Invariance as a basis for necessity and laws

Many philosophers are baffled by necessity. Humeans, in particular, are deeply disturbed by the idea of necessary laws of nature. In this paper I offer a systematic yet down to earth explanation of necessity and laws in terms of invariance. The type of invariance I employ for this purpose generalizes an invariance used in meta-logic. The main idea is that properties and relations in general have certain degrees of invariance, and some properties/relations have a stronger degree of invariance than others. The degrees of invariance of highly-invariant properties are associated with high degrees of necessity of laws governing/describing these properties, and this explains the necessity of such laws both in logic and in science. This non-mysterious explanation has rich ramifications for both fields, including the formality of logic and mathematics, the apparent conflict between the contingency of science and the necessity of its laws, the difference between logical-mathematical, physical, and biological laws/principles, the abstract character of laws, the applicability of logic and mathematics to science, scientific realism, and logical-mathematical realism.

• The laws of special relativity are invariant under changes of inertial reference frames. They're the same in all such frames. They're indifferent to, hence not affected by, replacement of one inertial frame by another. • The laws of universal grammar are invariant under variations in natural language. They hold in all natural languages. They don't distinguish between one natural language and another. They don't even distinguish between actual and linguistically possible (but not actual) natural languages.
Invariance, as these examples suggest, plays a central role in many theories and fields of knowledge. Its fruitfulness, explanatory power, and objectivity are widely recognized. Here are a few citations: [T]here is a structure in the laws of nature which we call the laws of invariance. ... [L]aws of nature could not exist without principles of invariance. (Wigner 1967: 29. My emphases) [E]xplanatory relations must be invariant relations, where a relation is invariant if it remains stable or unchanged as we change various other things. ... [L]aws describe invariant relationships. (Woodward, 1997: S26-7 Holton (1973: 380) reports that ''for the first two years Einstein, in his letters, preferred to call [Special Relativity] not 'relativity theory' but exactly the opposite: Invariantentheorie''. And in a (1921) letter, Einstein said that the name ''invariance theory'' captures the method of special relativity. 3 Kronecker exalted invariance in mathematics: ''when the concept of invariants … is tied … to the general concept of equivalence, … [it] reaches the most general realm of thought''. (Cited from Mancosu, 2016:15).
A well-known invariantist project is Klein's Erlangen program. (The geometrical example above is based on it.) This project classifies the different geometrical fields as more or less general by comparing the class of transformations under which they, or their notions, are invariant. Narrower geometries (such as Euclidean geometry) are invariant under fewer transformations than more general geometries (such as topology).
Generalizing Klein's project, Tarski proposed a demarcation of logical notions in terms of invariance: [C]onsider the class of all one-one transformations of the space, or universe of discourse, or 'world', onto itself. What will be the science which deals with the notions invariant under this widest class of transformations? Here we will have ... notions … of a very general character. I suggest that they are the logical notions, that we call a notion 'logical' if it is invariant under all possible one-one transformations of the world onto itself. [Tarski, 1966/86: 149. Last two emphases are mine] In contemporary terminology, Tarski says that a notion is logical iff (if and only if) it's invariant under all permutations. The so-called Tarski-Sher thesis says that a notion, constant, or property is logical iff it's invariant under all isomorphisms, or under all bijections on domains of individuals.
The type of invariance I discuss in this paper generalizes the Tarski-Sher invariance. Let me now turn to this invariance.

II
My starting point is a simple and common-sensical picture of the world, one that is independent of any specific philosophical or mathematical theory. According to this picture, there are objects of various levels in the world: individuals (level 0) and properties (levels 1,2,3,…). Individuals are definite and distinct. Individuals have properties (including relations), these properties themselves have properties, and so on, and it's determined for each property and each object in its range whether it holds of this object. This picture, which underlies much of our thinking in philosophy, mathematics, and science, is convenient for introducing invariance. 4 Having this picture in mind, we may proceed to a simple yet significant observation about properties: (Selectivity) Properties in general are selective in character. They ''pay attention to'', ''are attuned to'', ''notice'', ''discern'' some differences between individuals, but not others. Accordingly, they may distinguish between some individuals but not all For example, the 1st-level property is-a-human distinguishes between Alfred Tarski and Mt. Everest, but not between Tarski and Meryl Streep or between Everest and Grand Canyon. 5 You can replace Tarski by Streep or Everest by Grand 4 (i) In principle, though, the present idea of invariance can be formulated by reference to other conceptions of the world.
(ii) For the sake of simplicity, I limit my attention to non-vague and non-paradox-generating properties and I assume bivalence.
(iii) I don't elaborate the notions of individual and property beyond what is needed for the present paper. 5 Note: Throughout the paper I use words in italics separated by dashes (as in ''is-a-human'') for 1st-level properties (relations).
Invariance as a basis for necessity and laws 3949 Canyon and the property is-a-human won't notice. But if you replace Tarski by Everest it will. This can be expressed in terms of ''invariance'': is-a-human is invariant under a replacement of Tarski by Streep and of Everest by Grand Canyon, but it isn't invariant under a replacement of Tarski or Streep by Everest or Grand Canyon.
Although the language of the Selectivity observation is metaphorical, it's so just for the sake of stimulating our common-sense intuitions. There is nothing metaphorical about its content as captured by invariance. Given a (non-empty) domain of individuals, D, a 1st-level property P-say, is-a-human-divides it into two sub-domains. One contains all the individuals in D that have the property P, the other contains all the individuals in D that don't have P. P is invariant under all replacements of individuals that have P by individuals that have P and of individuals that don't have P by individuals that don't have P. It's not invariant under any replacements of individuals that have P by individuals that don't have P and vice versa.
Many properties have distinct extension and intension. 6 The selectivity of properties that have a distinct extension and intension is determined by their intension. For the purpose of this paper it is sufficient to assume an intuitive distinction between intension and extension, according to which the extension of a property is the set of actual objects falling under it and its intension has to do with its satisfaction conditions or the meaning of the concept corresponding to it (if there is one). Two properties can have the same extension without having the same intension, but not the other way around. A paradigm example is has-a-kidney and has-a-heart. Their extensions are the same, but their intensions differ.
Since ''invariance'', as we will presently see, is an extensional notion in the sense of being expressed in objectual terms (i.e., in terms of individuals and properties), to accurately express the selectivity of properties in terms of invariance, we have to ''extensionalize'' their intension. We do this by extending the extension of properties to counterfactual individuals. A counterfactual individual is a possible but not actual individual (where at this initial stage the notion of possibility is a merely intuitive notion, left unspecified). To indicate the difference in intension between has-a-kidney and has-a-heart we say that their actual-counterfactual extensions are the classes of all actual-counterfactual (actual and/or counterfactual) individuals that have them. This enables us to indicate that although they have the same actual extension they do not have the same actual-counterfactual extension. From now on I use this expanded notion of extension instead of ''intension''. I distinguish between actual, counterfactual, and actual-counterfactual extensions. Unless otherwise indicated (either explicitly or implicitly from the context) the intended use of ''extension'' is ''actual-counterfactual extension''. 6 It's more common to talk about the extension and intension of concepts (rather than properties), but because the present discussion focuses on properties rather than on concepts, it's more convenient to attribute extension and intension to properties. The underlying idea is that we can treat the concept ''has a heart'' as determining the property has-a-heart, and talk about the extension and intension of this property in much the same way that we usually talk about the extension/intension of the corresponding concept, saying, e.g., that the properties has-a-heart and has-a-kidney have the same extension but not the same intension. (We will shortly see how this works.).
It's important to indicate that our focus is on properties, and counterfactual individuals are introduced only in order to talk about the invariance of properties. By introducing counterfactual individuals, we're able to take account of differences between properties that are not reflected in their actual extension yet are real and important for understanding the aspects of necessity and laws that interest us here. This enables us to measure the selectivity of a given property in terms of invariance, by checking what replacements of actual-counterfactual individuals it's invariant under.
Introducing something like counterfactual individuals is indeed useful for understanding properties in general. Take the property has-a-mass (some mass, oneor-another). The selectivity of this physical property depends on its satisfaction conditions, and these do not distinguish between actual and counterfactual physical individuals. To get the selectivity/invariance of this property right, we have to realize that if Earth had a second moon, has-a-mass wouldn't have distinguished between it and its actual moon. Similarly, is-a-human is invariant not just under a replacement of Tarski by any actual human but also under a replacement of Tarski by any counterfactual human.
The same holds for higher-level properties. Consider the 2nd-level property IS-A-GEOLOGICAL-PROPERTY.
7 The selectivity of this property is reflected by the fact that it doesn't distinguish between any actual-counterfactual individuals that have 1stlevel geological properties, it doesn't distinguish between any actual-counterfactual individuals that don't have 1st-level geological properties, but it does distinguish between actual-counterfactual individuals that have and those that don't have 1stlevel geological properties. Thus, IS-A-GEOLOGICAL-PROPERTY doesn't distinguish between any mountains, between any canyons, between any mountain and canyon, between any humans, between any numbers, between any human and number, but it does distinguish between mountains and humans, between canyons and humans, and so on. In terms of invariance, this 2nd-level property is invariant under some replacements of actual-counterfactual individuals but not under others.
Although the notion of actual-counterfactual individual we use to identify the selectivity (invariance) of properties is completely intuitive and pre-theoretical, it's hard for philosophers steeped in the philosophical literature on counterfactuals to retreat to a pre-theoretical conception of counterfactuality. To accommodate readers who would like to know more about my use of ''actual'' and ''counterfactual'' let me add a few further clarifications 8 : Counterfactual Individuals. First, let me note that since our discussion of invariance is generalist in character, not only does the paper not need specialized accounts of its background notions-''individual '', ''property'', ''actual'', ''counterfactual'', etc.-but it is open to diverse precisifications of these notions. That is to say, most of what we say here is not dependent on any particular precisification of these notions. In fact, a detailed precisification of the background notions will stand in the way. The danger of precisifying beyond what is absolutely necessary to 7 Note: Throughout the paper I use words in small caps separated by dashes (as in ''IS-A-GEOLOGICAL-PROPERTY'') for 2nd-level properties. In contrast, ''Everest'' names an individual. 8 I'd like to thank an anonymous referee for comments that led to these clarifications.
Invariance as a basis for necessity and laws 3951 understand the general ideas expressed here is that a reader's prior objections to a particular precisification of some background notion might interfere with her openness to the discussion of invariance and necessity pursued here, which is independent of any particular precisification of this notion. Where needed, I further clarify certain notions as we go along. But for the most part, I rely primarily on readers' pre-theoretical understanding of the background notions. This said, here are a few clarifications: (i) To explain the general idea of invariance in terms of actualcounterfactual individuals, there's no need for a full-scale modal apparatus, and I don't use one here. (ii) There are two types of individuals: actual and counterfactual. When we abstract from the status of an individual as actual or counterfactual, we view it as an actual-counterfactual individual, or as an individual simpliciter. 9 Later on in the paper I will distinguish several types of possibility, with ramifications for counterfactuality. But at this (early) stage I assume a single, common-sensical, pre-theoretical, unspecified type of possibility. As an example of a counterfactual individual we can think of a second moon of Earth. (iii) All individuals have properties (stand in relations). An actual individual has the properties it actually has. A counterfactual individual is a non-actual individual, hence the cluster of properties it has isn't identical to the cluster of properties of any actual individual. (iv) There's no cross-actual-counterfactual identity: no actual individual is identical to any counterfactual individual. Informally, one may treat counterfactual individuals which are property-wise similar to a given actual individual in certain ways (to be specified outside invariance theory) as its ''counterparts'', but no specific theory of counterparts is assumed here. (v) At some point we introduce domains of individuals. Individuals are not domain relative. (vi) Domains are non-empty collections of actual-counterfactual individuals. (vii) Given a unary 1st-level property P, we may talk about its extension simpliciter, where by this we mean its extensions in the class of all actual-counterfactual individuals. We may also talk about its extension in domains D. The extension of P in domain D is the class of all actualcounterfactual individuals in D that have this property. These notions of extension are expanded to relational properties (an n-place property/ relation P) in the usual way, namely, by talking about n-tuples of individuals having (or standing in) it.
(viii) We also expand these notions to higher-level properties. For example, the extension of a unary 2nd-level property is the class of all 1st-level properties that have it. Later on we will explain how to discuss the extension of higher-level properties in a context involving domains.
Given the limitation of size on a journal paper, I leave further clarifications to future work. 10 To set the ground for the ensuing discussion of invariance, necessity, and laws let me offer a methodological remark.
Methodology. The methodology I use is holistic. ''Holism'' means different things to different people. My approach is holistic in a sense that is most naturally represented by the Neurath boat metaphor (on a realist interpretation). Knowledge is knowledge of the world, or some facet of the world, and this means that we always aim at truth (in a broad correspondence sense). But there's no Archimedean standpoint. To cognitively reach the world, we start from where we stand at the moment, use available tools, employ our critical and creative faculties, and begin, or continue, theorizing. Theorizing typically involves back-and-forth movement. Starting from elements we have, we develop new elements and then use these elements (together with others we've discovered or developed along the way) to turn back, re-examine the elements we started with, replace, revise, or keep these elements in place, and go on. Incorporating knowledge from various fields (both philosophical and others) is par de course. Partial circularity is also par de course. This, I believe, is the way progress in theorizing is usually achieved by humans.
We are now ready to define the notion of invariance associated with the selectivity of properties. I will call this notion ''property invariance''. To define ''property invariance'' without commitment to a particular mathematical background theory, I will use neutral terminology (not specific to a particular theory) and avoid terms of art as much as possible.
Let D be a domain-a non-empty class of actual-counterfactual individuals. As noted above, each 1st-level property P divides D into two sub-classes: the class of individuals (n-tuple of individuals) in D that have P, and its complement, the class of individuals (n-tuple of individuals) in D that don't have P.
10 Still I will consider a puzzle pointed out by an anonymous reviewer: Let P 1 , P 2 , P 3 be the properties is-Trump (x=t), is-Sanders (x=s), and is-the-only-president-of-the-USin-2019. Let a be an actual individual that has properties P 1 and P 3 . Let b be a counterfactual individual that has properties P 2 and P 3 . Given that a and b have property P 3 , they are identical. But given that a is an actual individual and b is a counterfactual individual, they are not identical according to our account.
This puzzle may be solved in different ways by different precisifications of the relevant notions. One way to solve it is to introduce an apparatus of possible worlds and index properties to worlds. There's an actual world, w 1 , in which there's an (actual) individual, a, which has the properties P 1 w 1 (P 1 indexed to w 1 ) and P 3 w 1 . There's also another, counterfactual, world, w 2 , with a (counterfactual) individual, b, which has the properties P 2 w 2 and P 3 w 2 . Since P 3 w 1 = P 3 w 2 , a and b need not be identical. I'd like to thank the anonymous reviewer for suggesting this way of solving the puzzle.
In the remainder of the paper, however, I will not assume that properties are indexed to possible worlds. I will assume that the puzzle can be resolved in one way or another and that there are ways to incorporate adequate solutions into our account.
Invariance as a basis for necessity and laws 3953 Let r be a replacement function, where a replacement function is a 1-1 function from a domain D 1 onto a domain D 2 (possibly D 1 = D 2 ). (Since the r's are 1-1 and onto, they apply only to pairs of equinumerous domains.) I will say that each r is indexed to a pair of (equinumerous) domains, \D 1 ,D 2 [. Sometimes, I will speak of r as indexed to a single domain, D, in which case the second domain will be the (exact) range of r.
Property-invariance or, for short, invariance, is then the relation P is invariant under r, where P is a property of any level and r is a replacement function (on individuals). To include higher-level properties in the definition of invariance, I introduce the notion of ''property P restricted to D''-''P D ''. If P is a 2nd-level property, P D is a 1st-level property whose actual-counterfactual extension is restricted to the domain D. Below, I will identify P D with its extension (i.e., the extension of P in D).
To make the definition of ''P is invariant under r''-''INV(P,r)''-clear, I will focus on three simple cases: P is an n-place 1st-level property, r is as above (i.e., indexed to D) Case 3: P is a unary 2nd-level property of unary 1st-level properties, r is as above INV(P,r) = Df (VP D )[P(P D ) $ P(r*(P D ))], where r*(P D ) is the image of P D under r 11 The full definition of INV(P,r) for all cases is a natural extension of the above definitions.
Examples Consider: 11 (i) Note that while r is a replacement of individuals of D, r* is a replacement, induced by r, of 1stlevel properties restricted to D (P D 's). The point is that since the P D 's are, as indicated above, identified with their extensions (in D), every replacement r of the individuals in D induces a replacement r* of the P D 's. For example: If r is a 1-1 function from D = D 1 = {a,b} onto D 2 = {c,d} and P D = {a}, then r*( P D ) = {r(a)}.
(ii) The fact that r-a function on individuals-induces a function on 1st-level properties-r*-means that we can determine the invariance of 2nd-level properties by focusing on individuals.
It's easy to see that: (i) P 1 is invariant under r 1 but not under r 2 . 12 (ii) P 2 is invariant under both r 1 and r 2 . 13 (iii) P 3 is invariant under r 2 but not under r 1 . 14 In a similar way we can determine whether any scientific and logical/mathematical property is invariant under a given r. For example, the property is-subjectto-gravity is invariant under r's that replace individuals that are subject to gravity by individuals that are subject to gravity and individuals that are not subject to gravity by individuals that are not subject to gravity. I.e., it's invariant under any replacement of physical individuals by physical individuals and under any replacement of non-physical individuals by non-physical individuals. But it is not invariant under any replacement of physical by non-physical individuals and vice versa.
This notion of property invariance is a generalization of the notion of invariance used in the Tarski-Sher thesis. The idea is that the latter notion, which has until now been used only in connection with logic, mathematics, and linguistics, can be fruitfully extended to explain the necessity of laws in general. 15 In the next section I 12 r 1 assigns humans to humans and non-humans to non-humans; r 2 does not. 13 a = b iff r i (a) = r i (b) for i = 1,2. (Here I use '' = '' for ''is-identical-to''). For questions concerning the necessity of identity statements see Sect. 3, including fn. 20, below. 14 D 1 has 3 individuals, so there are (up to extensional equivalence) 8 distinct 1st-level properties restricted to D 1 (P D1 's). Since geological individuals have geological properties and non-geological individuals don't, the only 1st-level geological properties on D 1 and D 2 are, extensionally, the P D1 's {Everest} and {Grand Canyon}. (For simplicity, I leave Ø out.) The image of {Everest} under r 2 is {Grand Canyon}. I.e., the image of each 1st-level geological property restricted to D 1 under r 2 is a 1stlevel geological property restricted to D 2 (and vice versa), and the image of each non-geological 1st-level property restricted to D 1 under r 2 is a non-geological 1st-level property restricted to D 2 (and vice versa). This is not the case with r 1 . 15 In the literature on logic there are a couple of alternative proposals for what I call here ''propertyinvariance'', due to Feferman (1999) and Bonnay (2008). The main difference between the alternative proposals and the Tarski-Sher proposal, expressed in the present terminology, is that the latter sets stronger requirements on r. (r has to be both 1-1 and onto. Feferman's proposal, for example, doesn't require r to be 1-1.) The reason I prefer to generalize the Tarski-Sher notion rather than the Feferman or Bonnay notion is my belief that the former is more fruitful and philosophically significant than the latter. Explaining this in detail here will divert us from our main subject, so let me just refer the reader to criticisms of Feferman's proposal in van Benthem (2002: 431), Sher (2008: 324-38), and Bonnay (2008: 42-4). I should add that (i) some of the criticisms of Feferman's proposal apply to Bonnay as well (for example, ad hocness and lack of theoretical philosophical justification), (ii) Feferman withdrew from his will state four theses that establish the connection between property-invariance and necessity (necessary laws).

III
Four Theses on Invariance, Necessity, and Laws: Thesis 1 Every property is invariant under some 1-1 and onto replacement(s) of individuals Thesis 2 Some properties have a higher degree of invariance than others; some, but not all, properties have maximal invariance Thesis 3 The higher the degree of invariance of a given property, the greater the degree of necessity of the laws/principles governing/describing it Thesis 4 The higher the degree of invariance of a given field of knowledge, the greater the degree of necessity of, or available to, its laws/principles Explanation: Thesis 1. Every property is invariant under some 1-1 and onto replacement(s) of individuals. This is trivial, since every property is invariant under the identity replacement of individuals, i.e., replacement of each individual by itself. Even properties as particular as is-Everest or is-Tarski are invariant under this replacement. But many properties, including those in the earlier examples, are invariant under nontrivial replacements of individuals as well. Has-a-mass, for example, is invariant not only under replacement of Earth by Earth, but also under replacement of Earth by Mars (and many other actual-counterfactual physical individuals). Thesis 2. Some properties have a higher degree of invariance than others; some, but not all, properties have maximal invariance. We've seen that all properties are invariant under some 1-1 and onto replacement(s) r of individuals. But are any properties invariant under all 1-1 and onto replacements of individuals? Yes. The 1st-level property is-identical-to is. (In one of the examples in Sect. 2 we saw that it's invariant under both r 1 and r 2 , but in fact, it's invariant under any r.) The 2nd-level property IS-A-NON-EMPTY-PROPERTY (the existential-quantifier property) 16 is also invariant under any r. 17 In this sense both identity and NON-EMPTINESS are maximally-invariant. I define: Footnote 15 continued proposal, adopting (Feferman 2015) an altogether different approach to logicality. Of course, nothing I say here should discourage other philosophers from trying to generalize any of the alternative proposals to non-logical properties and compare the different generalizations. 16 ''(Ax)Px'' says: ''P is non-empty''.
I.e., P is maximally-invariant iff P is invariant under every replacement function r.
Note: Although sometimes maximally-invariant properties P are universal (P holds of all individuals if it is a 1st-level property, of all 1st-level properties if it is a 2nd-level property), this isn't generally the case. For example, the property isnot-identical-to is maximally-invariant but it doesn't hold of any individual in any domain, and the property IS-A-NON-EMPTY-PROPERTY doesn't hold of any empty 1stlevel property in any domain. Similarly, the property EXACTLY-TWO 18 (which holds of a 1st-level property P in a given domain D iff P holds of exactly two individuals in D) is maximally-invariant, yet is not universal in any D.
Not all properties are maximally-invariant. As we've seen above, is-a-human, issubject-to-gravity, IS-A-GEOLOGICAL-PROPERTY, etc. are not maximally-invariant.
If P 1 is maximally-invariant and P 2 isn't maximally-invariant, we'll say that P 1 has a higher degree of invariance than P 2 . Here, having a higher degree of invariance means being invariant under more r's, where ''more'' is understood in terms of proper inclusion. (If the class of all r's under which P 1 is invariant properly includes the class of all r's under which P 2 is invariant, then P 1 has a higher degree of invariance than P 2 .) 19 Both is-identical-to and IS-A-NON-EMPTY-PROPERTY have a higher degree of invariance than is-a-human, is-subject-to-gravity, and IS-A-GEOLOGICAL-PROPERTY. These examples of maximally-and non-maximally-invariant properties are sufficient to establish the present thesis.
Thesis 3. The higher the degree of invariance of a given property, the greater the degree of necessity of the laws/principles governing/describing it. To understand this Thesis, let's start with maximally-invariant properties, say, is-identical-to and IS-NON-EMPTY, and let's consider some principles that govern/describe them, for example: (Id) Every individual is-identical-to itself and (NE) If P IS-NON-EMPTY and every individual that has P has Q, then Q IS-NON-EMPTY Let's start with (Id). Given that is-identical-to is invariant under any 1-1 and onto replacement of any actual-counterfactual individuals, i.e., doesn't distinguish between any actual-counterfactual individuals, the principles governing/describing it cannot distinguish between any actual-counterfactual individuals either. (If they did, they wouldn't accurately describe identity.) I.e., (Id) is a maximally-necessary principle. Since it holds of some actual-counterfactual individuals (e.g., Tarski), it holds of all. As such (Id) has the kind of necessity that is required for laws, indeed, for laws of the strongest modal force. As far as its necessity is concerned, (Id) is thus an admissible candidate for a maximally-necessary law. 20 This demonstrates that to the extent that necessity is concerned, laws are possible.
The same holds for (NE). Given that IS-NON-EMPTY is invariant under any 1-1 and onto replacement of any actual-counterfactual individuals, hence doesn't distinguish between non-empty properties that hold of individuals of different kinds, the principles governing/describing it, such as (NE), can't distinguish between such properties either. (NE) holds of non-empty properties of any individuals, including properties that are nonempty only in domains of extremely counterfactual individuals, such as the property isboth-all-red-and-yellow-at-the-same-time. (If (NE) didn't hold of non-empty properties of that kind, it wouldn't accurately describe IS-NON-EMPTY.) In Sect. 5 we will see how this result extends to laws governing/describing properties that are highly, but not maximally, invariant, such as the physical property is-subject-to-gravity.
For those philosophers of logic (e.g., nominalists) who are worried about the mysteriousness of necessity in logic, it's important to note that there is nothing mysterious about the way the present account explains the possibility of maximallynecessary principles/laws such as (Id) or (NE). The maximal necessity of these principles/laws is due to the non-mysterious fact that identity and non-emptiness are maximally-invariant, i.e., invariant under any r.
Thesis 4. The higher the degree of invariance of a given field of knowledge, the greater the necessity of, or the necessity available to, its laws/principles. The correlation between higher degrees of invariance and higher degrees of necessity (greater modal force) can be extended to fields of knowledge. We can characterize the degree of invariance of a given field as the degree of invariance of its most highly-invariant properties, and the degree of necessity of, or available to, its laws/principles as the degree of necessity of its most highly-necessary laws/principles. (Here, I mean by ''properties of a field X'' properties that are distinctive of it. For example, has-a-mass and is-subjectto-gravity are physical properties, but although logical and mathematical properties apply to physics, they are not distinctly physical, so they are not physical properties.) In Sect. 5 I will further discuss properties of, and necessity in, scientific fields.
Using ''DI'' for ''degree of invariance'', ''DN'' for ''degree of necessity'', ''1'' for the relation ''higher than'' between DI's, and ''['' for the relation ''higher than'' between DN's, it follows from Theses 3 and 4 that: In the next section I will discuss the application of these theses to logic and mathematics.
Footnote 20 continued in our sense, ''Hesperus is Phosphorus'' is not. (in this paper I leave it open whether it is necessary in some weaker sense.).

IV
The application of invariance to logic and mathematics yields rich results. Technically, these results employ the above definitions of ''P is invariant under r'' (INV(P,r)) and ''P is maximally-invariant'' (Max-INV), as well as the following definition of ''higher degree of invariance'' (1): (1) The degree of invariance of P 1 is higher than the degree of invariance of P 2 iff the class of all r's under which P 1 is invariant properly includes the class of all r's under which P 2 is invariant In symbols: DI(P 1 ) 1 DI(P 2 ) iff {r: INV(P 1 ,r)} . {r: INV(P 2 ,r)} Clearly, 1 is anti-reflexive, anti-symmetric, and transitive. I.e., it is a strong partial ordering. 21 In the existent literature, the rich results of the applications of invariance to logic are commonly formulated in terms of ''invariance under isomorphisms/bijections'', but this is just another way to express ''invariance under 1-1 and onto replacements of individuals'' as it's defined here. These results are closely related to the Tarski-Sher thesis. The most relevant works for the present discussion are Tarski (1966/86) and Sher (1991Sher ( , 2016. While Tarski (op.cit) doesn't view himself as concerned with the philosophical question ''What is logic?'', Sher (op.cit) does. The starting point of Sher (2016) is epistemic: given humans' epistemic aspirations on the one hand and their cognitive limitations on the other, they greatly benefit from a powerful method of inference applicable to all or most fields. In particular, they benefit from the development of a type of inference, or consequence, that transmits truth from premises to conclusion with an especially strong modal force. The question is how, theoretically, do we build such a system. The Tarski-Sher thesis approaches this question from the perspective of admissible choices of logical constants. Using the present terminology, the question is ''Which choice of logical constants yields a logical system that sanctions all and only logical consequences/principles/laws that are maximallynecessary?''. The demarcation of admissible logical constants is given in terms of (property-) invariance. 22 Here I limit myself to a brief discussion of a few theoretical results of the above theses for logic and mathematics. 21 Although 1 is transitive, in its present formulation it's only vacuously transitive, since the antecedent of ''(x1y & y1z) ? x1z'' is empty. (Only maximally-invariant properties have a higher degree of invariance than other properties.) This partial ordering is sufficient for our applications of invariance to logic and mathematics (where, as we'll see below, only differences in degree of invariance between maximally-invariant and non-maximally-invariant properties play a role). In Sect. 5 we'll refine the definition of 1 in a way that renders its transitivity non-vacuous (so its non-trivial applications are extended to pairs of non-maximally-invariant properties). 22 Note: Adherents of this demarcation focus on theoretical considerations, such as logic's role in knowledge, and have no prior stand on the relation between logic and mathematics. Critics usually focus on intuitive considerations, natural language, and/or prior commitments concerning the relation between Before turning to these results, however, let me address a thorny methodological point. It's natural to formulate the results of the above theses for logic and mathematics using mathematical terms-of-art, but this is likely to create the false impression that the results themselves involve commitment to a particular background mathematical theory. To avoid this, it's desirable to conduct the discussion in two steps. In the first step we limit ourselves to general philosophical and everyday terminology, in order to emphasize the mathematical neutrality of the discussion. The second step is a step of precisification, using the resources of a specific mathematical theory (yet still without commitment to either its unique or perfect adequacy for this task). In describing property-invariance above in terms of ''replacement of individuals'', I have used only general terminology. But limitations of space prevent me from a strict separation of the general and specific discussion in what follows. Readers, however, should keep in mind the intended division into two levels of explanation.
There are various orders in which one could present the ramifications of invariance for logic and mathematics. Given the concerns of this paper, I will start with the relation between invariance and formality.
It's natural to think of formality (in the objectual sense, which is relevant here, as opposed to the syntactic sense) as strong structurality. For example, it's natural to think of logic and mathematics as formal in the sense of being highly-structural. Strong structurality, in turn, is naturally characterized as invariance under all isomorphisms. But invariance under all isomorphisms is (as we've noted above) equivalent to maximal property-invariance, namely, invariance of properties under all 1-1 and onto replacements of individuals. Accordingly, maximal-invariance can be viewed as a criterion of formality. Note that this use of ''formality'' implies that there are no formal individuals, only formal properties.
Given this conception of formality we have: Result 1 A property is formal iff it is maximally-invariant; a law/principle is formal iff it governs/describes formal properties. It follows from Thesis 3 that formal laws/principles are maximally-necessary. Now, it has been shown (Tarski-Sher Thesis) that all the logical properties of standard logic (those denoted by its logical constants) are formal. Hence the logical principles/laws (those governing/describing these properties), whatever they are, are highly-necessary Proceeding to the next result, we note that by identifying maximal-invariance with formality, we are led to an additional fruitful idea, the idea of formal necessity/possibility. The point is that the category of formally-possible situations is extremely broad. It encompasses not just situations involving physically-possible individuals, but also situations involving physically-impossible individuals, such as individuals that are both all-red and yellow (at the same time). Formally necessary laws/principles hold in all such situations. Accordingly, formal necessity is extremely strong. Laws/ principles governing formal properties are formally necessary.
We can use the ''formality'' terminology to set important conditions on an adequate semantic or model-theoretic definition of logical consequence, such as the Tarskian definition, (T) A sentence S is a logical consequence of a set of sentences C iff there is no model in which all the sentences of C are true and S is false If we view a genuine logical consequence as one that transmits truth from sentences (premises) to a sentence (conclusion) with an especially strong modal force and based on the logical structure (distribution and identity of logical constants) of the sentences involved, then we have: Result 2 (Adequacy of Definition of Logical Consequence): To be adequate (i.e., to identify only formally-necessary consequences as logical), a Tarskian definition of logical consequence has to require that (i) logical constants denote formal properties, and (ii) the totality of models represents all formally-possible situations 23 This result has further ramifications for the scope of logic. If we think of logical consequence as dependent on the logical structure of the sentences involved and of logical structure as dependent on the division of constants into logical and nonlogical constants, then the scope of logic is significantly dependent on this division. Result 2 shows, theoretically, that any formal constant (one that denotes a formal property) can, in principle, be considered logical as far as the requirement of transmission-of-truth-with-an-especially-strong-modal-force is concerned. Of course, if we set additional requirements on an adequate logical system, some formal constants might be ruled out. But as far as the modal requirement is concerned, all formal constants are in principle admissible.
As an example, consider the 2nd-level property (quantifier)MOST. 24 This property, it's easy to see, is maximally-invariant, just like the existential-and universalquantifier properties, IS-NON-EMPTY and IS-UNIVERSAL. 25 Accordingly, the consequence (1) (Most x)Px; therefore (Ax)Px, transmits truth from premise to conclusion with the same modal force as (2) (Vx)Px; therefore (Ax)Px. 26 Put otherwise, the logical principle expressed by (1) has the same modal force as that expressed by (2). 23 This result explains why Tarskian models are not ''representational'' in Etchemendy's (1990) sense, which does not distinguish between formal and general-metaphysical possibility/necessity. (See Sher, 1996.). 24 P D has the property MOST iff the cardinality of P D is greater than the cardinality of P D -the property of not having the property P, restricted to D. 25 P D has the property UNIVERSAL iff P D is empty.

Result 3 (Scope of Formal Properties). All higher-level mathematical properties
and several 1st-level mathematical properties (e.g., equality-that is, identity-and non-equality) are formal (maximally-invariant) This result has several significant ramifications. First, it enables us to delineate in a precise way a maximalist conception of the scope of logic: as far as the maximal necessity of logic is concerned, any higherlevel and several 1st-level mathematical properties are admissible denotations of logical constants. Second, Result 3 yields a highly significant result for mathematics:

Result 4 All higher-level mathematical laws/principles are maximally-necessary
That is, the connection between maximal invariance and maximal modal force (Thesis 3), together with Result 3, establish the maximal necessity of a large portion of mathematics (mathematical principles/laws).
What about 1st-level/order mathematics? Results due to Tarski (1966/86) 27 show that 1st-level mathematical properties have a weaker degree of invariance than logical properties and higher-level mathematical properties. Two approaches to 1stlevel mathematics that are compatible with Result 4 are: (a) 1st-level mathematics is different from higher-level mathematics and has a weaker degree of necessity than the latter (and logic). (b) 1st-level mathematics represents higher-level mathematics: 1st-level arithmetic laws represent higher-level laws of finite cardinalities and 1st-level set-theoretical laws represent higher-level formal laws in general. The maximal necessity of higher-level mathematical laws is then extended to 1st-level mathematical laws in virtue of their representational role. Tarski (1966/86) is indifferent between the first approach and approaches that are more similar to the second approach. Sher (2016: Chapters 8, 10) favors the second approach for reasons that have to do with its ability to explain (i) mathematical truth in general, (ii) the strong necessity of mathematical laws, and (iii) the similarity between logic and mathematics in their applicability to science. On the second approach, the continuum hypothesis, for example, is either necessarily true or necessarily false in the strongest sense of necessity, but the current axioms of set theory are not sufficient to decide whether it is (necessarily) true or (necessarily) false. 28 I will conclude with a result concerning logical and mathematical realism. This result is based on the observation that property-invariance is worldly or objectual, rather than linguistic or conventional. It deals with properties of objects rather than with words or conventions. Objects have properties of many kinds, including maximally-invariant-formal-properties. Accordingly, the invariantist necessity of laws/principles governing/describing these properties-logical and mathematical laws/principles-is also objectual or worldly rather than linguistic or conventional. In this sense, the proper attitude toward mathematical and logical laws/principles is realist. We may be unsure what the laws/principles of formal properties are, yet sure, based on invariance results, that whatever they are, they have a very strong degree of necessity.
Result 5 (Logical-Mathematical Realism). Formal laws hold of objects in the world (individuals, properties of individuals, and so on), hence theories stating formal laws are true or false in the objectual sense, i.e., true or false about the world, broadly understood. Accordingly, the appropriate stance toward such theories is realist (rather than conventionalist, or purely pragmatist, or linguistic) Let me clarify that this realism is Aristotelian rather than Platonistic in character. The underlying picture is that of one world (rather than two, physical and Platonic), with objects that have both physical and formal properties. Talk of counterfactual individuals is a device used to identify the selectivity of different properties (or rather their non-selectivity, which is the basis for their invariance) and doesn't involve commitment to Platonic worlds. Accordingly, the laws/principles governing/describing different properties-including logical and mathematical properties, which are formal in character-are real in a (broad) Aristotelian sense.
Logic and mathematics, on this view, are closely interconnected yet not identical. Mathematics studies objectual laws/principles governing/describing formal properties, and logic studies laws/principles of reasoning (inference, consequence) grounded in formal laws/principles. The two fields develop in a back-and-forth process (of the kind described in my discussion of methodology above).
This Aristotelian realism has interesting ramifications for the relation between mathematics and physics as well. Among other things, it leads to a straightforward answer to Wigner's (1960) question about the applicability of mathematics to physics. Formal properties, being maximally-invariant, don't distinguish between individuals of any kind. Therefore, the laws governing them-both logical and mathematical laws-are applicable to individuals (and properties) of any kind, hence to physical individuals (and properties). There is nothing surprising or mysterious about their applicability to physics.
This concludes our discussion of invariance as a basis for logical and mathematical necessity. Let's turn to natural necessity (the necessity of laws of nature). V So far we've shown that formal, i.e., logical and higher-level mathematical, properties are maximally-invariant and that the connection between invariance and necessity implies that, and explains why, formal principles/laws have a maximal degree of necessity. We've also shown that natural (physical, biological, …) properties are not maximally-invariant, hence natural principles/laws have a lower degree of necessity than logical and mathematical principles/laws.
Can we use the simple ordering relation, 1, as it now stands, to compare the degrees of invariance of natural properties? No. Let me first show that this, indeed, is the case, then analyze why it's the case, and finally delineate a refinement of 1 that can perform such comparisons.
Consider two distinct non-maximally-invariant properties (distinct in the sense of not being invariant under exactly the same r's), say, is-subject-to-gravity and is-astar. Based on our informal considerations above, the degree of invariance of issubject-to-gravity is higher than the degree of invariance of is-a-star (because all replacements of stars by stars are also replacements of individuals that are subject to gravity by individuals that are subject to gravity but some replacements of individuals subject to gravity by individuals subject to gravity are replacements of stars by non-stars). The 1-test, as it now stands, misses this result: according to the present version of this test neither property has a higher degree of invariance than the other. To see this, it's sufficient to present two r's, r 1 and r 2 , such that one of these properties is invariant under r 1 but not r 2 and the other is invariant under r 2 but not r 1 . 29 Here are such r's: r 1 and r 2 are indexed to \D 1 ,D 2 [, where D 1 = {a 1 ,b 1 ,c 1 }, D 2 = {a 2 ,b 2 ,c 2 }, and where the a's are individuals that are subject to gravity but are not stars, the b's are both stars and subject to gravity, and the c's are neither (say the c's are mathematical individuals). r 1 (a 1 ) = b 2 , r 1 (b 1 ) = a 2 , r 1 (c 1 ) = c 2 , r 2 (a 1 ) = c 2 , r 2 (b 1 ) = b 2 , r 2 (c 1 ) = a 2 .
This result is generalizable. The present version of 1 suffices to show that (i) all maximally-invariant (logical) properties have a higher degree of invariance than all non-maximally-invariant (e.g., natural) properties and (ii) all maximally-invariant (logical) properties have the same degree of invariance. But for the most part it renders distinct non-maximally-invariant properties (including natural properties) invariance-wise incommensurate.
Why does the 1-test work properly when at least one of the properties is maximally-invariant but not when neither is maximally-invariant? Why does it work properly in comparing the degrees of invariance of identity and gravity but not of gravity and star?
The reason is that in all cases the 1-test, as it now stands, takes into account r's indexed to domains with any actual-counterfactual individuals, but only when (at least) one of the properties is maximally-invariant are all these r's relevant and should be taken into account. We may say that maximally-invariant properties are global; others are local, and to discern differences in degree of invariance between the latter, 1 has to ''localize'' the r's taken into account by weeding out replacements of ''irrelevant'' actual-counterfactual individuals. For example, to discern that DI(is-subject-to-gravity) is higher than DI(is-a-star), 1 should take into account only replacements of physically viable (actual-counterfactual) individuals, excluding other individuals such as mathematical individuals and individuals that are both all-red and yellow. It's the fact that we took into account mathematical individuals that skewed our result in the example above. Had we limited ourselves to replacements of physically viable actual-counterfactual individuals, we would have obtained the right result.
This suggests that to render the 1-test applicable to pairs of non-maximallyinvariant properties we need to refine it by adjusting the replacements taken into account. The decision of which replacements to take into account is done outside invariance theory, holistically, based on our interests and our current knowledge/ understanding of the properties in question.
Technically, there are a few ways to proceed. Without going into details, here is a brief outline of one of these. We proceed in two steps: A. For each pair of properties whose degree of invariance we wish to compare we identify a range of relevant r's reflecting the selectivity of the properties involved. For example, it's reasonable to limit the r's to domains of physicallyviable actual-counterfactual individuals when applying 1 to IS-A-PHYSICAL-PROPERTY or is-subject-to-gravity. It's reasonable to limit the relevant r's to domains of physically-viable disks of uranium and gold when comparing the degrees of invariance of properties of such disks (see example below). And so on. Based on our earlier results, we stipulate that all individuals simpliciter are relevant to formal (logical, higher-level mathematical) properties. B. We apply the 1-test using relevant r's.

31
We may also introduce the notions ''physically maximally-invariant property'' and ''biologically maximally-invariant property''. A property is physically/biologically maximally-invariant iff it is invariant under all r's relevant to is-a-physical/ biological-individual. Clearly, is-a-physical-individual is physically maximally-31 An anonymous reviewer of this paper suggested that 1* might be replaceable by a simpler notion. This is an open question. Two possibilities based on the reviewer's comments are: [1]: P 1 0*P 2 only if P 1 is a proper subset of P 2 in all domains consisting of relevant individuals.
[2]: P 1 0*P 2 only if P 1 is a proper subset of P 2 in the union of all domains consisting of relevant individuals (or in the class of all relevant individuals). First, let me note that we need an ''iff'' to replace 0* by another notion. Second, let me note that the above ''only if'' do not hold in all cases. To show this I present four cases. Both [1] and [2] don't hold in the first three cases; [1] also doesn't hold in the fourth case. In each case, I present two properties, P 1 and P 2 , choose (prima-facie) reasonable relevant individuals, and identify a specific domain D of such individuals, where P 1 0*P 2 yet P 1 is not a proper subset of P 2 in D / the class of all relevant individuals. I use ''m'', ''s'', ''f'', ''d'' (sometimes with subscripts) to indicate a mathematical individual, a star, a frog, and a dog.
Case 1: P 1 =IS-A-(1ST-LEVEL-UNARY)PHYSICAL-PROPERTY, P 2 =IS-A-(1ST -LEVEL-UNARY-) LOGICAL-PROPERTY. Relevant individuals: any actual-counterfactual individual. D={s,m}. In this case we have two 2nd-level properties, P 1 and P 2. P 1 0*P 2 . Explanation: There are exactly two logical 1st-level unary properties (properties satisfying P 2 )-the property that does not hold of any actualcounterfactual individual (we can refer to it as ''x=x'') and the property that holds of every actualcounterfactual individual (''x=x''). Every r indexed to any \D 1 ,D 2 [ takes (x=x) D1 to (x=x) D2 and (x=x) D1 to (x=x) D2 and it takes every P D1 that is different from both (x=x) D1 and (x=x) D1 to a P' D2 that is different from both (x=x) D2 and (x=x) D2 . So P 2 is maximally-invariant. But P 1 is not maximally invariant, because for some r's, the induced r*'s take 1st-level unary physical properties to 1st-level unary mathematical, hence non-physical, properties. Now, for [1], let's go to D above. There, P 1 ={{s}}, P 2 ={;,{s,m}}; therefore, P 1 is not a proper subset of P 2 in D. For [2], the intersection of logical properties and physical properties in the class of all actual-counterfactual individuals is empty, hence, P 1 is not a proper subset of P 2 in this class.
Case 2: P 1 =is-a-star, P 2 =is-not-self-identical (x=x). Relevant individuals: any actual-counterfactual individual. D={s,f}. In this case we have two 1st-level properties, one (P 1 ) non-logical, the other (P 2 ) logical. As such, P 1 0*P 2 . But since P 2 is empty in D / the class of all actual-counterfactual individuals and P 1 is not, P 1 is not a proper subset of P 2 in D / the class of all actual-counterfactual individuals.
Case 3: P 1 =is-a-frog, P 2 =is-immortal. Relevant individuals: any actual-counterfactual biological individual (assuming no actual-counterfactual biological individual is immortal). D={d,f}. This case is similar to Case 2, but P 2 is not a logical property.
Case 4: P 1 =is-a-star, P 2 =is-subject-to-gravity. Relevant individuals: any actual-counterfactual physical individual. D={s 1 ,s 2 }. Here we have two non-empty 1st-level properties. Given that only physical individuals are relevant, P 2 is invariant under all r's indexed to domains of relevant individuals, but P 1 is not. P 1 , however, is universal in the particular D we are considering. Hence it is not a proper subset of P 2 in that D.
While the above possibilities won't do as they stand, in principle, as I noted above, there could be a systematic connection between 1* and some simpler notion that could replace it. Due to limitations of space and for the sake of accessibility, I leave further discussion of this possibility for another paper. I should emphasize, though, that what this paper aims to show is that it's possible to explain significant aspects of necessity and laws based on the simple observation that properties are selective in character, that such an explanation can be systematized (in one way or another) using invariance, and that this reduces the mystery that is often attributed to necessity and laws. Whether it's possible to explain this in other terms, not involving invariance, is left open in this paper.
I'd like to thank the reviewer for a comment that led to this clarification.
invariant, but (given our current state of knowledge) is-subject-to-gravity is also physically maximally-invariant. Is-a-star, however, is not physically maximallyinvariant. Is-a-biological-individual is biologically maximally-invariant, but neither is-a-mammal nor IS-A-HEREDITARY-PROPERTY are. It's useful to introduce the notion ''close to being physically/biologically maximally-invariant'' as well. I won't undertake the task of defining this notion here, but an example of a physical property which (depending on how we demarcate it) is either physically maximally-invariant or close to being physically maximallyinvariant is has-a-mass.
We are now ready to apply Thesis 3, which connects invariance and necessity, to the natural sciences. The greater the invariance of a given property, the more replacements (''more'' in the sense of inclusion) of actual-counterfactual individuals it doesn't ''notice''. And if a property doesn't notice replacements of certain individuals, the laws/regularities governing/ describing it cannot notice them either. (If they do, they are not laws of this property.) If P doesn't distinguish between a and b, its laws (if any) cannot distinguish between them either. If they hold of one, they hold of both. In particular, principles/laws governing/describing physically maximally-invariant properties don't distinguish between a great many actualcounterfactual individuals, and so do their laws/regularities. They hold in the entire class of physically actual-counterfactual individuals. This means that the actualcounterfactual scope of the laws/regularities governing physically maximallyinvariant properties is very large. It's so large as to render these laws/regularities physically-highly-necessary. And the laws/regularities governing/describing properties that are close to being physically maximally-invariant render the laws/ regularities of many of these physically-fairly-highly-necessary. 32 An example of a principle/law that, if true, is highly or at least fairly highly necessary is Newton's law of gravity. This law, 33 connects the gravitational force between two bodies to their masses and the distance between the centers of their masses. In our terminology, Newton's law connects three properties that are at least close to being physically-maximal, and as such has at least a fairly high degree of necessity. Indeed, given that the properties in question have a higher degree of necessity than almost all physical properties, including both most physical properties ''proper'' and all natural properties outside physics proper, the degree of necessity of Newton's law of gravity is indeed very high-as high as we'd require the necessity of laws of nature to be. Applying Thesis 4 to the natural sciences, we have: DI(Physics)1*DI(Biology), 32 Assuming all the predicates appearing in a canonical formulation of these principles/laws denote properties that are at least close to being physically maximally-invariant. 33 Where: F-gravitational force between two bodies; G-gravitational constant; m 1 ,m 2 -masses of the two bodies; r-distance between centers-of-mass of these bodies.

A. Invariance and Metaphysics
Two questions concerning invariance and metaphysics naturally arise: (i) Is invariance theory a metaphysical theory? (ii) What is the relation between metaphysical invariance/necessity and other types of invariance/necessity, e.g., logical and physical invariance/necessity? (i) Invariance theory, as it is conceived here, does not belong to a single branch of philosophy. It belongs to a cluster of philosophical fields (subfields) which includes, in addition to metaphysics, also epistemology, philosophy of logic, philosophy of science, and possibly others. (ii) From the point of view of invariance, metaphysics is a highly heterogeneous field of knowledge, since it deals with heterogeneous topics, from the topic of objects simpliciter, associated with the maximally-invariant property is-an-object(-simpliciter), to such topics as causality and free will, which, in spite of the fact that they're associated with properties of far lower degrees of invariance, stand on a par with the former topic within metaphysics. It's thus difficult to assign a degree of invariance/necessity to metaphysics as a whole. It's tempting to place metaphysical properties/principles in the interval between formal properties/principles and physical properties/principles. But in fact this is a complex issue that requires a separate inquiry.
B. Invariance and Science 1. Invariance, Necessity, Natural Laws, and Humeanism (a) Humeans believe that the idea of laws of nature is mysterious or otherworldly. We can conjure up laws in our minds, but in nature, there is just a mosaic of particular physical objects, each with its own properties. The connection between invariance and necessity delineated in this paper shows, however, that if the natural mosaic itself is not mysterious, the idea of necessary principles governing/describing the behavior of some properties in this mosaic is not mysterious either. Given that (i) every property, including properties acceptable to Humeans, is invariant under some 1-1 and onto replacements of individuals, that (ii) some properties, including ones acceptable to Humeans, have a fairly high degree of invariance, that (iii) high degree of invariance is connected to, and explains, the necessity of principles/laws governing/describing highly-invariant properties, and that (iv) none of (i)-(iii) is mysterious, it's neither surprising nor mysterious that there are necessity-wise admissible candidates for necessary laws. The claim that, due to their necessity, to accept natural laws is to commit oneself to something mysterious or otherworldly, is unfounded. To the extent that individuals in the world have properties and that some significant properties have a high degree of invariance, there is an infrastructure in nature itself for necessary laws. What do I say to a Humean who accepts the claim that some physical properties have a high degree of invariance (though not maximal invariance) yet refuses to accept the consequence that natural laws associated with such properties can in principle be significantly necessary, on the ground that they do not hold of all formally (or logically) possible objects? My response has two parts: (i) The Humean of the last paragraph seems to recognize only maximal-logical-necessity as ''real'' necessity. In so doing, he discards Thesis 3 of this paper, which connects degrees of invariance to degrees of necessity on all levels, not just the maximal level, without offering any criticism, let alone refutation, of this thesis. (ii) The Humean seems to assume a rigid binary bifurcation which leaves room only for two modal states: logical-necessity and contingency. I am wary of rigid binary bifurcations of this kind, which include, in addition to the necessary-contingent bifurcation, also the analytic-synthetic and the apriori-aposteriori bifurcations. These bifurcations lump together everything that does not belong to one extreme case (the purely analytic, purely apriori, logically necessary), losing sight of theoretically significant differences between elements that are thus lumped together. In the case of necessity and contingency, our Humean loses sight of the considerable difference in modal force between such extremely modally weak facts as the fact that I am wearing a black sweater today and modally far stronger facts, such as the fact that I, along with all material bodies, am subject to gravity. In contrast, the invariantist offers a non-mysterious way to recognize, make sense of, and introduce structure into, such differences. (iii) In saying that laws of nature are not necessary since they don't hold of all logicallypossible objects, the humean doesn't realize how deeply different formally-but-not-physically-possible objects are from objects that are reasonably viewed as possible in scientific contexts. Think, for example, of an object that is both entirely red and yellow, of a person who is both dead and has a beating heart, a functioning brain, and so on, of a particle which is both sub-atomic and six feet long, of a body which is both a moon of Earth and a black hole, and so on. Logic doesn't distinguish between such physically/biologically impossible objects and physically/biologically possible objects. But the natural sciences do and should. The logical possibility of such objects isn't relevant to science. Natural necessity isn't applicability to such objects. Natural necessity is applicability to non-actual objects that are physically or biologically possible, such as a second moon of Earth or a species of humanoids with DNA instructions for six rather than five fingers in each hand. Physical necessity is, and ought to be recognized as, different from logical necessity, and one should not reject it on the ground that it doesn't satisfy the standards of logical necessity. (b) While the connection between invariance and the necessity of laws captures something basic and significant about laws, it's important to recognize its limits. First, this connection shows that the world is ''ready'', in certain significant ways, or has an appropriate ''infrastructure'' in place, for necessary physical laws, but whether there are, in fact, such laws is not determined by this connection. The fact that there's an infrastructure for necessary physical laws suggests that, and explains why, searching for laws of nature is reasonable. But it neither guarantees that scientists will find (or have found) such laws nor determines what these laws are or with which highly-invariant properties they're associated. Second, necessity may not be limited to principles governing highlyinvariant properties. If, and to the extent that, there are other types of necessity, the present account doesn't explain them. For example, if, and to extent that, some laws obtain their status by (mere) stipulation, their necessity is not explained by our account. If, and to the extent that, there are singular laws-laws describing singular features of nature-their necessity may not be explained by the present account.
Here the main question is whether the singular feature described by a putative law is an isolated feature or is related to features of highlyinvariant properties. This question arises in the case of the principle/ law stating that nothing moves faster than light, for example. What about necessary principles/laws of the kind ''Given the way the world actually is, X must be the case in the world''? Here the necessity is due to the connection between the way the world is and X. If the connection is formal-logical or mathematical-then its necessity is explained by our account. If it's based on a natural principle/law that governs/describes certain highly-invariant properties, its necessity is also explained by our account. But if it's based on something altogether different, it wouldn't be explained by our account.
Another limit of our account is that it doesn't distinguish between necessary principles and bona-fide laws. In this respect our account follows the common practice in logic, where all logical principles/ truths/consequences are on a par. For various reasons, some scientists and philosophers of science do distinguish between the two. In determining which principles should be presented as laws, they appeal to many considerations, only one of which is modal force. The additional considerations may be pragmatic, methodological, or based on scientific inquiry. They may focus on nature itself (''laws of nature'') or on what our theories should present as laws (''laws of theories of nature'').
A few examples of considerations other than modal force are projectability considerations (Goodman, 1955/65), naturalness considerations (Lewis, 1983), and invariance considerations that relate to other types of invariance than the one discussed here (Lange, 2000;Woodward, 2018). But at least some of the worries underlying these considerations, such as the worries about ''artificial'' properties, are irrelevant to the invariantist, who is concerned only with necessity-wise admissible candidates for laws. (c) The connection between necessity and property-invariance implies, as we've seen above, that necessity and possibility (of the kinds we focus on here) come in degrees. The higher the degree of invariance of a given property is, the higher the degree of necessity of laws governing/describing it. The degree of necessity of logical laws is higher than that available to physical laws, and the degree of necessity available to physical laws is higher than that available to biological laws/principles, such as evolutionary principles. This explains why survival of the fittest, for example, may be evolutionarily necessary, but is neither logically nor physically necessary. (d) The introduction of types or ''spaces'' of possibility enables the invariantist to account for the resolution of issues requiring a differentiation between necessity and accidental generality in a simple and systematic manner. Consider the widely discussed difference between the non-existence on Earth of spheres of gold one mile in diameter and the non-existence on Earth of spheres of uranium one mile in diameter. The former is a matter of the limited amount of gold on Earth, the latter of the inner structure of uranium (regardless of its location). But if there is just one type of possibility, logical possibility, there is no difference between the two. Logically, the non-existence-on-Earth of both is accidental. But as soon as we distinguish between physical and logical possibility, the explanation is simple. Whereas in the space of logical possibilities neither is-auranium-sphere nor is-a-gold-sphere is invariant under all 1-1 replacements of small-diameter spheres by large-diameter spheres, this is not the case in the space of physically-possible uranium and gold spheres. Here, no such replacement preserves the property is-auranium-sphere, but some such replacements preserve is-a-goldsphere.
Humeans have always recognized differences between outwardly similar phenomena, such as the absence of large gold spheres on Earth and the absence of large uranium spheres on Earth. But our account goes a step further by systematically explaining differences of this kind in terms of invariance, and through it, physical possibility.

Abstraction
The connection between invariance and laws relieves another worry about laws of nature, namely, their abstract character. The worry is that what the laws abstract from is just what is real. All there is in nature are particular things, and the abstractness of laws violates this particularity. To abstract is to falsify. What is abstraction? Abstraction is often characterized in a way that is similar to our characterization of selectivity and invariance.
To abstract is to ''leav[e] things out'', to ''ignor[e] things'' (Godfrey-Smith, 2009: 47). Those who associate abstraction with falsehood think that the only source of abstraction is the human mind. There is no abstraction in nature. It's we who abstract from natural objects when we think of nature in terms of laws. And in so doing we distort nature. Contrary to this view, invariance theory shows that it isn't we (or not just or primarily we) who are the source of abstraction. Nature itself is its source. Natural properties are selective in nature. They have a non-trivial degree of invariance. And to have a non-trivial degree of invariance-to be selective-is to abstract from some features of natural objects, i.e., to abstract from the particularity of natural objects. But if natural properties are abstract in nature, then abstraction is, in principle, true to nature. Abstraction resides in the world itself. What we do is, primarily, bring it into view. Natural properties such as is-a-star, is-an-electron, emits-light, is-subject-to-gravity, is-subject-to-evolution, IS-A-HEREDITARY-PROPERTY, IS-AN-ASTRONOMICAL-PROPERTY, differ from each other in what they pay attention to and what they overlook or abstract from. Some abstract from more features of objects than others (have a higher degree of invariance than others). This is a factual, objective, matter; not a human fabrication. If the truth of the matter is that a particular object has a property with a high degree of invariance, then it is neglecting this fact, rather than bringing it to light, that leads us to stray from truth. Since abstraction is built into nature, the possibility of abstract laws is built into it as well. 34 3. Scientific Realism Having a high degree of invariance is, as we've noted, an objectual or worldly feature of properties. As a result, the necessity of principles/laws governing/describing highly-invariant properties is also primarily worldly (rather than epistemic, conceptual, mental, or linguistic). Worldly necessity is sometimes called ''metaphysical necessity'', but ''metaphysical'' here is ambiguous. It can be understood as referring to worldly necessity as opposed to non-worldly (e.g., linguistic) necessity. But it can also be understood as referring to a type of necessity that is different from other types of necessity, say logical, physical, and/or biological necessity. (See discussion of metaphysics above.) For that reason I prefer the expression ''worldly necessity''. The worldly character of natural necessity is one of the building blocks of scientific realism. Psillos (2014) says that the reality of laws is based on the reality of patterns and regularities. The idea is that ''there is a network of natural patterns in nature'' (ibid.: 9), these natural patterns are the basis for natural regularities, and these worldly regularities are, in turn, the basis for natural laws. Invariance theory adds another layer to Psillos's conception of scientific realism. The high degree of invariance of some natural properties is the basis for the necessity of principles, and eventually laws, describing regularities in the patterns they exhibit.