Uploaded August 2021 | Updated September 2026, 6 hours ago
Recorded in the Carl Friedrich von Weizsäcker Colloquium on the 14th of July 2021
Prof. Dr. Federico Pailos (Buenos Aires): Why metainferences matter
In this talk, I will present new arguments that shed light on the importance of metainferences of every level, and metainferential standards of every level, when (semantically) characterizing a logic. This implies that a logician cannot be agnostic about metainferences, metametainferences, etc. The arguments I will introduce show why a thesis that Dave Ripley defends in [1] and [2] is false. This is how he presents it.
Note that a meta0counterexample relation X [i.e., a counterexample relation for infer- ences, which is (in most contexts) equivalent to a satisfaction relation for inferences], on its own, says nothing at all about validity of metaninferences for 0 ﹤ n. Despite this, there is a tendency to move quickly from X to [X] [i.e., a full counterexample relation for every metainferential level], at least for some purposes... For example, [3] (p. 360, notation changed) says “[A]bsent any other reasons for suspicion one should probably take [X] to be what someone has in mind if they only specify X.” I don’t think this tendency is warranted. Most of the time, when someone has spec- ified a meta0counterexample relation (which is to say an ordinary counterexample relation), they do not have the world of all higher minferences [i.e., metainferences of any level], full counterexample relations, etc, in mind at all. They are often focused on validity for meta0inferences (which is to say inferences). ([1], page 12.)
Though I do think that, in a sense, people do have in mind [X] when they say X, I will not argue for that. I just want to defend that they should have something like that in mind. Specifically, I will show why the following position should be revised:
As I’ve pointed out, an advocate of ST as a useful meta0counterexample relation has thereby taken on no commitments at all regarding metancounterexample relations for 1 ≤ n. ([1], page 16).)
Or, as Ripley puts in somewhere else:
... if someone specifies just a metanconsequence relation, they have not thereby settled on any particular metan+1 consequence relation. ([2]).)
If Ripley’s statements are true, then two different logicians may count as advocates of the same inferential logic (or any metainferential logic of level n), despite adopting quite different criteria regarding what counts as a valid metainference (or a valid metainference of level n+1). If Ripley is right, then not only can a supporter of a (non-transitive) logic like ST accept or reject the metainference corresponding to (some version of) the Cut rule, but also she can admit a metainferential counterexample relation that correspond to a trivial or an empty metainferential consequence relation. Moreover, this might have repercussions on the inferential level, as an
empty metainferential logic invalidates any metainference with an empty set of premises and a valid ST-inference as a conclusion. Thus, the only available option is to admit that inferences, on the one hand, and metainference with an empty set of premises and that inference as its only conclusion, on the other hand, are not only different, but also non-equivalent things. Something similar happens if we chose a trivial metainferential counterexample relation while adopting ST at the inferential level. In this case, there will be invalid ST-inferences that turns out to be valid in its metainferential form, forcing this logician to chose between one of the options that we have specified before.
This is a particular strong result, and it is even stronger than what might initially seem, in two senses: (1) it does not depend on the notion of metainferential validity being favoured—e.g., whether one thinks that the local way to understand it is better than the global, or the other way around; (2) it does not depend on the special features of the (mixed) inferential/metainferential relations, as this result can be replied for any pair of (mixed) metainferential relations of level n/n+1.
References
[1] D. Ripley. One step is enough. (Manuscript).
[2] D. Ripley. A toolkit for metainferential logics. (Manuscript).
[3] C. Scambler. Classical Logic and the Strict Tolerant Hierarchy. Journal of Philosophical Logic, page forthcoming, 2019. DOI: doi.org/10.1007/s10992-019-09520-0.
Recorded in the Carl Friedrich von Weizsäcker Colloquium on the 14th of July 2021
Prof. Dr. Federico Pailos (Buenos Aires): Why metainferences matter
In this talk, I will present new arguments that shed light on the importance of metainferences of every level, and metainferential standards of every level, when (semantically) characterizing a logic. This implies that a logician cannot be agnostic about metainferences, metametainferences, etc. The arguments I will introduce show why a thesis that Dave Ripley defends in [1] and [2] is false. This is how he presents it.
Note that a meta0counterexample relation X [i.e., a counterexample relation for infer- ences, which is (in most contexts) equivalent to a satisfaction relation for inferences], on its own, says nothing at all about validity of metaninferences for 0 ﹤ n. Despite this, there is a tendency to move quickly from X to [X] [i.e., a full counterexample relation for every metainferential level], at least for some purposes... For example, [3] (p. 360, notation changed) says “[A]bsent any other reasons for suspicion one should probably take [X] to be what someone has in mind if they only specify X.” I don’t think this tendency is warranted. Most of the time, when someone has spec- ified a meta0counterexample relation (which is to say an ordinary counterexample relation), they do not have the world of all higher minferences [i.e., metainferences of any level], full counterexample relations, etc, in mind at all. They are often focused on validity for meta0inferences (which is to say inferences). ([1], page 12.)
Though I do think that, in a sense, people do have in mind [X] when they say X, I will not argue for that. I just want to defend that they should have something like that in mind. Specifically, I will show why the following position should be revised:
As I’ve pointed out, an advocate of ST as a useful meta0counterexample relation has thereby taken on no commitments at all regarding metancounterexample relations for 1 ≤ n. ([1], page 16).)
Or, as Ripley puts in somewhere else:
... if someone specifies just a metanconsequence relation, they have not thereby settled on any particular metan+1 consequence relation. ([2]).)
If Ripley’s statements are true, then two different logicians may count as advocates of the same inferential logic (or any metainferential logic of level n), despite adopting quite different criteria regarding what counts as a valid metainference (or a valid metainference of level n+1). If Ripley is right, then not only can a supporter of a (non-transitive) logic like ST accept or reject the metainference corresponding to (some version of) the Cut rule, but also she can admit a metainferential counterexample relation that correspond to a trivial or an empty metainferential consequence relation. Moreover, this might have repercussions on the inferential level, as an
empty metainferential logic invalidates any metainference with an empty set of premises and a valid ST-inference as a conclusion. Thus, the only available option is to admit that inferences, on the one hand, and metainference with an empty set of premises and that inference as its only conclusion, on the other hand, are not only different, but also non-equivalent things. Something similar happens if we chose a trivial metainferential counterexample relation while adopting ST at the inferential level. In this case, there will be invalid ST-inferences that turns out to be valid in its metainferential form, forcing this logician to chose between one of the options that we have specified before.
This is a particular strong result, and it is even stronger than what might initially seem, in two senses: (1) it does not depend on the notion of metainferential validity being favoured—e.g., whether one thinks that the local way to understand it is better than the global, or the other way around; (2) it does not depend on the special features of the (mixed) inferential/metainferential relations, as this result can be replied for any pair of (mixed) metainferential relations of level n/n+1.
References
[1] D. Ripley. One step is enough. (Manuscript).
[2] D. Ripley. A toolkit for metainferential logics. (Manuscript).
[3] C. Scambler. Classical Logic and the Strict Tolerant Hierarchy. Journal of Philosophical Logic, page forthcoming, 2019. DOI: doi.org/10.1007/s10992-019-09520-0.
