Many-valued logic of informal provability: a non-deterministic strategy Paper forthcoming in The Review of Symbolic Logic Pawel Pawlowski Centre for Logic and Philosophy of Science, Ghent University, Belgium Rafal Urbaniak Centre for Logic and Philosophy of Science, Ghent University, Belgium Institute of Philosophy, Sociology and Journalism, University of Gdansk, Poland Abstract Mathematicians prove theorems in a semi-formal setting, providing what we’ll call informal proofs. There are various philosophical reasons not to re- duce informal provability to formal provability within some appropriate ax- iomatic theory (Leitgeb, 2009; Marfori, 2010; Tanswell, 2015), but the main worry is that we seem committed to all instances of the so-called reflection schema: B(ϕ) → ϕ (where B is the informal provability predicate). Yet, adding all its instances to any theory for which L¨ ob’s theorem for B holds leads to inconsistency. Currently existing approaches (Shapiro, 1985; Horsten, 1996, 1998) to for- malizing the properties of informal provability avoid contradiction at a rather high price. They either drop one of the Hilbert-Bernays conditions for the provability predicate, or use a provability operator that cannot consistently be treated as a predicate. Inspired by (Kripke, 1975), we investigate the strategy which changes the underlying logic and treats informal provability as a partial notion. We use non-deterministic matrices to develop a three-valued logic of informal prov- ability, which avoids some of the above mentioned problems. Keywords. Informal provability, many-valued logic, non-deterministic se- mantics, L¨ ob’s theorem, paradoxes of provability AMS classification. 03A05, 00A30 Acknowledgments. Research on this paper has been funded by the Re- search Foundation Flanders (FWO). The authors would like to express their gratitude to all those who commented on the earlier versions of this paper or contributed to discussions about the topic when the material was presented: Cezary Cie´ sli´ nski, Leon Horsten, Hannes Leitgeb, Frederik Van De Putte and Stanislav Speransky.