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By
Wansing, Heinrich
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20 Citations
In this paper nonnormal worlds semantics is presented as a basic, general, and unifying approach to epistemic logic. The semantical framework of nonnormal worlds is compared to the model theories of several logics for knowledge and belief that were recently developed in Artificial Intelligence (AI). It is shown that every model for implicit and explicit belief (Levesque), for awareness, general awareness, and local reasoning (Fagin and Halpern), and for awareness and principles (van der Hoek and Meyer) induces a nonnormal worlds model validating precisely the same formulas (of the language in question).
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By
Shramko, Yaroslav; Wansing, Heinrich
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3 Citations
The famous “slingshot argument” developed by Church, Gödel, Quine and Davidson is often considered to be a formally strict proof of the Fregean conception that all true sentences, as well as all false ones, have one and the same denotation, namely their corresponding truth value: the true or the false. In this paper we examine the analysis of the slingshot argument by means of a nonFregean logic undertaken recently by A.Wóitowicz and put to the test her claim that the slingshot argument is in fact circular and presupposes what it intends to prove. We show that this claim is untenable. Nevertheless, the language of nonFregean logic can serve as a useful tool for representing the slingshot argument, and several versions of the slingshot argument in nonFregean logics are presented. In particular, a new version of the slingshot argument is presented, which can be circumvented neither by an appeal to a Russellian theory of definite descriptions nor by resorting to an analogous “Russellian” theory of λ–terms.
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By
Wansing, Heinrich
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21 Citations
This is a purely conceptual paper. It aims at presenting and putting into perspective the idea of a prooftheoretic semantics of the logical operations. The first section briefly surveys various semantic paradigms, and Section 2 focuses on one particular paradigm, namely the prooftheoretic semantics of the logical operations.
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By
Wansing, Heinrich
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1 Citations
An extension L^{+} of the nonassociative Lambek calculus Lis defined. In L^{+} the restriction to formulaconclusion sequents is given up, and additional left introduction rules for the directional implications are introduced. The system L^{+} is sound and complete with respect to a modification of the ternary frame semantics for L.
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By
Wansing, Heinrich; Kamide, Norihiro
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2 Citations
A new combined temporal logic called synchronized lineartime temporal logic (SLTL) is introduced as a Gentzentype sequent calculus. SLTL can represent the nCartesian product of the set of natural numbers. The cutelimination and completeness theorems for SLTL are proved. Moreover, a display sequent calculus δSLTL is defined.
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By
Odintsov, Sergei P.; Wansing, Heinrich
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5 Citations
This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations,
$${\models_t}$$
and
$${\models_f}$$
, considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN_{3} (Shramko and Wansing, J Philos Logic, 34:121–153, 2005). The solution is based on the fact that a certain algebra isomorphic to SIXTEEN_{3} generates the variety of commutative and distributive bilattices with conflation (Rivieccio, 2010).
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By
Omori, Hitoshi; Wansing, Heinrich
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In this introduction to the special issue “40 years of FDE”, we offer an overview of the field and put the papers included in the special issue into perspective. More specifically, we first present various semantics and proof systems for FDE, and then survey some expansions of FDE by adding various operators starting with constants. We then turn to unary and binary connectives, which are classified in a systematic manner (affirmative/negative, extensional/intensional). Firstorder FDE is also briefly revisited, and we conclude by listing some open problems for future research.
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By
Olkhovikov, Grigory K.; Wansing, Heinrich
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3 Citations
In this paper we consider logical inference as an activity that results in proofs and hence produces knowledge. We suggest to merge the semantical analysis of deliberatively seeingtoitthat from stit theory (Belnap et al. in Facing the future: agents and choices in our indeterminist world, Oxford University Press, New York, 2001) and the semantics of the epistemic logic with justification from (Artemov and Nogina in Journal of Logic and Computation 15:1059–1073, 2005). The general idea is to understand proving that A as seeing to it that a proof of A is (publicly) available. We introduce a semantics of various notions of proving as an activity and present a number of valid principles that relate the various notions of proving to each other and to notions of justified knowledge, implicit knowledge, and possibility. We also point out and comment upon certain principles our semantics fails to validate.
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By
Kamide, Norihiro; Shramko, Yaroslav; Wansing, Heinrich
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1 Citations
In this paper, biintuitionistic multilattice logic, which is a combination of multilattice logic and the biintuitionistic logic also known as Heyting–Brouwer logic, is introduced as a Gentzentype sequent calculus. A Kripke semantics is developed for this logic, and the completeness theorem with respect to this semantics is proved via theorems for embedding this logic into biintuitionistic logic. The logic proposed is an extension of firstdegree entailment logic and can be regarded as a biintuitionistic variant of the original classical multilattice logic determined by the algebraic structure of multilattices. Similar completeness and embedding results are also shown for another logic called biintuitionistic connexive multilattice logic, obtained by replacing the connectives of intuitionistic implication and coimplication with their connexive variants.
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By
Odintsov, Sergei P.; Wansing, Heinrich
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The relationships between various modal logics based on Belnap and Dunn’s paraconsistent fourvalued logic FDE are investigated. It is shown that the paraconsistent modal logic
$$\mathsf{BK}^\Box $$
, which lacks a primitive possibility operator
$$\Diamond $$
, is definitionally equivalent with the logic
$$\mathsf{BK}$$
, which has both
$$\Box $$
and
$$\Diamond $$
as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with
$$\mathsf{BK}^\Box $$
without the absurdity constant. Moreover, a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a fourvalued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic
$$\mathsf{BK}^\Box \times \mathsf{BK}^\Box $$
over the nonmodal vocabulary of MBL. On the way from
$$\mathsf{BK}^\Box $$
to MBL, the Fischer Servistyle modal logic
$$\mathsf{BK}^\mathsf{FS}$$
is defined as the set of all modal formulas valid under a modified standard translation into firstorder FDE, and
$$\mathsf{BK}^\mathsf{FS}$$
is shown to be characterized by the class of all models for
$$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$
. Moreover,
$$\mathsf{BK}^\mathsf{FS}$$
is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for
$$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$
. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that
$$\mathsf{BK}^\mathsf{FS}$$
and
$$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$
are weakly definitionally equivalent.
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By
Wansing, Heinrich
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4 Citations
A particularly appealing feature of intuitionistic propositional logic, IPL, is that it may be regarded as the logic of cumulative research, see [12]. It is sound and complete with respect to the class of all nonempty sets I of information states which are quasiordered by a relation ⊑ of ‘possible expansion’ of these states, and in which atomic formulas established at a certain state are also verified at every possible expansion of that state. There are thus three constraints which are imposed on the basic picture of information states related by ⊑ : (i) the persistence (alias heredity) of atomic information, (ii) the reflexivity, and (iii) the transitivity of ⊑ . The persistence of every intuitionistic formula emerges as the combined effect of (i) and (iii). Although a Kripke frame, that is, a binary relation over a nonempty set, admittedly provides an extremely simple model of information dynamics, and, moreover, each of the conditions (i) — (iii), as well as their combinations, may be of value for reasoning about certain varieties of scientific inquiry, it is nevertheless interesting and reasonable to consider giving up all or some of these conditions. Evidently, conceiving of information progress as a steady expansion of previously acquired insights is extremely idealized and the basic model of such a progress should leave room for incorporating revisions, contractions, and merges of information as well. If persistence is given up, ⊑ can no longer be understood as a relation of possible expansion. This reading, however, may be replaced by more generally thinking of ⊑ as describing a possible development of information states. Development thus need not imply the persistence of information.
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By
Wansing, Heinrich
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A fullcircle theorem^{1} for a given logical system ℒ says that certain proof systems S_{1}, ..., S_{4} for ℒ of the four most important types of inference systems (Hilbertstyle, natural deduction, tableaux, sequent calculi) are all equivalent in the following sense (cf. Figure 1):
Every proof of a wff A from wffs A_{1},..., A_{k} in S_{1} can be transformed into a proof of A from A_{1},..., A_{k} in S_{2};
every proof of A from A_{1},..., A_{n} in S_{4} can be transformed into a proof of A from A_{1},..., A_{k} in S_{1}.
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By
Wansing, Heinrich; Shramko, Yaroslav
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9 Citations
According to Suszko’s Thesis, there are but two logical values, true and false. In this paper, R. Suszko’s, G. Malinowski’s, and M. Tsuji’s analyses of logical twovaluedness are critically discussed. Another analysis is presented, which favors a notion of a logical system as encompassing possibly more than one consequence relation.
[A] fundamental problem concerning manyvaluedness is to know what it really is.
[13, p. 281]
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By
Wansing, Heinrich
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2 Citations
The paper provides a uniform Gentzenstyle prooftheoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem's modal perspective on firstorder logic are considered. The Gentzen systems for these logics augment Belnap's display logic by introduction rules for the existential and the universal quantifier. These rules for ∀x and ∃x are analogous to the display introduction rules for the modal operators □ and ♦ and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal ‘modal’ predicate logic to full firstorder logic, axiomatic extensions are captured by purely structural sequent rules.
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By
Ciabattoni, Agata; Ramanayake, Revantha; Wansing, Heinrich
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6 Citations
This paper presents an overview of the methods of hypersequents and display sequents in the proof theory of nonclassical logics. In contrast with existing surveys dedicated to hypersequent calculi or to display calculi, our aim is to provide a unified perspective on these two formalisms highlighting their differences and similarities and discussing applications and recent results connecting and comparing them.
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