Behavioral Mereology: A Modal Logic for Passing Constraints
DDavid I. Spivak and Jamie Vicary (Eds.):Applied Category Theory 2020 (ACT2020)EPTCS 333, 2021, pp. 276–288, doi:10.4204/EPTCS.333.19
Behavioral Mereology: A Modal Logic for PassingConstraints
Brendan Fong, David Jaz Myers, David I. Spivak
Mereology is the study of parts and the relationships that hold between them. We introduce a be-havioral approach to mereology, in which systems and their parts are known only by the types ofbehavior they can exhibit. Our discussion is formally topos-theoretic, and agnostic to the topos,providing maximal generality; however, by using only its internal logic we can hide the details andreaders may assume a completely elementary set-theoretic discussion. We consider the relationshipbetween various parts of a whole in terms of how behavioral constraints are passed between them,and give an inter-modal logic that generalizes the usual alethic modalities in the setting of symmetricaccessibility.
Many thinkers, from Heidegger to Isham and Döring have asked “What is a thing?” [Hei68; DI10].Heidegger for example says,From the range of the basic questions of metaphysics we shall here ask this one question:What is a thing? The question is quite old. What remains ever new about it is merely that itmust be asked again and again.In this article, our way of asking about things is focused on the mereological aspect of things, i.e. therelationship between parts and wholes. The point of departure is that, at the very least, a part affects awhole: “when you pull on a part, the rest comes with.” For example, wherever my left hand is, my righthand is not far away. A whole then, has the property that it coordinates constraints—or said anotherway, it enables constraints to be passed —between parts. In this article, we present a logic for constraintpassing.Our approach has roots in categorical logic, and in particular Lawvere’s observation that existentialand universal quantification can be characterized as adjoints to pullback, in any topos. In particular, asystem, or more evocatively a behavior type B S , will be a set which we imagine as the set of ways asystem can behave over time. If S is a dynamical system, then we may think of B S as the set of lawfultrajectories of this system. We are inspired here by Willem’s behavioral approach to control theory (see[Wil98; Wil07; WP13]).We work behavior-theoretically; to paraphrase Gump, “X is as X does”. We associate to a system S its set B S of possible behaviors — B S is the behavior type of S . If P is a part of our system S , then if weknow the total behavior of S we also know the behavior of P ; so, we have a function | P : B S → B P whichwe think of as “restricting” the behavior b ∈ B S of S to the behavior b | P ∈ B P of P . However, we are More generally, we may take a behavior type to be an object in any topos [MM92]. This allows behavior types whichwhere the behaviors may vary in time (or space). Since we don’t expect our audience to know any topos theory, and since allthe ideas we describe will make sense in any topos, we will use the language of sets through this paper, and leave experts tomake the topos-theoretic translation themselves. . Fong, D. Jaz Myers & D.I. Spivak
P as a part of S , so every behavior of P must come from some behavior of the whole system S ; so, the restriction map (cid:12)(cid:12) P : B S → B P must be surjective.We use this analysis of parthood to define a part of the system S to be a quotient of B S , i.e. a sur-jection B S (cid:16) B P from the behavior type. This surjection describes how a behavior of the entire systemdetermines a behavior of the part, and any behavior of the part qua part must extend to the behavior ofthe whole: for any behavior of my hand, there exists at least one compatible behavior of my whole body.Given a behavior on one part, we can consider all possible extensions to the whole, and subsequently askhow those extensions restrict to behaviors of other parts. In this way one part may constrain another.To describe the logic of these constraints, we introduce two new logical operators, or “inter-modalities”,closely related to the classical “it is possible that” and “it is necessary that” modalities, known as the alethic modalities [Kri63]. We view a constraint φ on a part P as a predicate on the behaviors of P — thepredicate “satisfies the constraint φ ”. We may ask whether satisfying this predicate allows, or whetherit ensures, various behaviors on another part B Q : the constraint φ is passed in these two ways from P to Q . To be a bit more explicit, the first new operator is called allows , written ♦ PQ φ . This describes the setof behaviors in B Q for which there exists an extension in B S that restricts to some behavior satisfying φ .The second is the operator ensures , written (cid:3) PQ φ . This describes the set of behaviors in B Q for which allextensions to B S restrict to some behavior satisfying φ in B P . Our goal in this paper is to describe theproperties of these inter-modalities (“inter” because they go from one part to another), and to demonstratetheir utility with some elementary examples.Our inter-modalities allow us to faithfully express concepts of the behavioral approach to controltheory as expressed in Willems’ Open Dynamical Systems and their Control [Wil98]. In particular:• A time-invariant system S is controllable if and only if for any two behaviors b and b , there is atime delay D such that the behavior b | < restricted to time before 0 and b | > D restricted to timeafter D are compatible in the sense of Definition 4.• If b is a behavior of a part P of S and b a behavior of part Q of S , then b is observable from b if and only if b determines b in the sense of Definition 9.• If C is the constraint a controller P places on the behavior of a plant Q , then the controlled behavior is the constraint ♦ PQ C of behaviors of Q which are allowed by C in the sense of Definition 14.• The control problem is the problem of choosing a constraint C on P so that a constraint φ of theplant Q is satisfied. The universal solution to this problem is given by the constraint (cid:3) QP φ ofbehaviors on P which ensure that Q behaves according to φ , in the sense of Definition 14.We believe the logic for constraint passing presented in this paper can be a useful tool for formalizingarguments in the behavioral approach to control theory. When we constrain a part of a system, we are constraining what it does . This suggests that we shouldmodel a system by its type of possible behaviors , its behavior type . Luckily, we won’t need to settle on what precisely a behavior type is, so long as we can reason about itlogically. For this, we need the behavior type of our system and its parts to be objects in a topos; then,78
Behavioral Mereology we can use the internal logic of the topos to reason about our behavior types. A topos can be understoodas a system of variable sets ; in our case, this allows us the freedom to have sets varying in time, in space,or according to different observers. We will just work in the topos of sets and functions, leaving it to theexperts to extend the theory to arbitrary toposes.So, a behavior type is simply a set whose elements are regarded as possible behaviors of a system.We can think of it as the “phase space” of our system, in a general sense. This terminology is inspired byWillem’s behavioral approach to control theory [WP13], which describes a dynamical system as a subset B ⊆ W T of lawful trajectories parameterized by time T in some value space W . The set B is the behaviortype of this system, where a behavior of the system is simply a lawful trajectory. Many of our examplesfollow this general form. Example 1.
We will present a few running examples of systems considered in terms of their behaviortypes. Let’s introduce them now.• Consider a bicycle. The bicycle pedals might be moving at some speed p , and the bicycle wheelsmight be moving at some speed w , both real numbers. If the pedal is pushing at a certain speed,then the wheels are moving at a least a constant multiple of that speed. Therefore, we will take thebehavior type of our bicycle to be B Bicycle : = { ( p , w ) ∈ R × R | w ≥ r p } for some fixed ratio r ∈ R .• Consider a glass of water placed in a room of temperature R . The glass of water has temperature T t ∈ R for every time t ∈ N . By Newton’s principle, the temperature of the water satisfies thefollowing simple recurrence relation: T t + = T t + k ( R − T t ) . Therefore, the behavior type of this glass of water is B Water : = { T ∈ N → R | T t + = T t + k ( R − T t ) } . • Consider an ecosystem consisting of foxes and rabbits. At any given time t ∈ N , there are f t foxesand r t rabbits, where we mask our uncertainty about the precise population by allowing these tobe arbitrary real values, rather than integer values. The population of the species at time t + t , according to the relation R t ( f , r ) given by the followingstandard recurrences: f t + = ( − d f ) f t + c f r t f t , and r t + = ( + b r ) r t − c r r t f t . This is known as the Lotka–Volterra predator–prey model, but we could use any model. Here d f is the death rate of the foxes, b r is the birth rate of rabbits, and c f and c r are rates at whichthe consumption of rabbits by foxes affect their respective populations. From here, we take thebehavior type of this ecosystem to be B Eco : = { ( f , r ) : N → R × R | ∀ t . R t ( f , r ) } . . Fong, D. Jaz Myers & D.I. Spivak If we know a whole system S (say, my body) is behaving like b , then we also know how any part P of S (say, my hand) is behaving: we just look at what P is doing while S does b . In other words, there shouldbe a restriction function, which we denote (cid:12)(cid:12) P : B S → B P , from the behavior type of the whole system tothe behavior type of the part.Moreover, every behavior of a part P will arise from some behavior of the whole system: how coulda part of the system do something if the system as a whole had no behaviors in which P was doing thatthing? Remember, we are considering the part P as a part of the system S , not on its own: my hand, nota severed hand. If we sever P from the system S , it may be able to behave in ways that have no extensionto S . But as a part of the system S , every behavior of the P must be restricted from a behavior of S . Wewill give examples below, but first a definition. Definition 2.
The behavior type of a part of a system S is a surjection (cid:12)(cid:12) P : B S → B P out of B S which wecall the “restriction from S to P ”. We define the category of parts of B S to have as objects the parts of S and as morphisms the commuting triangles B S B P B Q | P | Q | Q Note that if such a map B P → B Q exists, then it will be unique and a surjection. If there is such a map,we write P ≥ Q and say that Q is a part of P . This gives a preorder on parts, which we refer to as the lattice of parts of S . For example, suppose that a system S is divided into a plant P and a controller C . For example, B S might be { α : R → R p + c | L ( α ) } a set of p + c real variables satisfying a dynamical law f , thefirst p of which concern the plant P and the last c of which concern the controller C . Then B P wouldbe { ρ : R → R p | ∃ γ : R → R c . L ( ρ , γ ) } of p real variables for which there is some extension of c variables (the behavior of the controller) which is valid according to the dynamical law. The projectionmap R p + c → R p gives a surjection from B S (cid:17) B P , witnessing that the plant is a part of the whole system.In practice, we may have certain parts of S in mind, and so we may consider a sublattice of thatdefined in Definition 2. Remark 1.
Definition 2 may look a little backwards. Usually a “part” is a subset; here we have defineda part to be a quotient . What we have defined to be a part is often called a partition (of B S ).What is happening here is a well-known “space/function” duality: we are not considering the system S as some sort of object in space, but rather its type of behaviors B S . Often, the behaviors B S of a system S may be realized as functions on some sort of space S ; this gives us a contravariance in S , which we seein the definition of part Q is part of P if there is a map “going the other way,” B P → B Q . Example 3.
Here are some examples of parts.• The two parts of the bicycle under consideration are the pedal and the wheel. Explicitly, thebehavior types of these parts of the bicycle are the types of all possible behaviors which arise as We will see in Section 2.4 that it is indeed a lattice. Behavioral Mereology some behavior of the whole bicycle: B Pedal : = { p ∈ R | ∃ w . ( p , w ) ∈ B Bicycle } , B Wheel : = { w ∈ R | ∃ p . ( p , w ) ∈ B Bicycle } . In this case, every real number is a possible speed of the pedal, and every real number a possiblespeed of the wheel.• In the system
Water of the cup of water sitting in the room, there is just one thing we are consid-ering the behavior of: the cup. But, we can see this behavior at many different times. For everytime t ∈ N , we get a part Water t of the cup at time t with behaviors B Water i : = { x ∈ R | ∃ T ∈ B Water . T t = x } . In fact, for any set D ⊆ N of times, we get the behavior type of the cup during D : B Water D : = { x ∈ D → R | ∃ T ∈ B Water . ∀ d ∈ D . T d = x d } . • The ecosystem consisting of foxes and rabbits is more complicated than the cup, but the principleis the same. We can consider the system at different times, and restrict our attention to just foxesor rabbits as we please. In particular, we let Fox t be the system of foxes at time t , and Rabbit t bethe system of rabbits at time t . These have behavior types B Fox t : = { x ∈ R | ∃ ( f , r ) ∈ B Eco . f t = x } , B Rabbit t : = { x ∈ R | ∃ ( f , r ) ∈ B Eco . r t = x } . A surjection (cid:12)(cid:12) P : B S → B P out of a set may equally be presented by its kernel pair, the equivalencerelation on B S where b ∼ P b (cid:48) iff b (cid:12)(cid:12) P = b (cid:48) (cid:12)(cid:12) P . We may call this relation observational equivalence ; thebehaviors b , b (cid:48) with b ∼ P b (cid:48) are observationally equivalent relative to P . This is clearest when thinkingof P as some measuring device in a larger system; two behaviors of the whole system are observationallyequivalent relative to our measuring device when it measures them to be the same. Two behaviors ofmy body are hand-equivalent if they are indistinguishable by looking at my hand; and two times areclock-equivalent when they read the same on the clock face.There is essentially (i.e. up to isomorphism) no distinciton between a quotient of B S and an equiva-lence relation on B S . Thus we are defining parts of S to be equivalence relations on S -behaviors. Thisseems to be a novel approach to mereology, though we cannot claim to know the literature well enoughto be sure. Now we turn our attention to how behaviors of various parts of the system relate to one another. The mostbasic relation between behaviors of two parts is that of being simultaneously realizable by a behavior ofthe whole system. We call this relation compatibility . Definition 4. If P and Q are parts of S , then we say behaviors a ∈ B P and b ∈ B Q are compatible , denoted c ( a , b ) , if there is a behavior s ∈ B S of the whole system that restricts to both a and b , i.e. c ( a , b ) : ≡ ∃ s ∈ S . a = s (cid:12)(cid:12) P ∧ s (cid:12)(cid:12) Q = b . Generally, if a i ∈ P i is some family of behaviors indexed by a set I , then this family is said to be compatible if there is an s ∈ S such that s (cid:12)(cid:12) P i = a i for all i ∈ I . . Fong, D. Jaz Myers & D.I. Spivak Example 5.
Examples of compatible behaviors are easily obtained by restricting a single system behav-ior to two parts.• In the bicycle example, we see that a speed p of the pedal is compatible with a speed w of thewheel if and only if w ≥ r p : c ( p , w ) = w ≥ r p . • In the cup of water example, a temperature T ∈ B Water t at time t is compatible with a temperature T ∈ B Water t (cid:48) at a later time t (cid:48) are compatible if and only if T follows from T via the recurrencerelation. In particular, if t (cid:48) = t +
1, then c ( T , T ) = ( T = T + k ( R − T )) . • In the ecosystem example, we have a number a different comparisons to choose from. A foxpopulation f ∈ B Fox t at time t is compatible with f ∈ B Fox t + at time t + r so that f = ( − d f ) f + c f r f .Two simultaneous fox and rabbit populations are compatible if and only if there is some history ofthe ecosystem which achieves those population at that time. In particular, any two populations offoxes and rabbits at time 0 are compatible. We can express some of the parts-lattice operations in terms of the compatibility relation.
Proposition 6.
The meet P ∩ Q of parts P and Q of S has behavior type given by the following pushout. B S B P B Q B P ∩ Q (cid:113) In other words, a behavior of P ∩ Q is either a behavior of P or a behavior of Q , where these are consideredequal if they are compatible. B P ∩ Q ∼ = B P + B Q c . Here, B P + B Q is the disjoint union of these two sets, and we are quotienting out by the smallestequivalence relation for which p ∼ q whenever c ( p , q ) .Dually, the join P ∪ Q has behaviors given by the image factorization of the induced map B S → B P × B Q . B S B P B P ∪ Q B Q B P × B Q Behavioral Mereology
In other words, a behavior of P ∪ Q is a pair of compatible behaviors from P and from Q . B P ∪ Q ∼ = { ( a , b ) ∈ B P × B Q | c ( a , b ) } . Furthermore, the largest part (cid:62) is S , and the smallest part ⊥ is empty if S is empty and a singletonotherwise. Definition 7.
A part P is strongly disjoint from a part Q if every behavior of P is compatible with everybehavior of Q . The two parts P and Q are disjoint if their intersection P ∩ Q is the minimal part. Stronglydisjoint parts are disjoint: ∀ a ∈ B P . ∀ b ∈ B Q . c ( a , b ) ⇒ B P ∩ Q = ⊥ In general, we will be more interested in joins than in meets because joins are easier to work with(being subsets of a product, rather than quotients of a disjoint union by a non-transitive relation).
Example 8.
Let’s consider some examples of joins and meets of parts.• In the example of the bicycle, note that we have B Bicycle = B Pedal ∪ Wheel , since a behavior of the bicycle was defined precisely to be a behavior of a pedal and a wheelsatisfying a compatibility constraint.• In the example of the cup of water, the behaviors B Cup D over a duration D ⊂ N of times are theunion of the behaviors B Cup d for each time d ∈ D : B Cup D = B (cid:83) Cup d . • Similarly, in the ecosystem example, the parts of the ecosystem at various times are the join ofparts at particular times. More interestingly, recall that every behavior of Fox (starting populationof foxes) is compatible with every behavior of Rabbit (starting population of rabbits). Therefore, B Fox ∩ Rabbit = ⊥ This witnesses the fact that the parts
Fox and Rabbit do not at all mutually constrain each other,and so have no shared sub-parts. Definition 9. If P and Q are parts of S , and a ∈ B P and b ∈ B Q , then a determines b if every behavior s of the whole system S which restricts to a also restricts to b . d ( a , b ) : ≡ ∀ s ∈ S . s (cid:12)(cid:12) P = a ⇒ s (cid:12)(cid:12) Q = b . We say that a part
P determines a part Q if every behavior a ∈ B P , determines some behavior b ∈ B Q .This is a much stronger notion than compatibility, and we shall show in Section 11 that it can be usedto recover the original order relation ≥ on parts. . Fong, D. Jaz Myers & D.I. Spivak Lemma 10.
A behavior always determines uniquely: if d ( a , b ) and d ( a , b (cid:48) ) , then b = b (cid:48) . Proof.
We know there is some s ∈ S which restricts to a . Since a determines b and b (cid:48) , s restricts to both b and b (cid:48) ; but then b = b (cid:48) . Proposition 11.
For parts P and Q of S , the following are equivalent:1. Q is a part of P , i.e. there is a surjection B P (cid:16) B Q under B S .2. P c -determines Q , in the sense that for every a ∈ B P there is a unique b ∈ B Q such that a iscompatible with b . In other words, ∀ a ∈ B P . ∃ ! b ∈ B Q . c ( a , b ) .3. P d -determines Q , in the sense that for every a ∈ B P there is a b ∈ B Q such that a determines b . Inother words, ∀ a ∈ B P . ∃ b ∈ B Q . d ( a , b ) .4. For all a ∈ B P and b ∈ B Q , if a is compatible with b , then a determines b . In this section, we introduce our new logical operators, ♦ and (cid:3) , and prove some basic properties aboutthem. We shall see in Section 3.2 that these two operators pass constraints between parts. But first, whatis a constraint? We will identify a constraint φ on a part P with the predicate “satisfies φ ” on behaviors B P of P . In otherwords, we have the following definition.Let Prop be the two element set { true , false } of truth values. We think of functions φ : X → Prop as predicates concerning the elements of X — applied to x ∈ X , φ gives a truth value φ ( x ) which sayswhether or not x satisfies the predicate φ . Definition 12. A constraint on a part P is a map φ : B P → Prop . The type of constraints on P is Prop P .We write φ (cid:96) ψ to mean that φ entails ψ , that is, if φ ( b ) = true , then ψ ( b ) = true .For parts P ≥ Q , we get an adjoint triple that allows us to transform constraints on P to those on Q ,and vice versa, given by the logical quantifiers: Prop B P Prop B Q ∃ PQ ∀ PQ ∆ QP ⇒⇐ These functors are defined logically as follows: ∃ PQ φ ( q ) : = ∃ p ∈ B P . (cid:0) ( p (cid:12)(cid:12) Q = q ) ∧ φ ( p ) (cid:1) ∆ QP ψ ( p ) : = ψ ( p (cid:12)(cid:12) Q ) ∀ PQ φ ( q ) : = ∀ p ∈ B P . (cid:0) ( p (cid:12)(cid:12) Q = q ) ⇒ φ ( p ) (cid:1) Behavioral Mereology
The fact that they are adjoint means that ∃ PQ φ (cid:96) ψ ⇐⇒ φ (cid:96) ∆ QP ψ ∆ QP ψ (cid:96) ξ ⇐⇒ ψ (cid:96) ∀ PQ ξ We will write ∃ P , ∆ P , and ∀ P for ∃ SP , ∆ PS and ∀ SP respectively. As mentioned, these operations arefunctorial, meaning that ∃ PP ( φ ) = ∆ PP ( φ ) = ∀ PP ( φ ) = φ and for R ≤ Q ≤ P , ∃ QR ∃ PQ = ∃ PR , ∆ RQ ∆ QP = ∆ RP , ∀ QR ∀ PQ = ∀ PR . Lemma 13.
Recall that for part P of system S we write s ∼ P s (cid:48) for the relation s (cid:12)(cid:12) P = s (cid:48) (cid:12)(cid:12) P on B S . Thenfor any predicate φ on S we have:1. ∆ P ∃ P φ ( s ) = ∃ s (cid:48) . ( s ∼ P s (cid:48) ) ∧ φ ( s (cid:48) ) ∆ P ∀ P φ ( s ) = ∀ s (cid:48) . ( s ∼ P s (cid:48) ) ⇒ φ ( s (cid:48) ) φ (cid:96) ∆ P ∃ P φ and ∆ P ∀ P φ (cid:96) φ Thinking again of P as a way to observe behaviors, ∆ P ∃ P φ is the set of system behaviors s that ourobserver says plausibly satisfy φ : there is something P -equivalent to s that satisfies φ . And ∆ P ∀ P φ is theset of system behaviors that our observer can guarantee satisfy φ . Now we turn to the question of how constraints on the behavior of some part of the system constrain thebehavior of other parts. We discuss two ways to pass constraints between parts.
Definition 14.
A constraint φ on a part P induces a constraint on a part Q (of the same system S ) in twouniversal ways:• “Allows φ ”: ♦ PQ φ : = ∃ Q ∆ P φ ♦ PQ φ ( q ) = ∃ s ∈ B S . ( s (cid:12)(cid:12) Q = q ) ∧ φ ( s (cid:12)(cid:12) P )= ∃ p ∈ B P . c ( p , q ) ∧ φ ( p ) . • “Ensures φ ”: (cid:3) PQ φ : = ∀ Q ∆ P φ (cid:3) PQ φ ( q ) = ∀ s ∈ B S . ( s (cid:12)(cid:12) Q = q ) ⇒ φ ( s (cid:12)(cid:12) P )= ∀ p ∈ B p . c ( p , q ) ⇒ φ ( p ) . A behavior q of Q allows a constraint φ on P if Q can be doing q while P is satisfying φ ; we writethis as ♦ PQ φ ( q ) . A behavior q of Q ensures φ on P if whenever Q does q , P must satisfy φ ; we write thisas (cid:3) PQ φ ( q ) .These symbols are chosen due to their relation to the usual modalities of possibility ( ♦ ) and necessity( (cid:3) ) [Kri63]; a behavior q allows φ if it is possible that P satisfies φ while Q does q , and a behavior q . Fong, D. Jaz Myers & D.I. Spivak φ if it is necessary that P satisfies φ while Q does q . Indeed, in the case that the accessibilityrelation in the Kripke frame is an equivalence relation, we will be able to recover the usual possibilityand necessity modalities from our allowance and ensurance operators (see Section 3.3).Note that compatibility and determination appear as particular, pointwise cases of the allowance andensurance operators. For any p ∈ B P and q ∈ B Q , we have c ( p , q ) = ♦ PQ (= p )( q ) = ♦ QP (= q )( p ) d ( p , q ) = (cid:3) QP (= q )( p ) We write (= p ) for the map B P → Prop that sends p (cid:48) to true if and only if p = p (cid:48) . Example 15.
We return to our running examples to see our new operators in action.• In the example of the bicycle with gear ratio of r , we can ask what behavior of the pedal is ensuredby the wheels moving slower than w = (cid:3) PedalWheel ( ≤ ) is the constraint p ≤ r .• If the cup of water has temperature T ∈ Water at time 0, then it cannot have a temperature furtheraway from the ambient room temperature R at a later time. Therefore, | R − T t | > | R − T | (cid:96) ¬ ♦ Water Water t (= T )( T t ) . • Suppose that in the ecosystem example, one was given the goal of introducing a fox population attime 0 in order to keep the rabbit population in check after a given deadline d . Let’s say that beingkept in check means being between two fixed bounds, r t (cid:55)→ inCheck ( r t ) : = k < r t < k so that inCheck : B Rabbit t → Prop is a constraint on rabbits at time t . The constraint of being keptin check for all times after the deadline d is the constraint r (cid:55)→ ∀ t ≥ d . inCheck ( r t ) on the join (cid:87) t ≥ d Rabbit t . The goal may then be expressed as finding a starting fox population f which ensures that the rabbit population is kept in check at all times after the deadline: (cid:3) (cid:87) t ≥ d Rabbit t Fox ( ∀ t ≥ d . inCheck )( f ) . Example 16.
We can see a higher-order ensurance in the ecosystem example. If there are any rabbitsat time 0, and if the rabbit population is bounded independent of time, then the rabbits must ensure thatthere are foxes, and that the foxes ensure there are rabbits: r ≥ ∧ r < k (cid:96) (cid:3) FR ( f > ∧ (cid:3) RF ( r > )) . If there are no foxes, then the rabbit population is unbounded, and if there are foxes, then there mustbe rabbits for them to eat. We see that this ecosystem model exhibits a rudimentary form of symbiosis;though the foxes eat the rabbits, they counter-intuitively must ensure that the rabbits do not go extinct,lest they themselves go extinct.Assuming the law of excluded middle, our operators are inter-definable by conjugating with negation.86
Behavioral Mereology
Proposition 17.
Assuming Boolean logic, allowance and ensurance are de Morgan duals. That is, ¬ ♦ PQ ¬ = (cid:3) PQ . Proof.
The proof uses the law of excluded middle twice: ¬ ♦ PQ ¬ φ ( q ) = ¬∃ p . c ( p , q ) ∧ ¬ φ ( p )= ∀ p . ¬ ( c ( p , q ) ∧ ¬ φ ( p ))= ∀ p . ¬ c ( p , q ) ∨ ¬¬ φ ( p )= ∀ p . c ( p , q ) ⇒ φ ( p ) . Note that Proposition 17 does not generalize to arbitrary toposes, where the variation of the sets (intime or in space) means that one must reason constructively in general.
Finally, we describe the manner in which our intermodalities generalize the classical alethic modalitiesof possibility and necessity.
Proposition 18.
Let P and Q be parts of the system S and φ a constraint on P . Then:1. If a constraint φ entails ψ , then allowing φ entails allowing ψ , and ensuring φ entails ensuring φ .That is, ♦ PQ and (cid:3) PQ are monotone.2. If q ensures that P does φ , then q allows P doing φ . That is, (cid:3) PQ φ (cid:96) ♦ PQ φ .3. Allowing φ entails ψ if and only if φ entails ensuring ψ . That is, ♦ PQ is left adjoint to (cid:3) QP .4. ♦ PQ commutes with ∨ and ∃ , and (cid:3) PQ commutes with ∧ and ∀ .Fix a part P . Then for any part Q , we obtain two modalities on P by composing our intermodalitiesfrom P to Q with their corresponding intermodality from Q to P . Proposition 19.
The operators ♦ QP ♦ PQ and (cid:3) QP (cid:3) PQ are a pair of adjoint modalities on Prop B P . They arethe identity modality if and only if P ≤ Q . Proof.
By Proposition 18 item 3, these two modalities are the composites of adjoint pairs of operators,and hence are adjoint themselves. Moreover, ♦ QP ♦ PQ is the identity if and only if ♦ QP ♦ PQ φ (cid:96) φ for all φ ,which occurs if and only if ♦ PQ φ (cid:96) (cid:3) PQ φ for all φ , which occurs if and only if P ≤ Q .These modalities describe constraints on P as seen through the part Q . To obtain a description ofpossibility and necessity, assume that B S is inhabted — that there is some behavior of the system. Welet Q = ⊥ be the system whose behavior type B Q = ∗ consists of just a single element. Then the adjointmodalities ♦ ⊥ P ♦ P ⊥ left adjoint to (cid:3) ⊥ P (cid:3) P ⊥ describe possibility and necessity on B P .For example, for any constraint ϕ on B P , the constraint ♦ ⊥ P ♦ P ⊥ ( ϕ ) maps all elements of B P to true ifthere is some behavior p that satisfies ϕ , and maps all elements to false otherwise. Thus the modalitydetects whether ϕ is possible : that is, whether there is some behavior that satisfies ϕ . . Fong, D. Jaz Myers & D.I. Spivak (cid:3) ⊥ P (cid:3) P ⊥ ( ϕ ) is the constraint that maps all elements of B P to true if all behaviors p ∈ B P satisfy ϕ , and maps all elements to false otherwise; thus this modality detects whether ϕ isalways satisfied, and hence necessary .More generally, the usual semantics of the “it is possible that” and “it is necessary that” modalities ♦ and (cid:3) take place in a Kripke frame ( W , A ) , where W is a set, known as the set of worlds, and A isa binary relation on W known as the accessibility relation. The predicate then ♦ ( ϕ )( w ) holds if ϕ ( w (cid:48) ) for some w (cid:48) such that wAw (cid:48) , and (cid:3) ( ϕ )( w ) holds if ϕ ( w (cid:48) ) for all w (cid:48) such that wAw (cid:48) [Kri63]. If A is anequivalence relation, then we may equivalently describe A by an epi W (cid:16) W / A . In this case, we have ♦ = ♦ W / AW ♦ WW / A and (cid:3) = (cid:3) W / AW (cid:3) WW / A as modalities on Prop W . We have presented a logic that describes how constraints—restrictions on behavior—are passed fromone part of a system to another. While we have presented this from a set theoretic point of view, wehave taken care to use arguments that are valid in any topos (with the noted exception of Proposition17, which only holds in boolean toposes). As a consequence, our logic retains its character as a logic ofconstraint passing across a wide variety of semantics. One possibly valuable notion of semantics is onethat captures a notion of time.Indeed, behavior is best conceived as occurring over time, though of course the question of whattime is remains an issue. One can imagine that a system has, for any interval or “window” of time, a setof possible behaviors, and that each such behavior can be cropped or “clipped” to any smaller windowof time. This is the perspective of temporal type theory [SS18]. While that work uses topos theory in asignificant way, the main idea is easy enough.Whether we speak of a bicycle, an ecosystem, or anything else that could be said to exist in time, itis possible to consider the set of behaviors of that thing over an interval of time, say over the ten-minutewindow ( , ) . Above we often discussed an idea which can be generalized to any system S that existsin time. Namely, we can consider different parts of time as parts of S . Given any behavior s over the10-minute window, we can clip it to the first minute s (cid:12)(cid:12) ( , ) ; this gives a function S ( , ) → S ( , ) , whichis often called restriction , though we will continue to call it clipping. Let’s assume that every possiblebehavior at ( , ) extends to some behavior over the whole interval—i.e. that the universe doesn’t just endunder certain conditions on ( , ) -behavior—at which point we have declared that the clipping functionis surjective, and hence gives a part in the sense of section Definition 2. We call it a temporal part .What then does it mean to pass constraints between temporal parts? The idea begins to take on acontrol-theoretic flavor: behavioral constraints at one time window may allow or ensure constraints atother time windows. A mother could say “doing this now ensures no dessert tonight”. The child couldask “does our position on the road now allow me to play with Rutherford this afternoon?” A controlsystem could attempt to solve the problem “what values of parameter P can I choose, 5 seconds fromnow, that both allow current conditions and ensure that in 10 minutes we will achieve our target?”In any case, as mentioned in the introduction, our original goal was to understand what makes a thinga thing, e.g. what gives things like bricks the quality of being cohesive (not two bricks) and closed (notthe left half of a brick). We believe that a good logic of constraint passing between parts is essential forthat, but perhaps not sufficient. The question of what additional structures need to be added or consideredin order construct a viable notion of thing, remains future work.88 Behavioral Mereology
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