Abstract
Anti-exceptionalism about logic states that logical theories have no special epistemological status. Such theories are continuous with scientific theories. Contemporary anti-exceptionalists include the semantic paradoxes as a part of the elements to accept a logical theory. Exploring the Buenos Aires Plan, the recent development of the metainferential hierarchy of \({\mathbf {ST}}\)-logics shows that there are multiple options to deal with such paradoxes. There is a whole \({\mathbf {ST}}\)-based hierarchy, of which \({\mathbf {LP}}\) and \({\mathbf {ST}}\) themselves are only the first steps. This means that the logics in this hierarchy are also options to analyze the inferential patterns allowed in a language that contains its own truth predicate. This paper explores these responses analyzing some reasons to go beyond the first steps. We show that \({\mathbf {LP}}\), \({\mathbf {ST}}\) and the logics of the \({\mathbf {ST}}\)-hierarchy offer different diagnoses for the same evidence: the inferences and metainferences the agents endorse in the presence of the truth-predicate. But even if the data are not enough to adopt one of these logics, there are other elements to evaluate the revision of classical logic. Which is the best explanation for the logical principles to deal with semantic paradoxes? How close should we be to classical logic? And mainly, how could a logic obey the validities it contains (just like classical logic)? From an anti-exceptionalist perspective, we argue that \({\mathbf {ST}}\)-metainferential logics in general—and \({\mathbf {STT}}_{\omega }\) in particular—are the best available options to explain the inferential principles involved with the notion of truth.
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Notes
Of course, we could also consider the case of \(\mathbf {{K}_{3}}\) and the fixed-point Kripke construction (Kripke 1975). In this case, similar points connected with Excluded Middle and Contraction could be elaborated to show that the Buenos Aires Plan is also better than \(\mathbf {{K}_{3}}\). And, of course, the complaints would be similar to \(\mathbf {FDE}\).
For example, the three-valued standard for the propositional logic \(\mathbf {LP}\) says that a valuation that gives either value 1 or value \(\frac{1}{2}\) to every premise, and value 0 to every conclusion, is a counterexample to that inference. If the valuation does not behave that way, then it satisfies the inference. Validity in \(\mathbf {LP}\) is defined as satisfaction in every valuation. There is no circularity here. Of course, we say that some valuations are counterexamples to the validity of a certain inference, but no reference to the notion of validity is need in order to specify what it means for a valuation to either be a counterexample or satisfy a particular inference. Thus, there is no circularity here. We will like to thank an anonymous referee for pointing out this potential problem.
For more information about this notion, and the difference between a local and a global notion of metainferential validity, see Humberstone (1996). In a nutshell, global metainferential validity stands for preservation of validity from premises to conclusions, i.e., a metainference is globally valid if and only if, either some premise is not valid, or some conclusion is valid.
Specifically, those logics invalidates higher-level forms of Cut, which the authors labelled as \(\textit{Meta-Cut}_{n}\).
This logic is also equivalent to Chris Scambler’s twist logic \({\mathbb {T}}\) (Scambler 2020a).
This sequent calculus is based on Girard’s calculus \(\mathbf {G3SK}\), adapted to deal with metainferential levels—and labels. For more about \(\mathbf {G3SK}\), see Girard (1987).
This, for example, is the position defended in Ripley (2012).
Scambler (2020a) defends that a logic should also be characterized through what he calls its antivalidities, while we also argue (in Barrio and Pailos 2021) for including contingencies as part of the picture. In this sense, validities are associated with what a logic (or a logician who supports that logic) accepts (just by supporting that logic), antivalidities with what a logic rejects, and contingencies with the kind of things a logic neither accepts nor rejects. Another way to understand this is in terms of strong and weak assertions, put forward by David Ripley in many papers. In \({\mathbf {CL}}\) both notions collapse. But in \({\mathbf {ST}}_{\omega }\) they are not (extensionally) the same. In a nutshell, a formula being strongly accepted can be cash out as receiving value 1 (in a giving valuation), while being weakly accepted means either receiving value 1 or value \(\frac{1}{2}\) (in a given valuation). Thus, for example, any classical contradiction can be weakly (but not strongly) accepted in the same context (i.e., valuation) in \({\mathbf {ST}}_{\omega }\), but not in \({\mathbf {CL}}\). Finally, Cobreros et al. (2020) propose three other conditional properties closely related to both validities and antivalidities.
As an anonymous reviewer suggests, it might sound a little arbitrary to stop here. Can’t (or shouldn’t) we go one step forward, and explore a logic \(\mathbf {STT}_{\omega + 1}\)? Shouldn’t we have one \(\mathbf {ST}\)-metainferential logic for each ordinal level? We might. In fact, this is an option developed by Scambler in Scambler (2020b). Nevertheless, there are two peculiarities of this kind of approach that should be mentioned. The first one is that, in order to attain non-finite levels, Scambler admits metainferences with premises belonging to different levels. Thus, for example, a metainference of level \(\omega + 3\) might include as premises a metainference of level 45 and a metainference of level 2. This might be reasonable, but is at least non-standard. And second, there is still no sequent calculus for Scambler’s hierarchy. Nevertheless, there are many sequent calculi for \(\mathbf {ST}_{\omega }\) that have been developed by, e.g., Cobreros et al. (2021)—that uses a labeled sequent calculus—, Golan (2021) and Da Ré and Pailos (2021)—both unlabeled options. For these two reasons, we prefer to stick to \(\mathbf {ST}_{\omega }\) instead of going all the way up.
Recovered from Padró (2015, p. 113).
Cfr. (Leitgeb et al. 2020, Suppl. B).
Recently, Rossberg and Shapiro (2021) argue against AEL saying that the supporters of this view “disagree with one another regarding what logic and, indeed, anti-exceptionalism are, and they are at odds with naturalist philosophers of logic, who may have seemed like natural allies.” It is precisely because of this criticism that it is necessary to make explicit what type of AEL we are defending.
This approach should not be confused with Metaphysical anti-exceptionalism, which claims the content of logical theories is unexceptional, i.e. it is similar to the content of other scientific theories (Martin and Hjortland 2019) attributes Metaphysical-AEL to Williamson when saying that logical statements would be the most abstract statements, and to Graham Priest by saying that logical theories would be much like linguistic theories, they are generalizations about linguistic facts. We are not interested in the comparison between the content of logical theories and the content of other scientific theories, but just to the similarity between their justifications.
We are following to Colyvan considering that “normative models are not supposed to model actual behaviour or explain actual behaviour; rather, they are supposed to model how agents ought to act.” Colyvan (2013, p. 1338)
Recently Hlobil argues against this methodological AEL (Hlobil 2020). The rational theory choice in logic cannot be performed by abduction. According to him, abduction cannot serve as a neutral arbiter in logical disputes. The data are conditioned by each of the theories of truth that seek to validate them. Maybe the information about goodness and badness of particular inferences and metainferences could be affected by the theories that are comparing. However, even granting Hlobil’s point, we could minimally deflate the data. No one reasons (and should not reason) trivially in the presence of the truth and the assertion of S has to be equivalent to the assertion of the truth of S. This minimal core that all logic of truth should explain seems to be beyond all theoretical conditioning.
Of course, as an anonymous referee points out, there are many epistemic values to change theories. Not only the adequacy to data and the minimal mutilation must be taken in consideration. There’s also elegance, simplicity, internal coherence, universal coherence, beauty, simplicity, unification, durability, fruitfulness, and applicability... and on and on. We are not absolutely sure that by all the standards in theory choice we have shown the BA-Plan is the best. Of course, someone might think that some other standard is relevant to analyze the revision that we are proposing. In that case, new reasons must be provided to justify that what we propose is the best option. We thank an anonymous referee for pointing this out to us.
Furthermore, these logics would be considered worst, as they are not compatible with Ripley’s bilateralist reading of inferential validity—and metainferential validity as preservation of inferential validity.
We will later mention another reason against it, but as a way to understand obedience.
In the first case, \(\phi \) is an inference. (In Cobreros et al. 2021, it can also be a metainference of any level, but we do not need to be this general to make our point.) In the last one, it represents a sentence.
This can be generalized for higher levels, but, as we have already mentioned, for the present purposes, defining it for level-1 metainferences is enough.
Once again, we are being a little bit sloppy here, since the relevant standards are neither \({\mathbf {TS}}\) nor \({\mathbf {ST}}\), but the truth-theories based on them. But being explicit about it will make things more confusing.
Here is —not very appealing— one way in which this line of thought can be resisted. What we have written is not really Cut. At best, it is a possible labeled way to understand it. We will not initiate a discussion about the relation between labeled and unlabeled metainferences. We would just like to stress the following. It is hard to claim this, and at the same time argue that \({\mathbf {ST}}\) recovers every classical inferential validity. \({\mathbf {ST}}\)’s validities are also implicitly labeled—i.e. because inferences should be understood in a specific way. For example, \({\mathbf {ST}}\) does not validate Modus Ponens—i.e., \(A, A \rightarrow B \vDash B\)—, but, actually, this labeled version of it: \(\mathbf {S:} A, \mathbf {S:} A \rightarrow B \vDash \mathbf {T:} B\). (This is how inferences should be read, according to \({\mathbf {ST}}\).) Therefore, it is not possible to maintain that \({\mathbf {ST}}\) recovers every classically valid inference, which seems to be an assumption in this debate.
Where, for instance, \(\mathbf {X_n}:\Gamma \) is equivalent to \(\{\mathbf {X_n}:\gamma \ |\ \gamma \in \Gamma \}\)
One more thing against this criticism: Ripley’s—and also Scambler’s—demand seems equivalent to ask for local validities to be also globally valid. But why should we ask for that (even if we focus on schemes and not on instances of inferences)? We do impose that at the (traditional) inferential level. And, as metainferences are just inferences of a higher level, we do not impose that for metainferences either.
Or, if inferential logics does not fix how metainferences should be interpreted, how a supporter of (a truth-theory based on) \(\mathbf {ST/ST}\) interprets it.
More precisely, in both cases we should be talking about the truth-theories based on \({\mathbf {TS}}\) and \({\mathbf {ST}}\): \({\mathbf {TST}}\) and \({\mathbf {STT}}\). But, in order to improve readability, we preferred to use the names of the logics.
Nevertheless, we think that in fact there is a property related to both forms of obedience that a theory should satisfy. We claim that, in the context of mixed metainferential logics, that property is instantiated by Mixed Obedience.
Scambler and Porter have pointed out that the non-obedience of \({\mathbf {STT}}_{\omega }\) to its principles looks a lot like the infinite regress problem posed by Modus Ponens and the Carroll’s Turtle. But according to this mixed reading, the problem does not arise—because \({\mathbf {STT}}_{\omega }\) does obey the principles it contains. For more about the infinite regress of the justification of Modus Ponens, see Carroll (1895) and Padró (2015).
Graham Priest has elaborated on this complaint during the Workshop on Substructural Logics, Hierarchies Thereof, and Solutions to the Liar, which took place on CUNY, New York, and was hosted by The Logic and Metaphysics Workshop and the Saul Kripke Center, on October 30, 2020.
Everything that we will say about \({\mathbf {ST}}\) can be also said regarding \({\mathbf {ST}}_{\omega }\), as it shares with the former the standard for sentences and inferences.
What we say about it can also be extended to the rest of the \({\mathbf {ST}}\)-theories and higher than sentential validities in a fairly straightforward way.
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This article belongs to topical collection “Anti-Exceptionalism about Logic”, edited by Ben Martin, Maria Paola Sforza Fogliani, and Filippo Ferrari.
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Barrio, E.A., Pailos, F. & Calderón, J.T. Anti-exceptionalism, truth and the BA-plan. Synthese 199, 12561–12586 (2021). https://doi.org/10.1007/s11229-021-03343-w
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DOI: https://doi.org/10.1007/s11229-021-03343-w