## Abstracts

Monday August 13

14h-14h30: E.A-Anna Dietz and S. Hölldobler. Modeling the Suppression Task under Three-Valued Lukasiewicz and Well-Founded Semantics

Two three-valued approaches for modeling human conditional reasoning are presented. The suppression task, a study from psychological reasoning, formalized by Stenning and van Lambalgen, is examined. It is known that the suppression task is adequately solved using logic programs under the three-valued Lukasiewicz semantics. In this paper we examine this approach and compare it with well-founded semantics. We show that, by some modification, both approaches yield to identical results for a specic class of programs.

14h30-15h30: K. Schulz (Invited talk, TBA)

Tuesday August 14

14h-14h30: H. Burnett. A Multi-Valued Delineation Semantics for Absolute Adjectives

In this paper, I present a new solution to the puzzle of the gradability of absolute adjectives within the delineation approach, one that takes into account the empirical observation that these constituents can be used imprecisely or vaguely (cf. Lewis 1979, Pinkal 1995, Kennedy 2007, a.o.). I show that by integrating a simplified version of [8]'s comparison-class based logical system with the similarity-based multi-valued logical framework proposed by Cobreros et al. 2012 to model the vagueness/imprecision associated with these predicates, we can arrive at new logical framework that can treat the absolute/relative distinction without degrees in the ontology.

14h30-15h: J. Zehr and O. Percus. TCS for presuppositions

For several decades now, there have been attempts to account for vagueness and presuppositions in a uniform manner. These attempts are motivated by the fact that the two phenomena have something in common: the lack of a solid truth-value judgment that we need when a presuppositional statement is used in a presupposition failure situation, and when a vague predicate when used to describe a borderline case. In this paper, we extend this enterprise by considering the logical system named TCS3 and developed by Cobreros&al. (2012) to account for vagueness. Cobreros&al. introduce two notions of truth: tolerant truth and strict truth, that allow propositions to be characterized as both true and false, or as neither true nor false. We explore to what degree this system can be applied to treat presuppositional phenomena. In particular, we use the notion of strict truth to derive the presupposition associated with a statement; this way of deriving presuppositions extends naturally to complex sentences and thus leads to an account for the phenomenon of presupposition projection.

15h-15h30: P. Cobreros, P. Egre, D. Ripley and R. van Rooij. How many degrees of truth do we need for vague predicates?

The object of this talk is to make a case for the idea that the right semantic framework for vague predicates is three-valued logic. On the view we favor, classical bivalent logic is too coarse-grained to accommodate vagueness, while standard fuzzy logic is unnecessarily fine-grained. Our discussion will start from Smith's 2008 account of vagueness, in which he argues that 3 values are not enough. We present a 3-valued version of the strict-tolerant theory of vagueness (Cobreros et al. 2012) which we argue satisfies all of Smith's desiderata. One of the advantages of our 3-valued approach is its inclusion of similarity relation in the object-language. We consider the results of adding these relations to fuzzy theories of the sort Smith favors.

Wednesday August 15

14h-14h30: V. Degauquier. Are True and False Not Enough?

Trivalent logic is usually interpreted as a non-classical logic that violates the principle of bivalence. However, this logic can also be interpreted as a bivalent logic that ignores either the principle of completeness or the principle of consistency. In light of these bivalent interpretations, the three-valued interpretation can be understood either in a paraconsistent or paracomplete perspective. With this in mind, we investigate the semantic and proof-theoretic relationships between the paraconsistent and paracomplete conceptions of trivalent logic. More specifically, our purpose is to provide a unified framework for characterizing the notion of logical consequence specific to these conceptions. To do this, we propose a notion of validity and an associated hypersequent-inspired calculus for each of the two conceptions of trivalent logic.

14h30-15h30: A. Avron (invited talk). Using Trivalent Semantics for Paraconsistent Reasoning

We describe a general method for a systematic and modular generation of cut-free calculi for thousands of

paraconsistent logics. The method relies on the use of 3-valued non-deterministic semantics for these logics.

Thurday August 16

14h-14h30: T. Perkov. A trivalent logic that encapsulates intuitionistic and classical logic

A third truth-value is proposed with purpose to distinguish intuitionistic valid formulas within classical validities.

14h30-15h: S. Kuznetsov. Trivalent logics arising from L-models for the Lambek calculus with constants

We consider language models for the Lambek calculus that allows empty antecedents and enrich them with constants for the empty language and for the language containing only the empty word. No complete

calculi are known with respect to these semantics, and in this paper we consider several trivalent systems that arise as fragments of these models' logics.

15h-15h30: B. Konikowska and A. Avron. Reasoning about Rough Sets Using Three Logical Values

The paper presents a logic for reasoning about coveringbased rough sets using three logical values: the value t corresponding to the positive region of a set, the value f — to the negative region, and the undefined value u — to the boundary region of that set. Atomic formulas of the logic represent membership of objects of the universe in rough sets, and complex formulas are built out of the atomic ones using three-valued Kleene connectives. In the paper we provide a strongly sound and complete Gentzen-style sequent calculus for the logic.

Friday August 17

14h-14h30: P. Kulicki and R. Trypuz. Doing the right things – trivalence in deontic action logic

Trivalence is quite natural for deontic action logic, where actions are treated as good, neutral or bad. We present the ideas of trivalent deontic logic after J. Kalinowski and its realisation in a 3-valued logic of M. Fisher and two systems designed by the authors of the paper: a 4-valued logic inspired by N. Belnap’s logic of truth and information and a 3-valued logic based on nondeterministic matrices. Moreover, we com-

bine Kalinowski’s idea of trivalence with deontic action logic based on boolean algebra.

14h30-15h30: G. Malinowski (invited talk). Logical three-valuedness and beyond.

The modern history of many-valuedness starts with Ćukasiewicz’s construction of three-valued

logic. This pioneering, philosophically motivated and matrix based construction, first presented in

1918, was in 1922 extended to n-valued cases, including two infinite. Soon several constructions of

many-valued logic appeared and the history of the topic became rich and interesting. However, as it

is widely known, the problem of interpretation of multiple values is still among vexed questions of

contemporary logic. With the talk, which essentially groups my earlier settlements, I intend to put a

new thread into discussion on the nature of logical many-valuedness. The topics, touched upon, are:

matrices, tautological and non-tautological many-valuedness , Tarski’s structural consequence and

the Lindenbaum-Wójcicki completeness result, which supports the Suszko’s claim on logical two-

valuedness of any structural logic. Futher to that, two facets of many-valuedness – referential and

inferential – are unravelled.

The first fits the standard approach and it results in multiplication of semantic correlates of

sentences, and not logical values in a proper sense. It is based on the matrix approach and results in a

multiple-element referential extensionality. In that paradigm, the central concepts are: the tautological

many-valuedness and many-valued consequence.

The second many-valuedness is a metalogical property of quasi-consequence and refers to

partition of the matrix universe into more than two disjoint subsets, used in the definition of inference,

using the inference rules, which from non-rejected premises lead to the accepted conclusions.