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What is Truth?
Can linguistics, mathematics, and logic tell us anything about it?
In the late 1840s, the people of Ireland had it really, really bad.
Famine reared its ugly face on the landscape of the emerald isle. Starving families and crumbling communities became ever more desperate for something—anything—to eat.
In the end, a million Irish people starved to death. The direct cause was a devastating potato blight, but lurking behind that biological calamity was another sinister force: a notion of "Truth" rigidly held by British leadership.
In London, Prime Minister Lord John Russell was caught between mounting calls for intervention and the sacred text of Adam Smith's "The Wealth of Nations." To him, the doctrine of laissez-faire economics was more than an economic theory; it was an incontrovertible Truth, a moral code. Assisting the Irish would corrupt the market and, worse, the very souls of the suffering, he reasoned.
While farmers watched their children slowly waste away, British dogma stood firm.
A million died, and millions more suffered because a prevailing "Truth" overshadowed the tangible, horrifying reality before their eyes.
While other factors were certainly at play—longstanding British prejudice against the Irish (not universal, but widespread) and complicated laws surrounding trade in particular—devout faith in this Truth was undeniably a part of what went wrong.
Truth is one of, if not the single most important concept in human affairs. As individuals, we strive to understand the reality around us and shape it to our benefit. As a collective, we must agree on the most basic matters, from who’s who to what is or isn’t cool to do. All of our major institutions, from science to governments to religions, are predicated on the search for truth.
If you are reading this, you probably value reasoning, critical thinking, and truth. You probably also believe there is an objective reality that is at least approximately knowable using a combination of reasoning and observation. By objective reality, I mean the existence of a world outside our minds whose laws are independent of what we believe. And maybe you also believe this objective world is the primary cause of everything that happens inside our minds.
However, Truth with capital T is something really hard to define. Intuitively, we can say that Truth is the correspondence between our beliefs and reality. But are there different flavors of truth? Is it the same kind of thing saying “two plus two equals four,” “electrons have a negative charge,” “murder is bad,” or “coffee tastes better than chocolate”? Can the same thing be true sometimes and false in others? Can it be something other than true or false? Does your perspective matter, and if so, to what degree?
This article is an attempt to explore these questions. We will adopt a semantic approach grounded in formal logic, which is neither the only nor the most adequate point of view about truth always, but it is an interesting one in this case.
To begin our discussion, we must first agree on what we mean when we say something is true. Then, we will look at different types of truths and the systems that allow us to assert the truth of something. Finally, we’ll go back to formal logic and explore the implications of a theorem severely limiting our capacity to define Truth, with capital T, once and for all. This whole endeavor will take us on a tour around epistemology, logic, science, and the nature of reality.
As usual, I don’t pretend that you take my conclusions at face value, but rather to help you develop your own. Buckle up.
What types of things can be true?
Let’s start our discussion by analyzing the things that can be bearers of truth. I will suggest a few examples:
All prime numbers greater than 2 are odd.
The speed of light in space is approximately 300,000 kilometers per second.
All humans should be free to choose their goals in life.
Paper money has value.
Coffee tastes better with brown sugar.
All of the above are things any sensible person could believe are true. But they are all not the same. If you don’t believe #1, you’re definitely wrong. However, you can disagree with #5; I won’t hold it to you. Then there are cases like #2, which most people would agree, but there is a bit of a “depends” there. How approximate is approximately 300,000 km/s? In contrast, #4 seems true, but only because a sufficiently high number of people believe it to be so. And then we have #3, which is different because it’s telling how things should be, not necessarily how things are.
But all of these are examples of things we can say that are true or false. One is about numbers, an abstract construction; another about a physical phenomenon, something really out there; yet another is about the human condition; and another about something as mundane as coffee. These worlds are as different as any two things can be. So the question is, what do they have in common?
The answer is that these are all claims formulated in natural language; these are linguistic objects. We are talking about many different things, from numbers to particles to human rights, but the things we evaluate, whether true or not, are not the actual numbers, particles, or people but claims about those things.
And so we arrive at our first conclusion about the nature of Truth.
Truth is not a property of the things themselves but of claims about them.
Are there different types of claims?
To understand the nature of Truth, we must then study the nature of claims. What types of claims can we formulate? Let’s look at three fundamental differences between claims.
Subjective and objective claims
The first obvious distinction is that of subjective vs. objective claims. When we say something is objective, we mean it depends only on the reality of an object and not on the subjects that may be thinking about it and their beliefs.
More specifically, let’s call an objective claim whose truth value is independent of the person evaluating the claim, such as “The Earth is round.” In contrast, a subjective claim is a claim whose truth value depends on the subject(s) evaluating it, such as, “Chocolate ice cream tastes better than vanilla.” Not all subjective claims are mere opinions, though. When many believe something is true, it can become true, such as “This piece of paper is worth something.”
The nice thing about objective claims is that, in principle, they do not allow for meaningful disagreement. The shape of the Earth, whatever that is, should definitely be an objective property of the Earth, irrespective of the observer. If two different observers at different moments disagree on something about the shape of the Earth, then either one or both of them must be wrong, right?
Well, not exactly. Even if we all agree that some properties are objective, and thus, we cannot, in principle, disagree on their objective traits, how we talk about these properties is highly subjective. Thus, if we both interpret an objective claim the same way and disagree on its truth value, then at least one of us must be wrong. But we can still disagree on how we interpret the claim. Keep this thought for now.
Descriptive and normative claims
The second important distinction is between descriptive (is-a) claims and normative (should-be) claims.
A descriptive claim states the way things are. It can be an objective claim, such as “the sky is blue”or a subjective claim, such as “this is painful.” In contrast, a normative claim states how things should be. An example is “You should not steal.” Descriptive claims are factual, meaning they always refer to some state of reality —even if a subjective state. In contrast, normative claims are always judgemental; they can never be based on the grounds of how things are.
This fundamental distinction is called Hume's guillotine, a philosophical argument stating that no normative claim can be derived from a descriptive claim. This means there is no way to judge how things should be without introducing some normative axiom. This an issue we will cover in a future article.
Specified and quantified claims
The third interesting distinction is between specified and quantified claims.
A claim is specified when it refers to specific objects unambiguously, such as “The Earth is round,” or “Two is a prime number.” In contrast, a claim is quantified when it refers to some quantity of objects without specifying exactly which objects are those, such as “Seven dwarfs live in the mountains.”
The two most important quantified claims are universal and existential claims.
A universal claim is in the form, “For all X (such that Y), it is true that Z.” For example, “all electrons are negatively charged,” or “all prime numbers greater than two are odd,” or “all humans have equal rights.” These claims apply to all the elements of a given class, and thus, to assert their truth, we must commit to it for all elements of the given class, even if they are infinitely many or are unknown. This is usually a big leap requiring some theoretical framework to base the claim on.
In contrast, an existential claim is of the form “There exist (at least one) X such that Z.” For example, “There is one planet with life in the Solar System.” The easiest way to convince oneself that an existential claim is true is to find a specific object that satisfies the claim. However, sometimes, we can ensure an existential claim must be true even if we don’t know the exact object that satisfies it. Examples abound in math, but a real-life scenario could be something like, “This person dies of a bullet shot, so there must be a gun.”
The position of claims toward knowledge
Our desire to have robust world knowledge usually drives our interest in truth. So, comprehending the nature of the knowledge we obtain if a claim is true gives us clues on how to evaluate it and the nuances of its interpretation. Briefly explaining this intricate distinction is just the starting point.
A claim that depends only on the intrinsic characteristics of an object is an ontological claim. Ontology focuses on the core characteristics, origin, and properties. What is it? Where is it? What is it made of? How does it fit into a category of objects? These are examples of questions we answer with ontological claims.
Teleological claims, on the other hand, describe causality. For example, "It's raining because there is a storm" relates a phenomenon to an object and the properties we associate with it. This category is tricky since the relationship can adopt the guise of an ontological claim—"The existence of X is self-evident." Also, most of the fallacies follow this structure.
Claims regarding truth itself are a bit different from those about objects and reality. Epistemological claims focus on explaining knowledge, how we can obtain facts, and how we should reason about nature and other claims. Most of the rest of this article will further clarify this idea.
What makes a claim true?
If Truth is a property of claims, then what makes a claim true? How do we decide whether a given utterance such as “The Earth is approximately round,” “All even numbers can be decomposed into the sum of two primes,” or “You shall not commit adultery” is true or false?
To do that, we define an epistemic system, a set of rules determining which claims are true. For this discussion, consider an epistemic system —or an epistemology— as a function that takes a claim in a given language as input and outputs either true, false, or undecidable.
An epistemic system is the very definition of truth for the set of claims it can talk about —that is, for the set of claims that can be expressed in the language of that system. We call this whole set of claims the domain of the epistemic system. Trying to determine the truth value of a claim outside the domain of an epistemic system is an example of a category mistake.
So now, let's look at different epistemic systems.
A zoo of epistemic systems
First, we have formal or linguistic epistemologies. In these systems, the notion of truth is defined via a set of allowed syntactic transformations upon a set of initial axioms or a priori truths —which is why I call it "linguistic" as an alternative to "formal." The most famous formal epistemic system is mathematics, but we can see specific branches of logic and math as distinct epistemic systems. Formal systems are our strongest epistemologies. No ambiguity or nuance is left when we can say something is formally true or false.
Second, we have natural or empirical epistemologies. These are systems in which the notion of truth is defined (oversimplifying) in terms of how much a given claim is supported by empirical evidence, but there are a lot of nuances here. The scientific method —interpreted mostly as falsificationism— is the most famous empirical epistemology. Empirical claims like “Fotons have zero mass” are conferred a true status once there is an insurmountable pile of evidence. However, we never get an absolute formal proof, like in math. Also, empirical claims are often approximate as newer theories refine previously held truths to reveal special cases where they don’t hold, such as general relativity did to Newtonian mechanics.
Third, we have cultural or social epistemologies, in which truth is defined as that which most individuals accept in that culture or society. A given society's moral values can be considered a cultural epistemology.For example, the claim “It is rude to ask for someone’s age” is only true if people believe so. This has the unpleasant effect of making morality relative, which not everyone agrees with, but that discussion is beyond our scope. Religions are cultural epistemologies that often attempt to grant truth status to claims that fall outside their domain, thus involving a category error.
Finally, we have personal epistemologies, in which the notion of truth is defined by an individual's a priori stance on each claim. For example, the meaning of life or the best way to enjoy it is perhaps only expressible in personal epistemologies. It doesn’t matter what everyone else believes is the meaning of life, only what you believe.
Are some epistemic systems better?
At this point, it should be obvious that there is a thick line separating formal and empirical from social and personal epistemic systems. The former only formulate objective claims, while the latter only deal with subjective (or intersubjective) claims.
This does not mean that objective claims are “better.” We must agree on the truth value of subjective claims if we live as a society. For example, whether our system of government is desirable or not is an intersubjective claim of utmost importance, more than almost anything physics or math has to say.
However, there are some traits we may want all epistemic systems to have, regardless of whether they talk about things in reality or else. Ultimately, an epistemic system is a collection of claims and a method to assign a truth value to each claim. Two basic properties we can ask of such a system are consistency and completeness.
Consistency is simple to define. An epistemic system is consistent if it never produces a contradiction. We say there is a contradiction when two contradictory claims are assigned the same truth value (true or false). Contradictions are easy to formulate in formal epistemic systems —e.g., all prime numbers are odd, and there is at least one prime number that is even— but they can happen in any epistemic system —e.g., the economy is improving, and the quality of life for the average citizen is worsening.
An inconsistent epistemic system is not merely useless but actively dangerous for several reasons. Since two contradictory claims cannot be true and false simultaneously —if you subscribe to logical reasoning— that epistemic system clearly asserts some false claims. It is harmful on those grounds because you could believe false things.
However, it gets worse because clever people can use a contradiction to convince you of anything just by using intuitionist logic. This is called the principle of explosion, and it works like this.
Take any pair of contradictory claims P and !P, such as the Earth is round and the Earth is flat. Assume both are true.
Now take any claim Q you want to prove, such as Aliens have visited us.
Because P is true, logic tells us that P or Q is true.
Thus, the claim R = the Earth is round or Aliens have visited us must be true.
But since !P is true, we can reject the first part of the argument, so the Earth is not round; thus, the only way to make R true is that the second part is true.
Henceforth, we can know for sure aliens have visited us!
If this sounds contrived, it is because it is! But the deduction is logically sound. It is just based on a flawed premise that both some claim and its opposite are true. Now, laid out like this, you can see the obvious mistake at the beginning: we assume a contradiction. But smart people can squeeze a contradiction inside seemingly inoffensive claims and wrap the deduction in enough linguistic clutter to hide the faulty logic.
Completeness is a whole different monster. For an epistemic system to be complete, it must assert true or false for all the claims in its domain. That is, it must never assert that a claim is undecidable. We won’t talk too much about completeness now because that discussion deserves a whole article on its own, but let it suffice to say that we know sufficiently strong formal systems cannot be both consistent and complete simultaneously.
Thus, if you aim to have a consistent epistemic system strong enough to cover our existing math, at least, it cannot be complete. There are some undecidable claims in there. Otherwise, you must deal with bizarre concoctions like “This sentence is false.”
In any case, your epistemic system can be quite arbitrary as long as it is consistent. But you'll be dead pretty soon if it doesn't closely match objective reality —or at least the part of objective reality upon which your survival is contingent.
Hence, on pragmatic grounds, some epistemic systems can still be considered “better” than others because they confer truth status to claims that closely resemble objective reality, in the sense that when acting upon reality guided by those claims, you are more likely to get the desired outcome.Ugh, that was a mouthful.
If we agree that Truth is a property of claims always defined in a specific epistemic system, we must ask exactly how these epistemic systems define Truth. Can a given epistemic system objectively define Truth with capital T?
Take, for example, the Scientific Method, our most cherished epistemic system, which defines truth mostly in terms of falsifiability. A scientific claim is considered true if it makes sufficiently strong falsifiable predictions and doesn't get falsified.Thus, the more we fail to falsify a scientific hypothesis, the more of a truth status we confer it.
But is this criterion of falsifiability something we can objectively claim as true? The criterion of falsifiability itself is a claim, and thus, we can ask if this claim is true and in what epistemic system we can evaluate it. But here's the kicker: We cannot evaluate falsifiability in the scientific epistemic system. The claim that specifies which scientific claims are true is itself not a scientific claim. Falsifiability is not falsifiable!
Thus, science needs some meta-epistemic system to evaluate the meta-claims about the claims of science. This is unfortunate because once we go that way, we can find meta-meta-claims that cannot be evaluated in that meta-system. Does this make sense so far?
What we want is a self-contained epistemic system. One in which the rules that determine which claims are true are also expressible within that same system and also evaluated to be true. If we had this, we might attempt to claim we have found an objective definition of Truth.
However, this seems impossible to achieve with any sufficiently strong epistemic system. We can prove it in formal systems: no sufficiently strong formal system can define its own criteria for truth.
This result is called Tarki’s undefinability theorem. It is hard to say outside logic if it applies since it relies on specific formal semantics. If we want epistemic systems that at least cover science —and, by extension, math— we are severely limited in self-representation.
However, natural language is self-contained. All claims about natural language, and claims about those claims, and so on, are expressible in natural language. Can we find a good semantic definition of Truth in natural language that works for any claim?
It seems difficult because natural language contains claims like “This sentence is false” that cannot be evaluated consistently. This is a classic example of self-referential negation, the type of claim that leads to Russell's paradox. Any sufficiently self-contained system must be able to produce these types of claims.
To get rid of those, we can instead attempt a layered approach, in which self-referential claims can never exist, such as the following:
Take all claims that do not contain the phrase “is true” or “is false” and define that as layer zero.
Define Truth for claims in layer zero, using whatever epistemic system you prefer.
Take all claims X in layer zero, build the claims “X is true” and “X is false,” and define that as layer one.
Define Truth in layer one, saying, “The claim ‘X is true’ is true if and only if the claim ‘X’ is true according to the epistemic system used in layer zero.”
Carry on for layers two and beyond.
Layer zero will contain all non-meta claims. Things like regular logical, scientific, social, and ethical claims. We still haven't completely solved how to assign truth to those, but whatever you do, using any epistemic system will work for us.
Then, layers one and upward contain increasingly meta claims. But, crucially, no layer contains self-referential claims. That is because, by construction, layers one and above only refer to lower-layer claims, and layer zero doesn’t contain any self-referential claim. Claims like “This sentence is false” do not exist anywhere in this infinite ladder of layers because “This sentence” is not even a valid claim, so there’s no X from which to form “X is false.”
Have we solved the Truth, though? Well, only partially, it seems. For starters, we know there are many claims this layered system cannot represent, including all those self-referential ones. How do we know we didn’t throw away anything meaningful? If logic says anything about the real world, it seems to imply we can’t have our cake and eat it, too. We either have a weak system, an incomplete system, or an inconsistent one.
In summary, a claim is only true or false in a given epistemic system. There is no notion of truth outside an epistemic system and no universal epistemology. There are only useful epistemic systems in some contexts.
When we talk about “the truth” of something, that something is never directly an entity in objective or subjective reality but a claim about those entities. Claims are formulated in a given language, and all language requires at least an emitter and a receptor, two subjects. Thus, meaning in that language is always intersubjective, contingent on the interpretation that those subjects attach to each utterance.
To be clear, there can be objective truths (with lowercase t). Those are claims about objective reality that are (approximately) true in an agreed epistemology. Claims about math, natural laws, physics, and chemistry would be objective, as well as claims about how social entities behave under certain conditions, like humans or companies.
But there can’t be an objective definition of Truth with capital T.
That would imply there is a single, universal, self-contained epistemology on which to evaluate all possible claims. Tarski's undefinability shows that at least for formal systems —the ones where we would expect to be easier to demonstrate things— the definition of truth always needs a bigger epistemic system, which also cannot self-contain.
This doesn't mean all truth must be relative, though. There are grounds for evaluating different epistemic systems and deciding which ones are most sensible. Pragmatism is one of them. If your epistemic system fails to produce useful predictions, it's at best useless and, at worst, dangerous for yourself.
But pragmatism is just one possible meta-system. There is no a priori way to demonstrate that pragmatism is any more true than, say, revelation. It is only more useful.
Reality and Truth
As we peel back the layers of "Truth," we must confront a humbling yet inescapable limitation: the observation problem.
Simply put, we are all unreliable witnesses. Our senses are limited tools, and they’re not only reporting information to our minds, but they’re also interpreting.
Even when we employ instruments to extend our senses, we're still bound by frameworks and constructs we ourselves have devised.
“Inherent murkiness” might be a great way to frame how we see things. We're not just interpreting data; we're interpreting our interpretations of that data. All this interpretation takes place within a mind that is a product of subjective experiences, hardly a blank slate.
This brings us to a final unsettling epiphany: all truths are, to varying degrees, subjective. Gravity, our perception of time, and even concepts like existence are all filtered through our all-too-human lenses. In Truth, like in all matters, we are constrained by our empirical experiences and the words and symbols we've devised to describe them.
This is a semantic theory of truth, in which truth is analyzed from the linguistic point of view as a property of sentences in a given language. There are other theories of truth, not necessarily contradicting the semantic ones.
Well, you can make the pedantic argument that the sky is blue is subjective because blueness is not a quality of the objects but an emergent qualia of the way our perceptual systems work. And yes, you would be right. But you could interpret that claim as “the sky scatters light in the frequencies that most people associate with blueness.” And in that sense, it is an objective claim. Again, interpretation matters.
Epistemology is actually a branch of philosophy that studies, among other things, the nature of knowledge and how to attain it. It is way bigger than we can discuss here, so I apologize to my fellow philosophers for borrowing and oversimplifying the concept.
Note that social sciences, insofar as they make claims that can be falsified experimentally, are not social but empirical epistemologies.
Perhaps a more evident social truth is that “Paper money is worth something.” Money is only valuable as long as people believe it is. Once people believe money is worthless, it immediately loses its value. The same happens with stocks, bonds, and any other currency without material value, including —yes, you know— cryptocurrencies.
Maybe you believe that your definition of “the meaning of life” is universal, but I don’t, and that fact already makes it a personal epistemology.
If you don’t think these two claims are contradictory, then that shows we have a different interpretation of what the economy is improving means, underscoring this essay's whole point.
This is a crucial step that should be self-evident; if not, think about it for a second. The usual (and logical) meaning of “or” is literally that at least one of two claims is true. Thus, if I “or” a true claim with anything else, the composite claim is, by definition, also true.
This is the other crucial step of the proof. If “A or B” is true, then either A, B, or both must be true. Otherwise, the whole claim would be false. Thus, if I know that A is false, that automatically tells me that B has to be true.
And this can happen by accident, too. It gets even more concerning when you realize that P and Q don’t need any semantic relation.
This is called a pragmatic theory of truth: true claims are those that, when acting upon them, you obtain the best possible results.
Yes, there is a lot of nuance here. We’ll discuss science and falsifiability in greater depth in a future post.