In his essay The Relativity of Wrong, Isaac Asimov famously wrote:
When people thought the Earth was flat, they were wrong. When people thought the Earth was spherical, they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.
In mathematics and formal logic, at least according to every textbook I have read, a valid statement may hold exactly one of two truth values - true or false. However, just as "the earth is flat" and "the earth is spherical" are not equally wrong, there are mathematical statements which, though false, can be comparatively more or less false.
For example take the statements: $$(1)\qquad\forall x\in\mathbb{N}.\frac{x}{2}\in\mathbb{N}$$ $$(2)\qquad\forall x\in\mathbb{R}.\frac{x}{2}\in\mathbb{N}$$ $(1)$ is false if $\exists x\in\mathbb{N}:2\nmid x$, and $(2)$ is false if $\exists x\in\mathbb{R}:2\nmid x$. Since both of these conditions hold $(1)$ and $(2)$ are both false. However, $(1)$ is much closer to being true than $(2)$, on account of there being uncountably more counterexamples to $(2)$ than to $(1)$. Furthermore, for exactly half of all $x\in\mathbb{N}$, the statement "$2\mid x$" is true. It then seems reasonable to say that $(1)$ is exactly half true (by which it is also half false). This follows from the fact that every other natural number is divisible by two, which implies that as a subset $X\subset\mathbb{N}$ grows to encompass all $\mathbb{N}$, the ratio between the even numbers $x\in X:2\mid x$ and odd numbers $x\in X:2\nmid x$ converges to $\frac{1}{2}$.
Intuitively, it would follow that any statement which can be phrased as "Object (domain) - Relation - Object" (i.e. $x\mid y,\quad a\subset b\iff x=3,\quad a\ast b=b\ast c\implies a=c$, etc.) can be assigned a truth value between $0$ (completely false, not at all true), and $1$ (completely true, not at all false).
If we permit this, then statements which could not otherwise be evaluated logically, including certain nonsense statements like the liar's paradox, become amenable to formal logic. Likewise, should a false statement have an underlying truth to it (as indicated by a nonzero truth value), the domain can be adjusted until the truth value reaches $1$.
I find this an incredibly useful notion, both in the steps leading up to a proof and in connecting mathematics to everyday life.
However, I am not sure how to make the notion of fractional truth values rigorous. Is there a good way to formalize partial truth so that it is consistent with itself and with accepted mathematical logic? Ideally, I would like to extend the notion of truth values to encompass all potential propositions, rather than challenge extant logic.
As a side note, I think this is very similar - though not quite identical - to probability, and certain ideas carry over quite well. For example the truth value of $A\land B$ (assuming $A$ and $B$ are independent of each other) is equal to the product of the truth values of $A$ and $B$ (you can test this with propositions about objects in one or more finite sets) - similar to how the probability of two independent events occurring simultaneously is the product of there probabilities.
As a side side note, if there is an airtight way to formalize this, it might provide a valid basis for challenging Tarski's undefinability theorem - although the definition of truth would be invariably circular. I think it might also require using real or extended real numbers to encode for statements rather than standard Gödel numbering, but I don't know enough about incompleteness to say for certain.
