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## Logic, Language, Information, and Computation: 26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, Proceedings

Language and relational models, or L-models and R-models, are two natural classes of models for the Lambek calculus. Completeness w.r.t. L-models was proved by Pentus and completeness w.r.t. R-models by Andréka and Mikulás. It is well known that adding both additive conjunction and disjunction together yields incompleteness, because of the distributive law. The product-free Lambek calculus enriched with conjunction only, however, is complete w.r.t. L-models (Buszkowski) as well as R-models (Andréka and Mikulás). The situation with disjunction turns out to be the opposite: we prove that the product-free Lambek calculus enriched with disjunction only is incomplete w.r.t. L-models as well as R-models. If the empty premises are allowed, the product-free Lambek calculus enriched with conjunction only is still complete w.r.t. L-models but in which the empty word is allowed. Both versions are decidable (PSPACE-complete in fact). Adding the multiplicative unit to represent explicitly the empty word within the L-model paradigm changes the situation in a completely unexpected way. Namely, we prove undecidability for any L-sound extension of the Lambek calculus with conjunction and with the unit, whenever this extension includes certain L-sound rules for the multiplicative unit, to express the natural algebraic properties of the empty word. Moreover, we obtain undecidability for a small fragment with only one implication, conjunction, and the unit, obeying these natural rules. This proof proceeds by the encoding of two-counter Minsky machines.

The Lambek calculus was introduced as a mathematical description of natural languages. The original Lambek calculus is NP-complete (Pentus), while its product-free fragment with only one implication is polynomially decidable (Savateev). We consider Lambek calculus with the additional connectives: conjunction and disjunction. It is known that this system is PSPACE-complete (Kanovich, Kanazawa). We prove, in contrast with the polynomial-time result for the product-free Lambek calculus with one implication, that the derivability problem is still PSPACE-complete even for a very small fragment (∖,∧), including one implication and conjunction only. PSPACE-completeness is also provided for the (∖,∨) fragment, which includes only one implication and disjunction. Categorial grammars based on the original Lambek calculus generate exactly the class of context-free languages (Gaifman, Pentus). The class of languages generated by Lambek grammars extended with conjunction is known to be closed under intersection (Kanazawa), and therefore includes all finite intersections of context-free languages and, moreover, images of such intersections under alphabetic homomorphisms. We show that the same closure under intersection holds for Lambek grammars extended with disjunction, even for our small (∖,∨) fragment.

We present a sequent calculus for the weak Grzegorczyk logic 𝖦𝗈 allowing non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs.

Aristotle was the founder not only logics but also of ontology which he describes in Metaphysics and Categories as a theory of general properties of all entities and categorical aspects they should be analyzed. Meanwhile it is commonly accepted that we inherited from him not one but two different logics: early dialectical logoi of Topics and later formal syllogistic of Prior Analytics. The last considers logics the same way as the modern symbolic logic do. According to J. Bocheński the symbolic logic is “a theory of general objects” (by apt turn in phrase, a "physics of the object in general”) hence logics, as it is interpreted now, has the same subject as ontology. But does Aristotle himself counts that ontology (as it is accepted to speak now) is just a kind of “prolegomenon” to logic? In the paper some aspects of this issue are studied at length.

Contemporary science to date is featuring by an interdisciplinary approach that is claimed in many newest scientic programs. Interdisciplinary interaction according to V. S. Stepin is based \on 'paradigmatic grafting' { transfer of notions of the special scientic picture of the world, as well as investigation ideals and norms, from one scientic discipline to another" [1, p. 307].

Chapter 1 contains 25 mathematical an logical sophisms; the reader is encouraged to find errors in the arguments "proving" the absurd assertions. In Chaper 2, we analyze these sophisms.

The Tbilisi Symposium on Language, Logic and Computation is an interdisciplinary conference at the interface of logic, linguistics and computer science with the goal of sharing new results and developing mutually beneficial relationship between these fields. The Symposium is held biennially in different parts of Georgia. It is organized by the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam in conjunction with the Centre for Language, Logic and Speech and Razmadze Mathematical Institute of the Tbilisi State University.

Carnap took Heidegger to task for the production of ‘philosophical nonsense’. Carnap’s criterion for classifying Heidegger’s assertions as nonsense is rooted in the Logical Positivists' 'principle of verification’. According to this principle, a sentence has literal meaning if and only if the proposition it expresses is either analytic or empirically verifiable. The most obvious (or ironic) criticism of the verification principle is the extent to which it is nonsense on its own terms (i.e. it is neither analytic nor empirically verifiable), but from a Heideggerian point of view the most fruitful critique of the verification principle comes from WVO Quine and Wilfird Sellars: namely, that the verification principle assume words and sentences have a direct relation to a given empirical reality without explaining how that reality is given. And in this essay, I argue that Heidegger's so-called 'philosophical nonsense' represents a concrete attempt to explain the conditions through which empirical reality is presented to human beings such that our signs can meaningfully correspond to it.

The author teaches to awaken creativity in oneself, using emotions as a factor of motivation, explains the concept of critical thinking, gives the reader tools to add / edit publications to increase the clarity and rationality of their own judgments, and also shows where a particular theory is applicable

Peirce aspired for the completeness of his logic cum the theory of signs in his 1903 Lowell Lectures and other late manuscripts. Semeiotic completeness states that everything that is a consequence in logical critic is derivable in speculative grammar. The present paper exposes the reasons why Peirce would fall short of establishing semeiotic completeness and thus why he would not continue seeking a perfect match between the theories of grammar and critic. Some alternative notions are then proposed.