# Is there such a thing as N-valued logic?

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.

• N-valued logic and even infinite-valued logic has philosophical implication: In the study of logic itself, infinite-valued logic has served as an aid to understand the nature of the human understanding of logical concepts. Kurt Gödel attempted to comprehend the human ability for logical intuition in terms of finite-valued logic before concluding that the ability is based on infinite-valued logic... Jun 29 at 2:22
• @DoubleKnot What is a reference for Godel's comments about infinite-valued logic? Jun 29 at 3:15
• You can generate Logics with 2^n values by taking power set operations on the set {True, False}. You could see this as representing successively higher orders of combinations of truth values being assigned to single statements. For example you could have a truth value in 8-value logic that represents a sentence being both neither true and false and yet also true. This can model hypothetical truth values assigned by different agents under uncertainty. E.g he thinks she thinks it's true, but she thinks it's neither true nor false. Jun 29 at 3:20
• In fuzzy logic, there are Continuum-many truth values, so even more than the |omega|-many truth values in the Lukasiewicz precursor theory. Perhaps extraordinarily more truth values, even (depends on if the Continuum's cardinality is supposed to the first aleph after the zeroth, or if it's some monstrously greater aleph, like the first fixed point of the aleph function that also is cofinal with one of the alephs beyond the zeroth, say). Jun 29 at 4:32
• "The general problem of finding an intuitive understanding of the truth degrees occasionally has a nice solution: one can consider them as comprising different aspects of the evaluation of sentences" SEP, Many-Valued Logic. Jun 29 at 5:20

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 20-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 20 = 21, then the Continuum's cardinality is also not going to be any aleph cofinal with ℵ1. Worse, we could also force that 20 = 21 = 22 = 23, 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.

• All of the operations used to define the logical operators in fuzzy logic are closed under the real interval [0,1]. Why would fuzzy logic need more numbers than are in that interval? Jun 29 at 8:06
• I'll add that what Łukasiewicz is missing in his quote is that there may be not only degrees of the possible, but also different ways in which a statement may fail to be assignable a value of true or false. For example, "this statement is false" as no consistent assignment of truth value but "this statement is true" has two consistent assignments of truth value. To account for this difference, we need a four-valued logic t/f/b/n. Jun 29 at 8:10
• The real interval [0, 1] is equinumerous with the entire Continuum, so it's just that there might be extraordinarily many real numbers even in that interval. Re: Łukasiewicz's indicated modal logic, I myself (for now!) accept the propositional operator theory of modality as such, so I wouldn't see the extra truth values as "degrees of possibility," at least not quite in the same sense as Łukasiewicz. I have some odd obsessive opinions about the liar sentence, too, so not sure how I'd relate it to this immediate question... My only sense of > 2 truth values is as percentages of truth, I guess. Jun 29 at 8:40
• My point is that the extra truth values have different meanings for different logics and applications. In the 3-valued logic of SQL, the third value means "unknown" or more precisely "no data available". In another 3-valued logic, it is used to handle paradoxical sentences, so it means there is no consistent and unique assignment of truth values. An infinite number of truth values can be used for imprecision as in fuzzy logic, for uncertainty as in probability, or for confirmability as in Popper's logic of confirmation. Jun 29 at 11:02

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).

• I don't agree that the question has anything to do with the distinction, if there is one, between mathematical logic and philosophical logic. There are many logics, and often it is possible to map one logic into another. As you say, it is possible to map n-valued logic into binary logic, but it doesn't follow that this makes n-valued logic less fundamental in some way. Classical logic can be mapped into intuitionistic logic, and vice versa, but both are logics, and neither is more mathematical or philosophical than the other. There is nothing peculiarly philosophical about the number two. Jun 29 at 22:54

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.