A Taxonomy of C-Systems

In the dawn of the XXI century, debates on the statute of contradiction in logic, philosophy and mathematics are still likely to raise the most diverse and animated sentiments. And this is an old story, whose first dramatic strokes can be traced back to authors as early as Aristotle (for the defense of non-contradiction), or Heraclitus (for the contrary position). Be that as it may, the fact is that in the beginning of the last century essentially the same dispute was still taking place, this time contraposing Russell to Meinong. And so it could still proceed, for centuries, if only the philosophical aspects of the dispute were touched. Even on more technical grounds, logicians of caliber, such as Alfred Tarski, would eventually speculate about that (cf. [106]): Against such suspicions, the philosopher Wittgenstein, who had devoted almost half of his late work to the philosophy of mathematics and used to refer to it as his &s;main contribution&s; (cf. the entry Mathematics, in [63]), would have had something to say. Indeed, he often felt puzzled about &s;the superstitious fear and awe of mathematicians in face of contradiction&s; (cf. [109], Ap.III-17), and asked himself: &s;Contradiction. Why just this one spectre? This is surely much suspect.&s; (id., IV-56). His point was that &s;it is one thing to use a mathematical technique consisting in the avoidance of contradiction, and another thing to philosophize against contradiction in mathematics&s; (id., IV-55), and that it was necessary to remove the &s;metaphysical thorn&s; stuck here (id., Vn-12). In this respect, the philosopher described his own objective as precisely that of altering the attitude of mathematicians concerning contradictions (id., HI-82).