ABSTRACT. Epistemic logic is known as logic that captures the knowledge and beliefs of agents and has undergone various developments since Hintikka (1962). In this paper, we propose a new logic called agent-knowledge logic by multiplying the individual knowledge structure with the relationships among agents. This logic is based on the Facebook logic proposed by Seligman et al. (2011) and the Logic of Hide and Seek Game proposed by Li et al. (2021). We show that this logic can embed the epistemic logic we know well. We also discuss various sentences and inferences that this logic can express.

ABSTRACT. The method Nils Kürbis used to formalise definite descriptions with a binary quantifier I, such that Ix[F,G] indicates `the F is G', is examined and improved upon in this work. Kürbis first looked at I in an intuitionistic logic and its negative free form. It is well-known that intuitionistic reasoning approaches truth constructively. We also want to approach falsehood constructively, in Nelson's footsteps. Within the context of Nelson's paraconsistent logic N4 and its negative free variant, we examine I. We offer an embedding function from Nelson's (free) logic into intuitionistic (free) logic, as well as a natural deduction system for Nelson's (free) logic supplied with I and Kripke style semantics for it. Our method not only yields constructive falsehood, but also provides an alternate resolution to an issue pertaining to Russell's interpretation of definite descriptions. This comprehension might result in paradoxes. Free logic, which is often used to solve this issue, is insufficiently powerful to produce contradictions. Instead, we employ paraconsistent logic, which is made to function in the presence of contradicting data without devaluing the process of reasoning.

ABSTRACT. In this paper we present a new algebraic formulation of Morrill's CatLog system, a Type-Logical Calculus which accounts for grammaticality in Natural Languages (NL) in terms of theoremhood. CatLog has been studied in depth in several papers from a proof-theoretic point of view. Crucially the CatLog's algebraic account of non-linearity linguistic phenomena in this article is done with no S4 modalities, only with Moortgat style substructural modalities and (modal) structural rules. We prove Cut-elimination and completeness with respect a well-defined algebraic semantics. Moreover, the proof of Cut-elimination is algebraic, not proof-theoretic. In the paper we correct the proposed type-assigments of relative pronouns (in several papers over the years, at least from 2015), avoiding thus undergeneration of very basic relative clause phenomena in NL. Finally, our new algebraic framework is used to improve a publeshed Soft Linear Logic account of Catlog.

ABSTRACT. McCall’s logic CC1 [3], one of the earliest and best known connexive systems in the literature, has been often criticised for possessing some implausible validities (and for lacking some plausible ones), as well as for the absence of a convincing intuitive interpretation. Inspired by some work by Herzberger [1] and Song et al. [4], we suggest a new account of CC1, aimed at vindicating its naturalness. Under this construal, sentences are characterised both by a truth value (true or false) and by a content polarity (positive or negative). In particular, in full accord with the ideas underlying many other connexive logics, a CC1 conditional is true if and only if 1) it comes out true as a material conditional; and 2) the contents of its antecedent and of its consequent are compatible, i.e., they have the same polarity. After showing that Angell and McCall’s matrix for CC1 can be read along these lines, we prove the completeness of the system w.r.t. a class of algebraic models obtained via a certain construction on Boolean algebras. Finally, we discuss the prospects for a relating semantics in the style of [2].

References

[1] H.G. Herzberger, Dimensions of truth, Journal of Philosophical Logic, 2(4):535-556, 1973.

[2] T. Jarmużek, Relating semantics as fine-grained semantics for intensional propositional logics, In A. Giordani, J. Malinowski (eds.), Logic in High Definition. Trends in Logical Semantics, vol. 56 of Trends in Logic, 13–30, Springer, 2021.

[3] S. Mc Call, Connexive implication, Journal of Symbolic Logic, 31:415-433, 1966.

[4] Y. Song, H. Omori, J.R.B. Arenhart, and S. Tojo, A generalization of Beall’s off-topic interpretation, Studia Logica, forthcoming.

ABSTRACT. The sentential calculus with identity, SCI in short, is the most simplified version of R. Suszko’s non-Fregean logic and can be obtained by adding the sentential identity connective ≡ to classical logic [1]. SCI has the following identity axioms:

(E1) A ≡ A

(E2) A ≡ B →B ≡A

(E3) A ≡ B∧B ≡C →A≡C

(C1) A ≡ B →¬A≡¬B

(C2) A ≡ B∧C ≡D→(A%C)≡(B%D), where %∈{∧,∨,→,≡}

(SI) A ≡ B →(A→B)

We will consider to deal with a simple Liar sentence : “This sentence is not true” in SCI. Let’s define A=”This sentence is true”, then we have an equation of A ≡ ¬A which means the referent of two sentences A and ¬A are identical, but it is logically falsehood by (SI), i.e., ¬(A ≡ ¬A) holds in SCI. To solve the matter, we have introduced a referential relation of pair-sentence ((_)ⁱ ,(_)^j) form , where i,j are some stage numbers, as the similar way to identity connective, i.e., ¬(A ≡ ¬A) ⇐⇒ (A⁰,¬A¹). More precisely speaking, we assume that for any formulas A appear in pair-sentence (A⁰,¬A¹) form, if Aⁱ holds in some situation with superscript i then its successor situation Aⁱ⁺¹ is referred to Aⁱ⁺¹ := ¬Aⁱ. We have proposed a system PSC that just rejects the principle of identity “A is A”. This treatment is similar to Gupta’s sentencedefinition with revision stage number [4], but the difference is our formalization was based on Suszko’s SCI [5]. When doing logical reasoning, it is usually assumed that several fundamental postulates implicitly hold by a priori. These postulates are called Aristotle’s classical three principles for thinking. The first principle of identity says that “A is always A and not being ¬A”, the second principle of contradiction says that “A is not both A and ¬A”, and the third principle of excluded middle says that “either A is B or A is ¬B”. Then we get the following schemata from Aristotle’s three postulates.

(AT1) ¬(¬A →A)

(AT1’) ¬(A → ¬A)

(AT1”) (A → A)

(AT2) ¬((A →B)∧(¬A→B))

(AT2’) ¬(A∧¬A)

(AT3) (A →B)∨(A→¬B)

(AT3’) A∨¬A

If we do not admit some of them, we will get several kinds of non-classical reasoning. But some postulates of (AT),(AT’) and (AT2) are at all non-theorem of classical logic. Nowadays the standard notion of connexive logic can be characterized by the logical reasoning with external negation ¬ and connexive implication → (as non-symmetric) which satisfy Aristotle’s non-classical postulates (AT1),(AT1’) and also additionally the following similar Boethius’ theses [7].

(BT) (A →B)→¬(A→¬B)

(BT’) (A → ¬B) →¬(A→B)

To simulate connexive reasoning in PSC, we have introduced the following interpretation of external negation for each connectives: for two stage numbers 0 and 1,

(1) ¬(¬A) ⇐⇒ A

(2) ¬(A∧B) ⇐⇒ ¬A∨¬B

(3) ¬(A∨B) ⇐⇒ ¬A∧¬B

(4) ¬(A →B) ⇐⇒ A⁰→B¹

(5) ¬(A,B) ⇐⇒ (A⁰,B¹)

Then we get some extensions of PSC which admit the requirements in connexive logic and also can be seen not as one of four-valued logic, but as a classical two-valued logic according to Suszko’s Thesis of bivalence [2].

References

[1] S. L. Bloom and R. Suszko, Investigations into the sentential calculus with identity, Notre Dame Journal of Formal Logic,vol.XIII, No. 3, (1971), pp.289–308.

[2] C. Caleiro, W. Carnielli, M. E. Coniglio and J. Marcos, Two’s company: “The humbug of many logical values”, Logica Universalis, eds. by J.Y. Beziau, Birkhäuser, Germany(2005), pp. 169–189.

[3] C. Fiore, Classical Logic is Connexive, Logic,vol.21(2) (2024), Article no.3, pp.91–99. Australasian Journal of Logic,vol.21(2) (2024), Article no.3, pp.91–99.

[4] A. Gupta and N. Belnap, The Revision Theory of Truth, MIT Press, Cambridge, 1993

[5] T. Ishii, A syntactical comparison between pair sentential calculus PSC and Gupta’s definitional calculus Cn, Bulletin of NUIS, Niigata University of International and Information Studies, 2016.

[6] T. Ishii, SCI for pair-sentence and its completeness, Non-Classical Logics, Theory and Applications, Vol. 8, 2016, pp.61–65.

[7] H. Wansing, Connexive logic, In E. N. Zalta(ed.), The Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/logic-connexive/.

ABSTRACT. Sextus Empiricus in his books Against the Logicians and Outlines of Pyrronism, represents the difference of opinion between Diodorus Cronus and Filo, on the conditions for the truth of a conditionals, which we use in our everyday thinking. Philo understands conditional exactly as classical implication is understood today. Diodorus criticizes this by giving a number of examples refuting the assumption that everything follows from falsity. Also, the assumption of the truth of an implication when both its elements are true is unacceptable to Diodorus. His counter-examples are quite convincing. Diodorus agrees with Philo only in that an implication is false when its antecedent is true and its consequent is false. His conditions for the truthfulness of the implication are more complex and closer to our thinking, although not entirely consistent with it. He considers an implication to be true when, by necessity, if the antecedent is true, then the consequent is also true. Unfortunately, Diodorus understood necessity temporally: now and always in the future when the antecedent is true, the consequent is true. This has its undesirable consequences. Therefore, a better understanding of necessity is that of Chrysippus: the truthfulness of the predecessor cannot occur simultaneously with the falsity of the successor. This Chrysippian understanding is believed to apply to contradiction. The choice of implication of the Philo type underpinned the remarkable development of mathematics and the unique role of classical logic as a meta-logic for developing non-classical logics. Although it seems that the implication cannot be extensional. Otherwise it is a simple disjunction. The actual nature of human implication is intensional.

The presentation will propose such a formalization of implication that seems to satisfy the suggestions of Diodorus (without temporal understanding of necessity) and Chrysippus. Moreover, it also conforms to Aristotle’s postulates, which are nowadays considered the basis of connexive logics– the actual human implication should be connexive. It is likely that the insightful Aristotle, who lived many years before Philo, would not have recognized Philo’s conditions of the truth of implication as a correct definition of implication. His "connexive" postulates for implication preclude its material, i.e. extensional, flat understanding.

We will attempt to reconstruct the logic of content to a form that is as close as possible to both our thinking and classical logic. Among the classical connectives, some seem close to our thinking, others foreign to it. The former include connectives of negation, conjunction and disjunction. The latter, the connectives of implication and equivalence. Therefore, the connectives of classical implication and equivalence are replaced by the intensional T-implication and T-equivalence. The semantics is Fregean i.e. with the content implication and the synonymy. Thus every model interprets negation, conjunction, disjunction, T-implication, T-equivalence, content implication and synonymy (i.e. a conjunction of two mutually inverse content implications). Our semantics consists of a class of (Fregean) models and one (Fregean) mapping. This one mapping assigns a content to each sentence and that is why it is called sentence understanding. Since it is one, each sentence has its own specific, unchanging content in our semantics. Naturally, sentences having the same logical value of true or false, i.e. T-equivalent, may have different contents, i.e. they may not be synonymous. Negation, conjunction and disjunction are interpreted in a classical way. These connectives are used to define T-implication and T-equivalence. However, none of these two just defined connectives is extensional because in their definitions some specific subclass of the class of all models is used. Thus, if T-implication is satisfied in one model, then it is satisfied in every model of this selected subclass.

References

[1] K. Ajdukiewicz, Okres warunkowy a implikacja materialna, Studia Logica 4, 117-134, 1956.

[2] R. B. Angell, Deducibility, Entailment and Analytic Containment [in] J. NormanandR.Sylvan (eds.), Directions in Relevant Logic, Kluwer Academic Publishers, 119-143, 1989

[3] Diogenes Laertius, Lives and Opinions of Eminent Philosophers.

[4] P. Łukowski, Paradoxes, Trends in Logic 31, Springer, 2011.

[5] P. Łukowski, Contentual Approach to Negation, Studies in Logic, Grammar and Rhetoric 54 (67), 47-60, 2018.

[6] P. Łukowski, A “Distrubitive” or a “Collective” Approach to Sentences?, Logic and Logical Philosophy 28, 331-354, 2019.

[7] B. Mates, Stoic logic, University of California Press, 1961.

[8] Sextus Empiricus, Against the Logicians.

[9] Sextus Empiricus, Against the Mathematicians.

[10] Sextus Empiricus, Outlines of Pyrrhonism.

[11] H. Wansing, Connexive Logic, The Stanford Encyclopedia of Philosophy, 2023.

[12] M. White, The Fourth Account of Conditionals in Sextus Empiricus, History and Philosophy of Logic 7, 1-14, 1986.

ABSTRACT. Connexive logic is closely linked to the history of logic, in particular to the ancient logicians Aristotle and Chrysippus, and the early medieval Boethius. According to Storrs McCall, connexive implication was first defined by a Stoic logician whose basic idea had been described as follows:

And those who introduce the notion of connection say that a conditional is sound when the contradictory of its consequent is incompatible with its antecedent.

While the concurring conceptions of material and strict implication were attributed to Philo and Diodorus, respectively, Sextus Empiricus didn’t mention the name of the logician who defended the third conception of connexive implication. According to the Kneales, however, this person most likely was Chrysippus. The main difference between Diodorean and Chrysippean implication can be illustrated by the example

DIOD If atomic elements do not exist, then atomic elements do exist.

Diodorus accepted DIOD as “sound” because the existence of physical atoms is (at least in the Stoics’ opinion) necessary. Hence, the antecedent of DIOD is impossible while its consequent is necessary, so that it can never happen that the antecedent be true and yet the consequent false. For Chrysippus, however, DIOD fails to be “sound” because the contradictory of its consequent is identical with its antecedent and hence there is no real conflict, no “incompatibility” between these two propositions.

As has been argued in [7], what makes Chrysippean implication connexive is not the basic idea of the above quoted definition which might be formalized, with ‘I’ abbreviating the relation of incompatibility, as follows:

CHRYS 1 (p → q) ⇔ I(p, ¬q).

The real source of the connexivity of Chrysippus’s conception rather lies in the assumption that the relation ‘I’ is strictly anti-reflexive. That is, for Chrysippus no proposition whatsoever is incompatible with itself:

CHRYS 2 ¬I(p, p).

Substituting ‘¬q’ for ‘p’ in CHRYS 2 yields ¬I(¬q, ¬q), which, according to CHRYS 1, means that (¬q → q) is always false. Following McCall, the latter principle shall be referred to as Aristotle’s first thesis:

ARIST 1 ¬(¬q → q).

As a matter of fact, Aristotle used this principle in order to prove another characteristic law of connexive law, saying “that two implications of the form ‘If p then q’ and ‘If not-p, then q’ cannot both be true”([11], p.415). In accordance with McCall’s terminology, this principle shall be referred to as Aristotle’s second thesis:

ARIST 2 ¬((p → q) ∧ (¬p → q)).

ARIST 1 is often formulated “passively” by saying that a proposition cannot be implied by its own negation. Similarly, ARIST 2 might be paraphrased as saying that no proposition q can be implied (or entailed) by both of two contradictory propositions p and ¬p. Let us, however, also consider corresponding “actively” formulated variants saying that no proposition implies its own negation, and that no proposition p implies both of two contradictory propositions q and ¬q:

ABEL 1 ¬(q → ¬q)

ABEL 2 ¬((p → q) ∧ (p → ¬q)).

While ABEL 2 is usually referred to as ‘Boethius’ Thesis’, it shall here be called Abelard’s (second) thesis because the “Palatine master” explicitly defended these principles (together with their Aristotelian counterparts) in his Dialectica, [2]. The aim of this paper is to examine the views of medieval logicians not only concerning the connexive principles ARIST 1,2, ABEL 1,2, CHRYS 1,2, but also concerning “anti-connexive” principles like “Ex impossibili quodlibet”, “Necessarium ad quodlibet”, and “Ex contradictione quodlibet”.

References

[1] Burley, Walter: On the Purity of the Art of Logic, New Haven 2000.

[2] De Rijk, Lambert M. (Ed.): Petrus Abaelardus Dialectica, Assen2 1970.

[3] Duns Scotus, John: Opera Omnia, vol. 2., Paris 1891.

[4] Hughes, George E.: John Buridan on Self-Reference, Cambridge 1982.

[5] Hughes, George E.: Paul of Venice Logica Magna, Part II Fascicule 4, Oxford 1990.

[6] Kneale, William & Martha: The Development of Logic, Oxford 1962.

[7] Lenzen, Wolfgang: “The Third and Fourth Stoic Account of Conditionals”, in M. Blicha & I. Sedlár (eds.), The Logica Yearbook 2020, London 2021, 127–146.

[8] Lenzen, Wolfgang: “Rewriting the History of Connexive Logic”, in Journal of Philosophical Logic 51 (2022), 525–553.

[9] Lenzen, Wolfgang: “Abelard and the Development of Connexive Logic”, in I. Sedlár (ed.), The Logica Yearbook 2022, London 2023, 55–78.

[10] Martin, Christopher: “Embarrassing arguments and surprising conclusions in the development of theories of the conditional in the 12th century”, in J. Jolivet & A. de Libera (eds.), Gilbert de Poitiers et ses Contemporains, Naples 1987, 377–400.

[11] McCall, Storrs: “A History of Connexivity”, in D. M. Gabbay, F. J. Pelletier & J. Woods (eds.), Handbook of the History of Logic, vol. 11: Logic: A History of its Central Concepts, Elsevier 2012, 415–449.

[12] Paulus Venetus: Logica Parva, translated by A. Perreiah, Munich 1984.

[13] Thom, Paul & Scott, John (eds.): Robert Kilwardby, Notule libri Priorum. Oxford 2015.

[14] Wright, Thomas (ed.): Alexandri Neckham– De Naturis Rerum Libri Duo. London 1863.

ABSTRACT. Our paper will discuss the main topics of research in Boolean connexive logic (BCL) and some possible directions for further research development (for an introduction to BCL, see [5, 6]). We will try to consider the comments and suggestions of researchers interested in the broadly understood topic of connexive logics (for current trends in and an introduction to connexive logic, see [14, 15]). And that applies both to problems related to connexive logic in general (e.g., the problem of the origins of connexive logic, see [10, 9], cf. [13]) and to BCL in particular. The main topics of BCL research include:

1. philosophical motivations and applications of BCL (cf. [2, 5, 8, 11]),
2. connexive adaptation of philosophical relating logics (cf. [5, 8, 11, 16]),
3. proof theory for BCL (cf. [5, 6, 7]),
4. linguistic and semantic modification of the basic BCL: modal systems, systems of combined semantics, hyperconnexive systems, etc. (cf. [6, 7, 12, 16]),
5. comparison of BCL with other connexive logics (cf. [1, 12]).

Topics 2 and 3 are especially interesting to us, so our paper focuses on these two topics.

BCL might be considered a subfamily of relating logic (see [3, 4]), and topic 2 concerns the problem of modifications of philosophically motivated relating logics in such a way as to obtain connexive logic. Usually, such modification is related to an analysis of relationships of sentences due to, for instance, a relationship between the contents of sentences or a situation dependence. As part of the research on this topic, we examine to what extent a given relationship of sentences can constitute a special form of connexivity relation (“connection” or “coherence” relation, cf. [10]). This topic is, of course, strongly related to topics 1 and 4. If analyzing a given relationship between sentences leads to connexive logic, then such logic can be considered philosophically correct, i.e., well-motivated philosophically. Additionally, modifying a given philosophical logic usually requires modifying its semantic structure. Topic 3 is interesting for us, as it is a natural supplement to topic 2 and topic 4. It is also related to topic 5 since an axiomatic or sequent presentation often makes comparing a given logic with other formal systems easier.

References

[1] L. Estrada-González, 2022, ‘An analysis of poly-connexivity’, Studia Logica 110:925–947.

[2] A. Giordani, 2024, ‘Situation-based connexive logic’, Studia Logica 112:295323.

[3] T. Jarmużek and F. Paoli, 2021, ‘Relating logic and relating semantics. History, philosophical applications and some of technical problems’, Logic and Logical Philosophy 30(4):563–577.

[4] T. Jarmużek and F. Paoli, 2022, ‘Applications of relating semantics. From non-classical logics to philosophy of science’, Logic and Logical Philosophy.

[5] T. Jarmużek and J. Malinowski, 2019a, ‘Boolean connexive logics. Semantics and tableau approach’, Logic and Logical Philosophy 28(3):427–448.

[6] T. Jarmużek and J. Malinowski, 2019b, Modal Boolean connexive logics. Semantic and tableau approach, Bulletin of the Section of Logic 48(3):213243.

[7] M. Klonowski, 2021, Axiomatization of some basic and modal Boolean connexive logics, Logica Universalis 15:517–536.

[8] M. Klonowski and L. Estrada-González, 2024, Boolean connexive logic and content relationship, Studia Logica 112:207–248.

[9] W. Lenzen, 2022, Rewriting the history of connexive logic, Journal of Philosophical Logic 51:525–553.

[10] S. McCall, 2012, A history of connexivity, In D. M. Gabbay et al. (eds.), Logic: A History of its Central Concepts, Handbook of the History of Logic, Vol. 11, 415–449, Elsevier.

[11] J. Malinowski and R. Palczewski, 2021, ‘Relating semantics for connexive logic’, In A. Giordani, J. Malinowski (eds.), Logic in High Definition. Trends in Logical Semantics, Trends in Logic, Vol. 56, 49–65, Springer.

[12] J. Malinowski and R. A. Nicolás-Francisco, 2023, ‘Relating semantics for hyper-connexive and totally connexive logics’, Logic and Logical Philosophy.

[13] E. Mares and F. Paoli, 2019, ‘C. I. Lewis, E. J. Nelson, and the modern origins of connexive logic’, Organon F 26(3):405–426.

[14] H. Omori and H. Wansing, ‘Connexive logics. An overview and current trends’, Logic and Logical Philosophy 28(3):371–387.

[15] H. Omori and H. Wansing, 2024, ‘Connexive logic, connexivity, and connexivism: Remarks on terminology’, Studia Logica 112:1–35.

[16] N. Zamperlin, 2023, ‘Generalized Epstein semantics for connexive logic’, In M. Klonowski, M. Oleksowicz (eds.), The book of abstracts: Trends in Logic XXIII, 58–59. https://sites.google.com/view/ trends-bless-2023/home/abstracts