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# Modal and Many-Valued Logics

**Modal and Many-Valued Logics**

Workshop

January 31, 2018

École Normale Supérieure

Salle Théodule Ribot

29, rue d'Ulm

75005 Paris

Workshop organized by École Normale Supérieure and Institut d’Histoire et de Philosophie des Sciences et des Techniques, with funding from the University Paris 1 Panthéon-Sorbonne and the project “New Ideas in Mathematical Philosophy” (DEC-ENS).

*Program*

9:00 – 9:50 – Andreas Herzig (CNRS, IRIT), A poor man’s epistemic logic based on propositional assignment and higher-order observation (joint work with Emiliano Lorini and Faustine Maffre)

9:50 – 10:40 – Paul Égré (CNRS, ENS), Varieties of logical consequence and Suszko's Problem (joint work with Emmanuel Chemla)

10:40 – 10:50 – coffee break

10:50 – 11:40 – Allard Tamminga (University of Groningen, Utrecht University), Two-sided sequent calculi for FDE-like four-valued logics

11:40 – 12:30 – Ekaterina Kubyshkina (Paris 1, IHPST), On modal translation of many-valued logics

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*Abstracts*

Andreas Herzig (joint work with Emiliano Lorini and Faustine Maffre)

A poor man’s epistemic logic based on propositional assignment and higher-order observation

We introduce a dynamic epistemic logic that is based on what an agentcan observe, including joint observation and observation of what other agents observe. This generalizes van der Hoek,Wooldridge and colleague’s logics ECLPC(PO) and LRC where it is common knowledge which propositional variables each agent observes. In our logic, facts of the world and their observability can both be modified by assignment programs. We show how epistemic operatorscan be interpreted in this framework and identify the conditions under whichthe principles of positive and negative introspection are valid. Finally, we show how public and private announcements can be expressed and illustrate the latter by the gossip spreading problem.

--

Paul Égré (joint work with Emmanuel Chemla)

Varieties of logical consequence and Suszko's Problem

Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence (following Cobreros et al. 2012’s terminology), allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on the characterization of mixed consequence relations more generally. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic counterparts. Based on those we derive and strengthen maximum-rank results proved recently by French and Ripley (2017), and by Blasio, Wansing and Marcos (2017) in a different setting for logics with various structural properties (reflexivity, transitivity, none, or both). We use those results to discuss the foundational problem of what to admit as a bone fide consequence relation in logic.

--

Allard Tamminga

Two-sided sequent calculi for FDE-like four-valued logics

We present a general modular method to generate cut-free, two-sided sequent calculi for four-valued logics like first degree entailment (FDE). Our method relies on correspondences between truth-table entries and sequent rules. First,we show that for every truth-functional n-ary operator * every truth-table entry f*(x1, . . . ,xn) = y can be characterized in terms of two sequent rules. Consequently, every truth-functional n-ary operator can be characterized in terms of 2x4^n sequent rules. Secondly, we build sequent calculi on the basis of these characterizing sequent rules and prove completeness for every sequent calculus with respect to its particular semantics. Lastly, we show that the 2x4^n sequent rules that characterize an n-ary operator can be systematically reduced to at most four sequent rules.

--

Ekaterina Kubyshkina

On modal translation of many-valued logics

Kooi and Tamminga (2013) define a Translation Manual that converts any formula of any three-valued propositional logic into a modal formula. This Translation Manual permits to show that every three-valued valuation has an equivalent S5-model and vice versa. The authors remark that an analogical result can be introduced for any four-valued propositional logic and modal logic interpreted on minimal models M = <W, N, V>, where N is a universal neighborhood. The universality of the neighborhood imposed on the model, as well as the restriction to a S5-model in a three-valued case, are strong and not always desirable conditions. In our talk we modify the Translation Manual for the case of four-valued logics. This shows that the universality of neighborhoods may be replaced by the D-condition (also known as the consistency condition, it corresponds to the D-axiom). The D-condition is weaker than the universality condition and in some cases (especially if the modal operators are interpreted epistemically) is more desirable. We exemplify our proposal on a specific four-valued logic introduced by Kubyshkina and Zaitsev (2016). Our technique may be also applied to the three-valued case and it provides a way to avoid the restriction to S5-models.

--

*Organizers*

Paul Égré (paul.egre@ens.fr)

Ekaterina Kubyshkina (ekaterina.kubyshkina@univ-paris1.fr)

École Normale Supérieure

Salle Théodule Ribot

29, rue d'Ulm

75005 Paris

Workshop organized by École Normale Supérieure and Institut d’Histoire et de Philosophie des Sciences et des Techniques, with funding from the University Paris 1 Panthéon-Sorbonne and the project “New Ideas in Mathematical Philosophy” (DEC-ENS).

*Program*

9:00 – 9:50 – Andreas Herzig (CNRS, IRIT), A poor man’s epistemic logic based on propositional assignment and higher-order observation (joint work with Emiliano Lorini and Faustine Maffre)

9:50 – 10:40 – Paul Égré (CNRS, ENS), Varieties of logical consequence and Suszko's Problem (joint work with Emmanuel Chemla)

10:40 – 10:50 – coffee break

10:50 – 11:40 – Allard Tamminga (University of Groningen, Utrecht University), Two-sided sequent calculi for FDE-like four-valued logics

11:40 – 12:30 – Ekaterina Kubyshkina (Paris 1, IHPST), On modal translation of many-valued logics

--

*Abstracts*

Andreas Herzig (joint work with Emiliano Lorini and Faustine Maffre)

A poor man’s epistemic logic based on propositional assignment and higher-order observation

We introduce a dynamic epistemic logic that is based on what an agentcan observe, including joint observation and observation of what other agents observe. This generalizes van der Hoek,Wooldridge and colleague’s logics ECLPC(PO) and LRC where it is common knowledge which propositional variables each agent observes. In our logic, facts of the world and their observability can both be modified by assignment programs. We show how epistemic operatorscan be interpreted in this framework and identify the conditions under whichthe principles of positive and negative introspection are valid. Finally, we show how public and private announcements can be expressed and illustrate the latter by the gossip spreading problem.

--

Paul Égré (joint work with Emmanuel Chemla)

Varieties of logical consequence and Suszko's Problem

Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence (following Cobreros et al. 2012’s terminology), allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on the characterization of mixed consequence relations more generally. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic counterparts. Based on those we derive and strengthen maximum-rank results proved recently by French and Ripley (2017), and by Blasio, Wansing and Marcos (2017) in a different setting for logics with various structural properties (reflexivity, transitivity, none, or both). We use those results to discuss the foundational problem of what to admit as a bone fide consequence relation in logic.

--

Allard Tamminga

Two-sided sequent calculi for FDE-like four-valued logics

We present a general modular method to generate cut-free, two-sided sequent calculi for four-valued logics like first degree entailment (FDE). Our method relies on correspondences between truth-table entries and sequent rules. First,we show that for every truth-functional n-ary operator * every truth-table entry f*(x1, . . . ,xn) = y can be characterized in terms of two sequent rules. Consequently, every truth-functional n-ary operator can be characterized in terms of 2x4^n sequent rules. Secondly, we build sequent calculi on the basis of these characterizing sequent rules and prove completeness for every sequent calculus with respect to its particular semantics. Lastly, we show that the 2x4^n sequent rules that characterize an n-ary operator can be systematically reduced to at most four sequent rules.

--

Ekaterina Kubyshkina

On modal translation of many-valued logics

Kooi and Tamminga (2013) define a Translation Manual that converts any formula of any three-valued propositional logic into a modal formula. This Translation Manual permits to show that every three-valued valuation has an equivalent S5-model and vice versa. The authors remark that an analogical result can be introduced for any four-valued propositional logic and modal logic interpreted on minimal models M = <W, N, V>, where N is a universal neighborhood. The universality of the neighborhood imposed on the model, as well as the restriction to a S5-model in a three-valued case, are strong and not always desirable conditions. In our talk we modify the Translation Manual for the case of four-valued logics. This shows that the universality of neighborhoods may be replaced by the D-condition (also known as the consistency condition, it corresponds to the D-axiom). The D-condition is weaker than the universality condition and in some cases (especially if the modal operators are interpreted epistemically) is more desirable. We exemplify our proposal on a specific four-valued logic introduced by Kubyshkina and Zaitsev (2016). Our technique may be also applied to the three-valued case and it provides a way to avoid the restriction to S5-models.

--

*Organizers*

Paul Égré (paul.egre@ens.fr)

Ekaterina Kubyshkina (ekaterina.kubyshkina@univ-paris1.fr)