Is there an established method for assigning a truth value $0<t<1$ to undecidable statements?

Question

Is there an established method for assigning a truth value $0<t<1$ to undecidable statements?

Classical logic assumes every proposition is either true or false, i.e., the law of excluded middle. This is not true of fuzzy logic, where the truth of propositions typically takes on some real number in $[0,1]$.

I thought it seemed intuitive that statements which are neither true nor false should be considered to be somewhere in between. The question is how to decide what truth values to give them. My thinking is that allowing our axioms to have non-integer truth values is a start.

Answer 1

Fuzzyfied Logic

Let $\,x,y,z\,$ be sets (set theory) or propositions (mathematical logic). For these elements, the axioms of boolean algebra are supposed to be valid: $$ \begin{array}{lll} z \cap z = z & ; & z \cup z = z \\ y \cap z = z \cap y & ; & y \cup z = z \cup y \\ x \cap (y \cap z) = (x \cap y) \cap z & ; & x \cup (y \cup z) = (x \cup y) \cup z \\ x \cap (y \cup z) = (x \cap y) \cup (x \cap z) & ; & x \cup (y \cap z) = (x \cup y) \cap (x \cup z) \\ 0 \cap z = 0 & ; & 1 \cup z = 1 \\ 1 \cap z = z & ; & 0 \cup z = z \\ \overline{y \cap z} = \overline{y} \cup \overline{z} & ; & \overline{y \cup z} = \overline{y} \cap \overline{z} \\ z \cap \overline{z} = 0 & ; & z \cup \overline{z} = 1 \\ \overline{\overline{z}} = z & & \end{array} $$ Let the above axioms are valid for numbers $x,y,z$ which assume values from the set $\,\{0,1\}$ . We seek a generalization of the axioms for numbers which assume values between $0$ and $1$ , that is from the interval $\,\left[0,1\right]$ . Define for $0\le a\le 1$ and $0\le b\le 1$ : $$ \begin{array}{lllll} a \cap b := a.b & ; & a \cup b := a + b - a.b & ; & \overline{a} := 1 - a \end{array} $$ Elementary number algebra reveals that the following axioms are still valid: $$ \begin{array}{lll} y \cap z = z \cap y & ; & y \cup z = z \cup y \\ x \cap (y \cap z) = (x \cap y) \cap z & ; & x \cup (y \cup z) = (x \cup y) \cup z \\ 0 \cap z = 0 & ; & 1 \cup z = 1 \\ 1 \cap z = z & ; & 0 \cup z = z \\ \overline{y \cap z} = \overline{y} \cup \overline{z} & ; & \overline{y \cup z} = \overline{y} \cap \overline{z} \\ \overline{\overline{z}} = z & & \end{array} $$ But other axioms are no longer valid: $$ \begin{array}{lll} z \cap z \ne z & ; & z \cup z \ne z \\ x \cap (y \cup z) \ne (x \cap y) \cup (x \cap z) & ; & x \cup (y \cap z) \ne (x \cup y) \cap (x \cup z) \\ z \cap \overline{z} \ne 0 & ; & z \cup \overline{z} \ne 1 \end{array} $$

Fuzzy Equality

The following truth table (: logic) and Karnaugh diagram (: set theory) are valid for equivalent (read: equal) boolean elements.

    a | b | a = b             ---------
   ---------------         a |    | XX |
    0 | 0 |   1               ---------
    0 | 1 |   0              | XX |    |
    1 | 0 |   0               ---------
    1 | 1 |   1                     b

Thus it is seen that fuzzy equality corresponds with the following non-trivial arithmetical expression: $$ (a \cap b) \cup (\overline{a} \cap \overline{b}) = (a.b) \cup (1-a)(1-b) = $$ $$ a.b + (1-a)(1-b) - a.b(1-a)(1-b) = a.b + (1-a)(1-b)(1-a.b) = $$ $$ 1 - a - b + a.b + a^2.b + a.b^2 - a^2.b^2 $$ In classical logic the intersection between $(a \cap b)$ and $(\overline{a} \cap \overline{b})$ is zero: the sets $(a \cap b)$ and $(\overline{a} \cap \overline{b})$ are disjoint. Thus $a.b.(1-a)(1-b) = 0$. This means that a suitable fuzzyfication could also be given by: $$ (a \cap b) \cup (\overline{a} \cap \overline{b}) = a.b + (1-a)(1-b) $$ That is: a bilinear interpolation between
$\;\left[\; (0,0) , (1,1) \; \mbox{with equal} = 1 \;\right]\;$ and $\;\left[\; (1,0) , (0,1) \; \mbox{with not equal} = 0 \;\right]\;$ .
There does also exist a 'standard' version of Fuzzy Logic, as can be found at:

• Fuzzy set operations

Union, intersection and complement are defined in this 'standard' version as: $$ \begin{array}{lllll} a \cap b := \min(a,b) & ; & a \cup b := \max(a,b) & ; & \overline{a} := 1 - a \end{array} $$ The Fuzzy Equality is then given by $\;\max(\min(a,b),\min(1-a,1-b))$ .

Find the Differences

An application of Fuzzy Equality as a part of Fuzzy Logic is the detection of differences between almost equal pictures:

Programming notes. After loading the two pictures, it must be found out what pixels share the same position (approximately). This is an art in itself which shall not be described here. Having done that, all (grey-valued) pixel colors are converted to equivalent number values in the range $\,\left[0,1\right]$ , giving numbers $\,a\,$ for picture $(1)$ and numbers $\,b\,$ for picture $(2)$. These are fed into the Fuzzy Equality routine, with numbers in the same $\,\left[0,1\right]\,$ range as the output:

function Fuzzy_Equality(method : integer; a,b : double) : double;
{
  Fuzzy Equality
}
var
  v : double;

function max(x,y : double) : double;
var
  m : double;
begin
  m := x; if m < y then m := y;
  max := m;
end;

function min(x,y : double) : double;
var
  m : double;
begin
  m := x; if m > y then m := y;
  min := m;
end;

begin
  v := 0;
  case method of
{ Probablistic }
  0 : v := a*b + (1-a)*(1-b)*(1-a*b);
{ Bilinear }
  1 : v := a*b + (1-a)*(1-b);
{ Standard }
  2 : v := max(min(a,b),min(1-a,1-b));
  end;
  Fuzzy_Equality := v;
end;

The outputs are converted to equivalent $\color{red}{\mbox{reddish}}$ colors: the more unequal the pixel greys, the more reddish. In this way, the pixel positions with most different grey values are emphasized as the most reddish spots in the leftmost picture:

It can be shown experimentally that any version of the above Fuzzy Equalities seems to work equally well. So the fuzzyfication of boolean logic is not quite unambiguous. This is a common feature with "going back to reality", as is exemplified numerous times with e.g. Numerical Analysis: there exist many numerical approximations for the same differential equation.

Update. Maybe the picture example as stated is not (yet) the logic as expected by the OP.
How then about the following logic (yes?) question: How much equal are the two pictures?
An answer is produced by the program if we add all fuzzy equality numbers together and divide by the number of pixels. Resulting in:

 8.23370043338440E-0001 : Fuzzy Logic method 0
 8.28104562088338E-0001 : Fuzzy Logic method 1
 8.31005249431013E-0001 : Fuzzy Logic method 2

The outcomes are not much different from each other, as expected. So the two pictures are $\,82.3\%$ , $82.8\%$ , $83.1\%$ equal, dependent on the fuzzy equality method chosen.

Answer 2

"Established" may be a strong word, but one option you have in first-order logic is to pick a probability measure over models of your theory and assign to each statement the probability that it's true in some model. This is done, for example, in Christiano-Yudkowsky-Herreshoff-Barasz's Definability of Truth in Probabilistic Logic; there it's shown as a lemma that an assignment of probabilities to statements satisfying some reasonable axioms must have the above form. Of course now one has to ask where to get such probability measures...