# Partial truth values?

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.

## 1 Answer

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.

I think Asimov was just being provocative here. Perfectly flat and perfectly spherical by no means exhausts all of the infinite possibilities. Clearly, neither is the case. Both are false. That in itself does not mean that some weird new logic of "partial truths" or degrees of truth and falsity is called for. Classical logic (the basis for most if not all of modern mathematics, science an technology) is more than up to the job of making sense of the constantly changing shape of the earth with its flattened poles, drifting continents, erosion, etc.

• Well, it depends, actually - topologically speaking, Earth is indeed a sphere (up to homeomorphism, which is as good as $=$ in topology). As for flat - if we consider Earth a smooth manifold, the it is at least locally flat (ie diffeomorphic to $\mathbb{R}^2$). Sorry, couldn't resist teasing a little ;-) – j4nd3r53n Dec 17 '18 at 10:40
• This comment made my day. – Algebra geek Dec 17 '18 at 11:22
• @DanChristensen - The Asimov quote is only meant to get the general idea across; the point of the question is that while two mathematical statements may both be false, they need not be equally so. For example consider $\forall x\in\mathbb{R}.x^2=a\implies x=\sqrt{a}$ and $\forall x\in\mathbb{R}.x^2=a\implies x=\frac{a}{2}$. Any mathematician would agree that both statements are technically incorrect, but few would assert that the first is just as incorrect as the second. What I am looking for is a way to make this distinction rigorous. – R. Burton Dec 17 '18 at 14:10
• @R.Burton We are not assigning part-marks on some pop quiz here. I really doubt that any mathematician would talk about degrees of incorrectness in this case. Both statements are false, period. – Dan Christensen Dec 17 '18 at 23:29
• @DanChristensen I'm not arguing that both statements are not false, nor am I arguing that mathematics doesn't get along fine relying on binary truth values. What I am saying is that by regarding both statements as totally false, without any further qualification, you are are throwing away useful information. It isn't incorrect to say they are both false, it's just less useful than specifying the extent and manner in which each is false. – R. Burton Dec 18 '18 at 2:46