Truth = Utility
By: Naim Kabir
Tags: Science, Philosophy, Mathematics, Truth, Knowledge

Truth = Utility

Definition and implications of a new kind of epistemology

Source: Todd Quackenbush at Unsplash

We have officially entered the post-truth era.

With the rise of deep-fakes, lying politicians, and Surkovian disinformation campaigns, it’s hard to get a handle on what truth even is.

For a few months I was deep in a skeptical hole where I had truly lost grip on what I considered “real”, and I had to claw my way out by getting real silly and coming up with a formal definition that we might all agree with. Truth, I propose, is given by this expression:

That’s it. That’s truth.

I’ll define terms shortly and it’ll be clear that I’m abstracting away many messy details, but I will try to convince you that this basic structure agrees with many of our informal intuitions and is useful for decision-making — potentially even serving as an objective function for the automated scientists of the near future. Even better: perhaps it can be used as an objective function for search-systems or newsfeeds, only returning the top results as ranked by their truth-value.

If you buy the definition wholesale, I’ll show that it has interesting implications that aren’t immediately obvious. If you don’t buy the definition at all, I invite you to put all critiques, counterexamples, and challenges in the comments so I can update accordingly.

The tau truth-function

The first thing you’ll notice is that truth is a function.

This basic framing already aligns with our intuition in important ways: truth isn’t a value that floats out in the ether all by its lonesome, it’s a property of a statement — which we write here as s. Statements themselves don’t have a truth-value on their own, either: they are always grounded by some set of contexts, D.

Some basic validation by intuition: the statement “the boy runs” doesn’t have any truth-value assigned to it until you give it a context. If the context involves a boy running, then the statement is true. If the context does not include a boy running, then it is false. You need both a statement s and a set of contexts D to evaluate truth, even if D means “all possible contexts” (for example, tautologies are true in all contexts!).

It can be argued that we can do away with the notion of contexts if statements were fully qualified (i.e “The boy runs and the time and place is such where there is a boy running”), but since contexts are an easy-to-use shorthand, it’s what we’ll go with.

The context set, D

D is a Domain: a set of contexts d.

d is a set of self-consistent true statements. This captures the idea that the truth of s, in part, is a function of its consistency with established true statements.

However, it is possible for some d₀ to contain statements that disagree with statements in a d₁, even though an s is consistent with both. For example, take d₀ to be all of Newtonian classical mechanics and d₁ to be Einstein’s General Relativity. The s “Gravity exists” is consistent with both contexts and its truth-value should be evaluated in each of them piecemeal, even though the two d fundamentally disagree elsewhere.

Truth-values are numbers

It has likely not escaped you that the result of the 𝛕 function is a sum.

This implies some weird things:

Truth-values are composed of subcomponents. Truth-values can be operated on by addition.

I want to enforce one other constraint, too: truth-values are numbers. This implies that when you add a positive truth-value to another positive truth-value, the result is something more true. Negative truth-values make something less true.

When you add a positive truth-value to another positive truth-value, the result is something more true.

Why is this a useful notion? Well, we’ve already established that a statement s can be evaluated within the contexts d₀ and d₁, your chosen D. Let’s say we have three statements s₀, s₁, and s₂, with the following properties:

s ₀ is not consistent with d ₀ nor d ₁

₀ is not consistent with ₀ nor ₁ s ₁ is consistent with d ₀ but not with d ₁

₁ is consistent with ₀ but not with ₁ s₂ is consistent with both d₀ and d₁

It seems natural to say that 𝛕ᴅ(s₀) < 𝛕ᴅ(s₁) < 𝛕ᴅ(s₂) where D={d₀, d₁}. To get a concrete example, let’s once again borrow from physics.

Luminiferous aether, courtesy of Skitterphoto.

Let’s take the statements to be:

s ₀: light is a wave in the luminiferous aether

₀: light is a wave in the luminiferous aether s ₁: light is a particle

₁: light is a particle s₂: light is a wave and a particle

With the domain D composed of:

d ₀: {“The Michelson-Morley experiment was conducted”, “Light excites electrons in discrete quanta”}

₀: {“The Michelson-Morley experiment was conducted”, “Light excites electrons in discrete quanta”} d₁: {“The Michelson-Morley experiment was conducted”, “Light excites electrons in discrete quanta”, “Light produces diffraction patterns”}

In this case, s₀ is inconsistent with the Michelson-Morley experiment in both contexts, s₁ is consistent with d₀ but not the “diffraction pattern” statement in d₁, and s₂ is consistent with both. Again, it feels natural to me to say 𝛕ᴅ(s₀) < 𝛕ᴅ(s₁) < 𝛕ᴅ(s₂) where D={d₀, d₁}.

This ordering couldn’t happen if truth-values were booleans — there’s something more numeric (or at the very least, ordinal) at play, here. As the next section will make clear, I’ll assert that truth-values are not only numbers but that they are real numbers.

Truth is marginal utility

This one’s probably going to be controversial but bear with me.

The truth value of s within a context d is determined, fundamentally, by how much more utility we get than if it were not included.

Utility here is defined by a utility function (AKA objective function) U𝐝. Baked into our notation is the idea that U𝐝 is specified by a context d, meaning, not only is d a collection of statements, but also a task that the statements in d and s should be used to perform with a goal to optimize U𝐝.

It’s worth emphasizing that we aren’t concerned with absolute utility U𝐝 as much as we are with marginal utility, ΔU𝐝.

This definition has counterfactual thinking buried at its core, because we have to estimate what Ud would have been in a world where s was (or wasn’t) used for some task and compares it to the measured Ud in the actual case.

Again, let’s go through an example. Let’s say we’re evaluating the statement s=“You should wear a mask” in the context d={“There is a global viral pandemic”, “Masks mitigate the spread”, “Failure to mitigate spread leads to deaths”}. The context d also comes with a task to maximize the utility function Ud = -VirusDeaths(s,d).

Seriously though, wear a mask. Source: CDC

If Ud(d)=-220,000, but Ud(s,d)=-100,000, then we can say that ΔU𝐝(s,d)=120,000. If d is the only context in our Domain D, we can say that 𝛕ᴅ(s) = 120,000 — a positive, and thus true, value.

You should wear a mask.

I remember being very uncomfortable with the idea that truth was something necessarily task-relevant. But there is no better test of a statement’s truth than if it can be productively used for some task, whether or not that’s predicting the position of the Moon or launching a successful social media campaign.

The implication that truth must be useful might remind you of angry old men yelling about National Science Foundation budgets in Congress and how we’re “wasting money” studying mantis shrimp or whatever. But it’s worth noting that the domain D in which we estimate truth values can be vast and future-looking: there is certainly some future context where studies on shrimp unlock enhanced performance in materials science or medicine, making blue skies research an engine of truth even if utility is low in present contexts.

True statements don’t have to be useful right now — but they must be useful in a plausible world, past or imagined future.

True statements don’t have to be useful right now — but they must be useful in a plausible world, past or future.

Where’s the “reality” term?!

Let’s reiterate our formulation of the truth.

There is a term for a domain of contexts, a utility function, and a statement to evaluate — but there’s no mention of “facts” or “reality”. But isn’t truth “fidelity to reality”? How can anything be true if it isn’t… real?

First of all, “reality” is ill-defined and I know of precisely 0 ways to formulate a distance metric between a statement and reality. Most definitions of reality also invoke truth within them, so framing truth as a function of reality would be circular at best.

Secondly, we never explicitly seek out to represent “reality” as accurately as possible in our day-to-day, anyway. It is overwhelmingly likely that our own sensory and perceptual system builds our representation of the world by means of heuristics and approximations — just good enough to help us survive and reproduce.

Third: assuming there were a precise distance metric to quantify the fidelity of a statement to reality, you could bake that distance metric into a domain d as part of a utility function. The task, in this case, might be “represent reality as accurately as possible”, making the “fidelity to reality” definition of the truth a special case of the marginal utility definition I present here.

I’ll repeat, though: representing reality as accurately as possible isn’t something we ever want to actually do. It’s just not helpful. Remember that a map is not the territory and that a map is only useful because it’s a compact representation of a land that is too vast to comprehend. We use maps precisely because they are not accurate: they are useful.

We use maps precisely because they are not accurate: they are useful.

Photo by Ylanite Koppens from Pexels

Consensus does not matter, except for conventions

It is true that you should stop when you see this sign. Photo by Davis Sanchez from Pexels

Another thing you’ll notice is that there is no term for how much a statement agrees with the beliefs of others. The standard sophomoric answer for what the “truth” is from freshmen in Epistemology 101 is rebutted pretty handily here.

This should be a satisfying result: cults and re-education camps can be quite marvelous at manufacturing consensus, but very few people would make the claim that they are truth-makers.

There is a case where consensus matters: but it is baked entirely into the utility function Ud. When it is useful for many people to be in agreement about something otherwise arbitrary, it becomes true. For example, if your objective function Ud is to stay alive as long as possible in the contexts of city-streets while driving a vehicle, it is true that you should stop at stop signs. In New York, it is also true that you should drive on the right side of the road. A little deeper: if your objective function is to run a functioning economy, it is true that fiat currency has value. Because it is useful for us all to agree on that statement.

Wide truths are better truths

The truth-function involves summing marginal utilities over several separate and potentially unrelated domains. The implication here is that if a statement causes enhanced performance in many separate areas under scrutiny, it is true-er than statements that only enhance performance in one or two domains. Intuitively I wouldn’t have come up with this, but on deeper reflection, it actually maps on pretty well to how we rank propositions in science.

Think about the difference between a Law and a Theory. A Law is a straightforward rendering of a deterministic relationship. For example, the Ideal Gas Law relates moles of a gas to its temperature, pressure, volume, and so on. The Ideal Gas Law is true and useful in the many contexts of predicting how various properties of a gas will change.

But what about the Kinetic Molecular Theory of gas? It is also useful, but I’d argue it is more useful than the strict functional recording of the Ideal Gas Law, mainly because its insights can now be used in contexts other than predicting the changes in various properties of a gas. It can now be used to predict how diffusion might occur, how dense the atmosphere is at various altitudes, etc. — all applications that transcend the strict recording of relationships in the original context of gases in a closed container. I would argue that the Theory is more true than the Law because it is an explanation that can be ported to multiple contexts beyond where it was developed.

To take it to another extreme — instead of scientific laws, let’s instead consider the weights learned by a neural network when it’s presented with a large volume of inputs and outputs. I’d call those weights true, insofar as they actually do model relationships between those inputs and outputs. But if an alternative approach were to simply write a small dense equation to model that same relationship, I would call that alternative more true — simply because it is portable across multiple different contexts and directly manipulable, whereas the neural network’s weights would only be useful in the context it was deployed, on a specific operating system, in a particular format, etc. etc.

Deep truths are the truest truths

We’ve seen that a statement is true if widely useful in a variety of contexts, which speaks to portability and manipulability being a heavy driver of truth-value. Another insight I’d like to consider needs me to impose some structure.

Let’s say that contexts — collections of consistent true statements, remember — don’t exist as islands, but rather can be related by dependencies. The reason why you’d want to have this structure is because at each branching level, you’d want contexts to be able to disagree, even though they have some shared parent context that they both are consistent with.

Now let’s evaluate the truth-value of a statement that can be added to d₀. If it turns out to enhance utility and is still consistent with child nodes, then it should also enhance utility or at the very least be 0-valued for those child nodes.

Now if we instead insert the statement in d₃, our upper bound for truth-value is much smaller. It can only ever hope to enhance the utility of itself and child contexts.

This strikes me as an explanation for why we find certain statements to be profound or fundamental. When something is a “deep truth”, it is actually deeper down the dependency tree of a particular Domain.

So, if you were looking for a way to maximize the leverage of your truth-seeking: go deep.

Be humble: truth lives on a surface, not a point

Another consequence of this framing is that there are no privileged statements. A statement s₀ could produce some positive truth value, but a statement s₁ that contradicts it could produce an equivalent truth value if it had the same impact on marginal utility in as many contexts.

Source: Jacopo Bertolotti, Wikimedia Commons

A ready example of this is the right-hand rule when describing chirality or electromagnetism — there’s no particular reason we couldn’t have changed this convention in some way and achieved just as much with science and technology. A more controversial example: the Copenhagen interpretation of material objects — which says they do not have certain properties until they are measured and a wave-function collapses — is well and good and can certainly be useful in thinking through problems. But the Broglie-Bohm pilot-wave theory is probably equivalent in its usefulness in a variety of contexts and thus equivalently true. Neither of them has a monopoly on truth except in contexts where a single unified convention is itself the main added value.

There is a full surface of statements mapping onto truth-values, and plenty of them will be on a level given the same context, even if they disagree with each other.