Salieri

The Scientific Need For Lawhood: Is Necessity Necessary?

May 2024 · Writing sample

Abstract

Humans excel best in noticing patterns and making connections to scientific generalizations. However, deriving universal laws from finite observations poses philosophical and scientific challenges. How can we be certain that our axiomatic or inductive generalizations about nature's behavior will hold true in all instances, including unobserved cases? Our laws appear to be missing a certain element of necessity that would allow us to have full confidence in applying a particular law to real-world instances and cases. In this essay, I will be explaining why regularities under Simple Regularity Theory (SRT) fail to explain different counterfactual variations of instances of laws and explore Nomic Necessity as a possible solution to solidify our understanding of the laws and its merits over SRT.

Introduction & Motivation

It is no exaggeration to say that the ultimate goal of sciences is to provide a rigorous explanation of how different phenomena occur and relate to each other in the world. However, a lot of times science does not explain why things work or behave the way they do. We know that it is a law of nature that the sun necessarily rises from the east every single day and we understand why and how that happens on the mechanical level. But why the East? Is it impossible to conceive of a world where every single thing in the universe is exactly the same, yet the sun rises from the west? Or a world where electricity is conducted through insulators like rubber instead of copper? It is also challenging to determine if every instantiation of a single law will work in the same way. We can easily imagine a scenario where one arbitrary wire made of copper behaves differently from the regular behavior exhibited by copper. This singular anomalous case of copper, along with all other cases of copper that follow the regular pattern, are known as regularities. Regularities are our only means, as beings adept at pattern recognition, to make logical generalizations about the laws that govern the workings of the universe. This shows that there is a need to necessitate some facts that constitute laws to make them stronger than regularities in a universal way. In this essay, I will explore the concept of nomic necessity, the relationship between first and second-order universals, and the merits and demerits of these concepts as a metaphysical explanation of the nature of laws. The central question I aim to address is: "Is necessity necessary to explain laws and regularities?" in other words, "Do laws have to have some form of logical obligation to behave in the same manner once the law has been posited?"

The Need for Necessity

Let us start by hypothesizing a simple law from observations of regularities. We notice that some creatures are warm-blooded, egg-laying, and distinguished by the possession of feathers, wings, a long beak, and typically by the ability to fly. We call those beings birds. We notice some instantiations of this creature with black feathers that share the other qualities of birds, and we call those ravens. It would not be a bold claim to shout loudly the statement: "All ravens are black birds," for we have searched hard enough and found that all ravens we have observed share the same qualities with each other and other birds. Call this Law δ.

Now, after some years, we find a creature that shares all the qualities of birds and ravens yet has white feathers. An error in our initial generalization has occurred because, as we have defined it, a raven possesses black feathers. What we can do now is either go back and redefine the color of feathers as an accidental property of ravens to keep our law true, or we completely change law δ to become: "All black ravens are birds" until we study the mischievous white (ravens?) and try to uncover their true nature.

We can have the same speculative story for ravens of different colors or nose sizes that would drastically affect the reputation of our law δ. Bird (1998) states that regularities cannot "explain their instances in the way a law of nature ought to." Something has to bind the regularities together in order to give the law a certain absolute lawhood. Simple laws like "A liquid boils at a temperature at which its vapor pressure is equal to the pressure of the gas above it" or "A fair coin flip will always have a probability of ½ under the Bernoulli distribution" are both essential laws that can fall under the same regularity problem that law δ fell for. This is because we can always assume a factual exception, an alternative to the generalization of the law that has been hypothesized under the observation of a finite number of occurrences.

This is, of course, not a mere exasperating nitpicky philosophical attempt at radical skepticism. It is necessary to do so in the process of formulating laws in an attempt to cover all imaginable edge cases that could fail the righteous status of a law. An ancient European could claim, "A human being is necessarily white-skinned," just because they have only seen white people. A valid factual exception made by an ancient philosopher would be, "What if there are black-skinned people? Or brown-skinned? Purple-ish-haired ones?" For an ancient European, that could be as odd a claim as us hypothesizing red swans or flying elephants, yet it is a necessary process for verifying the scientific law. We need something to necessitate the properties or relations between all possible instantiations of a law without falling into the regularity trap.

The Nature of Necessity

For us to be able to analyze regularities from a god's eye logical perspective, we need to have a more abstract version of our law δ. This can be represented with the Simple Regularity Theory (SRT), which states that it is a law that F's are G's if and only if all F's are G's. SRT, as a general abstraction of law δ, suffers from all the problems that law δ suffers from. It is also in the form of ∀x(Fx ⊃ Gx) or ∀x(Fx ≡ Gx). The first means that any x in G is a subset of all x in F (all ravens are subsets of birds), and the second means that all x in G are equivalent to all x in F (all birds are birds)1. SRT tries to deal with more universal versions of regularities under laws. Those universal versions are the main tool of conceptual thinking. We cannot think of a bird or a raven without having an idea of a vision of the "Bird," an abstract picture in our heads of what any random bird could look like. We also cannot have Newton's laws of motion if we do not have the ability to imagine what an abstract object is or how a force feels like. Universals work as the primary tool for being able to formulate general law-like accounts on any agent to categorize universal versions of such agents.

Then, we can have two types of universals: first-order universals and second-order universals. A first-order universal is a property of or relation among particular things. For instance, the blackness of ravens is a first-order universal, represented by the first formulation ∀x(Fx ⊃ Gx), where F represents the property of being a raven, and G represents the property of being black. Similarly, the property of a bird being a bird is also a first-order universal, represented by the second formulation ∀x(Fx ≡ Gx), where both F and G represent the property of being a bird. Hence, we have two formal formulations to capture the different types of first-order universals.

On the other hand, a second-order universal is a property of or relation among first-order universals. In other words, second-order universals are properties that properties can have. For example, consider the property of being a property of birds. This is a second-order universal because it is a property that the first-order universal "being a bird" can have. To illustrate further, being an even integer is a property of numbers (a first-order universal). On an upper level, being a type of number is something that first-order universal properties like being even or being prime can have, making it a second-order property.

The nomic necessity view that David Armstrong introduced posits necessity as a second-order universal property between first-order universals that binds them together. It is not that having the property of birdness and the property of ravenness, instantiated in a singular particular of a raven, is what makes it a bird. It is the presence alone of both properties at the same time that necessarily brings about the property of blackness under necessity as a second-order relation of those two properties that bind them together like glue. Every F will be a G if there is a law requiring Fness to be G. This is due to the fact that if x is F, then x's Fness will lead x to become G. Thus, a deterministic law's presence will bring on the same regularity in the absence of the inverse holding Bird (1998). We no longer have to worry about accidental regularities of law δ as having the first-order universal of being black necessitated by being a raven does not mean that the opposite must hold. This solves many of the problems that SRT presents because processes of induction on universals are now thoroughly distinct from induction on particulars. The law of a raven being both a raven, black, and a bird, and the general fact of all ravens being black, are both quite distinct from the fact of ravenness necessitating birdness or blackness. The former are facts about individual ravens, while the latter is a fact about the universals of ravenness, birdness, and blackness. We can better formulate our conception of induction under necessity.

So, instead of saying:

All observed ravens are black.

∴ All unobserved ravens are black.

This can be expressed as a two-stage process:

All observed ravens are black.

∴ Ravenness necessitates blackness.

∴ All unobserved ravens are black.

Yet, necessitation itself sounds (at first glance) complicated and vague. When we say that we can deduce our laws by noticing observable regularities and then make inductive generalizations over them to assume that the unobservables would work the same, despite the problems that we could come out of this approach, it is easy to understand. It provides an adequate mix between the simplicity of the method and the strength of the results and applicability, while necessity does not offer the same. No matter how hard one can look, they will not find necessitation acting on concepts of ravenness or birdness. It looks like when we say that Fness necessitates Gness, it does not differ much from when we assume the occurrence of a law by observing the regularities. For a believer in SRT, necessity is offered as a backdoor metaphysical solution to seem stronger than regularities despite being almost the same. Then, we have two accusations against nomic necessity:

  1. Necessity is the same as regularities.
  2. Necessity has to support factual exceptions and counterfactual conditionals.

Armstrong tries to provide a list of properties that distinguish necessity from SRT and give it a more varying flavor to become its standalone recipe. I will be explaining the properties and the accusations in the warm-blooded vertebrate language of blackness and ravenness in the next section.

The Problem with Necessity

The following list of properties for necessity is not completely new, as they can be formalized from our previous analysis of the nature of necessity and how it is distinguished from SRT.

  1. If ravenness necessitates blackness, then this entails that everything that is a raven is also black [If N(F,G) as a second-order relation holds, then it implies that ∀x(Fx ⊃ Gx)].
  2. The reverse entailment does not follow. Instances of blackness may only coincidentally also be instances of ravenness, without there being any necessitation [∀x(Fx ⊃ Gx) as a first-order relation does not imply N(F,G)].
  3. Since necessitation is a relation, it is a universal. Furthermore, since necessitation is a relation among universals like ravenness and blackness, it is a second-order universal [N(F,G) is a second-order relation among first-order universals F and G].
  4. Since necessitation is a universal, it has instances. Its instances are cases of, for example, a raven's being black because it is a raven, its ravenness necessitates its blackness [(Fa ⊃ Ga) is an instance of ∀x(Fx ⊃ Gx) from the necessitation relation N(F,G)].

Yet, those conditions do not suffice to distinguish necessity from regularities. This is because we can use it to also explain an accidental regularity, like the Ramsey-Lewis view of laws as the simplest systemization of regularities Bird (1998). According to the Ramsey-Lewis (RL) view, the relation RL between ravenness F and blackness G holds precisely when:

(a) All ravens are black.

(b) This is part of the axiomatic system that best captures the complete history of the universe in the simplest way.

This RL relation satisfies the properties of necessitation offered by Armstrong:

  1. If F and G are RL related, then by (a) all Fs (ravens) are Gs (black).
  2. But the reverse doesn't hold, due to (b) requiring maximal simplicity.
  3. RL is a relation between the properties F and G, so a second-order relation.
  4. We can treat "this raven being black because it's a raven" as an instance of RL, since it contributes to the simple systemization in (b).

The problem with Armstrong is that despite trying to distinguish necessity from regularities, as he still tries to preserve the possibility of counterfactuals, being able to make conditional statements that do not match the reality of law to be able properly analyze edge-cases of the law, implying that nomic necessity has to have some sort of soft modal force. By soft, I mean that he tries to maintain the ability to make counterfactual statements like "If ravens were pink, they would have been more beautiful." Armstrong wants to make necessity modally contingent Armstrong (1993); that would allow the law to be completely true yet always modally contingent by the necessity relation in our world, so it might be false in any other possible world, thus allowing counterfactual conditionals and factual exceptions. That comes with the price of the ability to make statements like RL statements or any highly systematic laws that could blur the line between regularities and necessity.

Final Remarks & Conclusion

It is not surprising that Armstrong's nomic necessity has this tension within it. It makes me think of the question of why not let laws necessitate regularities. In this case, we can keep the same modal contingent relation between first-order universals that constitute a certain law. The problem with that is that it goes against the actual role of a law. A law must be able to explain its instances without depending on them, and at the same time, those facts have to count as evidence to formulate the law while allowing for some instances (regularities) to diverge from the law, even if there is a systematic pattern of those instances. Thus, nomic necessity cannot be in the form: (Necessity = Regularity + Something). It has to be independent. For nomic necessity to resolve its tensions, Armstrong would have to accept a form of pragmatism where necessity has to necessitate itself (second-order universals necessitating second-order universals) and disregard the 'soft' modality of the laws. In this case, necessity itself would have to be a hard fact where the use of RL statements or even the problems of Law δ are not allowed by the system of necessity. And like most pragmatic points of view, it limits the ontological freedom given by the system but perhaps increases the simplicity of the system.

Notes

  1. However, this formulation is rarely discussed as I assume it could be considered to have a trivial modal character. That means that such a statement is always true in all possible worlds and thus is not worthy of discussion unlike statements of the first formulation.

References

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