Logics, interpretations and informal rigor

The development of non-classical logics brought to light the question of their philosophical interpretation. Given a non-classical logic L, what is the informal/philosophical interpretation of its vocabulary? We find in the literature arguments defending that logical systems do not have a canonical philosophical interpretation, so the same logical system L has different philosophical interpretations. Although the thesis that a logic can be philosophically interpreted in different ways is well known, the thesis that a specific philosophical interpretation can be captured by different logics has not been widely explored. In this paper, we argue that if Kreisel’s informal rigor method is adequate to show that formal notions of logical consequence capture informal notions of logical consequence, then a specific philosophical interpretation only correspond to a unique logic.