## New Perspectives on the Objects and Grounds of Structural Rules (OGSR)

**Principal Investigator:**

Co-PIs: Bogdan Dicher, Elia Zardini

**Funding Institution:**FCT Project 2022.03194.PTDC

**Date:**2023-2024

#### Description

This project concerns the interpretation of the role of structural rules in logic. Logic is concerned with what follows from what—that is, at its heart lies the analysis of logical consequence. Showing that a conclusion is a logical consequence of some premises typically requires two distinct kinds of rules. To wit, it requires operational rules, which codify the behaviour of the logical operations (negation, conjunction, universal quantification etc.). But it also requires structural rules, which govern instead the ways in which the premises and conclusions of an argument are structured, independently of the logical operations which appear in them (an example is the structural rule of Commutativity, which states, roughly, that the order of the premises does not matter). On the basis of this distinction, it is possible to characterise uniformly a multitude of non-classical logics that have in effect in common the feature of rejecting some structural rules of classical logic (but that may actually agree with classical logic concerning instead its basic operational rules). These logics are nowadays knows as “substructural logics”: logics that are weaker than classical logic in that they reject at least one of its structural rules.

Now, traditionally, structural rules are interpreted as codifying the fundamental properties of the relation of logical consequence. Contrary to this traditional interpretation, the project explores two hypotheses on which structural rules are understood in a radically different manner, either regarding their objects or regarding their grounds.

On one hypothesis to be explored, structural rules codify certain properties of the basic materials of reasoning: sentences, propositions, information-tokens, assertions/denials etc. For example, on this hypothesis, the structural rule of Contraction (which states, roughly, that the number of occurrences of the same premise does not matter) codifies the property that the basic materials of reasoning one is employing do not admit of different tokens of the same type. Such a property would indeed seem to be enjoyed by propositions, so that the structural rule of Contraction would be acceptable when reasoning with propositions (but not, say, when reasoning with information-tokens). Moreover, on this hypothesis, the basic materials of reasoning always occur already embedded in networks of inferential relations: while structural rules therefore codify properties pertaining to the configuration of those relations within a single network, logical consequence itself is a relation among different networks, and so it operates at a different level from that of structural rules.

On the other hypothesis to be explored, structural rules do codify properties of the relation of logical consequence, but such properties are no longer fundamental and are instead derivative on the properties of certain logical operations that are definable in the broad framework of the logic in question. Underlying this hypothesis is the idea that the main structural components of logical consequence (premise combination, conclusion combination and entailment) actually consist in certain kinds of logical operations (conjunction, disjunction and the conditional respectively). For example, on this hypothesis, premise combination consists in a certain kind of conjunction, and so the structural rule of Contraction is derivative on the operational rule of Idempotency for conjunction (according to which A entails the conjunction A&A).

Substructural logics have had important philosophical applications in several areas (logical pluralism, rivalry between logics, paradoxes etc.). In these applications, structural rules are typically understood from the standpoint of the traditional interpretation, both in the sense of codifying properties of the relation of logical consequence and in the sense of such properties’ being fundamental with respect to the logical operations. In virtue of challenging this traditional interpretation, the project has therefore the potential for provoking a fertile debate on the conceptual foundations of substructural logics and their philosophical applications.

#### Research Team

Members: Bruno Jacinto (FCUL), Francesco Paoli (University of Cagliari), Lucas Rosenblatt (University of Buenos Aires), Ricardo Santos (FLUL), Pilar Terres-Villalonga (Louvain), Carlos benito Monsalvo (Postdoctoral researcher, FLUL)

Consultants: Eduardo Barrio (Buenos Aires), Sara Negri (Genoa), Greg Restall (St Andrews), Heinrich Wansing (Bochum)

#### Publications

Dicher, B. (2023). Substructural heresies. Inquiry, 1–25. https://doi.org/10.1080/0020174X.2023.2254816

Dicher, B. Non-foundationalist bases for logical metainferentialism (under review, R&R)

Dicher, B (with Francesco Paoli). Metainferential levels and inferential networks (under review, R&R)

Dicher, B. (with Francesco Paoli). Metainferential harmony (under review, R&R)

Zardini, E. ‘Paradox and Substructurality’, forthcoming in E. Brendel, M. Carrara, F. Ferrari, O. Hjortland, G. Sagi, G. Sher, F. Steinberger (eds), *Oxford Handbook of Philosophy of Logic*, Oxford University Press, Oxford.

Zardini, E. ‘Verdict Exclusion, Higher-Order Vagueness, Cross-Order Vagueness and Borderlineness. Reflections on Footnote 41 of Crispin Wright’s ‘On Being in a Quandary’ ’, forthcoming in M. Petrolo, G. Venturi (eds), Paradoxes between Truth and Proof, Springer, Dordrecht.

Zardini, E. ‘Neoclassical Logic, Cloisonnist Vagueness, Nonobjectual Truth’, forthcoming in S. Leuenberger, A. Rieger (eds), *Themes from Alan Weir*, Springer, Dordrecht.

Zardini, E. ‘Seconde naïveté’, forthcoming in A. Abasnezhad, O. Bueno (eds), *On the Sorites Paradox*, Springer, Dordrecht.

Zardini, E. ‘The Underdetermination of the Meaning of Logical Words by Rules of Inference’, forthcoming in D. Dodd, E. Zardini (eds), *The A Priori: Its Significance, Sources, and Exten*t, Oxford University Press, Oxford.

Zardini, E. (with D. Dodd), ‘Three Questions about the A Priori’, forthcoming in D. Dodd, E. Zardini (eds), *The A Priori: Its Significance, Sources, and Extent*, Oxford University Press, Oxford.

Zardini, E. (2023), ‘Logical Particularism’, in C. Llorente, P. Hernando (eds), *Workshop on Language and Thought. Contributions*, Complutense University of Madrid, Madrid, pp. 43–45.

Zardini, E. (with D. Dodd) eds, *The A Priori: Its Significance, Sources, and Extent*, Oxford University Press, Oxford.

#### Events

2nd Workshop on the Objects and Grounds of Structural Rules Workshop New Perspectives on the Objects and Grounds of Structural Rules Talks: Dicher, B. Substructural heresies. Symposium on Substructural Logic and Metainferences [SuLoMet] @ 17th CLMPST, SADAF and the University of Buenos Aires. Dicher, B. Two ST conditionals? July 31--August 1, 4, 2023,Argentinean Society of Philosophical Analysis. Buenos Aires, Argentina Dicher, B. Logical metainferentialism, Workshop on Rules, Proofs, and Meanings, Arche, University of St Andrews, 11-12 April 2024. Talks by Zardini, E in the scope of OGSR: ‘Changing without Contra(di)ction (II)’, Research Seminar, Federico Villarreal National University, 23/01/2024. - ‘Changing without Contra(di)ction (I)’, Research Seminar, Federico Villarreal National University, 22/01/2024. - ‘Changing without Contra(di)ction’, Research Seminar, National University of Saint Anthony the Abbot in Cuzco, 18/01/2024. - ‘Paradox and Substructurality’, SALOME 1, National University of Saint Anthony the Abbot in Cuzco, 15/01/2024. - ‘Paradox and Substructurality’, Current Debates in the Philosophy of Logic Seminar, University of Bologna, 22/12/2023. - ‘Ápeiron’, Conference Formal Philosophy 2023, Higher School of Economics, Moscow, 12/10/2023. - ‘Totality=Every; Dependence=Some; Choice=Any; Chance=A Certain’, 1st COMP Meeting in Philosophy of Logic and Mathematics, University of Santiago de Compostela, 09/10/2023. - ‘Changing without Contra(di)ction’, Workshop Processes and Changes, Complutense University of Madrid, 25/09/2023. - ‘Paradox and Substructurality’, ECAP 11, Central European University/University of Vienna, 25/08/2023. - ‘Paradox and Substructurality’, Workshop New Perspectives on the Objects and Grounds of Structural Rules 1, University of Lisbon, 15/06/2023. - Response to E. Barrio’s ‘Why Do Meta-Inferences Matter?’, LOGOS Colloquium, University of Barcelona, 23/05/2023. - ‘Becoming Something Else’, Workshop Logical Investigations into Being and Becoming, Complutense University of Madrid, 19/05/2023. - ‘Paradox and Substructurality’, Workshop Beyond Classical Operations, Complutense University of Madrid, 20/04/2023. - ‘Changing without Contra(di)ction’, LLC Seminar, University of Turin, 30/03/2023. - ‘Unstable Knowledge’, Epistemology & Mathematics Seminar Series, IUSS, Pavia, 29/03/2023.