Lecture series

The Logics of Metainferences

Practical information
24 June 2022

Lectures by Eduardo Barrio on The Logics of Metainferences

Eduardo Barrio (CONICET, Buenos Aires) will be Invited Professor at ENS for one month from June 10 to July 10. Professor Barrio is invited by the Philosophy Department of ENS, with also the support of CNRS (summer school on Conditionals) and of the team CoLoR of Institut Jean-Nicod. He will give a series of lectures on the Logics of Metainferences. The lectures will take place:


The week of June 13 to June 17: 
- from 5.30 to 6.30 pm every day, in the context of the CNRS summer school Conditionals 2022 held at INALCO, 65 rue des Grands Moulins, 75013 Paris. 

The week of June 20 to 24, on:
June 20 at ENS, 29 rue d’Ulm, Institut Jean-Nicod, salle de réunion du pavillon Jardin, 2pm-4pm. 
June 24 at ENS, 29 rue d’Ulm, Salle Ribot, 2pm-4pm.

 The lectures are open to all interested in logic and proof theory. ENS students and faculty are also welcome to attend other lectures in the CNRS summer school on conditionals. Registration mandatory (no fee).



Metainferences have recently come into focus as a useful way of distinguishing between various substructural solutions to semantic paradoxes. as a new way to characterize a logic, as a way to analyze the debate between global and local validity, as a toolkit for understanding abstract features of consequence relations, and as a key for a new version of the collapse argument against logical pluralism. Moreover, there are some paraconsistent elements that connect Priest’s Logic of Paradox (LP) and the Strict-Tolerant approach (ST) of Cobreros, Egré, Ripley and van Rooij: giving up Cut in the latter has as a consequence the loss of other metainferences, closely connected with Modus Ponens and Explosion in the former (aka. Meta-modus Ponens and Meta-explosion). There is, then, a suggestive match between the valid metainferences of ST and the set of inferential validities of LP. This result can be generalized to recapture more and more classical metainferences. As a result, a new hierarchy of metainferential logics based on the non-transitive logic ST can be presented. In this course, we will analyze this hierarchy to recover full classical logic in Strong Kleene models. We will also present the BA-plan: the project to apply full classical logic to deal with semantic paradoxes. From an anti-exceptionalist perspective, we will argue that the adoption of the BA-plan is better than its rivals as theories of transparent truth.

Week 1

What is a Logic? (June 13, 14)

Different notions of logical consequence in many-valued models.  LP – K3. Paraconsistent and Paracomplete logics. Mixed logical consequences: ST and TS. Substructural properties. 

Logics and Metainferences (June 15, 16, 17)

What is a metainference? Local and global validity. The Cut Rule. The cases of Meta-explosion and Meta-Modus Ponens.  Internal and external notions of logical consequence. The method of traductions. Structural properties as a logic. 

Week 2

What is Classical Logic? (June 20)

Metainferential validities. The hierarchy of classical logics. Different definitions of logics: extensional and intensional criteria.  

The BA-plan (June 24)

Applying logics to paradoxes. Could Classical logic deal with semantics paradoxes? Which is the best explanation of the logic of truth? 



  • Barrio, Eduardo, Federico Pailos, and Damian Szmuc. "What is a paraconsistent logic?" In Contradictions, from consistency to inconsistency, pp. 89-108. Springer, Cham, 2018.
  • Barrio, Eduardo, Federico Pailos, and Damian Szmuc. "A hierarchy of classical and paraconsistent logics." Journal of Philosophical Logic 49, no. 1 (2020): 93-120.
  • Barrio, Eduardo, Federico Pailos, and Damian Szmuc. "(Meta) inferential levels of entailment beyond the Tarskian paradigm." Synthese 198, no. 22 (2021): 5265-5289.
  • Barrio, Eduardo, Lucas Rosenblatt, and Diego Tajer. "The logics of strict-tolerant logic." Journal of Philosophical Logic 44, no. 5 (2015): 551-571.
  • Barrio, Eduardo, Federico Pailos, and Joaquín Toranzo Calderón. "Anti-exceptionalism, truth and the BA-plan." Synthese 199, no. 5 (2021): 12561-12586.
  • Chemla, Emmanuel, Paul Égré, and Benjamin Spector. "Characterizing logical consequence in many-valued logic." Journal of Logic and Computation 27, no. 7 (2017): 2193-2226.
  • Chemla, Emmanuel, and Paul Égré. "Suszko’s problem: Mixed consequence and compositionality." The Review of Symbolic Logic 12, no. 4 (2019): 736-767.
  • Chemla, Emmanuel, and Paul Égré. "From many-valued consequence to many-valued connectives." Synthese 198, no. 22 (2021): 5315-5352.
  • Cobreros, Pablo, Paul Égré, David Ripley, and Robert van Rooij. "Tolerant, classical, strict." Journal of Philosophical Logic 41, no. 2 (2012): 347-385.
  • Cobreros, Pablo, Paul Égré, David Ripley, and Robert Van Rooij. "Reaching transparent truth." Mind 122, no. 488 (2013): 841-866.
  • Mares, Edwin, and Francesco Paoli. "Logical consequence and the paradoxes." Journal of Philosophical Logic 43.2 (2014): 439-469.
  • Priest, Graham. "The logic of paradox." Journal of Philosophical logic (1979): 219-241.
  • Ripley, David. "One step is enough." Journal of Philosophical Logic (2021): 1-27.
  • Scambler, Chris. "Classical logic and the strict tolerant hierarchy." Journal of Philosophical Logic 49, no. 2 (2020): 351-370.


Organization and contact at the ENS: Paul Égré (paul.egre@ens.fr)

Acknowledgements: Patrick Caudal & Ghanshyam Sharma (Summer school “Conditionals 2022”)