Here’s something that keeps me up at night: How do you prove you love someone? Not the easy kind of love after years together, but that raw, vulnerable moment when you’re falling hard and they’re still figuring out if you’re for real.

Every new relationship starts with this weird information gap. You know exactly how you feel, but your person can only guess based on what you say and do. They’re analyzing everything. Is this grand gesture genuine, or just manipulation? When you say “I love you,” do you mean it, or are you just saying what you think they want to hear?

It’s like being a detective and a suspect at the same time.

The Math Behind Trust#

Okay, bear with me here. This gets a little nerdy, but stick around because it actually makes sense of something we all go through.

Think about how we update our beliefs about someone. Psychologists call this Bayesian inference, which sounds fancy but is basically how we process new information about whether someone’s being real with us.

Let’s formalize this. Define the type space
$$T = {\text{Sincere}, \text{Insincere}}$$
representing the lover’s true intentions. The beloved starts with a prior belief
$$p_0 = P(T = \text{Sincere})$$
based on initial impressions, reputation, or context.

When the lover sends a signal $s$ from the signal space $S$ (could be words, actions, gifts, time spent), the beloved observes this and updates their belief using Bayes’ theorem:

$$ P(T = \text{Sincere} \mid s) = \frac{P(s \mid T = \text{Sincere}) \cdot P(T = \text{Sincere})}{P(s)} $$

where the marginal probability of the signal is:

$$ P(s) = P(s \mid T = \text{Sincere}) \cdot P(T = \text{Sincere}) + P(s \mid T = \text{Insincere}) \cdot P(T = \text{Insincere}) $$

The key insight is in the likelihood ratio:

$$ \Lambda(s) = \frac{P(s \mid T = \text{Sincere})}{P(s \mid T = \text{Insincere})} $$

If $\Lambda(s) > 1$, the signal is more likely to come from a sincere lover, increasing trust. If $\Lambda(s) < 1$, it’s more likely from someone insincere, decreasing trust.

But here’s what this really means. Imagine your new partner throws you a surprise birthday party. If you think a sincere person would have an 80% chance of doing this
$$P(s \mid \text{Sincere}) = 0.8$$
but someone who’s just playing games only has a 20% chance
$$P(s \mid \text{Insincere}) = 0.2$$
then with a starting belief of 50-50, your trust level jumps from 50% to about 80%.

The gesture worked because
$$\Lambda(s) = \frac{0.8}{0.2} = 4$$
it’s four times more likely to come from someone genuine.

Here’s the catch though. And this is where it gets interesting. As long as there’s even a tiny chance that someone fake could pull off the same move, you can never be 100% certain. You can get really, really close, but never all the way there.

Formal Result:
For any finite sequence of signals $s_1, s_2, \ldots, s_n$, if
$$P(s_i \mid T = \text{Insincere}) > 0 \quad \text{for all } i$$
then:
$$ P(T = \text{Sincere} \mid s_1, \ldots, s_n) < 1 $$

No matter how many signals someone sends, if there’s any possibility of faking it, you’ll never reach complete certainty about their intentions.

The Trust Game We All Play#

We can think about this whole thing as a signaling game. Not in a manipulative way, but in the sense that there are players with different information and strategies.

Game Setup:

  • Players: Sender (lover) and Receiver (beloved)
  • Types: Sender knows their type $t \in {S, I}$ (Sincere or Insincere), Receiver doesn’t
  • Strategies:
    • Sender chooses signal $m \in M$ (messages/actions)
    • Receiver chooses response $a \in A$ (Trust or Don’t Trust)
  • Payoffs: $U_{\text{sender}}(t, m, a)$ and $U_{\text{receiver}}(t, m, a)$

For this to work out well, we need what game theorists call a separating equilibrium. Basically, a situation where sincere and insincere people naturally choose different behaviors. This only happens when the signal costs something.

Mathematical Condition for Separating Equilibrium:
A strategy profile $\sigma^*(t), \alpha^*(m)$ where
$$\sigma^*: T \to M \quad \text{and} \quad \alpha^*: M \to A$$
constitutes a separating equilibrium if:

  1. Incentive Compatibility: For each type $t$, $$ \sigma^*(t) \in \arg\max_{m \in M} U_{\text{sender}}(t, m, \alpha^*(m)) $$

  2. Sequential Rationality: For each message $m$, $$ \alpha^*(m) \in \arg\max_{a \in A} \sum_{t \in T} P(t \mid m) \cdot U_{\text{receiver}}(t, m, a) $$

  3. Separation: $$ \sigma^*(S) \neq \sigma^*(I) $$

Think about it. If saying “I love you” was enough, everyone would do it. But planning a thoughtful surprise, being there during tough times, or making real sacrifices requires effort. The cost has to hit that sweet spot where:

$$ U_{\text{insincere}}(m) - C(m) < 0 < U_{\text{sincere}}(m) - C(m) $$

Where:

  • $U_{\text{sincere}}(m)$ is the utility a genuinely loving person gets from sending signal $m$
  • $U_{\text{insincere}}(m)$ is the utility someone just playing around gets
  • $C(m)$ is the cost of the signal

The Single-Crossing Property:
For separation to work, we need the marginal rate of substitution between the signal and other goods to differ across types. Formally, if $c(m,t)$ is the cost function:

$$ \frac{\partial^2 c(m,t)}{\partial m , \partial t} < 0 $$

This means the marginal cost of signaling decreases with sincerity. Genuine lovers find it relatively less costly to make grand gestures.

If planning that surprise party costs more effort than someone fake would invest but less than someone serious would happily spend, it becomes a credible signal.

Why You Can Never Actually Prove It#

Here’s where things get philosophical. Even with all these costly signals, can you ever truly prove your love? Logic says no, and here’s why.

This reminds me of something called Gödel’s Incompleteness Theorems. Without getting too deep into the weeds, these theorems basically say that in any sufficiently complex formal system, there are true statements that can’t be proven from within that system.

Gödel’s First Incompleteness Theorem (simplified):
For any consistent formal system $F$ that can express basic arithmetic, there exists a statement $G$ such that neither $G$ nor $\neg G$ is provable in $F$.

Your relationship is like a formal system. Let $R$ be the “relationship system” with:

  • Language: Actions, words, gestures that can be observed
  • Axioms: Basic assumptions about human behavior
  • Inference Rules: How the beloved interprets signals

The statement “I genuinely love you” has this weird self-referential quality. If you’re lying, it’s false. If you’re sincere, it’s true. But you can’t use the statement itself to prove it’s true.

The Self-Reference Problem:
Consider the statement $L$: “The sender of this statement loves the receiver.” This creates a logical structure similar to the Liar Paradox:

  • If $L$ is sent by someone insincere, then $L$ is false but appears true
  • If $L$ is sent by someone sincere, then $L$ is true
  • But the receiver cannot determine which case applies using only $L$

Any evidence you provide: every loving text, every thoughtful gesture, every sacrifice… could theoretically be part of an incredibly elaborate performance. There’s always some logical possibility that it’s all an act.

Formal Impossibility Result:
Let $\mathcal{E}$ be the set of all possible evidence sequences, and let $\text{Sincere}(\mathcal{E})$ be the proposition that the sender is sincere given evidence $\mathcal{E}$. Then:

$$ \forall \mathcal{E} \in \mathcal{E}_{\text{finite}}: P(\text{Sincere}(\mathcal{E})) < 1 $$

It’s like trying to prove a system is consistent from within itself. You need something outside the system to do that. But in love, there is no outside referee.

By analogy to Gödel’s Second Incompleteness Theorem:
A relationship system $R$ cannot prove its own sincerity using only the evidence available within $R$.

So trust becomes this foundational choice. You decide to believe someone’s sincerity as an axiom and build from there.

Making It Work in Real Life#

All this theory actually leads to some pretty practical insights for anyone trying to build real trust:

Stop Demanding Impossible Proof:
Nobody can prove their love with mathematical certainty. The impossibility theorem shows that expecting complete certainty sets everyone up to fail. Real trust means accepting there’s always going to be a small gap of uncertainty.

Let Your Actions Do the Talking:
Since costly signals work best according to signaling theory, show your love through things that actually require sacrifice. The separating equilibrium requires differential costs. Not necessarily big dramatic gestures. Consistent small efforts often work better. Time is usually the most valuable thing you can give someone.

Build Slowly:
One perfect date won’t convince anyone. Trust builds through sequential updating. Each positive interaction provides new evidence:

$$ P(\text{Sincere} \mid s_1, \ldots, s_n) = \frac{\prod_{i=1}^n P(s_i \mid \text{Sincere}) \cdot P(\text{Sincere})}{\sum_{t \in T} \prod_{i=1}^n P(s_i \mid t) \cdot P(t)} $$

Each time you follow through on something small, you’re making deposits in the trust bank.

Talk About the Uncertainty:
Sometimes the most powerful thing you can do is acknowledge how vulnerable it feels to trust someone. Making trust-building an explicit shared project, rather than a secret test, can actually strengthen your connection. This is like making the belief updating process transparent.

The beautiful paradox is that love can’t be proven with absolute certainty. But that’s exactly what makes choosing to trust so meaningful. It is not a logical conclusion. It is an act of faith. When someone decides to trust you despite the uncertainty, they are not just accepting your love. They are choosing to build something real together.

Maybe that leap of faith is what transforms two people figuring each other out into something that actually looks like love.