculus, called LLMS (for Linear Logic Model of State), to model state manipulation via the notions of sequential composition and ‘regenerative values’. His calculus introduces the connective ‘before’ ▹ and an associated modality †, for the storage of objects sequentially reusable. Earlier and independently de Paiva introduced a (collection of) dialectica categorical models for (classical and intuitionistic) Linear Logic, the categories Dial2Set. These categories contain, (...) apart from the structure needed to model linear logic, an extra tensor product functor and an extra comonad structure corresponding to a modality related to the extra tensor product. It is surprising that these works arising from completely different motivations can be related in a meaningful way. In this article, following joint work with Corrêa and Haeusler, we first adapt Reddy's system LLMS providing a commutative version of the connective ‘before’ and its associated modality and then construct a dialectica category on Sets , which we show is a sound model for the modified version of Reddy's the system LLMSc. Moreover, following the work of Tucker, we provide another variant of the Dialectica categories with a non-commutative tensor and its associated modality, which models soundlyLLMS itself. We conclude with some speculation on future applications. (shrink)

In this paper we give a brief treatment of a theory of proofs for a system of Full Intuitionistic Linear Logic. This system is distinct from Classical Linear Logic, but unlike the standard Intuitionistic Linear Logic of Girard and Lafont includes the multiplicative disjunction par. This connective does have an entirely natural interpretation in a variety of categorical models of Intuitionistic Linear Logic. The main proof-theoretic problem arises from the observation of (...) Schellinx that cut elimination fails outright for an intuitive formulation of Full Intuitionistic Linear Logic; the nub of the problem is the interaction between par and linear implication. We present here a term assignment system which gives an interpretation of proofs as some kind of non-deterministic function. In this way we find a system which does enjoy cut elimination. The system is a direct result of an analysis of the categorical semantics, though we make an effort to present the system as if it were purely a proof-theoretic construction. (shrink)

Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the ‘structural’ rules of contraction and weakening, adding a modal operator and adding finer versions of the propositional connectives. Linear logic has fundamental logical interest and applications to computer science, particularly to Petri nets, concurrency, storage allocation, garbage collection and the control structure of logic programs. In addition, there (...) is a direct correspondence between polynomial-time computation and proof normalization in a bounded form of linear logic. In this paper we show that unlike most other propositional logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes PSPACE-complete. We also establish membership in NP for the multiplicative fragment, NP-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for fragments of noncommutative propositional linear logic. (shrink)

A new characterization of all the MV-algebras embedded in a CL-algebra has been presented. A new sequent calculus for Lukasiewicz ℵ0-valued logic is introduced. Some links between this calculus and the sequent calculus for multiplicative additive linear logic are established. It has been shown that Lukasiewicz ℵ0-valued logic can be embedded in a suitable extension of MALL.

We present a game semantics in the style of Lorenzen for Girard's linear logic . Lorenzen suggested that the meaning of a proposition should be specified by telling how to conduct a debate between a proponent P who asserts and an opponent O who denies . Thus propositions are interpreted as games, connectives as operations on games, and validity as existence of a winning strategy for P. We propose that the connectives of linear logic can be (...) naturally interpreted as the operations on games introduced for entirely different purposes by Blass . We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective is weaker than the interpretation implicit in Girard's proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Gödel's Dialectica interpretation , which was connected to linear logic by de Paiva , fits with game semantics. (shrink)

Linear Logic was introduced by Girard as a resource-sensitive refinement of classical logic. In this paper we establish strong connections between natural fragments of Linear Logic and a number of basic concepts related to different branches of Computer Science such as Concurrency Theory, Theory of Computations, Horn Programming and Game Theory. In particular, such complete correlations allow us to introduce several new semantics for Linear Logic and to clarify many results on the (...) complexity of natural fragments of Linear Logic. As a main corollary, any non-deterministic and concurrent computation is proved to be simulated directly within the framework of each of these systems. (shrink)

The main concern of this paper is the design of a noetherian and confluent normalization for LK 2. The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD, delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programming-with-proofs' to classical logic ; it is painless: since we reduce strong normalization and confluence to the same properties for (...) class='Hi'>linear logic using appropriate embeddings ; it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ brings to the fore the latter's defects for these `deconstructive purposes'. (shrink)

Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.

We present a sequent calculus for the modal logic S4, and building on some relevant features of this system we show how S4 can easily be translated into full propositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities only in correspondence with S4 modalities. We discuss the complexity of the decision problem for several classes of linear formulas naturally arising from the proposed (...) translations. (shrink)

We present a semantic study of a family of modal intuitionistic linear systems, providing various logics with both an algebraic semantics and a relational semantics, to obtain completeness results. We call modality a unary operator on formulas which satisfies only one rale (regularity), and we consider any subsetW of a list of axioms which defines the exponential of course of linear logic. We define an algebraic semantics by interpreting the modality as a unary operation on an IL-algebra. (...) Then we introduce a relational semantics based on pretopologies with an additional binary relationr between information states. The interpretation of is defined in a suitable way, which differs from the traditional one in classical modal logic. We prove that such models provide a complete semantics for our minimal modal system, as well as, by requiring the suitable conditions onr (in the spirit of correspondence theory), for any of its extensions axiomatized by any subsetW as above. We also prove an embedding theorem for modal IL-algebras into complete ones and, after introducing the notion of general frame, we apply it to obtain a duality between general frames and modal IL-algebras. (shrink)

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of (...) Kleene's intuitionistic theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems. (shrink)

The logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: γ, xn → y / γ, xm → y. It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such (...) class='Hi'>logic has the finite model property with respect to its algebraic semantics and hence that the logic is decidable. (shrink)

This paper presents a formulation and completeness proof of the resolution-type calculi for the first order fragment of Girard's linear logic by a general method which provides the general scheme of transforming a cutfree Gentzen-type system into a resolution type system, preserving the structure of derivations. This is a direct extension of the method introduced by Maslov for classical predicate logic. Ideas of the author and Zamov are used to avoid skolomization. Completeness of strategies is first (...) established for the Gentzen-type system, and then transferred to resolution. The propositional resolution system was implemented by T. Tammet. (shrink)

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class (...) of Kripke resource models extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the l.. (shrink)

This article presents an analysis of Gödel's dialectica interpretation via a refinement of intuitionistic logic known as linear logic. Linear logic comes naturally into the picture once one observes that the structural rule of contraction is the main cause of the lack of symmetry in Gödel's interpretation. We use the fact that the dialectica interpretation of intuitionistic logic can be viewed as a composition of Girard's embedding of intuitionistic logic into linear (...) class='Hi'>logic followed by de Paiva's dialectica interpretation of linear logic. We then investigate the various properties of the dialectica interpretation, such as the characterisation theorem, and variants of Gödel's interpretation within the linear logic context. The role of contraction in extensions to classical logic, arithmetic and analysis is also discussed. (shrink)

Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally (...) due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics. (shrink)

This article presents an analysis of Gödel's dialectica interpretation via a refinement of intuitionistic logic known as linear logic. Linear logic comes naturally into the picture once one observes that the structural rule of contraction is the main cause of the lack of symmetry in Gödel's interpretation. We use the fact that the dialectica interpretation of intuitionistic logic can be viewed as a composition of Girard's embedding of intuitionistic logic into linear (...) class='Hi'>logic followed by de Paiva's dialectica interpretation of linear logic. We then investigate the various properties of the dialectica interpretation, such as the characterisation theorem, and variants of Gödel's interpretation within the linear logic context. The role of contraction in extensions to classical logic, arithmetic and analysis is also discussed. (shrink)

I present a semantics for the language of first-order additive-multiplicative linear logic, i.e. the language of classical first-order logic with two sorts of disjunction and conjunction. The semantics allows us to capture intuitions often associated with linear logic or constructivism such as sentences = games, SENTENCES = resources or sentences = problems, where “truth” means existence of an effective winning strategy.The paper introduces a decidable first-order logic ET in the above language and gives (...) a proof of its soundness and completeness with respect to this semantics. Allowing noneffective strategies in the latter is shown to lead to classical logic.The semantics presented here is very similar to Blass's game semantics. Although there is no straightforward reduction between the two corresponding notions of validity, my completeness proof can likely be adapted to the logic induced by Blass's semantics to show its decidability, which was the main problem left open in Blass's paper.The reader needs to be familiar with classical logic and arithmetic. (shrink)

RASP is a recent extension to Answer Set Programming (ASP) that permits declarative specification and reasoning on the consumption and production of resources. ASP can be seen as a particular case of RASP. In this paper, we study the relationship between linear logic and RASP problem specification. We prove that RASP programs can be translated into (a fragment of) linear logic, and vice versa. In doing so, we introduce a linear logic representation of default (...) negation as understood in ASP. We are also able to establish a link between linear logic and here-and-there (HT) logic. (shrink)

There is an intimate connection between proofs of the natural deduction systems and typed lambda calculus. It is well-known that in simply typed lambda calculus, the notion of formulae-as-types makes it possible to find fine structure of the implicational fragment of intuitionistic logic, i.e., relevant logic, BCK-logic and linear logic. In this paper, we investigate three classical substructural logics (GL, GLc, GLw) of Gentzen's sequent calculus consisting of implication and negation, which contain some of (...) the right structural rules. In terms of Parigot's -calculus with proper restrictions, we introduce a proof term assignment to these classical substructural logics. According to these notions, we can classify the -terms into four categories. It is proved that well-typed GLx--terms correspond to GLx proofs, and that a GLx--term has a principal type if stratified where x is nil, c, w or cw. Moreover, we investigate embeddings of classical substructural logics into the corresponding intuitionistic substructural logics. It is proved that the Gödel-style translations of GLx--terms are embeddings preserving substructural logics. As by-products, it is obtained that an inhabitation problem is decidable and well-typed GLx--terms are strongly normalizable. (shrink)

A wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes a non-classical truth-value that is ‘contaminating’ in the sense that whenever the value is assigned to a formula φ⁠, any complex formula in which φ appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple (...) contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behavior is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper. (shrink)

Coherence with respect to Kelly–Mac Lane graphs is proved for categories that correspond to the multiplicative fragment without constant propositions of classical linear first-order predicate logic without or with mix. To obtain this result, coherence is first established for categories that correspond to the multiplicative conjunction–disjunction fragment with first-order quantifiers of classical linear logic, a fragment lacking negation. These results extend results of [K. Došen, Z. Petrić, Proof-Theoretical Coherence, KCL Publications , London, 2004 ; (...) K. Došen, Z. Petrić, Proof-Net Categories, Polimetrica, Monza, 2007 ], where coherence was established for categories of the corresponding fragments of propositional classical linear logic, which are related to proof nets, and which could be described as star-autonomous categories without unit objects. (shrink)

After the seminal work of Kraus, Lehmann and Magidor (formally known as the KLM approach) on conditionals and preferential models, many aspects of defeasibility in more complex formalisms have been studied in recent years. Examples of these aspects are the notion of typicality in description logic and defeasible necessity in modal logic. We discuss a new aspect of defeasibility that can be expressed in the case of temporal logic, which is the normality in an execution. In this (...) contribution, we take Linear Temporal Logic (LTL) as case study for this defeasible aspect. LTL has found extensive applications in Computer Science and Artificial Intelligence, notably as a formal framework for representing and verifying computer systems that vary over time. However, some systems may presents exceptions at some innocuous time points where they can be tolerated, or conversely, exceptions at other crucial time points where they need to be addressed. In order to ensure the reliability of such systems, we study a preferential extension of LTL, called defeasible linear temporal logic (LTL ~ ). In the first part of this paper, we show how semantics of KLM's preferential models can be integrated with LTL. We also discuss the addition of non-monotonic temporal operators as a way to formalise defeasible properties of these systems. The second part of this paper is a study of the satisfiability problem of LTL ~ sentences. Based on Sistla and Clarke's work on the complexity of the classical LTL language, we show the bounded-model property of two fragments of LTL ~ language. Moreover, we provide a procedure to check the satisfiability of sentences in both of these fragments. (shrink)

In this paper, we study temporal logic for finite linear structures and surjective bounded morphisms between them. We give a characterisation of such structures by modal formulas and show that every pair of linear structures with a bounded morphism between them can be uniquely characterised by a temporal formula up to an isomorphism. As the main result, we prove Kripke completeness of the logic with respect to the class of finite linear structures with bounded morphisms (...) between them. (shrink)

We present an overview of linear-time temporal logics with Presburger constraints whose models are sequences of tuples of integers. Such formal specification languages are well-designed to specify and verify systems that can be modelled with counter systems. The paper recalls the general framework of LTL over concrete domains and presents the main decidability and complexity results related to fragments of Presburger LTL. Related formalisms are also briefly presented.

We give the precise correspondence between polarized linear logic and polarized classical logic. The properties of focalization and reversion of linear proofs are at the heart of our analysis: we show that the tq-protocol of normalization for the classical systems and perfectly fits normalization of polarized proof-nets. Some more semantical considerations allow us to recover LC as a refinement of multiplicative.

This work contributes to the theory of judgement aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgement aggregation to cope with non-classical logics, we discuss in particular results for the case of Intuitionistic Logic, the Lambek calculus, Linear Logic and Relevant Logics. The motivation for studying judgement aggregation in non-classical logics is that they offer a number of modelling choices to represent agents’ reasoning in aggregation problems. By (...) studying judgement aggregation in logics that are weaker than classical logic, we investigate whether some well-known impossibility results, that were tailored for classical logic, still apply to those weak systems. (shrink)

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.

For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the (...) intuitionistic logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummett's intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences. (shrink)

Pretabularity is the attribute of logics that are not characterised by finite matrices, but all of whose proper extensions are. Two of the first-known pretabular logics were Dummett’s famous super-...

Substructural logics are obtained from the sequent calculi for classical or intuitionistic logic by suitably restricting or deleting some or all of the structural rules (Restall, 2000; Ono, 1998). Recently, this field of research has come to encompass a number of logics - e.g. many fuzzy or paraconsistent logics - which had been originally introduced out of different, possibly semantical, motivations. A finer proof-theoretical analysis of such logics, in fact, revealed that it was possible to subsume them under (...) the previous definition (see e.g. Aguzzoli and Ciabattoni, 2000).Although proof systems for substructural logics are currently being investigated with remarkable success, their algebraic models do not seem equally satisfactory. In fact:(i) such structures are often very weak, i.e. they do not possess many interesting algebraic properties;(ii) as a consequence, their theories of ideals, congruences, and representation are as a rule scarcely developed, or even lacking.In this paper, we address these difficulties. (shrink)

We introduce non-associative linear logic, which may be seen as the classical version of the non-associative Lambek calculus. We define its sequent calculus, its theory of proof-nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.

ABSTRACT The question of axiomatizability of first-order temporal logics is studied w.r.t. different semantics and several restrictions on the language. The validity problem for logics admitting flexible interpretations of the predicate symbols or allowing at least binary predicate symbols is shown to be ?1 1-complete. In contrast, it is decidable for temporal logics with rigid monadic predicate symbols but without function symbols and identity.

ABSTRACT A new type of calculi is proposed for a first order linear temporal logic. Instead of induction-type postulates the introduced calculi contain a similarity saturation principle, indicating some form of regularity in the derivations of the logic. In a finitary case we obtained the finite set of saturated sequents, showing that ?nothing new? can be obtained continuing the derivation process. Instead of the ?-type rule of inference, an infinitary saturated calculus has an infinite set of saturated (...) sequents, showing that only a ?similar? sequents can be obtained continuing the dérivation process. The saturation calculi have some resemblance with resolution-like calculi. (shrink)

Lawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety of non-classical predicate (...) logics. And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural algebraic logic and hyperdoctrinal categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics. (shrink)

We develop a timeout extension of propositional linear temporal logic to specify timing properties of timeout-based models of real-time systems. A timeout is used to model the execution of an action marking the end of a delay. With a view to expressing such timeout constraints, ToLTL uses a dynamic variable to abstract the timeout behaviour in addition to a variable which captures the global clock and some static timing variables which record time instances when discrete events occur. We (...) propose a corresponding timeout tableau for satisfiability checking of the ToLTL formulas. Further we prove that the validity checking for such an extended logic remains PSPACE-complete. Under discrete time semantics, with bounded timeout increments, the model-checking problem which verifies if a ToLTL-formula holds in a timeout Kripke structure is also shown to be PSPACE complete. We further prove that ToLTL, when interpreted over discrete time, can be embedded in the monadic second-order logic with.. (shrink)

We show that logic has more to offer to ontologists than standard first order and modal operators. We first describe some operators of linear logic which we believe are particularly suitable for ontological modeling, and suggest how to interpret them within an ontological framework. After showing how they can coexist with those of classical logic, we analyze three notions of artifact from the literature to conclude that these linear operators allow for reducing the ontological (...) commitment needed for their formalization, and even simplify their logical formulation. (shrink)

The authors of [MOC 00] conjectured that intuitionistic and classical logics are asymptotically identical. Their conjecture concerns the implicational parts of these logics over k variables and is trivially true for k = 1, because implicational parts of intuitionistic and classical logics over one variable are identical. So, it seems to be interesting to investigate the appropriate fragments of these logics for k = 2. The result is obtained by reducing the problem to the same one of Dummett's (...) intermediate linear logic of two variables. Actually, this paper shows the existence of the density of this logic and demonstrates that the linear calculus covers a substantial part of classical propositional one. (shrink)