Uploaded October 2021 | Updated September 2026, 18 hours ago
Recorded in the Carl Friedrich von Weizsäcker Colloquium on the 20th of October 2021
Prof. Dr. Jan von Plato (University of Helsinki): How Gödel discovered his incompleteness theorems
Gödel surprised the mathematical world by his famous incompleteness theorems of arithmetic of 1931. The way he arrived at these results is described against two shorthand notebooks of his that have been recently transcribed.
Recorded in the Carl Friedrich von Weizsäcker Colloquium on the 20th of October 2021
Prof. Dr. Jan von Plato (University of Helsinki): How Gödel discovered his incompleteness theorems
Gödel surprised the mathematical world by his famous incompleteness theorems of arithmetic of 1931. The way he arrived at these results is described against two shorthand notebooks of his that have been recently transcribed.


![Prof. Dr. Federico Pailos (Buenos Aires): Why metainferences matter
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. Prof. Dr. Federico Pailos (Buenos Aires): Why metainferences matter](https://i.ytimg.com/vi/xK5iHT9wK9M/mqdefault.jpg)

![Michel Janssen - COI Stories II: Inferences or Incentives?
Recorded as part of the CFvW Colloquium on 04. December 2024
COI Stories II: Inferences or Incentives? - Prof. Dr. Michel Janssen (Lichtenberg Group for History and Philosophy of Physics, University of Bonn & School of Physics & Astronomy, University of Minnesota)
Abstract:
In 2002*, I introduced COI (Common Origin Inference) as a subspecies of IBE (Inference to the Best Explanation). Several of my examples of COIs from the history of science, however, were not really about inference. In these examples, COI served not so much to increase the degree of belief in the inferred explanation as to establish the pursuit-worthiness of that explanation. In view of this (and sticking to the acronyms), I argue that we should allow the ‘I’ in both COI and IBE to stand not only for ‘Inference’ but also for ‘Incentive’ (to pursue some [common-origin] explanation). In science, the ‘I’ in most cases stands for ‘Incentive’. Switching from ‘Inference’ to ‘Incentive’ provides an answer to an objection to IBE articulated most forcefully by Wesley Salmon in 2001: Why should an elegant explanation be more likely than an ugly one? Or, to rephrase Salmon’s question in the terminology introduced by Peter Lipton, a staunch defender of IBE: why should likeliness track loveliness? Interpreting the ‘I’ in COI and IBE as ‘Incentive’, one can take the position that pursuit-worthiness tracks loveliness, which is more plausible.
* COI Stories: Explanation and Evidence in the History of Science. Perspectives on Science. The MIT Press Volume 10, Number 4, Winter 2002 pp. 457-522, muse.jhu.edu/article/43975
Explore our colloquium schedule on our website: https://uni-tuebingen.de/en/research/centers-and-institutes/carl-friedrich-von-weizsaecker-center/news-and-events/carl-friedrich-von-weizsaecker-colloquium Michel Janssen - COI Stories II: Inferences or Incentives?](https://i.ytimg.com/vi/xy1SlsbWzDc/mqdefault.jpg)



