The inferences of contraposition (A ⇒ C ∴ ¬C ⇒ ¬A), the hypothetical syllogism (A ⇒ B, B ⇒ C ∴ A ⇒ C), and others are widely seen as unacceptable for counterfactual conditionals. Adams convincingly argued, however, that these inferences are unacceptable for indicative conditionals as well. He argued that an indicative conditional of form A ⇒ C has assertability conditions instead of truth conditions, and that their assertability ‘goes with’ the conditional probability p(C|A). To account for inferences, Adams developed the notion of probabilistic entailment as an extension of classical entailment. This combined approach (correctly) predicts that contraposition and the hypothetical syllogism are invalid inferences. Perhaps less well-known, however, is that the approach also predicts that the unconditional counterparts of these inferences, e.g., modus tollens (A ⇒ C, ¬C ∴ ¬A), and iterated modus ponens (A ⇒ B, B ⇒ C, A ∴ C) are predicted to be valid. We will argue both by example and by calling to the results from a behavioral experiment (N = 159) that these latter predictions are incorrect if the unconditional premises in these inferences are seen as new information. Then we will discuss Adams’ (1998) dynamic probabilistic entailment relation, and argue that it is problematic. Finally, it will be shown how his dynamic entailment relation can be improved such that the incongruence predicted by Adams’ original system concerning conditionals and their unconditional counterparts are overcome. Finally, it will be argued that the idea behind this new notion of entailment is of more general relevance.
对立推理(A⇒C∴C⇒¬A)、假设三段论(A⇒B, B⇒C∴A⇒C)和其他推理被广泛认为是反事实条件句不可接受的。然而，亚当斯令人信服地指出，这些推论对于指示性条件句来说也是不可接受的。他论证了形式A⇒C的指示性条件具有可断言性条件而不是真条件，并且它们的可断言性与条件概率p(C|A)“一致”。为了解释推理，亚当斯发展了概率蕴涵的概念，作为经典蕴涵的延伸。这种结合的方法(正确地)预测对位和假设三段论是无效的推论。然而，也许不太为人所知的是，该方法还预测了这些推理的无条件对应物，例如，模态(A⇒C, C θ θ A)和迭代模态(A⇒B, B⇒C, A θ θ C)被预测为有效的。我们将通过实例和调用行为实验(N = 159)的结果来论证，如果这些推论中的无条件前提被视为新信息，则后一种预测是不正确的。然后，我们将讨论亚当斯(1998)的动态概率蕴涵关系，并认为它是有问题的。最后，将展示如何改进他的动态蕴涵关系，从而克服亚当斯的原始系统所预测的关于条件及其无条件对应物的不一致。最后，我们将论证这种新的蕴涵概念背后的思想具有更普遍的相关性。
{"title":"Dynamic Probabilistic Entailment. Improving on Adams' Dynamic Entailment Relation","authors":"R. van Rooij, Patricia Mirabile","doi":"10.12775/llp.2021.022","DOIUrl":"https://doi.org/10.12775/llp.2021.022","url":null,"abstract":"The inferences of contraposition (A ⇒ C ∴ ¬C ⇒ ¬A), the hypothetical syllogism (A ⇒ B, B ⇒ C ∴ A ⇒ C), and others are widely seen as unacceptable for counterfactual conditionals. Adams convincingly argued, however, that these inferences are unacceptable for indicative conditionals as well. He argued that an indicative conditional of form A ⇒ C has assertability conditions instead of truth conditions, and that their assertability ‘goes with’ the conditional probability p(C|A). To account for inferences, Adams developed the notion of probabilistic entailment as an extension of classical entailment. This combined approach (correctly) predicts that contraposition and the hypothetical syllogism are invalid inferences. Perhaps less well-known, however, is that the approach also predicts that the unconditional counterparts of these inferences, e.g., modus tollens (A ⇒ C, ¬C ∴ ¬A), and iterated modus ponens (A ⇒ B, B ⇒ C, A ∴ C) are predicted to be valid. We will argue both by example and by calling to the results from a behavioral experiment (N = 159) that these latter predictions are incorrect if the unconditional premises in these inferences are seen as new information. Then we will discuss Adams’ (1998) dynamic probabilistic entailment relation, and argue that it is problematic. Finally, it will be shown how his dynamic entailment relation can be improved such that the incongruence predicted by Adams’ original system concerning conditionals and their unconditional counterparts are overcome. Finally, it will be argued that the idea behind this new notion of entailment is of more general relevance.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48201035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present the history of and the research directions in relating logic. For this purpose we will describe Epstein's Programme, which postulates accounting for the content of sentences in logical research. We will focus on analysing the content relationship and Epstein's logics that are based on it, which are special cases of relating logic. Moreover, the set-assignment semantics will be discussed. Next, the Torunian Programme of Relating Semantics will be presented; this programme explores the various non-logical relationships in logical research, including those which are content-related. We will present a general description of relating logic and semantics as well as the most prominent issues regarding the Torunian Programme, including some of its special cases and the results achieved to date.
{"title":"History of Relating Logic. The Origin and Research Directions","authors":"Mateusz Klonowski","doi":"10.12775/llp.2021.021","DOIUrl":"https://doi.org/10.12775/llp.2021.021","url":null,"abstract":"In this paper, we present the history of and the research directions in relating logic. For this purpose we will describe Epstein's Programme, which postulates accounting for the content of sentences in logical research. We will focus on analysing the content relationship and Epstein's logics that are based on it, which are special cases of relating logic. Moreover, the set-assignment semantics will be discussed. Next, the Torunian Programme of Relating Semantics will be presented; this programme explores the various non-logical relationships in logical research, including those which are content-related. We will present a general description of relating logic and semantics as well as the most prominent issues regarding the Torunian Programme, including some of its special cases and the results achieved to date.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41359124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.
{"title":"S5-Style Non-Standard Modalities in a Hypersequent Framework","authors":"Y. Petrukhin","doi":"10.12775/llp.2021.020","DOIUrl":"https://doi.org/10.12775/llp.2021.020","url":null,"abstract":"The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48765944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article aims to examine Koellner’s reconstruction of Penrose’s second argument – a reconstruction that uses the DTK system to deal with Gödel’s disjunction issues. Koellner states that Penrose’s argument is unsound, because it contains two illegitimate steps. He contends that the formulas to which the T-intro and K-intro rules apply are both indeterminate. However, we intend to show that we can correctly interpret the formulas on the set of arithmetic formulas, and that, as a consequence, the two steps become legitimate. Nevertheless, the argument remains partially inconclusive. More precisely, the argument does not reach a result that shows there is no formalism capable of deriving all the true arithmetic propositions known to man. Instead, it shows that, if such formalism exists, there is at least one true non-arithmetic proposition known to the human mind that we cannot derive from the formalism in question. Finally, we reflect on the idealised character of the DTK system. These reflections highlight the limits of human knowledge, and, at the same time, its irreducibility to computation.
{"title":"Analysis of Penrose’s Second Argument Formalised in DTK System","authors":"A. Corradini, S. Galvan","doi":"10.12775/llp.2021.019","DOIUrl":"https://doi.org/10.12775/llp.2021.019","url":null,"abstract":"This article aims to examine Koellner’s reconstruction of Penrose’s second argument – a reconstruction that uses the DTK system to deal with Gödel’s disjunction issues. Koellner states that Penrose’s argument is unsound, because it contains two illegitimate steps. He contends that the formulas to which the T-intro and K-intro rules apply are both indeterminate. However, we intend to show that we can correctly interpret the formulas on the set of arithmetic formulas, and that, as a consequence, the two steps become legitimate. Nevertheless, the argument remains partially inconclusive. More precisely, the argument does not reach a result that shows there is no formalism capable of deriving all the true arithmetic propositions known to man. Instead, it shows that, if such formalism exists, there is at least one true non-arithmetic proposition known to the human mind that we cannot derive from the formalism in question. Finally, we reflect on the idealised character of the DTK system. These reflections highlight the limits of human knowledge, and, at the same time, its irreducibility to computation.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49006038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mo Liu, Jie Fan, H. van Ditmarsch, Louwe B. Kuijer
In this paper, we propose three knowability logics LK, LK−, and LK=. In the single-agent case, LK is equally expressive as arbitrary public announcement logic APAL and public announcement logic PAL, whereas in the multi-agent case, LK is more expressive than PAL. In contrast, both LK− and LK= are equally expressive as classical propositional logic PL. We present the axiomatizations of the three knowability logics and show their soundness and completeness. We show that all three knowability logics possess the properties of Church-Rosser and McKinsey. Although LK is undecidable when at least three agents are involved, LK− and LK= are both decidable.
{"title":"Logics for Knowability","authors":"Mo Liu, Jie Fan, H. van Ditmarsch, Louwe B. Kuijer","doi":"10.12775/llp.2021.018","DOIUrl":"https://doi.org/10.12775/llp.2021.018","url":null,"abstract":"In this paper, we propose three knowability logics LK, LK−, and LK=. In the single-agent case, LK is equally expressive as arbitrary public announcement logic APAL and public announcement logic PAL, whereas in the multi-agent case, LK is more expressive than PAL. In contrast, both LK− and LK= are equally expressive as classical propositional logic PL. We present the axiomatizations of the three knowability logics and show their soundness and completeness. We show that all three knowability logics possess the properties of Church-Rosser and McKinsey. Although LK is undecidable when at least three agents are involved, LK− and LK= are both decidable.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45333880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents the topological arrangements in four geometrical figures of modal propositions and their derivative relations by means of Peirce's gamma graphs and their rules of transformation. The idea of arraying the gamma graphs in a geometric and symmetrical order comes from Peirce himself who in a manuscript drew two cubes in which he presented the derivative relations of some (but no all) gamma graphs. Therefore, Peirce's insights of a topological order of gamma graphs are extended here backwards from the cube to the line and the square; and then forwards from the cube to the four-dimensional polyhedron.
{"title":"Topology of Modal Propositions Depicted by Peirce’s Gamma Graphs: Line, Square, Cube, and Four-Dimensional Polyhedron","authors":"Jorge Alejandro Flórez","doi":"10.12775/llp.2021.017","DOIUrl":"https://doi.org/10.12775/llp.2021.017","url":null,"abstract":"\u0000\u0000\u0000This paper presents the topological arrangements in four geometrical figures of modal propositions and their derivative relations by means of Peirce's gamma graphs and their rules of transformation. The idea of arraying the gamma graphs in a geometric and symmetrical order comes from Peirce himself who in a manuscript drew two cubes in which he presented the derivative relations of some (but no all) gamma graphs. Therefore, Peirce's insights of a topological order of gamma graphs are extended here backwards from the cube to the line and the square; and then forwards from the cube to the four-dimensional polyhedron.\u0000\u0000\u0000","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43147755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BAT is a logic built to capture the inferential behavior of informal provability. Ultimately, the logic is meant to be used in an arithmetical setting. To reach this stage it has to be extended to a first-order version. In this paper we provide such an extension. We do so by constructing non-deterministic three-valued models that interpret quantifiers as some sorts of infinite disjunctions and conjunctions. We also elaborate on the semantical properties of the first-order system and consider a couple of its strengthenings. It turns out that obtaining a sensible strengthening is not straightforward. We prove that most strategies commonly used for strengthening non-deterministic logics fail in our case. Nevertheless, we identify one method of extending the system which does not.
{"title":"Informal Provability, First-Order BAT Logic and First Steps Towards a Formal Theory of Informal Provability","authors":"Pawel Pawlowski, R. Urbaniak","doi":"10.12775/llp.2021.016","DOIUrl":"https://doi.org/10.12775/llp.2021.016","url":null,"abstract":"BAT is a logic built to capture the inferential behavior of informal provability. Ultimately, the logic is meant to be used in an arithmetical setting. To reach this stage it has to be extended to a first-order version. In this paper we provide such an extension. We do so by constructing non-deterministic three-valued models that interpret quantifiers as some sorts of infinite disjunctions and conjunctions. We also elaborate on the semantical properties of the first-order system and consider a couple of its strengthenings. It turns out that obtaining a sensible strengthening is not straightforward. We prove that most strategies commonly used for strengthening non-deterministic logics fail in our case. Nevertheless, we identify one method of extending the system which does not.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48106192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Luis Estrada-González, A. Giordani, Tomasz Jarmużek, Mateusz Klonowski, I. Sedlár, A. Tedder
In this paper we discuss whether the relation between formulas in the relating model can be directly introduced into the language of relating logic, and present some stances on that problem. Other questions in the vicinity, such as what kind of functor would be the incorporated relation, or whether the direct incorporation of the relation into the language of relating logic is really needed, will also be addressed.
{"title":"Incorporating the Relation into the Language?","authors":"Luis Estrada-González, A. Giordani, Tomasz Jarmużek, Mateusz Klonowski, I. Sedlár, A. Tedder","doi":"10.12775/llp.2021.014","DOIUrl":"https://doi.org/10.12775/llp.2021.014","url":null,"abstract":"In this paper we discuss whether the relation between formulas in the relating model can be directly introduced into the language of relating logic, and present some stances on that problem. Other questions in the vicinity, such as what kind of functor would be the incorporated relation, or whether the direct incorporation of the relation into the language of relating logic is really needed, will also be addressed.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43169189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical Logic in terms of appropriate matrix bundles and as semilattice-based logics, showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. Finally, we use Płonka sums of matrices to investigate the pure variable inclusion companions of an arbitrary finitary logic.
{"title":"Pure Variable Inclusion Logics","authors":"F. Paoli, M. Pra Baldi, D. Szmuc","doi":"10.12775/llp.2021.015","DOIUrl":"https://doi.org/10.12775/llp.2021.015","url":null,"abstract":"The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical Logic in terms of appropriate matrix bundles and as semilattice-based logics, showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. Finally, we use Płonka sums of matrices to investigate the pure variable inclusion companions of an arbitrary finitary logic.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45590791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The functionalist theory of mind proposes to analyze mental states in terms of internal states of Turing machine, and states of the machine’s tape and head. In the paper, I perform a formal analysis of this approach. I define the concepts of behavioral equivalence of Turing machines, and of behavioral individuation of internal states. I prove a theorem saying that for every Turing machine T there exists a Turing machine T’ which is behaviorally equivalent to T, and all of whose internal states of T’ can be behaviorally individuated. Finally, I discuss some applications of this theorem to computational theories of mind.
{"title":"A Formal Analysis of the Concept of Behavioral Individuation of Mental States in the Functionalist Framework","authors":"Maciej Malicki","doi":"10.12775/llp.2021.012","DOIUrl":"https://doi.org/10.12775/llp.2021.012","url":null,"abstract":"The functionalist theory of mind proposes to analyze mental states in terms of internal states of Turing machine, and states of the machine’s tape and head. In the paper, I perform a formal analysis of this approach. I define the concepts of behavioral equivalence of Turing machines, and of behavioral individuation of internal states. I prove a theorem saying that for every Turing machine T there exists a Turing machine T’ which is behaviorally equivalent to T, and all of whose internal states of T’ can be behaviorally individuated. Finally, I discuss some applications of this theorem to computational theories of mind.","PeriodicalId":43501,"journal":{"name":"Logic and Logical Philosophy","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46883589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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