Abstract
Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set ℐ of the sentences that get limit probability 1 is a Π3—theory, in fact ℐ can be a Π3—complete set. We prove incompleteness results for this setting, by showing for instance that for every k > 0 there are true Π3—sentences that get limit probability less than 1/2k. No set ℐ as above can contain the set of all true Π3—sentences, although we exhibit some ℐ containing all the true Σ2—sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Billingsley, P.: Probability and Measure. New York: Wiley 1986
Gaifman, H., Snir, M.: Probabilities over rich languages, testing and randomness. J. Symb. Logic 47, 495–548 (1982)
Gold, E.M.: Limiting recursion. J. Symb. Logic 30, 28–18 (1965)
Hájek, P.: Experimental logics and Π 03 theories. J. Symb. Logic 42, 515–522 (1977)
Jeroslow, R.G.: Experimental logics and δ 02 theories. J. Philosoph. Logic 4, 253–267 (1975)
Jockush, C.G.Jr.: Semirecursive sets and positive reducibility. Trans. Amer. Math. Soc. 131, 420–436 (1968)
Magari, R.: Su certe teorie non enumerabili. Ann. Mat. Pura Appl. 48, 119–152 (1974)
Mostowski, A.: Examples of sets definable by means of two and three quantifiers. Fund. Math. 42, 259–270 (1957)
Putnam, H.: Trial and error predicates and the solution of a problem of mostowski. J. Symb. Logic 30, 49–57 (1965)
Rogers, H.Jr.: Theory of Recursive Functions and Effective Computability. New York: McGraw-Hill 1967
Scott, D., Krauss, P.: Assigning probabilities to logical formulas. In: Hintikka, J., Suppes, P. (eds) Aspects of inductive logic, pp. 219–264. Amsterdam: North-Holland 1966
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Roberto Magari
This research was partially supported by MURST and the Human Capital and Mobility network Complexity, Logic and Recursion theory CHRXCT930415. We are greatly indebted to Roberto Magari. He was our teacher and friend. He is the one who suggested to try a probabilistic approach to experimental logics and formal systems in general. Unfortunately we will never know whether or not we worked out his ideas satisfactorily
Rights and permissions
About this article
Cite this article
Montagna, F., Simi, G. & Sorbi, A. Logic and probabilistic systems. Arch. Math. Logic 35, 225–261 (1996). https://doi.org/10.1007/s001530050043
Received:
Issue Date:
DOI: https://doi.org/10.1007/s001530050043