This volume contains a collection of articles applying methods of logic or, more generally, of ma... more This volume contains a collection of articles applying methods of logic or, more generally, of mathematics to solve problems, some of which come from logic itself, others from other sciences. Its range of subjects is far from complete, but broadly representative. The first group of papers in this volume consists of contributions to pure and applied modal logic. The problems discussed here range from the structure of lattices of normal and other modal propositional logics to modal proof theory and to the semantics of quantified modal logic. The second group of papers deals with Many-valued logics - an extensive domain of strictly logical investigations rooting in philosophical questions concerning the nature of logical values. Logical investigations in cognitive science have successfully utilized methods and systems of belief revision, non-monotonic logic and dynamic epistemic logic. Towards Mathematical Philosophy deals with focal issues of belief revision. The volume concludes with contributions which may be seen to belong to the field of formal epistemology, the area applying logical, probabilistic, game-theoretic and other formal methods to problems and issues in epistemology and philosophy of science, such as those concerning anti-realism, skepticism, theory comparison and theory choice, justification, sources of knowledge and learning theories.
In this paper we elaborate a conception of entailment based on what we call the Ackermann princip... more In this paper we elaborate a conception of entailment based on what we call the Ackermann principle, which explicates valid entailment through a logical connection between (sets of) sentences depending on their informational content. We reconstruct Dunn’s informational semantics for entailment on the basis of Restall’s approach, with assertion and denial as two independent (primary) speech acts, by introducing the notion of a ‘position description’. We show how the machinery of position descriptions can effectively be used to define the positive and the negative information carried by sentences of a given language and to present a formalized version of the Ackermann principle as an inclusion relationship between the informational contents of the conclusions and the premises of a valid entailment. Being so interpreted, the entailment relation exhibits certain properties, including the property of transitivity (and, more generally, admissibility of the cut rule). Whereas properties such as Anderson and Belnap’s variable sharing property or Parry’s proscriptive principle are normally presented as imposing a relevance requirement on valid entailment, the suggested formalization of the Ackermann principle supports all of Gentzen’s structural rules, including weakening, a rule that is normally given up in sequent-style proof systems for relevance logics. In this way we propose an Ackermann-inspired explication of the nature of entailment as a relation between the informational contents of sentences.
Livre: Truth and falsehood: an inquiry into generalized logical values (hardback) (series: trends... more Livre: Truth and falsehood: an inquiry into generalized logical values (hardback) (series: trends in logic) SHRAMKO Yaroslav, WANSING Heinrich.
In our everyday life we are holding each other responsible for what we believe and for how we for... more In our everyday life we are holding each other responsible for what we believe and for how we form beliefs. Since there are many similarities between holding an agent responsible for her actions and holding an agent responsible for her beliefs, some epistemologists think that we can model epistemic responsibility, i.e., responsibility for a belief, along the lines ofmoral responsibility, i.e., responsibility for performing an action. Research onmoral responsibility suggests that a person can be held responsible for an action she has performed only if she had control over performing that action. Even though our practice of holding each other responsible for our beliefs is similar to our practice of holding each other responsible for our actions, our formation of a belief appears to be different to our performance of an action when it comes to our capacity to exercise control over it. Beliefs are mostly taken to be more or less passive responses to how the world appears to us, while in performing an action we shape the world, often as we intend it to be. This difference between actions and beliefs raises a problem when it comes to establishing a viable notion of epistemic responsibility along the lines of the notion of moral responsibility. Thus, we need to rethink our notion of epistemic responsibility, the kind of control necessary for responsibility, as well as the connection between epistemic responsibility and doxastic control. This special issue partially documents results of the two-day workshop “Doxastic Agency and Epistemic Responsibility” which took place at the Ruhr University Bochum in June 2014. It addresses questions such as: do we have the same kind of control over our doxastic attitudes as we have over our actions, what kind of control do we have over our doxastic attitudes and how is epistemic responsibility related to notions such as epistemic justification and practical responsibility?Most of the authors
In this introductory note, we place the new essays on Belnap-Dunn logic, FDE, of the present volu... more In this introductory note, we place the new essays on Belnap-Dunn logic, FDE, of the present volume against the background of the development of FDE. This note is an invitation to study the volume. It presents a chronological perspective on Belnap-Dunn logic and a slightly idiosyncratic list of further research topics.
Francesco Berto proposed a logic for imaginative episodes. The logic establishes certain (in)vali... more Francesco Berto proposed a logic for imaginative episodes. The logic establishes certain (in)validities concerning episodic imagination. They are not all equally plausible as principles of episodic imagination. The logic also does not model that the initial input of an imaginative episode is deliberately chosen.Stit-imagination logic models the imagining agent’s deliberate choice of the content of their imagining. However, the logic does not model the episodic nature of imagination. The present paper combines the two logics, thereby modelling imaginative episodes with deliberately chosen initial input. We use a combination ofstit-imagination logic and a content-sensitive variably strict conditional à la Berto, for which we give a Chellas–Segerberg semantics. The proposed semantics has the following advantages over Berto’s: (i) we model thedeliberate choiceof initial input of imaginative episodes (in a multi-agent setting), (ii) we showframe correspondencesfor axiomatic analogues of ...
Abstract In this paper it is suggested to generalize our understanding of general (structural) pr... more Abstract In this paper it is suggested to generalize our understanding of general (structural) proof theory and to consider it as a general theory of two kinds of derivations, namely proofs and dual proofs. The proposal is substantiated by (i) considerations on assertion, denial, and bi-lateralism, (ii) remarks on compositionality in proof-theoretic semantics, and (iii) comments on falsification and co-implication. The main formal result of the paper is a normal form theorem for the natural deduction proof system N2Int of the bi-intuitionistic logic 2Int. The proof makes use of the faithful embedding of 2Int into intuitionistic logic with respect to validity and shows that conversions of dual proofs can be sidestepped.
The method used in Chapter 4 to show that every displayable logic enjoys strong cut-elimination w... more The method used in Chapter 4 to show that every displayable logic enjoys strong cut-elimination was derived from the proof of strong normalizability of typed λ-terms. It does not only apply to display calculi. The present chapter is devoted to a proof of strong cut-elimination in a labelled tableau calculus for the (constant domain) modal predicate logic QS5. Modal tableau calculi which build in the accessibility relation of possible worlds models were first introduced by Kripke [93] and were later ‘linearized’ by various authors, notably Fitting [62], [63], [64] and Mints [114]. As in Gabbay’s [68] theory of labelled deductive systems, the basic declarative unit of these tableau calculi is not just a formula A, but rather a formula plus label (σ, A). In the case of the modal logic S5 the label σ may just be a single positive integer, whereas in general it is a non-empty finite sequence of positive integers. Moreover, for S5 the accessibility relation between labels may be universal and hence neglected. In contrast to labelled tableaux, the modal tableau systems of, for example, Rautenberg [137] and Gore [74] do not use labelled formulas. For a general survey on tableau methods for modal and tense logics, see [79]. The use of labels allows to formulate tableau calculi for certain extensions of the minimal normal modal logic K by imposing constraints on accessibility and on occurrences and the shape of labels on tableau branches. These constraints may be regarded as structural in the sense of not referring to any connectives. In order to emphasize the relation to sequent calculi, we shall work with a tableau calculus TQS5 based on the ordinary notion of a sequent. By defining suitable mappings on cut-free closed tableaux it can easily be shown that the result of dropping cut from TQS5 is equivalent to Fitting’s tableau calculus for first-order S5 with respect to provable formulas (see Section 7.5). Usually modal tableau calculi are formulated without a cut rule. The admissibility of cut is, however, of interest for constructive proofs of equivalence with Hilbert-type systems; compare [93, p. 82]. Moreover, non-constructive proofs of cut-elimination are of little appeal when it comes to extending the notion of formuals-as-types to modal logic (see, for instance, [28], [106]). It is this respect in which the present chapter may be seen to have significance.
The aim of this chapter is twofold. First of all, we shall take a closer look at the strong, cons... more The aim of this chapter is twofold. First of all, we shall take a closer look at the strong, constructive negation ~ introduced in Chapter 8. We shall consider ~ from the point of view of a proof-theoretic characterization of negation and argue that negation may be seen as a connecting link between provability and disprovability (refutability). This notion of negation as falsity will be developed against the background of N. Tennant’s [165] considerations of negation in intuitionistic relevant logic, where Tennant also attends to disproofs in addition to proofs. It is shown that negation in intuitionistic relevant logic is a negation as syntactical inconsistency in the sense of Gabbay [66], and that every such negation as inconsistency is a negation as falsity, while the converse is not true. Secondly, we shall consider semantics-based nonmonotonic reasoning as introduced by, again, Gabbay [65]. In this approach, nonmonotonic inference is defined using a modal consistency operator that is interpreted as possibility with respect to the information order in semantical models of a monotonic base logic. It will be shown that certain anomalies of Gabbay’s approach can very naturally be avoided using David Nelson’s constructive three-valued system N3 [119] as the monotonic base system instead of intuitionistic logic, IPL or Kleene’s three-valued logic, 3. The counterintuitive features of semantics-based nonmonotonic reasoning based on IPL or on 3 also disappear if certain properties of the information order in Kripke models for IPL and model structures for 3 are given up. In the case of IPL this leads to subintuitionistic logics. In this way, the present chapter prepares the ground for Chapter 10. Whereas Chapter 10 is devoted to display sequent calculi for subintuitionistic logics, [195] solves the problem of providing a sound and complete proof-system for the modal logic of consistency over Nelson’s four-valued logic N4.
Fine-Grained Theories of Time (P Blackburn) Revision Sequences and Computers with an Infinite Amo... more Fine-Grained Theories of Time (P Blackburn) Revision Sequences and Computers with an Infinite Amount of Time (B Lowe) On Frege's Nightmare: A Combination of Intuitionistic, Free and Paraconsistent Logics (S Rahman) Truthmakers, Entailment and Necessity (S Read) Global Definability in Basic Modal Logic (M de Rijke & H Sturm) Ackermann's Implication for Typefree Logic (K Robering) Why Dialogical Logic? (H Ruckert) Semantics for Constructive Negations (Y Shramko) Recent Trends in Paraconsistent Logic (M Urchs) Obligations, Authorities, and History Dependence (H Wansing).
In this paper we will consider the existing notions of bilateralism in the context of proof-theor... more In this paper we will consider the existing notions of bilateralism in the context of proof-theoretic semantics and propose, based on our understanding of bilateralism, an extension to logical multilateralism. This approach differs from what has been proposed under this name before in that we do not consider multiple speech acts as the core of such a theory but rather multiple consequence relations. We will argue that for this aim the most beneficial proof-theoretical realization is to use sequent calculi with multiple sequent arrows satisfying some specific conditions, which we will lay out in this paper. We will unfold our ideas with the help of a case study in logical tetralateralism and present an extension of Almukdad and Nelson’s propositional constructive four-valued logic by unary operations of meaningfulness and nonsensicality. We will argue that in sequent calculi with multiple sequent arrows it is possible to maintain certain features that are desirable if we assume an un...
Despite the tendency to be otherwise, some non-classical logics are known to validate formulas th... more Despite the tendency to be otherwise, some non-classical logics are known to validate formulas that are invalid in classical logic. A subclass of such systems even possesses pairs of a formula and its negation as theorems, without becoming trivial. How should these provable contradictions be understood? The present paper aims to shed light on aspects of this phenomenon by taking as samples the constructive connexive logic C, which is obtained by a simple modification of a system of constructible falsity, namely N4, as well as its non-constructive extension C3. For these systems, various observations concerning provable contradictions are made, using mainly a proof-theoretic approach. The topics covered in this paper include: how new contradictions are found from parts of provable contradictions; how to characterise provable contradictions in C3 that are constructive; how contradictions can be seen from the relative viewpoint of strong implication; and as an appendix an attempt at ge...
This volume contains a collection of articles applying methods of logic or, more generally, of ma... more This volume contains a collection of articles applying methods of logic or, more generally, of mathematics to solve problems, some of which come from logic itself, others from other sciences. Its range of subjects is far from complete, but broadly representative. The first group of papers in this volume consists of contributions to pure and applied modal logic. The problems discussed here range from the structure of lattices of normal and other modal propositional logics to modal proof theory and to the semantics of quantified modal logic. The second group of papers deals with Many-valued logics - an extensive domain of strictly logical investigations rooting in philosophical questions concerning the nature of logical values. Logical investigations in cognitive science have successfully utilized methods and systems of belief revision, non-monotonic logic and dynamic epistemic logic. Towards Mathematical Philosophy deals with focal issues of belief revision. The volume concludes with contributions which may be seen to belong to the field of formal epistemology, the area applying logical, probabilistic, game-theoretic and other formal methods to problems and issues in epistemology and philosophy of science, such as those concerning anti-realism, skepticism, theory comparison and theory choice, justification, sources of knowledge and learning theories.
In this paper we elaborate a conception of entailment based on what we call the Ackermann princip... more In this paper we elaborate a conception of entailment based on what we call the Ackermann principle, which explicates valid entailment through a logical connection between (sets of) sentences depending on their informational content. We reconstruct Dunn’s informational semantics for entailment on the basis of Restall’s approach, with assertion and denial as two independent (primary) speech acts, by introducing the notion of a ‘position description’. We show how the machinery of position descriptions can effectively be used to define the positive and the negative information carried by sentences of a given language and to present a formalized version of the Ackermann principle as an inclusion relationship between the informational contents of the conclusions and the premises of a valid entailment. Being so interpreted, the entailment relation exhibits certain properties, including the property of transitivity (and, more generally, admissibility of the cut rule). Whereas properties such as Anderson and Belnap’s variable sharing property or Parry’s proscriptive principle are normally presented as imposing a relevance requirement on valid entailment, the suggested formalization of the Ackermann principle supports all of Gentzen’s structural rules, including weakening, a rule that is normally given up in sequent-style proof systems for relevance logics. In this way we propose an Ackermann-inspired explication of the nature of entailment as a relation between the informational contents of sentences.
Livre: Truth and falsehood: an inquiry into generalized logical values (hardback) (series: trends... more Livre: Truth and falsehood: an inquiry into generalized logical values (hardback) (series: trends in logic) SHRAMKO Yaroslav, WANSING Heinrich.
In our everyday life we are holding each other responsible for what we believe and for how we for... more In our everyday life we are holding each other responsible for what we believe and for how we form beliefs. Since there are many similarities between holding an agent responsible for her actions and holding an agent responsible for her beliefs, some epistemologists think that we can model epistemic responsibility, i.e., responsibility for a belief, along the lines ofmoral responsibility, i.e., responsibility for performing an action. Research onmoral responsibility suggests that a person can be held responsible for an action she has performed only if she had control over performing that action. Even though our practice of holding each other responsible for our beliefs is similar to our practice of holding each other responsible for our actions, our formation of a belief appears to be different to our performance of an action when it comes to our capacity to exercise control over it. Beliefs are mostly taken to be more or less passive responses to how the world appears to us, while in performing an action we shape the world, often as we intend it to be. This difference between actions and beliefs raises a problem when it comes to establishing a viable notion of epistemic responsibility along the lines of the notion of moral responsibility. Thus, we need to rethink our notion of epistemic responsibility, the kind of control necessary for responsibility, as well as the connection between epistemic responsibility and doxastic control. This special issue partially documents results of the two-day workshop “Doxastic Agency and Epistemic Responsibility” which took place at the Ruhr University Bochum in June 2014. It addresses questions such as: do we have the same kind of control over our doxastic attitudes as we have over our actions, what kind of control do we have over our doxastic attitudes and how is epistemic responsibility related to notions such as epistemic justification and practical responsibility?Most of the authors
In this introductory note, we place the new essays on Belnap-Dunn logic, FDE, of the present volu... more In this introductory note, we place the new essays on Belnap-Dunn logic, FDE, of the present volume against the background of the development of FDE. This note is an invitation to study the volume. It presents a chronological perspective on Belnap-Dunn logic and a slightly idiosyncratic list of further research topics.
Francesco Berto proposed a logic for imaginative episodes. The logic establishes certain (in)vali... more Francesco Berto proposed a logic for imaginative episodes. The logic establishes certain (in)validities concerning episodic imagination. They are not all equally plausible as principles of episodic imagination. The logic also does not model that the initial input of an imaginative episode is deliberately chosen.Stit-imagination logic models the imagining agent’s deliberate choice of the content of their imagining. However, the logic does not model the episodic nature of imagination. The present paper combines the two logics, thereby modelling imaginative episodes with deliberately chosen initial input. We use a combination ofstit-imagination logic and a content-sensitive variably strict conditional à la Berto, for which we give a Chellas–Segerberg semantics. The proposed semantics has the following advantages over Berto’s: (i) we model thedeliberate choiceof initial input of imaginative episodes (in a multi-agent setting), (ii) we showframe correspondencesfor axiomatic analogues of ...
Abstract In this paper it is suggested to generalize our understanding of general (structural) pr... more Abstract In this paper it is suggested to generalize our understanding of general (structural) proof theory and to consider it as a general theory of two kinds of derivations, namely proofs and dual proofs. The proposal is substantiated by (i) considerations on assertion, denial, and bi-lateralism, (ii) remarks on compositionality in proof-theoretic semantics, and (iii) comments on falsification and co-implication. The main formal result of the paper is a normal form theorem for the natural deduction proof system N2Int of the bi-intuitionistic logic 2Int. The proof makes use of the faithful embedding of 2Int into intuitionistic logic with respect to validity and shows that conversions of dual proofs can be sidestepped.
The method used in Chapter 4 to show that every displayable logic enjoys strong cut-elimination w... more The method used in Chapter 4 to show that every displayable logic enjoys strong cut-elimination was derived from the proof of strong normalizability of typed λ-terms. It does not only apply to display calculi. The present chapter is devoted to a proof of strong cut-elimination in a labelled tableau calculus for the (constant domain) modal predicate logic QS5. Modal tableau calculi which build in the accessibility relation of possible worlds models were first introduced by Kripke [93] and were later ‘linearized’ by various authors, notably Fitting [62], [63], [64] and Mints [114]. As in Gabbay’s [68] theory of labelled deductive systems, the basic declarative unit of these tableau calculi is not just a formula A, but rather a formula plus label (σ, A). In the case of the modal logic S5 the label σ may just be a single positive integer, whereas in general it is a non-empty finite sequence of positive integers. Moreover, for S5 the accessibility relation between labels may be universal and hence neglected. In contrast to labelled tableaux, the modal tableau systems of, for example, Rautenberg [137] and Gore [74] do not use labelled formulas. For a general survey on tableau methods for modal and tense logics, see [79]. The use of labels allows to formulate tableau calculi for certain extensions of the minimal normal modal logic K by imposing constraints on accessibility and on occurrences and the shape of labels on tableau branches. These constraints may be regarded as structural in the sense of not referring to any connectives. In order to emphasize the relation to sequent calculi, we shall work with a tableau calculus TQS5 based on the ordinary notion of a sequent. By defining suitable mappings on cut-free closed tableaux it can easily be shown that the result of dropping cut from TQS5 is equivalent to Fitting’s tableau calculus for first-order S5 with respect to provable formulas (see Section 7.5). Usually modal tableau calculi are formulated without a cut rule. The admissibility of cut is, however, of interest for constructive proofs of equivalence with Hilbert-type systems; compare [93, p. 82]. Moreover, non-constructive proofs of cut-elimination are of little appeal when it comes to extending the notion of formuals-as-types to modal logic (see, for instance, [28], [106]). It is this respect in which the present chapter may be seen to have significance.
The aim of this chapter is twofold. First of all, we shall take a closer look at the strong, cons... more The aim of this chapter is twofold. First of all, we shall take a closer look at the strong, constructive negation ~ introduced in Chapter 8. We shall consider ~ from the point of view of a proof-theoretic characterization of negation and argue that negation may be seen as a connecting link between provability and disprovability (refutability). This notion of negation as falsity will be developed against the background of N. Tennant’s [165] considerations of negation in intuitionistic relevant logic, where Tennant also attends to disproofs in addition to proofs. It is shown that negation in intuitionistic relevant logic is a negation as syntactical inconsistency in the sense of Gabbay [66], and that every such negation as inconsistency is a negation as falsity, while the converse is not true. Secondly, we shall consider semantics-based nonmonotonic reasoning as introduced by, again, Gabbay [65]. In this approach, nonmonotonic inference is defined using a modal consistency operator that is interpreted as possibility with respect to the information order in semantical models of a monotonic base logic. It will be shown that certain anomalies of Gabbay’s approach can very naturally be avoided using David Nelson’s constructive three-valued system N3 [119] as the monotonic base system instead of intuitionistic logic, IPL or Kleene’s three-valued logic, 3. The counterintuitive features of semantics-based nonmonotonic reasoning based on IPL or on 3 also disappear if certain properties of the information order in Kripke models for IPL and model structures for 3 are given up. In the case of IPL this leads to subintuitionistic logics. In this way, the present chapter prepares the ground for Chapter 10. Whereas Chapter 10 is devoted to display sequent calculi for subintuitionistic logics, [195] solves the problem of providing a sound and complete proof-system for the modal logic of consistency over Nelson’s four-valued logic N4.
Fine-Grained Theories of Time (P Blackburn) Revision Sequences and Computers with an Infinite Amo... more Fine-Grained Theories of Time (P Blackburn) Revision Sequences and Computers with an Infinite Amount of Time (B Lowe) On Frege's Nightmare: A Combination of Intuitionistic, Free and Paraconsistent Logics (S Rahman) Truthmakers, Entailment and Necessity (S Read) Global Definability in Basic Modal Logic (M de Rijke & H Sturm) Ackermann's Implication for Typefree Logic (K Robering) Why Dialogical Logic? (H Ruckert) Semantics for Constructive Negations (Y Shramko) Recent Trends in Paraconsistent Logic (M Urchs) Obligations, Authorities, and History Dependence (H Wansing).
In this paper we will consider the existing notions of bilateralism in the context of proof-theor... more In this paper we will consider the existing notions of bilateralism in the context of proof-theoretic semantics and propose, based on our understanding of bilateralism, an extension to logical multilateralism. This approach differs from what has been proposed under this name before in that we do not consider multiple speech acts as the core of such a theory but rather multiple consequence relations. We will argue that for this aim the most beneficial proof-theoretical realization is to use sequent calculi with multiple sequent arrows satisfying some specific conditions, which we will lay out in this paper. We will unfold our ideas with the help of a case study in logical tetralateralism and present an extension of Almukdad and Nelson’s propositional constructive four-valued logic by unary operations of meaningfulness and nonsensicality. We will argue that in sequent calculi with multiple sequent arrows it is possible to maintain certain features that are desirable if we assume an un...
Despite the tendency to be otherwise, some non-classical logics are known to validate formulas th... more Despite the tendency to be otherwise, some non-classical logics are known to validate formulas that are invalid in classical logic. A subclass of such systems even possesses pairs of a formula and its negation as theorems, without becoming trivial. How should these provable contradictions be understood? The present paper aims to shed light on aspects of this phenomenon by taking as samples the constructive connexive logic C, which is obtained by a simple modification of a system of constructible falsity, namely N4, as well as its non-constructive extension C3. For these systems, various observations concerning provable contradictions are made, using mainly a proof-theoretic approach. The topics covered in this paper include: how new contradictions are found from parts of provable contradictions; how to characterise provable contradictions in C3 that are constructive; how contradictions can be seen from the relative viewpoint of strong implication; and as an appendix an attempt at ge...
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