ABSTRACT. H. Wansing’s system C [5] is a connexive logic that has (positive) intuitionistic logic as its basis. As such, one natural way to motivate the logic would be to conceive it as an expansion of the Brouwerian system with a more constructive negation. If intuitionistic negation is retained, then this story leads to a slight variant of C with the falsity constant ⊥; such a system has been investigated by D. Fazio and S.P. Odintsov [2] under the name C^⊥.

A chief rival of C^⊥ in this narrative would be the system N4^⊥ [4] that expands the Almukdad-Nelson system N4 [1] by the falsity constant. C^⊥ and N4^⊥ have different conditions for refuting an implication, but the classical-looking condition for N4^⊥ might be seen as more intuitive and thus more favourable, than the condition for C^⊥ that induces connexivity.

Against this picture, In this talk I will discuss a constructive criterion, according to which C^⊥ can be a better candidate than N4^⊥, when one seeks to augment the constructivity of intuitionistic logic. I will then examine the relationship between this constructive criterion and some extensions of C^⊥.

References

[1] A. Almukdad and D. Nelson. Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49, 231–233, 1984.

[2] D. Fazio and S.P. Odintsov, An Algebraic Investigation of the Connexive Logic C. Studia Logica, 112, 37–67, 2024

[3] S. Niki, Intuitionistic views on connexive constructible falsity. Journal of Applied Logics, 11(2), 125–157, 2024.

[4] S.P. Odintsov, Constructive Negations and Paraconsistency. Dordrecht: Springer, 2008.

[5] H. Wansing. Connexive modal logic. In R. Schmidt, I. Pratt-Hartmann, M. Reynolds, and H. Wansing, (eds), Advances in Modal Logic. Volume 5. London, King’s College, 367–383, 2005.

ABSTRACT. In the articles [1, 2] we introduced the notions of Boolean connexive (BCL) and Boolean modal connexive logic (MBCL). We also defined the smallest systems of BCL and MBCL that satisfy the definition of connexive logic: they contain the principles of Aristotle, Boethius, and are closed under Modus Ponens [4].

The systems were defined according to the principle of Minimal Change Strategy (MCS), which states that as few changes as possible should be made. This is a version of Occam’s principle, a widely accepted principle of economic formulation of scientific theories [3].

Therefore, although the implication behaves in these systems as a connexive implication, the remaining functors behave in a Boolean and therefore classical way. So we have made a small change to the classical logic to obtain connexive logics.

However, this change is still large, since we have rejected all classical laws for the implication of →. Although we can define the material implication and all classical laws in our logics, the main implication of → is very weak: the set of principles of → is reduced to the connexive principles and what can be inferred from them by means of Modus Ponens.

The natural question then is, how do we extend the smallest BCL (or MBCL)? This is just a starting point for richer sets of formulas. Of course, one can add further counter-classical laws for →. Obviously, there is the continuum of such extensions — as many as there are subsets of sets of formulas.

During one of the previous WCL we asked the question how to extend BCL to get as close as possible to classical logic (7th Workshop on Connexive Logics, 2022, UNAM, Mexico). Some extensions of that kind have been already proposed, for example in [5], [3].

The question then is how to extend the smallest BCL, getting closer to classical logic, i.e. making BCL more and more classical, but not to the point of trivializing it. This problem will be the central problem of this paper.

References

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

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

[3] T. Jarmużek, J. Malinowski, A. Parol and N. Zamperlin, Axiomatization of Boolean Connexive Logics with syncategorematic negation and modalities, accepted in 2024 to Logic Journal of IGPL.

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

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

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

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

ABSTRACT. Two claims will be made.

1. It will be argued that the intuitive appeal of the quote from Meyer and Slaney:

"[We] see little reason to claim that such theories are all regular: there is no more compulsion for physicists or gymnasts to assert truths of logic than for logicians to learn gymnastics [12, p. 277]."

disappears if logical pluralism is taken into consideration. According to Roy Cook [3, p. 496], substantial logical pluralism holds that given a formal language L and an identification of L’s logical vocabulary, there exist distinct consequence relations ⊢₁ and ⊢₂ such that a certain correctness principle holds for the pairs ⟨L,⊢₁⟩ and ⟨L,⊢₂⟩. The correctness principle may be a matter of debate, but the main point to be made here is that substantial logical pluralism acknowledges that there may be at least two consequence relations over one and the same language that represent justified options for choosing between them. It may be held that the theorems of a logic are uninformative insofar as they are true in each and every model from a class of models with respect to which the logic in question is ideally sound and complete. A choice between ⟨L,⊢₁⟩ and ⟨L,⊢₂⟩ will, however, involve a choice between classes of models or even between kinds of classes of models. In view of the availability of such a choice, the theorems of a logic matter. If the pluralism is substantial, then so are the differences between the logics one may choose between. The available choice may give a physicist a reason to assert the theorems of ⟨L,⊢₁⟩ instead of those of ⟨L,⊢₂⟩ if the latter differ with respect to their sets of theorems.

2. It will be maintained that connexive counterpart theory is an apt example to illustrate that regular theories are relevant because (the theorems of) the underlying logic may have a significant impact on the meaning of the theoretical, non-logical vocabulary.

QC^=,= is a non-trivial negation inconsistent logic; its propositional fragment Cis negation inconsistent already, see also [9], [14]. In CT based on QC^=,=, CCT, additional contradictions are provable, e.g. the pair of formulas ∃x(∼A(x) → A(x)) and ∼∃x(∼A(x) → A(x)). Moreover, in CT the axioms Ax3 and Ax4 are provably equivalent with

Ax3′ ∀x∀y(C(x,y) → ∼∀z∼I(x,z))

Ax4′ ∀x∀y(C(x,y) → ∼∀z∼I(y,z)).

In CCT, the latter are provably strongly equivalent, and hence replaceable for each other, with

Ax3∗ ∀x∀y∼(C(x,y) → ∀z∼I(x,z))

Ax4∗ ∀x∀y∼(C(x,y) → ∀z∼I(y,z)).

In CT the latter trivialize the counterpart relation in the sense that everything is a counterpart of everything else.

References

[1] Ahmed Almukdad and David Nelson, Constructible falsity and inexact predicates. Journal of Symbolic Logic 49 (1984), 231–233.

[2] Sebastian Bauer and Heinrich Wansing, Consequence, Counterparts and Substitution, The Monist 85 (2002), 483–497.

[3] Roy T. Cook, Let a thousand flowers bloom: A tour of logical pluralism, Philosophy Compass 5 (2010), 492–504.

[4] Dirk van Dalen, Logic and Structure, 4th edition 2004, Springer, Berlin.

[5] Ned Hall, Brian Rabern, and Wolfgang Schwarz, David Lewis’s Metaphysics, The Stanford Encyclopedia of Philosophy (Spring 2024 Edition), Edward N. Zalta and Uri Nodelman (eds), URL = ⟨https://plato.stanford.edu/archives/spr2024/entries/lewis-metaphysics/⟩.

[6] David Lewis, Counterpart Theory and Quantified Modal Logic, Journal of Philosophy 65 (1968), 113–26.

[7] Paolo Maffezioli and Luca Tranchini, Equality and apartness in biintuitionistic logic, Logical Investigations 27 (2021), 82–106.

[8] Franci Mangraviti and Andrew Tedder, Consistent Theories in Inconsistent Logics, Journal of Philosophical Logic 52 (2023), 1133–1148.

[9] Satoru Niki and Heinrich Wansing, On the provable contradictions of the connexive logics C and C3, Journal of Philosophical Logic 52 (2023), 13551383.

[10] Grigory Olkhovikov, On the completeness of some first-order extensions of C, Journal of Applied Logics- IfCoLog Journal 10 ( 2023), 57–114.

[11] Hitoshi Omori and Heinrich Wansing, Varieties of negation and contraclassicality in view of Dunn semantics. In Katalin Bimbó (ed.), Relevance Logics and other Tools for Reasoning. Essays in Honour of Michael Dunn, London, College Publications, 2022, 309–337.

[12] Robert K. Meyer and John Slaney, Logic from A to Z, in: G. Priest, R. Routley, and J. Norman (eds) Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, Munich, 1989, 245–288.

[13] Heinrich Wansing, Connexive modal logic, in R. Schmidt, I. PrattHartmann, M. Reynolds, and H. Wansing, (eds), Advances in Modal Logic. Volume 5. London, King’s College, 2005, 367–383. See also ⟨http://www.aiml.net/volumes/volume5/⟩.

[14] Heinrich Wansing and Zach Weber, Quantifiers in connexive logic (in general and in particular), to appear in: Logic Journal of the IGPL, 2024.

ABSTRACT. This extended abstract reads more like a plan for a research project. It lists a fairly large number of claims many of which will be controversial and need to be substantiated.

1. Connexive logic is special in that it has historical motivations.

2. Modern connexive logic started in the middle of the 20th century, sometime between 1930 and 1962 (after connexive logic had been down for some 800 years).

3. Various formal representations of incompatibility can be distinguished (for the purposes of connexive logic). They are significant for Boethius’s Thesis, Aristotle’s Second Thesis and Abelard’s Thesis. (Aristotle’s First Thesis only needs a non-relational notion of impossibility.)

4. Connexive logic can be considered as a theory of conditionals.

5. At least two kinds of conditionals need to be distinguished (for the purposes of connexive logic). Let us call them entailment-representing conditionals and ordinary conditionals.

6. At least two kinds of connection or relevance need to be distinguished (for the purposes of connexive logic). Let us call them content relevance and status relevance.

7. Traditionally, connexive logic has focussed mainly on entailment-representing conditionals.

8. If connexive logic is (considered as) a theory of conditionals, then it should focus on ordinary conditionals, too.

9. In any case, one should take the utmost care in explaining what exactly a given connexive logical system is supposed to represent or model.

10. First case study: (Conjunctive) Simplification (A ∧ B)→A and (A ∧ B)→B

11. Second case study: Aristotle’s Second Thesis

(Because of time constraints, the two case studies are unlikely to be covered in the talk.)

References

[1] Francez, N., Natural Deduction for Two Connexive Logics, IFCoLog Journal of Logics and their Applications 3(3), 479–504, 2016.

[2] Gibbard, A., Two Recent Theories of Conditionals, in W.L. Harper, R. Stalnaker and G. Pearce (eds.), Ifs, 211–247. Dordrecht: Reidel, 1981.

[3] Martin, C.J., Embarrassing Arguments and Surprising Conclusions in the Development of Theories of the Conditional in the Twelfth Century, in J. Jolivet and A. de Libera (eds.), Gilbert de Poitiers et ses contemporains, pp. 377–400. Naples: Bibliopolis, 1987.

[4] Priest, G., An Introduction to Non-Classical Logic, Cambridge: Cambridge University Press, 2008.

[5] Priest, G., Tanaka, K. and Weber, Z., Paraconsistent logic, in E.N. Zalta (ed.), Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, edition Spring 2022.

[6] Quine, W.V.O., Word and Object, Cambridge, Mass.: MIT Press 1960. (New edition 2013.)

[7] Sylvan, R., Bystanders’ Guide to Sociative Logics (A Short Interim Edition), Australian National University, Canberra, Department of Philosophy, Research School of Social Sciences, 1989.

[8] Wansing, H., Connexive Logic, in E.N. Zalta (ed.), Stanford Encyclopedia of Philosophy, Stanford University, Metaphysics Research Lab, edition Summer 2023.

[9] Wansing, H. and Omori, H., Connexive Logic, Connexivity, and Connexivism: Remarks on Terminology, Studia Logica 112(1–2), 1–35, 2024.

ABSTRACT. In this talk I want to investigate possible applications of queer feminist views on (philosophy of) logic with respect to contradictory logics and especially bilateral representations of these. Thereby I want to show that, on the one hand, the formal set-up of contradictory logics makes them well-suited from the perspectives of feminist logic and, on the other hand, that queer feminist theories provide a relevant, and so far undeveloped, conceptual motivation for contradictory logics. Thus, applying contradictory logics to reasoning about queer feminist issues may prove fruitful both as a ‘real-life’ motivation for these rather marginalized logical systems and as a formal basis for a philosophical field that is still characterized by a distrust of formalism.

References

[1] R. T. Cook and A. Yap, Feminist philosophy and formal logic, University of Minnesota Press, in press.

[2] M. Eckert and C. Donahue, Towards a feminist logic: Val Plumwood’s legacy and beyond, In D. Hyde (Ed.), Noneist explorations II: The Sylvan jungleVolume 3, 424–448, Dordrecht, 2020.

[3] L. Eichler, Sacred Truths, Fables, and Falsehoods: Intersections between Feminist and Native American Logics, APA Newsletter on Native American and Indigenous Philosophy, 18(1): 4–11, 2018.

[4] T. M. Ferguson, From excluded middle to homogenization in Plumwood’s feminist critique of logic, The Australasian Journal of Logic, 20(2): 243–277, 2023.

[5] S. Gorelick, Contradictions of feminist methodology, Gender and Society, 5(4): 459–477, 1991.

[6] S. Haack, Epistemological reflections of an old feminist, Reason Papers, 18: 31–43, 1993.

[7] S. Harding, Words of power: A feminist reading of the history of logic, New York: Routledge, 1991.

[8] A. Nye, Whose science? Whose knowledge?- Thinking from women’s lives, Ithaca, NY: Cornell University Press, 1990.

[9] V. Plumwood, The politics of reason: Towards a feminist logic, The Australasian Journal of Philosophy, 71(4): 436–462, 1993.

[10] R. Routley and V. Routley, Negation and contradiction, Revista Columbiana de Mathematicas, XIX: 201–231, 1985.

[11] G. K. Russell, From anti-exceptionalism to feminist logic, Hypatia: 1–18, 2024.

[12] C. Saint-Croix and R. T. Cook, (What) is feminist logic? (What) do we want it to be?, History and Philosophy of Logic, 45(1): 20–45, 2024.

[13] H. Wansing, Connexive modal logic, In R. Schmidt, I. Pratt-Hartmann, M. Reynolds, H. Wansing (eds.), Advances in modal logic Vol. 5, 367–383, London: College Publications, 2005.

[14] H. Wansing, Natural deduction for bi-connexive logic and a two-sorted typed lambda-calculus, IFCoLog Journal of Logics and their Applications, 3(3): 413–439, 2016.

[15] H. Wansing, Beyond paraconsistency. A plea for a radical breach with the Aristotelean orthodoxy in logic, In A. Rodrigues, H. Antunes, A. Freire (eds.), Walter Carnielli on reasoning, paraconsistency, and probability, Springer, in press.

[16] H. Wansing and S. Ayhan, Logical multilateralism, Journal of Philosophical Logic, 52: 1603–1636, 2023.

ABSTRACT. We begin by considering the following problem. Given a first-order logic QL and its propositional fragment L such that QL and L are subsystems of the classical first-order logic QCL and the classical propositional logic CL, respectively, how do we find a minimal normal (w.r.t. an appropriate relational semantics) conditional operator for this logic? One strategy would be to emulate the success of the classical conditional logic CK introduced in [1] and to look for its analogues on the basis of a given logic L. It often happens, however, that several different systems have a claim to provide such an analogue. We will argue for an approach (somewhat loosely inspired by [2]) where the ultimate touchstone for our choice in these cases is given by the standard translation ST of CK into QCL. Namely, the conditional extension LCK of L is the right analogue of CK on the basis of L iff ST embeds LCK into QL.

In our talk, we will show that this strategy works for both intuitionistic propositional logic IL and the paraconsistent variant of Nelson’s logic of strong negation N4.

Turning next to the question of realizing connexive principles in a conditional logic extending a non-classical propositional basis L, we observe that the very nature of our criterion prevents the minimal conditional from displaying any connexive properties in case the implication of L fails to be connexive. Therefore, connexive conditional operators quite generally cannot be obtained as the minimal conditional operators on the basis of subsystems of CL. In case one is specifically interested in connexive conditionals, two strategies naturally suggest themselves: (1) one may try to realize connexive principles by extending LCK to some nonminimal conditional logic, and (2) one may look for analogues of CK on the basis of some logic L that has a connexive implication.

In both (1) and (2) one has to deal with additional challenges. As for (1), subsystems of classical logics often impose their own specific constraints on admissible extensions that are absent in CL; one example is Disjunction Property. As for (2), note that connexive logics are contra-classical and thus cannot be subsystems of CL. This means that our motivation for the choice of the right minimal conditional operator, which is based on the search for the correct analogue of the classical conditional logic CK must be put into a new perspective and at least somewhat generalized, which is not easy to do systematically, given the current state of research on conditional logics.

We will briefly assess the potential of following the strategy (1) in connection with CL, IL, and N4, and tentatively explore option (2) taking the connexive logic C, introduced in [3] as our main example.

References

[1] B. Chellas, Basic conditional logic, Journal of Philosophical Logic, 4:133–153, 1975.

[2] A. Simpson, The Proof Theory and Semantics of Intuitionistic modal Logic, PhD Thesis, University of Edinburgh, 1994.

[3] H. Wansing, Connexive modal logic, Advances in Modal Logic 5, 367–383, 2005.