The Concepts of Proof and Ground
Convinced that the concepts of proof and ground should be a main concern of proof theory as mathesis universalis, I discuss in this paper how one may try to develop Bolzano’s ideas about grounds in the light of Gentzen’s ideas about the justification of inferences. At the end, I briefly consider whether the result can give some insights about the concepts of proof and ground. I proceed in a somewhat experimental fashion, starting with proposals for how to specify grounds for propositions of various forms that stay fairly close to Bolzano’s ideas. I then apply results from present day proof theory and show how one has to depart successively from Bolzano in order to get a tenable concept of ground. The final definition of ground that I arrive at may be seen as a combination of ideas from Bolzano and Gentzen and later results in proof theory.
I am indebted to professor Göran Sundholm for helpful comments to the initial part of the manuscript.
- Berg, J. (1962). Bolzano’s logic. Uppsala: Almqvist & Wicksell.Google Scholar
- Buhl, G. (1961). Ableitbarket und Abfolge in der Wissenschaftslehre Bolzanos. In Kantstudien Ergänzungshefte, 83. Köln: Kölner Universitetsverlag.Google Scholar
- Dummett, M. (1991). The logical basis of metaphysics. London: Duckworth.Google Scholar
- Prawitz, D. (1965). Natural deduction. A proof-theoretical study. Uppsala: Almqvist & Wicksell. (Republished by Dover Publications, New York, 2006.).Google Scholar
- Prawitz, D. (1973). Towards a foundation of general proof theory. In P. Suppes et al. (Eds.), Logic, methodology and philosophy of science IV (pp. 225–250). Amsterdam: North-Holland.Google Scholar
- Prawitz, D. (2011). Proofs and perfect syllogisms. In C. Cellucci et al. (Eds.), Logic and knowledge (pp. 385–402). Newcastle upon Tyne: Cambridge Scholars Publishing.Google Scholar
- Prawitz, D. (2019). The fundamental problem of general proof theory. Studia Logica, 107, 11–29. Google Scholar
- Sundholm, G. (2008). A century of judgement and inferences, 1837–1936: Some strands in the development of logic. In L. Haaparanta (Ed.), The development of modern logic. Oxford: Oxford University Press.Google Scholar
- Sundholm, G. (2011). A garden of grounding trees. In C. Cellucci et al. (Eds.), Logic and knowledge (pp. 57–74). Newcastle upon Tyne: Cambridge Scholars Publishing.Google Scholar
- Wedberg, A. (1966). Filosofins historia. Från Bolzano till Wittgenstein. Stockholm: Bonniers. (English translation: A History of Philosophy, Volume 3. From Bolzano to Wittgenstein. Oxford: Clarendon Press, 1984.).Google Scholar