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It seems to me that arguments about logical theories itself are often done using classical logic. For example, one says that a theorem is provable or not provable, which is not automatically valid without law of excluded middle.
Are there references for study of logic from the viewpoint of intuitionism? Or maybe there are immediate problems and such an approach wouldn't make any sense?
Intuitionistic logic is currently studied... and "it works".
See Intuitionistic Logic and The Development of Intuitionistic Logic.
But also Luitzen Egbertus Jan Brouwer and Intuitionism in the Philosophy of Mathematics and Constructive Mathematics.
Some books dedicated to intuitionism :
Arend Heyting, 1956, Intuitionism: An Introduction, Amsterdam: North-Holland Publishing (3rd revised ed, 1971)
Anne Sjerp Troelstra and Dirk van Dalen, 1988, Constructivism in Mathematics: An Introduction, Amsterdam: North-Holland Publishing
Michael Dummett, 1977, Elements of Intuitionism (Oxford Logic Guides, 39), Oxford: Clarendon Press (2nd edition, 2000)
Grigori Mints, 2000, A Short Introduction to Intuitionistic Logic, KLUWER ACADEMIC PUBLISHERS.
For a "standard" textbook in math log with a chapter about intuitionistic logic, see :
Dirk van Dalen, Logic and Structure (5th ed - 2013), Chapter 6: Intuitionistic Logic.
