Induction problem

One problem with induction is the dependence on the language used in its formulation. Fortunately by placing reasonable restrictions on acceptable languages these dependencies can be resolved and the problem of induction be made well defined.

K.F. Samokhvalov considered formal definitions of induction methods and the requirements they must satisfy. While linguistic invariance - the invariance of induction methods with the respect to the formulating language - is an obvious requirement, he concluded that no induction methods satisfy it, except for some vacuous cases. (Samokhvalov K. F. Logics of Discovery. In: Inductive logic and scientific knowledge formation. Moscow, 1987, "Science", pp.173-184). This result was verified in another way in the paper
Vityaev E.E., Novikov V.F. Induction methods paradoxicality // International J. of Pattern Recognition and Artificial Intelligence (Proceedings of the International workshop on Expert Systems and Pattern Recognition (ESPR), 26-29 Oct., 1987, Novosibirsk, USSR), v.3, N.1, 1989, p. 147-157. where observational results were represented by points in a features space.

N. Goodman in "The New Riddle of Induction" (Journal of Philosophy 63 (1966): 281-331) was the first to notice this dependence of induction methods on language, a dependence that leads to paradoxes.
Generalization of the Goodman's paradox presented in the papers.

Evgenii Vityaev, Irina Khomicheva. Goodman's paradox generalizations. In: Proceedings of the 12th International Congress of Logic Methodology and Philosophy of Science, (12 International Congress of Logic, Methodology and Philosophy of Science Oviedo (Spain), August 7-13 2003), 2003, pp.146-147.

Vityaev E., Khomicheva I, Stating of induction problem using second level laws of nature In: Probabilistic ideas in science and philosophy, Novosibirsk, 2003, pp.72-86.

Vityaev E., Khomicheva I. Goodman's induction paradox. In: Methodological problems of cognitive processes. Proceedings of the Institute of mathematics SD RAS (Computer systems, #170), Novosibirsk, 2002 (in Russian)

Витяев Е.Е., Хомичева И.В. Универсальность парадокса Гудмена - возникновение парадокса Гудмена в каждом акте усиления гипотез. In: Информационные технологии работы со знаниями: обнаружение, поиск, управление (Вычислительные системы, вып. 175), Новосибирск, 2008, стр. 107-121

One way to avoid Goodman's paradox is to extend the notion of language/ontology with more properties concerning experimental procedures. By formulating a property of experiment - "the inheritance of experimental results" - it can be proven that experimental dependency is logically equivalent to a set of universal formulas. This property of experiment restricts the possible transformations of the language but gives a class of hypotheses which express the experimental dependence. Only hypotheses from this class need to be tested by the inductive method. This makes the problem of induction well defined.
"The inheritance of experimental results" is fulfilled for most of physics. If we have set of observational results A in a model M, then the observational result of a subset B of A will be submodel of the model M. Using these ideas an inductive method "Discovery" has been developed and used successfully in several areas.