When the word logic is pop uped in any mathematics or computer science textbooks it usually means to denote the Boolean functions.
Is logical operations are restricted to only Boolean functions in these textbooks (say in most of the cases)?
Is logical operations is restricted to only two states?Iam asking this because Wikipedia says "Leibniz established that, by using the binary system, the principles of arithmetic and logic could be combined."So I tend to think logic means dealing only with two states?
This is a bias in the field. Even classical logic is not categorical.
This result had been shown in 1999 by Pavicic and Megill. Eric Schechter's book on classical and non-classical propositional uses subsets of a three-set to interpret the nodes of the O6 ortholattice in what is called the hexagon intepretation of classical logic. As shown by Schechter, the hexagon interpretation has the same inference rules and tautologies as two-valued logic. The hexagon interpretation has one truth valuation and five falsity valuations.
In so far as logic concerns itself with proof, truth invariance across syntactic transformations makes a unique truth valuation the focus of the subject. So, a semantic theory with five falsity valuations would not interfere with this emphasis.
In so far as logic concerns itself with reasoning, there are many possible variations which have been formulated and studied. Schechter's book is one of only a few with this explicit pluralistic approach to logic in mathematics. He also points out how mathematicians regularly apply principles of relevant logic, although this is not included in standard accounts of so-called mathematical logic.
If one moves to first-order accounts of logic, the principle of indiscernibility of non-existents from negative free logic is ubiquitous in mathematics as a means to establish well-formedness -- it is essential to the understanding of partial functions in the theory of recursive functions. But this goes unrecognized in received views.