Three-valued and four-valued logic are topics I have very little background knowledge about. An afternoon of internet searching reveals that the world of three-valued logic is a bit older and bigger than four-valued logic, but in both cases, the list of who’s who and what’s what is not very long. I don’t want to take the time to summarize what’s out there, but none of seems to do exactly what I’m trying to do, which is this: In physics education research and curriculum development, we have a well-established tradition of reasoning about whether this is greater than, less than, or equal to that. What is valid logic at this level of precision? (As an aside, this style of reasoning is often called “qualitative reasoning,” but I think that is a poor name. Instead, it should be called “coarse-grained quantitative reasoning” or something similar, but “greater than” etc. are quantitative concepts, albeit with dramatically reduced precision.
In this universe, for two quantities, (assuming they are comparable – i.e., they have the same units), there are four possibilities: A1 is greater than A2, less than A2, equal to A2, or cannot be determined in relation to A2. I have invented a notation to try to capture these ideas, but I don’t know yet what the use may be for it. We’ll see, I guess. First, a proposition like A1 is greater than A2 takes the form of a variable “A12” having a value, which will be L, E, G, or U. For example, if A12 = L, then A1 is less than A2. If A12 = U, then it cannot be determined which of A1 or A2 is greater, or if they are equal.
Let’s invent a basic logical function to capture how we think about “LEGU” comparisons in relation to addition or multiplication. Suppose that algebraically, C = A * B. In this case, we know that if A1 is greater than A2 and B1 is greater than B2, then the product C1 = A1 * B1 must be greater than C2 = A2 * B2. Strangely, at this level of precision, the function is so abstract that addition and multiplication are indistinguishable. That is, if you were to define C = A + B, the logic would still hold. The definition of this function is its truth table.
TRUTH TABLE FOR C12 = A12 * B12 (or C12 = A12 + B12)
The values of A12 are in the leftmost column, and the values of B12 are along the top row. The resulting values of C12 for each combination of A12 and B12 are in the array.
| L | E | G | U | |
| L | L | L | U | U |
| E | L | E | G | U |
| G | U | G | G | U |
| U | U | U | U | U |
TRUTH TABLE FOR B12 = C12 / A12 (or B12 = C12 – A12)
The values of A12 are in the leftmost column, and the values of C12 are along the top row. The resulting values of B12 for each combination of A12 and C12 are in the array.
| L | E | G | U | |
| L | U | G | G | U |
| E | L | E | G | U |
| G | L | L | U | U |
| U | U | U | U | U |
I inspected the tables to see whether they were truly independent, and I saw they are not, or rather, if there was a simple way in which they were related. It’s this: The first and third row of each table is swapped when we change whether we multiply or divide by A. I realize we could construct the second table from the first if we also had this bit, a kind of inversion or negation(?) table. Negation comes to mind because it looks like the negation operator in binary logic, which converts T to F, and vice versa.
INVERSION / NEGATION ~
| A12 | ~ A12 |
| L | G |
| E | E |
| G | L |
| U | U |
Note that this form of inversion is not to be interpreted as “not”. The inversion of “E” for equal doesn’t mean that now the quantities are not equal; it means that if A1 and A2 are equal, then the inverse of A1 and A2 are also equal. If you are thinking of multiplication, then it’s a multiplicative inverse (1/x), and if addition, it’s an additive inverse (-x). Either way, the point is that if the value of A12 is E, then the value of ~A12 is also E, since the inverses of A1 and A2 are equal when A1 and A2 are equal. This all makes sense mathematically because you can always think of division and subtraction as multiplication and addition of the inverse.
Next, I thought about how U (“undeterminateness”) might spread through logical operations. Let’s imagine that we know something about A and C, but we don’t know yet about B. We reason about B on the basis of A and C and come to some conclusions about B. Suppose we then turn around and use what we’ve concluded about B to make conclusions for C’, as though we don’t already know C. Imagine like: “OK, we know 1161 / 27 = 43; what will happen if we multiply 27 * 43? Will it equal 1161? Oh my goodness, look, it does!” To the sufficiently mature mathematical student, this effort is wasted, since the student already believes math to be logically consistent. But there is a stage (or stages) of development in which the student enjoys seeing things come together because they are not sure they will. Perhaps it’s more an expression of the student’s uncertainty about whether they personally can honor and perform the logical consistency of math through their own efforts. Anyway. Let’s try this with coarse quantitative reasoning.
The leftmost two columns have all 16 combinations of input truth values for A and C. The third column values are determined by how the division function above is defined, given those two inputs. The fourth column takes the first and third columns as inputs and shows the output according to the multiplication function. All outputs that are not U are bold.
| A12 | C12 | B12 = C12 / A12 | C’12 = A12 * B12 |
| L | L | U | U |
| L | E | G | U |
| L | G | G | U |
| L | U | U | U |
| E | L | L | L |
| E | E | E | E |
| E | G | G | G |
| E | U | U | U |
| G | L | L | U |
| G | E | L | U |
| G | G | U | U |
| G | U | U | U |
| U | L | U | U |
| U | E | U | U |
| U | G | U | U |
| U | U | U | U |
Summarizing this table, we find that almost all (13 of 16) rows are undetermined. The only rows that aren’t are the ones for which A12 is E, that is, when A1 equals A2. In this case, the value for B matches the value of C, and the value for C’ in turn matches that for B again. I see this as a trivial result. So, we see that U spreads almost completely when these inverse functions are mutually composed.
Just as an example to explain how U spreads: First, if U is at least one of two inputs, it’s the output. So U instantly contaminates the logic for these functions. Furthermore, U is generated by the operations G * L, L * G, G / G, and L / L.
(The context of this investigation is to understand better why students would ever say G * L = E or similar, and how to help them learn to identify G * L = U reliably and to look for other paths of reasoning when they reach the logical dead end that is U.)