Many-Valued Logic

Introduction

Cast your mind back to the example that you looked at of the sea battle in the future. There we were looking at a statement: 'there will be a sea battle tomorrow.' On the assumption that all such statements are either true or false we were considering what it could tell us about the possibility of  de re necessity - i.e. whether the 'sea battle' was fated to occur or not. if the statement about the future is true then it is always true and the claim is that there must be a sea battle. (Similarly if the statement is false.) A common complaint against such arguments is that, as a statement about the future, there is nothing in the actual present world that makes the statement true or false, and therefore it can not be evaluated as either true or false. This suggests that there is a third truth value - 'Indeterminate' - and here the statement has that value.

Łukasiewicz's 3-Valued Logic

Inspired by arguments like that, the Polish logician Łukasiewicz defined a 3-valued logic with truth values T, F, and I. In his new logic, PŁ₃, the truth functions of PL are still truth functions (and we'll use the same symbols) but they are now defined thus:

Negation

Conjunction

Disjunction

Implication

You'll notice that the table for p É q will be the same as a table for ~p v q except that I É I is T. The reason for this is that p É p is a tautology in PL and we want it also to be a tautology in PŁ₃ (i.e. a formula that is true no matter what assignment of truth values is given to its parts,) so we want it to be true even if p is of indeterminate truth.

Equivalence

p º q is equivalent to (~p v q) & (p v ~q)

You'll notice that if we remove the columns and rows of those truth tables we are left with just the standard 2-valued logic, which is probably what we'd want. This means that any tautology in PŁ₃ will also be a tautology in PL. On the other hand not every PL-tautology will be a PŁ₃-tautology. Try the following, for example:

                         p v ~p                                      Law of Excluded Middle (LoEM)

                         ~(p & ~p)                                Law of Contradiction (LoC)

As a matter of interest let's also note that Łukasiewicz thought he could also define a truth functional version of a modal logic. This is also inspired by the issues around the sea-battle type arguments. If his 3-valued logic could deal with future contingencies there then it could work as a modal logic. He defined the modalities thus:

Modalities

One of the good features of this modal logic it that it gives us some welcome truths such as []p º ~<>~p, p É <>p, []p É p, and []p É <>p.

Łukasiewicz's n-Valued Logics

There are lots of other 3-valued logics, but we won't explore them here. Instead we'll see how the Łukasiewicz system can be extended to more than 3 values. 

We'll start by going back to the classical logic. Most of the time we haven't been using T and F for our truth values, we've been using 1 and 0. Partly that's because it's easier to read, and partly it's because we're infected by the habits of computer scientists. And one of the things that computer scientists and mathematicians going back to George Boole noticed about the truth functions is that they have a reasonably obvious arithmetical interpretation.

Let's take the truth values to be 1 and 0. Define the following functions on V₂ = {1, 0}:

~(p) = 1- p 

&(p, q) = min(p, q)

v(p, q) = max(p, q)

from which we could easily derive (given the obvious motivation here) that

É(p, q) = max(1-p, q)

º(p, q) = min(max(1-p, q), max(p, 1-q))

then we'll get the same tables as we used to define the corresponding truth functions in the introductory course.

If the set of truth values is instead V₃ = {1, ½, 0} then the same rules can be applied to get a 3-valued system. This wouldn't however be quite the same as PŁ₃ (compare the result of I É I and É(½, ½). But if we modified the faulty É function to

É(p, q) = 1, if p £ q

                 1 - p + q, otherwise

this would still give the right result for V₂, and for V₃ the table is just the same shape as the table given above for É in PŁ₃

But now we've got a set of arithmetical functions for 'truth values' that can easily be generalised to 4, 5, ..., n-valued logics. All we have to do is select n evenly spaced points from the interval [0, 1] and apply the rules to them. Here's PŁ₄ for example:

We see again that not all PL tautologies are going to be PŁ₄ tautologies. The two formulae LoEM and LoC which were examples of classical tautologies that weren't PŁ3 tautologies are examples of this too. 

For example; when p = ¹/₃ then p v ~p  = ¹/₃ v ~¹/₃ = ¹/₃ v ²/₃ =  ²/₃. 

Łukasiewicz's Infinitely-Valued Logic

Of course it's got to be a bit difficult to think of a use for n-valued logics for n between 4 and any very large number. What would you want a 349-valued logic for? On the other hand lots of people have thought that there might very well be a use for an infinitely-valued logic. The nice thing about the Łukasiewicz arithmetical truth functions is that they're easily extendable to infinite sets. We can have:

PŁ_À0: V_À0 = [0, 1] Ç Q (all the rational numbers, which can be made from fractions like a/b, between 0 and 1 inclusive)

PŁ_À1: V_À1 = [0, 1] Ç R (all the real numbers between 0 and 1 inclusive)

You can imagine that it's a bit difficult to use truth tables to establish that some formula is a tautology/contingency/contradiction. Filling in an infinitely sized table is not possible. Instead more indirect methods are used. Consider, for example, the formula p É (q É p).

Only a few possibilities exist:

    1.    p  £ (q É p)

    2.    p > (q É p)

           2a.  q £ p

           2b.  q > p

If 2a. is the case then (q É p) = 1, so not p > (q É p)

If 2b is the case then (q É p) = 1 - q + p, and since p > (q É p) we get p > 1 - q + p, so 1-q < 0, so q > 1, which is impossible.

Therefore only 1. is possible, in which case p É (q É p) = 1, and we've shown that it's a tautology.

Now this is all very interesting as a generalization of a formal system, but are these things really 'logics'? That depends on what you think a logic is supposed to be and we'll talk about that later. There are those who think that logics are really all about arguments and there really are no other truth values than T or F. Any system which uses more than 2 values is not a logic but a calculus. It's a tool for working out some problems in maths or provability theory or electronic circuits, etc. but it's not a logic. To be a logic requires at least some concern with truth values. Rescher quotes Jordan (p. 108):

The difficulty with the n-valued systems does not consist so much in technical problems, considerable as these are, as in finding an interpretation of the n ‘truth-values’ involved in the system. Without an interpretation assigning a definite logical meaning to the n ‘truth-values’ and given n-valued calculus remains an abstract structure. The importance of studying such structures cannot be denied. But according to the accepted doctrine, coming ultimately from Wittgenstein, the value of a logical system consists in providing a set of rules (possessing definite properties) for transforming expressions of a given meaning into other expressions, and thus in revealing their hidden properties and relations. This requirement is not satisfied by an abstract n-valued calculus.

On the other hand, there are ways to use many-valued 'logics' which do suggest that they are involved with directly understanding the relationship of some sentences and arguments to truth. Consider the Liar Paradox for example. In a two valued logic the statement

L₂: This statement is false

will yield a paradox, because if it is true then it is false and if it is false then it's true. We could try to fix this by saying that there are really three truth values, including one, N₁, for neither T nor F (to be interpreted as undecidable, ungrammatical, nonsensical, ...) and L₂ is to be assigned that value. That would fix L₂, but the problem recurs if we propose

L₃: This statement is false and is not N₁

because if L₃ is T, then L₃ is F, so it isn't T; and if it's N₁, then it's F, and so it's not N₁; and if it's F then it's T, so it isn't F. It's pretty clear that the problem won't go away if we just add more truth values like N₂, N₃, ... But if we added an infinite number of truth values then we could make it non-paradoxical. 

Rescher, N. Many-Valued Logic (NY: McGraw-Hill, 1969)