Laws and Phenomena

This post continues the conversation from this other post.

Let’s use our framework to formalize the idea that for any worlds that differ in physical laws, they also differ in perceptual character, that is, the character that they present to the senses. We can think of this as there being a correspondence between propositions about physical laws and propositions about perceptual character, such that, if a proposition about physical laws makes two worlds equivalent (both worlds give the same value to all individuals), then the corresponding propositions about perceptual character must also make those worlds equivalent.

Below is one way to capture, in our matrix-based semantics, the idea that whenever worlds differ in physical laws, they also differ in perceptual character. Informally, we will say there is a correspondence between propositions about physical laws and propositions about perceptual character such that any two worlds that are indistinguishable (i.e., assign the same truth-values to all individuals) with respect to the physical-laws proposition must also be indistinguishable with respect to the corresponding perceptual-character proposition.

---

1. Background and Setup

Recall our framework:

• $W$ is a set of worlds.
• $D$ is a set of individuals.
• A (unary) proposition $P$ is represented by a matrix $$M_P:W\times D\to \{0,1\}$$ where $M_P[w,d]=1$ means $P$ is true of individual $d$ in world $w$, and $M_P[w,d]=0$ otherwise.

Equivalence of worlds relative to a proposition

Given a proposition $P$, we say that two worlds $w$ and $w^{\prime }$ are $P$-equivalent (written $w{\sim }_P{w}^{\prime }$) just in case $P$ assigns the same truth values to all individuals in those worlds. Formally,

$$w{\sim }_P{w}^{\prime }⟺\mathrm{\forall }d\in D,M_P[w,d]=M_P[w^{\prime },d].$$

Intuitively, $w{\sim }_P{w}^{\prime }$ means that there is no individual $d$ about whom $P$ differs in $w$ and $w^{\prime }$.

---

2. Propositions About Physical Laws vs. Perceptual Character

Let us divide our unary propositions into two families:

1. Physical-Laws Propositions: Denote these by $\mathcal{L}=\{L_i\mid i\in I\}$.

   • Each $L_i$ is intended to capture some aspect of the “laws of nature” holding (or not) for each individual in each world.
   • Matrix representation: $M_{L_i}[w,d]\in \{0,1\}$.

2. Perceptual-Character Propositions: Denote these by $\mathcal{C}=\{C_i\mid i\in I\}$.

   • Each $C_i$ is intended to capture a particular aspect of “perceptual character” (how things seem perceptually to each individual in each world).
   • Matrix representation: $M_{C_i}[w,d]\in \{0,1\}$.

We assume there is some mapping (or “correspondence”) $\varphi :\mathcal{L}\to \mathcal{C}$ associating each physical-laws proposition $L_i$ with a perceptual-character proposition $C_{\varphi (i)}$.

In many philosophical discussions, physical laws (broadly construed) determine or constrain the phenomenal experiences or perceptual characters that individuals can have. Our aim is to say: if two worlds truly differ with respect to a law $L_i$, then they must also differ in the corresponding perceptual property $C_{\varphi (i)}$. Equivalently, if two worlds appear indistinguishable regarding $L_i$ (they give the same truth values for all individuals under $L_i$), they must also be indistinguishable for $C_{\varphi (i)}$.

---

3. Formal Statement of the Dependence

3.1 Basic Statement

For each physical-laws proposition $L_i$ and its corresponding perceptual proposition $C_{\varphi (i)}$, we require that:

$$\mathrm{\forall }w,w^{\prime }\in W:(\underset{w\text{ and }{w}^{\prime }\text{ are }{L}_i\text{-equivalent}}{\underbrace{\mathrm{\forall }d\in D,M_{L_i}[w,d]=M_{L_i}[w^{\prime },d]}})⟹(\underset{w\text{ and }{w}^{\prime }\text{ are }{C}_{\varphi (i)}\text{-equivalent}}{\underbrace{\mathrm{\forall }d\in D,M_{C_{\varphi (i)}}[w,d]=M_{C_{\varphi (i)}}[w^{\prime },d]}}).$$

This condition says: whenever $L_i$ fails to distinguish two worlds $w$ and $w^{\prime }$, the corresponding proposition $C_{\varphi (i)}$ also fails to distinguish them. Equivalently, if $w$ and $w^{\prime }$ are in the same equivalence class under $L_i$, they must also lie in the same equivalence class under $C_{\varphi (i)}$.

3.2 Contraposition

By contraposition, we can read it as:

$$\mathrm{\forall }w,w^{\prime }\in W:(\mathrm{\exists }d\in D,M_{C_{\varphi (i)}}[w,d]\ne M_{C_{\varphi (i)}}[w^{\prime },d])⟹(\mathrm{\exists }d\in D,M_{L_i}[w,d]\ne M_{L_i}[w^{\prime },d]).$$

In words: if two worlds differ in the perceptual-character proposition $C_{\varphi (i)}$, they must differ in the physical-laws proposition $L_i$.

---

4. Intuitive Meaning

Putting it more colloquially:

• Physical difference $⟹$ Perceptual difference. If you pick a particular way in which physical laws might differ across worlds (encoded by $L_i$), and you find that there are indeed two worlds $w$ and $w^{\prime }$ that produce different truth values under $L_i$ for some individual, then you should also find a corresponding difference in their perceptual character ($C_{\varphi (i)}$) for at least one individual.
• No difference in perceptual character $⟹$ no difference in physical laws. If you cannot find any difference in the perceptual character assigned by $C_{\varphi (i)}$ to two worlds $w$ and $w^{\prime }$, then you cannot find any difference in the physical laws assigned by $L_i$ either.

This formalizes (at least in a simplified, first-order modal setting) the philosophical view that how things appear cannot diverge unless the underlying physical facts (laws) also diverge, and conversely, that any difference in the laws must manifest itself in some difference of appearance.

---

5. Beyond Unary Propositions

If one wanted to incorporate more complex statements—e.g., relational claims (binaries or $n$-ary predicates) about physical laws or perceptual states—the same matrix approach generalizes naturally to higher-dimensional arrays:

• For a binary predicate $P$, one works with $M_P:W\times D\times D\to \{0,1\}$.
• The definitions of world-equivalence relative to $P$ and the requirement that $P$-equivalence imply (or be implied by) $Q$-equivalence follow a similar pattern, only now one checks all relevant tuples $(d_1,d_2)$.

---

6. Conclusion

In summary, to say that “for any worlds that differ in physical laws, they also differ in perceptual character” is to impose on our two-dimensional (worlds $\times$ individuals) semantics the constraint:

$$\mathrm{\forall }i\in I,\mathrm{\forall }w,w^{\prime }\in W:(w{\sim }_{L_i}{w}^{\prime })⟹(w{\sim }_{C_{\varphi (i)}}{w}^{\prime }).$$

This axiom captures, in the language of matrices and equivalence classes of worlds, a philosophical principle linking differences in the laws of nature to differences in phenomenal or perceptual aspects of experience.