Is there such a thing as N-valued logic, N being above 5 since there exist 3-valued and 4-valued logic. I am asking, because after true, false and neither, the additional truth value basically don't make any sense. I am not sure if these truth values make sense in philosophy. They may have application in technology and computer science, but I don't see how they could be relevant in philosophy.
I'll start with a section on this very question in the SEP article on Jan Łukasiewicz:
In proposing logics with infinitely many values, Łukasiewicz was thus the inventor of what was much later (43 years later, to be exact) to be called ‘fuzzy logic’. Commenting on these systems in 1930, Łukasiewicz wrote
it was clear to me from the outset that among all the many-valued systems only two can claim any philosophical significance: the three-valued one and the infinite-valued ones. For if values other than “0” and “1” are interpreted as “the possible”, only two cases can reasonably be distinguished: either one assumes that there are no variations in degrees of the possible and consequently arrives at the three-valued system; or one assumes the opposite, in which case it would be most natural to suppose, as in the theory of probabilities, that there are infinitely many degrees of possibility, which leads to the infinite-valued propositional calculus. I believe that the latter system is preferable to all others. Unfortunately this system has not yet been investigated sufficiently; in particular the relations of the infinite-valued system to the calculus of probabilities awaits further inquiry. (SW, 173)
But this precursor to fuzzy logic was supposed to have ℵ0-many truth values; fuzzy logic proper has 2ℵ0-many, which is at least ℵ1-many, but perhaps almost anything under the first uncountable strongly inaccessible cardinal (if that "exists"). (Generally, the only cardinals ruled out in this interval are ones cofinal with the zeroth aleph, though if we go with forcings that make 2ℵ0 = 2ℵ1, then the Continuum's cardinality is also not going to be any aleph cofinal with ℵ1. Worse, we could also force that 2ℵ0 = 2ℵ1 = 2ℵ2 = 2ℵ3, and so on and on, filtering out arbitrarily more singular cardinals from the mapping into the Continuum.)
So yes, there are "N-valued logics." There are also infinitary logics with countably or uncountably long conjunctions/disjunctions/quantifiers sets, and also things like ωth-order logic. So you could have an ωth-order infinitary logic ℒ(ω1, ω) that also has, say, ℵω1-many truth values to its name, I believe.
The answer depends on whether you are talking about philosophical or mathematical logic.
In the mathematical world, a binary logic engine (or call it a binary gate) outputs only (true, yes, or 1) or (false, no, or 0) upon operating on some binary input condition. It is also possible to allow the input states and the output states to be assigned (0, 1, 2, 3, 4, 5, ..., n) which in the computer science world is called n-state logic. An example of this is a mechanical adding machine where the logic engine can accept 10 different discrete input states and yield 10 different discrete output states, making for example base-10 arithmetic particularly easy to implement. All antique mechanical "adding machines" work in this way.
But it is possible to emulate any base n logic scheme with the right combination of purely binary (base-2) operators and states which themselves are particularly easy to construct out of switches which represent either yes (1) or no (0). For this reason, calculators that operate on decimal numbers convert those inputs into binary numbers, operate logically upon them with purely binary logic, and then convert the result back into decimal numbers. This trick works because base n numerals are simply a way of economically encoding base 2 numerals, where every possible base n digit is in 1:1 correspondence with an equivalent base 2 digit. `
Now, is it possible to implement, for example, 7-state logic in the nonmathematical world? One possible example would be a logic engine which answers the question "what color is this sample of visible light?" with one of 7 possible output states (red, orange, yellow, green, blue, indigo, violet).
The analogy I use for N valued logic is paint and it's various shades. The paint colors like ivory, cream, egg shell, vanilla, snow etc. can all be classified as white but are clearly not equivalent. Likewise black contains shades like onyx, charcoal, graphite, etc. So now Black and white which are typically considered Logic 1 and 0, can replaced with N values (one for each shade) that range from pure black to pure white with a set of rules or probabilities that govern the logic.