Program & Abstracts

Sara Ayhan (presenting work by Edoardo Canonica):

Inferring from Negated Conditionals Connexively: An Experimental Investigation of a Boethian Inference

In his sixth century commentary on Cicero’s Topics Boethius presents four examples of what he takes to be valid inferences involving a negated conditional, the form of which we generalise informally as “if ∼(A → B) and A, then ∼B”. It will be argued that Boethius’ endorsement of these inferences provides evidence of his likely endorsement of reversed variants of Boethius’ Thesis, e.g., ∼(A → B) → (A → ∼B), a principle which is validated by some connexive logics, however is classically invalid. Furthermore, an empirical study will be presented, testing the ‘connexivity’ vs. ‘classicality’ of human reasoning, the results of which are clearly in favor of connexivity. Other studies  have already provided evidence of naive reasoners endorsing connexive theorems. The purpose of the present study was to investigate whether inference patterns, specifically related to negated conditionals, which mirror these connexive principles reflect how reasoners behave when required to draw inferences, or whether reasoners' behaviour is better characterised according to classical alternatives to these principles. In light of the results, a connexive logic validating variants of reversed Boethius' Thesis seems much more suitable than classical logic with respect to representing naive human reasoning.

Maria Beatrice Buonaguidi: 

Paraconsistent arithmetics and classical strength

Abstract: JC Beall (2013) argues that, since we can add to non-classical (including paraconsistent) arithmetics rules restoring classicality, we can effectively recover classical arithmetical reasoning in non-classical systems. According to Halbach and Nicolai (2018), however, the move to non-classical arithmetic comes at the expense of proof-theoretic strength. Then how can paraconsistent arithmetics be said to recover classical strength? It is not sufficient to prove classical arithmetical consequences in a fragment of the language, as done for example by Friedman and Meyer (1992), since this would yield strictly weaker theorems.   

To gesture towards answering the question of whether some paraconsistent arithmetics could be as strong as their classical counterparts, I put forward a so-called recapture result for Zach Weber's paraconsistent arithmetic SUBDLQ-A, based on the logic SUBDLQ (Weber 2021). When dealing with paraconsistent systems of arithmetic, the notions of coding and recusion, which are essential to obtain a proof-theoretic analysis (at least for standard lower-bound techniques) need to be adapted: I reconstruct a notion of coding and recursion for this paraconsisent arithmetic, and show that the theory, if supplemented with additional forms of induction and classical axioms for identity, supports Gentzen's classical lower bound proof. This shows that SUBDLQ-A is at least as strong as classical Peano Arithmetic.

Beall, Jc (2013). LP+, K3+, FDE+, and their 'classical collapse'. Review of Symbolic Logic 6 (4):742-754.

Friedman, Harvey & Meyer, Robert K. (1992). Whither relevant arithmetic? Journal of Symbolic Logic 57 (3):824-831.

Halbach, Volker & Nicolai, Carlo (2018). On the Costs of Nonclassical Logic. Journal of Philosophical Logic 47 (2):227-257.

Weber, Zach (2021). Paradoxes and Inconsistent Mathematics. New York, NY.

Paul Egré: 

Conditional Logics and Presupposition

(Joint work with Lorenzo Rossi and Jan Sprenger)

Abstract: Trivalent logics of indicative conditionals based on either de Finetti's table or Cooper's table state that a conditional of the form "If A then C" is undefined when the antecedent A is false. This paper focuses on the logic OL first put forward by Cooper (1968), also called system C by Egré, Rossi, Sprenger (2025b, "Certain and Uncertain Inference with Indicative Conditionals") in relation to probability and to certainty preservation. I will discuss two related issues of the trivalent semantics and corresponding logic. The first concerns the presuppositions of indicative conditionals. Because of how it handles negation, it has been argued that the trivalent semantics of conditionals wrongly predicts conditionals to presuppose the truth of their antecedent. In the first part of this paper, I use a recent extension of system C to modalities (Egré, Rossi, Sprenger 2025b, "Trivalent Conditionals, Krater style'') to derive a weaker and more adequate presupposition, namely that indicative conditionals only presuppose their antecedent to be compatible with the context of assertion. The derivation is based on a specific version of Stalnaker's Bridge principle between semantic and pragmatic presuppositions, which we call ``Avoid Void". In the second part of the paper, the same principle is used to filter out some instances of the paradoxes of material implication that remain valid in OL, but that appear to violate presuppositions in the same way. Vanilla OL is a connexive, contradictory logic, but neither relevantist, nor paraconsistent. I present two strengthenings of it, called C# and C-x, which come closer to satisfying both of these features on presuppositional grounds.

Luis Estrada González: 

Inconsistent metatheory through uniformity, and what one can find there

Abstract: I develop previous unpublished work towards an argument for inconsistent metatheory and its effects on (apparently not so) well-known logics. The gist of the proposal to be presented here is as follows. Someone skpetical of the Dunn-Belnap-Weber arguments to connect internal and external negations (for example, “A is false” is logically the same as “A is not true”; “not-A belongs to a set” is the logically same as “A does not belong to the set”) may nonetheless be sensitive to the idea that core logical notions such validity and invalidity should be formulated uniformly. Whereas validity, i.e. 

• An argument is valid iff, if whenever the premises are true, the conclusion is true as well.

is (positively) homogeneous, for the Dunn clauses involved are all positive (since the copulas are all positive: “B is true”, for every premise B; “A is true”, for the conclusion), invalidity, i.e.

• An argument is invalid iff in some case the premises are true but the conclusion is not.

is not homogeneous, for some Dunn clauses are positive (those for the premises) and others are negative (that for the conclusion, for the copula is negative). Thus, the core logical notions are not uniformly formulated. A (positively) uniform formulation would require invalidity to be defined thus:

• An argument is invalid iff in some case the premises are true but the conclusion is false.

With this little change made upon some logics like FDE and LP, one obtains logics where some arguments are both valid and invalid (because in fact all arguments are invalid).

Note that in this case one does not “lift an inconsistent set of logical truths to external inconsistency”, for neither FDE and LP, even under the reformulation, contain inconsistent logical truths. But inconsistent validities can be related to external inconsistency as follows:

• If the argument is L-valid then it belongs to the set Val(L).

• If the argument is L-invalid then it does belong to the set Val(L).

Then there are arguments that belong and do not belong to Val(L). If L = Val(L), this means that there are arguments that belong and do not belong to L. However, this last move could be resisted, for there are reasons not to accept L = Val(L).

After this rather long preamble on another route to an inconsistent metatheory, I show how deep this would affect LP. For example, due to the implicative nature of validity, and since there is no other implication in LP but extensional implication, which is a disjunction in disguise, the definition of validity is equivalent to this one:

• An argument is valid iff, in every case, either the premises are false or the conclusion is true.

LP, or, better, Ԁ⅂, gets at once closer to classical logic (because, for example, it has exactly the same valid arguments as classical logic) but also farther from it (because, for example, all arguments are invalid). I end up discussing some potential avenues to have arguments that belong and do not belong to L even if one does not buy L = Val(L).

Satoru Niki: 

Provable contradictions and synonymy of formulas 

Abstract: Wansing's connexive logic C is one prominent example of contradictory logics, which is a family of logics which prove contradictory pairs of formulas. In order to better understand this phenomenon, it is desirable to be able to categorise contradictory formulas into different classes. The usual notion of equivalence in terms of biconditional is however of little use here, as the formulas being dealt with are theorems. An alternative can be found in a stricter notion (also introduced by Wansing) called synonymy, which takes derivations into account. In this talk, I will discuss some observations concerning provable contradictions in C from the viewpoint of synonymy. In addition, I will provide some elementary considerations towards the type-theoretic understanding of the notion of synonymy.

Grigory Olkhovikov: 

Constructive, Connexive, and Contraclassical: Conditionals over C

Abstract: In this talk, we consider several expansions of the propositional logic C with modal and conditional operators. This logic, originally introduced in [5], is a close relative of Nelson's logic of strong negation and, in particular, inherits the constructive character of the latter logic. In addition, C is connexive in the sense that it verifies principles like Aristotle's Thesis and Boethius' Thesis which fail in classical logic. Thus, C is also contraclassical.

In the first part of the talk, we consider a C-based modal logic CnK, which we also compare with several other modal logics (including C-based modal logic introduced in [5] together with C itself, N4-based logic FSK^d [2], and the intuitionistic modal logic IK [1]). We argue that CnK faithfully reflects the special features of C in its properly modal part and can be called a right C-based analogue of the classical modal logic K.

In the second part of the talk, we consider a minimal C-based conditional logic CnCK and its reflexive extension CnCK_R. We also touch upon the extensions of these logics given by 1-layered conditional semantics, named CnCK' and CnCK'_R. These logics are again compared to several other conditional logics, including intuitionistic conditional logic IntCK (introduced in [3]) and N4-based conditional logics N4CK and N4CK' (introduced in [4]). The most important among these comparisons, however, is with the group of conditional logics introduced in [6], as the approach of the latter paper serves as the inspiration source for the introduction of the logics in the group {CnCK, CnCK', CnCK_R, and CnCK'_R}.

We also quickly survey the connexivity profile of all the logics introduced in the talk and isolate some sets of provable negation-inconsistencies for each of them; in particular, we show how these provable negation-inconsistencies reflect the inconsistency structure of their base logic C at the level of modal (resp. conditional) language.

References

1. G. Fischer-Servi. Semantics for a class of intuitionistic modal calculi. M.L. Dalla Chiara, editor. Italian Studies in the Philosophy of Science. Studies in the Philosophy of Science, Vol. 47, 59-72. Dordrecht: Springer. (1981)

2. S. Odintsov, H. Wansing. Constructive predicate logic and constructive modal logic. Formal duality versus semantical duality. V. Hendricks et al., eds, First-Order Logic Revisited. 269-286, Berlin, Logos. (2004).

3. G. Olkhovikov. An Intuitionistically complete system of basic intuitionistic conditional logic. Journal of Philosophical Logic,  53:1199-1240 (2024)

4. G. Olkhovikov. A basic system of paraconsistent Nelsonian logic of conditionals. Journal of Logic, Language and Information,  33:299-337, (2024)

5. H. Wansing. Connexive modal logic. Advances in Modal Logic 5, 367-383 (2005).

6. H. Wansing, M. Unterhuber. Connexive conditional logic. Part I. Logic and Logical Philosophy, 28:567-610 (2019).

Hitoshi Omori: tba

Graham Priest: 

Dialetheism and Dimathematism 

Abstract: In 'One Heresy and One Orthodoxy: on Dialetheism, Dimathematism, and the Normativity of Logic' (Erkenntnis 89:181–205) Heinrich Wansing defines a notion he calls dimathematism, which, he says, is an alternative to dialetheism. Dimathematism is a perfectly coherent view, but it is not, I think, an alternative to dialetheism. In the talk I will explain why.

Yaroslav Shramko: 

Minimal Abelian Logic

Abstract: Summarizing Johansson's idea, one can conceive of a minimal negation with respect to some positive logic by defining it through implication (→) and some constant (say, ι), which is intuitively understood as a "falsity constant", so that ~ is introduced by A→ ι, and no additional assumptions about ι are made. The properties of such negation must then be determined solely by the properties of the corresponding implication. The implication of Abelian logic can be obtained by adding the Abelian axiom ((A → B) → B) → A to the so called BCI-logic. However, the standard extension of implicational Abelian logic by means of negation is usually carried out in a "non-minimal way", since its definition through a propositional constant presupposes certain conditions imposed on the constant itself. One of the consequences of these conditions is that conventional Abelian logic turns out to be contradictory with respect to negation (negation inconsistent) in the sense that there are statements that can be derived in this logic together with their negations. In my talk I propose Abelian logic in which negation is genuinely minimal. This logic turns out to be non-contradictory (negation consistent), and moreover it determines a full-fledge intensional logic in which intensional conjunction (fusion) and disjunction (fision) are distinct from each other. It can also be equipped with a suitable ternary relational semantics. 

Heinrich Wansing:  

Dimathematism and Dialetheism

Abstract: tba