The following three lectures were given in the form of a short course at the meeting Teoria della Dimostrazione e Filosofia della Logica, organized in Siena, 6–9 April 1983, by the Scuola di Specializzazione in Logica Matematica of the Universit`a degli Studi di Siena. I am very
grateful to Giovanni Sambin and Aldo Ursini of that school, not only for recording the lectures on tape, but, above all, for transcribing the tapes produced by the recorder: no machine could have done that work.
This written version of the lectures is based on their transcription. The changes that I have been forced to make have mostly been of a stylistic nature, except at one point. In the second lecture, as I actually gave it, the order of conceptual priority between the notions of proof and immediate inference was wrong. Since I discovered my mistake later the same month as the meeting was held, I thought it better to let the written text diverge from the oral presentation rather than possibly confusing others by letting the mistake remain. The oral origin of
these lectures is the source of the many redundancies of the written text. It is also my sole excuse for the lack of detailed references.