Implication as Inclusion and the Causal Asymmetry

: How does causation in the physical world relate to implication in logic? This article presents implication as fundamentally a relation of inclusion between propositions. Given this, it is argued that an event cannot “causally imply” another, also given the laws of nature. Then, by applying the notion of inclusion to physical objects, a relation “within the possibilities of” is developed, which generates a partial order on sets of entities and is independent of time. Based on this, it is shown that changes of physical objects in time (at any rate, a great many of them) imply, and thus counterfactually depend on, what we call “causes” – an asymmetric dependence which is robust despite the perspectival nature of the concept of “cause”.

incomplete -outline of the history of the problem of causation and implication in the last few centuries: Galileo, Hobbes, and Spinoza each thought that there is an effect if, and only if, there is a cause [see the quotes in Ingthorsson (2021), pp. 152-3], so that cause and effect, though distinct objects, are related by a biconditional.Hume's [(2000), VII, 2, 29] definition of causation reads: . . .we may define a cause to be an object followed by another, and where all the objects, similar to the first are followed by objects similar to the second.Or, in other words, where, if the first object had not been, the second never had existed.
Thus, Hume equates a notion whereby causes are sufficient for their effects (in the first sentence), with the notion that they are necessary for their effects, that is, with a counterfactual dependence of the effects on the causes (in the second), thereby simply confusing the two possible directions of the implication arrow [cf. the remark in Menzies and Beebee (2019), Section 1].In classical physics, a complete specification of a system at a given time t, i.e. of generalized coordinates q(t) and conjugate momenta p(t) of all particles in the system, together with the dynamic laws, uniquely determines -and hence, implies -the system's state for some other time t' [cf.e.g.Morrison (1990), pp. 3-5], where t' may be earlier or later than t.McTaggart (1915) is, to my knowledge, the philosopher who most clearly states that causation is a relation of implication from the cause to the effect (p. 326), and that the laws of nature are that which accounts for the connection between the two (p.335).A similar view is upheld in the classical deductive-nomological scheme of Hempel and Oppenheim (1948), where antecedent conditions in combination with laws entail a particular consequence.David Lewis, in his earlier work on causation (1973), carefully distinguishes between both directions of his strict implication operator "□→" (see e.g.p. 563), although somewhat oddly, he refers to statements of the form "A implies C" as relations of counterfactual dependence of C on A (p. 561 and pp. 563-4), rather than vice versa, a characterization reminiscent of Hume's confusion.Early Lewis, famously, defines the implication "□→" in terms of possible worlds (p.560), and then further specifies that it holds "in virtue of" propositions about laws, plus propositions about "particular fact" (p.563).Later counterfactual accounts of causation [as discussed e.g. in the contributions in Beebee, Hitchcock, Price (2017)] have developed and refined Lewis' account in ways to which I cannot do justice here, but broadly follow Lewis in that they explicate the causal relation in terms of implication (in one direction or the other), possible worlds, and laws.J. L. Mackie provides an original view of the relationship between implication and causation through his well-known " INUS" account (1965INUS" account ( , 1974)): here, conditions consisting of an aggregate of constituents are sufficient, but unnecessary, to bring about the effect, and "causes" are understood to be insufficient, but necessary -though later corrected to "non-redundant" (1974, p. 62) -parts of these sufficient conditions.Finally, for the purpose of this article it is important to highlight the work of Huw Price, who makes a strong case that the notion of an asymmetric, one-sided causal dependence between events is fundamentally perspectival, something deeply engrained in the worldview of us agents in spacetime with our inextricably temporal perspective, not a mind-independent feature of the world (e.g. 1996, 2007).
This brief overview illustrates the following challenges for a satisfactory account of the relationship between causation and implication: First, what differentiates causal implication, if it exists, from implication simpliciter?If the causal nexus comes with an implication arrow heading in one direction or the other (or both), it is clear that it cannot be equated to such an arrow, simply because implication operators -whatever their type -also connect causally unrelated propositions.Standard material implication, for example, holds between non-causal statements by which an object is placed into a given set in the antecedent, and into a superset in the consequent (e.g."Fred is a thrush" ⇒ "Fred is a bird", "r is real number" ⇒ "r is a complex number"), but also between some pairs of intuitively unrelated contingent statements (e.g."It is raining" ⇒ "Macron is president of France", an implication which holds at the time of writing, whether or not it really is raining).Strict implication, understood as the combination of material implication with the necessity operator, while it does not hold between pairs of unrelated contingent statements, does hold between intuitively unrelated statements whenever the antecedent is necessarily false (e.g."2+2 = 5" □⇒ "Macron is president of France").Non-logicians will point out that you cannot draw inferences about the president of France from the weather or from mathematical statements, and will protest -in my view rightly so -that such use of the term "implication", even if an established standard, is an abuse [cf.Anderson and Belnap (1975), pp. 17-18;Priest (2008), pp. 12-13].It has been seen that, very often, natural law and initial conditions were held to be that which accounts for the validity of causal implications [cf. the discussion of naturgesetzliche Implikation in Weizsäcker (2002), pp. 86-88], though this view has not gone completely unchallenged, having been criticized by Wittgenstein [(1995Wittgenstein [( ), 5.135-5.1361]],and Anscombe (1971).Lewis, too, thoroughly revised his counterfactual account in his later work (2000).
Second, if causal implication exists, which way does the arrow point?As shown, there has been confusion on this point, in that propositions of the form "cause (plus extra clauses) implies effect" are often misread as a counterfactual dependence of the effect on the cause.I call this a misreading because states of affairs depend counterfactually on what is necessary for them, not on what is sufficient.Being human depends counterfactually on being an animal (you cannot be human unless you are an animal), not vice versa.That aside, let the principal problems facing both possible directions be briefly outlined: The view whereby a cause, in some sense, implies an effect, faces the problem that, given any particular object taken to be a cause, its effect can be prevented.Conversely, the view whereby the absence of a given cause implies the absence of an effect faces the objection that this particular cause could, at least in many cases, be replaced by something else.
Third, the intuitive concept of an arrow of causation seems to be strongly linked to the temporal arrow, so that it is threatened by perspectivalism.Without going into detail, how strong this link is can be appreciated from the simple consideration that we typically associate causes with "sources", and effects with "receivers": the sun causes the earth to warm through its "outgoing" radiation, not vice-versa.This view of causation is perhaps most clearly enshrined in Suárez' [(1965), XII, 2, 4) definition causa est principium per se influens esse in aliud ("a cause is an origin which per se infuses being into something else").But reverse the arrow of timewind the film backwards -and the flow is reversed too, so that "receivers" and "sources" swap their roles.Hence, at least in this type of cases, the cause-effect distinction depends crucially on time's arrow, so that, if this arrow is perspectival, so is said distinction.
All these challenges are so severe that there seem to be good reasons to altogether give up on causal asymmetry, declare it to be an anthropocentric illusion, and dump it on the heap of other such illusions, like anisotropic space or global simultaneity.The consequences of such a move are hard to overstate, for it would mean that the very business of finding the causes on which phenomena dependa central motive of the scientific endeavour also today [cf.Illari and Russo (2014)]would turn out to be based on a naïve intuition.Alternatively, we could attempt to save causal asymmetry from this threat by declaring it to be something basic, not in need of further justification [cf.Weaver (2019), ch.3; a critique of causal apriorism can be found already in Geyser (1933), ch. 3].
I will go neither route, but instead propose a new derivation for counterfactual dependence in causation.To this end, after some brief preliminaries (Section 1), I will first examine the implication operator itself, and argue that it is based on a relation of inclusion between propositions (2).On this basis, I will discuss whether a notion of causal implication based on natural law is tenable (3).It will then be shown that there is indeed a relation of counterfactual dependence in causal interactions, and that, strikingly, this dependence is itself independent of the direction of time (4).

Preliminaries
Some notes on terminology before we move on: I will use collections, denoted by square brackets [. . .], rather than sets.Collections work like sets, with two important differences: 1.Only a collection of two or more entities is an entity in its own right, whereas a singleton collection is identical to its constituent, i.e. [x] = x, for any x. 2. There is a relation "included in"1 which, unlike the set-theoretical "element of", is transitive.Thus, given, say, the collection [y, z]], both x and [y, z] are included in M, as also are y and z, whereas [x, y] is not.In addition, given a collection C, a "sub-collection" will be a collection C ′ of objects which are included in C. For example, [x, y] is a sub-collection of M.
The notions of collection and inclusion can be extended, in particular, to propositions: By point (1), any atomic proposition is included in itself.In addition, any conjunction Q: = q 1 ∧ . . .∧ q n is a collection of propositions, which can therefore also be written as [q 1 , . . ., q n ].Also, the operators "∧", "∨", and "¬" will be understood to have their standard meanings.In particular, p ∨ q will be taken to be equivalent to ¬(¬p ∧ ¬q) and (p ∧ q) to ¬(¬p ∨ ¬q)."True" and "false" will also be used as basic, and mutually exclusive, attributes of propositions.
Finally, let x be a concrete particular, such as a physical object, or an event.Then, "x" will denote "x exists".This device will allow switching easily from the level of concrete particulars to that of propositions in order to use the tools of propositional logic.For example, "x implies y" is not a well-formed expression (concrete particulars cannot imply one another), but we can write "x implies y".

Implication as Inclusion
There is an obvious connection between inclusion and implication via the notion of sub-collections (or alternatively, subsets): for any sub-collection G of a collection F, and for any x, if x is in G, then it is in F. I will go beyond this and assume that implication just is a relation of inclusion between propositions.This leads to the fundamental principle: This extremely simple interpretation of implication is consonant with its etymological sense of "enfolding".I will use the symbol "→" for implication thus understood.The advantage of this interpretation is that no unrelated statements are connected by →, a motif shared by relevance logic [see Ferguson (2017), ch. 1;and again Anderson and Belnap (1975), pp.17-18], and connexive logic [see e.g.McCall (1966)].But doesn't FP limit implication to abstract relations of inclusion between propositions, thereby making it too strong to account for causal connections?Before addressing this point (in Sections 3 and 4), I will first spell out what this interpretation means, without however attempting a complete characterization of it.

Any atomic proposition implies only itself. Any conjunction
(where the q i 's are atomic propositions) will be taken to be a proposition only if it is non-contradictory, i.e. for all i and j, q i ≠ ¬q j , since I assume, for broadly Aristotelian reasons (cf., Metaphysics, 1005b(cf., Metaphysics, , 35 -1009a, 5, 5) that otherwise it is nonsensical.Any composite proposition of the form P = p 1 ∧ . . .∧ p n implies only and all of the 2 n − 1 conjunctions which can be built from the p i 's, i.e. each of the = n atomic propositions, each of the pairs of form p i ∧ p j (where i ≠ j), and so on, all the way up to the = 1 conjunction which is P itself.For example, "Anna knows that it is raining" → "it is raining", because "Anna knows that it is raining" is a composite proposition which includes three atomic ones: 1. that it really is raining; 2. that there is a mental representation of (1) in Anna; 3. that an epistemic condition is satisfied which qualifies this representation as knowledge, rather than belief or conjecture (a condition which is notoriously hard to specify and which gives headaches to epistemologists).
2. A disjunction has the form R: = r 1 ∨ . . .∨ r l .It implies that there is an i such that r i is true, and implies no more than this.R is true when this is satisfied, and false otherwise.Hence, disjunctions, unlike conjunctions, can include contradictory propositions r j = ¬r k without thereby themselves losing their status as propositions.3. A general proposition P is some combination of conjunctions and disjunctions, such as "q ∧ (s ∨ t)", "q ∨ (s ∧ t)", or such like.Each individual element q, s, t may itself be a conjunction or a disjunction, or it may be atomic.However, we can write any proposition as a conjunction of elements, i.e. in the general form P = p 1 ∧ . . .∧ p n .n may of course be 1, and this P = p 1 may be a pure disjunction (whose elements, once again, may be conjunctions or disjunctions).Thus, all possible combinations of ands and ors are subsumed in this general form.4. It is now straightforwardly seen that "→" has the following properties: -A truth never implies a falsehood.This is because a truth is a proposition -What is known as Boethius' thesis, whereby a proposition P cannot imply a proposition S as well as its negation ¬S, also holds.For once again, P is p 1 ∧ . . .∧ p n , and any S implied by it is p i or p j ∧ p k ∧ . . . .Suppose P also implied ¬S: ¬S is ¬p i or ¬p j ∨ ¬p k ∨ . . ., respectively.But each of these contradicts P, so that P would contradict itself.2 5. We can, in addition, add to our inventory a "blocking-off" operator |. . .| defined by the property that, for any p (whether atomic or composite) |p| means "only p is true", that is, p is then taken to represent the "totality of facts" [Wittgenstein (1995), 1.1] which obtain.Then, analytically (and trivially), |p| → ¬q for any q distinct from p, where "distinct from" means "neither identical with, nor included in". 6.Despite what has been said so far, we are very often justified in making claims of the form "p → q" also for q distinct from p.The reason is simply that we can assume the truth of further propositions constituting background knowledge that need not be stated explicitly, and which together with p imply q.We often make inferences of this type in everyday life and in science.For example, in the claim "you can't live on Mars, because there's no oxygen there", it is assumed as known that some living beings need oxygen, and that "you" is such a being.Thus, FP is not violated by claims of this type.7. The operator "→" admits addition: for any P and any q distinct from P, if p is a proposition included in P, P → p ∨ q.It has been objected that such explosion of the content of a proposition leads to any two propositions sharing some content [see e.g.Ferguson (2017), pp. 4-8;Gemes (1994); cf. also Stepanov (2004), p. 1].To this, I answer that addition must be admitted, simply because p ∨ q is of course ¬(¬ p∧¬q), which is satisfied given that P implies p. Nor does this lead to an undesirable explosion: only p ∨ q, but not q, is implied by P, so that, by point (2), we cannot deduce from the truth of P whether q is true or false.Thus, while the closure of P under "∨" explodes, the truth-values deducible from P are wellbehaved.The symbols "∨ q" therefore do not add content to the p implied by P any more than the addition of "+ 0" changes an algebraic expression (all that "∨ q" does is to produce in the hearer a mental representation of q's subject matter, but it asserts nothing whatsoever about that subject matter).But if there is no extra content, it is legitimate to view p ∨ q as included in P.
8. Let F be a variable denoting a real intrinsic property, I a collection containing indices for these properties written as i, and J a sub-collection of I whose members are written as j.Then, if S: = [x:∩ i F i x] and S ′ : = [y:∩ j F j y], for all z, the proposition ∩ i F i z, by which z is in S, includes the proposition ∩ j F j z, by which z is in S ′ .Thus, for example, "Fred is a thrush" includes "Fred is a bird", since "thrush" includes "bird" in its definition.In this way, a relation of inclusion between property collections translates into a relation of inclusion, and hence of implication, between propositions.Implication as inclusion therefore accommodates modus ponens, and also modus tollens: any w which doesn't have each F j required to be in S ′ (in this example, the collection of birds), also doesn't have each F i required to be in S (the thrushes).9. Implication as inclusion can also be applied to concrete particulars, such as objects or events, in the following ways: Suppose that some concrete particular V is by definition a collection of well-defined parts [v 1 , . . ., v n ].We can assert V's existence by writing the proposition V, which is [v 1 , . . ., v n ].Thus, for each v i , the proposition v i is also asserted, and V implies (in virtue of ( 1)) propositions such as v i , or v j ∧ v k ∧ . . . .Particulars with such well-defined parts are typical of our everyday, macroscopic world.The state vectors of quantum theory, on the other hand, have the general form Ψ = ∑ i c i  i , where the  i 's are eigenstates and the c i 's are complex probability amplitudes [see any textbook on quantum physics, e.g.Morrison (1990), p. 470].They can be read as pure disjunctions weighted by probabilities ("the system is in eigenstate  1 with probability mod(c 1 ) 2 , or . . .") and (in virtue of ( 2)) imply only that the system is in some eigenstate  k .Of course, the indices of the c's and the 's may also come from a continuous, rather than discrete, set.
Our implication operator "→" has some nice properties: It connects only related statements.It avoids the "paradoxes" which occur in material implication "⇒" and also in strict implication "□⇒", and which rightly strike non-logicians as absurd.
In this way, "→" is closer to the everyday meaning of implication.It could be objected, however, that all implications under "→" are tautological, and therefore uninteresting.I answer that the implications discussed so far are indeed tautological, in consonance with Wittgenstein's proposition [(1995), 5.133] that "all inference is done a priori".It will be seen in Section 4, however, that "→", when combined with the operator "|. . .|" also allows us to do more, namely to account for the counterfactual dependence of effects upon causes.

Is Sufficient Causation Possible?
According to sufficient causation, there are distinct concrete particulars c and econceived of as "cause" and "effect", respectively -such that c implies e, where "implies" refers generically to some type of implication.c and e may be events, or objects.Is sufficient causation tenable?The situation is, at first sight, ambiguous: On the one hand, cases of sufficient causation do seem to exist: a car close to the edge of a cliff and running towards it at high speed will certainly fall off; a rock which has fallen into the sun will certainly melt; or, to use the brutal, but paradigmatic, example of sufficient causation: beheading guarantees death [cf.McTaggart (1915), passim; Hume (2000), VIII, 1, 19].On the other hand, prevention of e can never, at least in principle, be ruled out until e is actually in place.In the case of prevention, we get c ∧ ¬e, which ruins the implication, whatever its type.Simple as this point is, it is in my view still often not enough appreciated how far-reaching it is [cf.Anscombe (1971), p. 147].For example, even death as the consequence of beheading -seemingly an unproblematic case of sufficient causation -could in principle be prevented through an extremely fast medical intervention [cf.Shewmon (2007)].
If implication is understood in terms of inclusion (as described in Section 2), the situation, I submit, presents itself as follows: First, sufficient causation is, strictly speaking, impossible: For given that c and e are distinct, the description of c (e.g."match lit under paper") does not include that of e (e.g."burning paper").Hence, nor does the proposition c include e, and so does not imply it.Of course, the same is true conversely.This reasoning applies also if c is, à la Mackie (1965Mackie ( , 1974)), an entire causal condition (e.g."match lit under paper, oxygen, low humidity"), rather than just a little local particular.By the same token, natural law implication is impossible: Let L be a true conjunction including only laws of nature, either currently known ones (such as Maxwell's equations, or the rules of quantum chromodynamics), or "ultimate" ones (if such there be), perhaps even all of them.Natural law implication would then give "c ∧ L implies e".
But L does not include e any more than c does -natural law statements are, after all, general statements, and do not refer to particulars -so that the left-hand side again cannot imply the right-hand side.Viewed in this way, natural law implication would mean getting something from nothing: something pops out on the right-hand side which is not included in the left-hand side [once again, cf.Wittgenstein (1995Wittgenstein ( ), 5.135-5.1361].The problem of prevention is, I propose, the empirical symptom of this logical gap between cause and effect.
How then to account for the cases of what seems to be sufficient causation in nature?I propose that they are best explained by realizing that, often, the system under consideration can in practice be considered isolated for a certain time interval.Given this, we can assume an equation of motion or a constant of the motion which, together with a specified initial condition includes the states of the system for all times in this interval.The simplest such case is uniform motion of a point particle m along a coordinate axis x at velocity v.The proposition asserting the existence of this system can be written as m: x(t = 0) = x 0 , x(t) = x 0 + vt, where 0 ≤ t ≤ t final .Since we can insert any such t, this proposition can be regarded as shorthand for a conjunction with continuously many elements (assuming that t is continuous), which includes, and therefore implies, the position x for each such t.A further example is an isolated, elastic and nonrelativistic collision between two particles such as billiard balls.To predict the final velocities from the initial ones, all that needs to be done is to impose constant total momentum of the system for before and after the collision, likewise for the kinetic energy, and then solve these two equations for the final velocities.These are already included in the equations, it's just that they need to be isolated algebraically.A final example is the, it seems, guaranteed destruction of structured arrangements of matter (e.g.crystals), and in particular of living beings, in high energy surroundings.This can be accounted for through the rule that any system of a given entropy S 0 and temperature T 0 will increase its entropy by an infinitesimal amount dS when energy dQ is added: dS = dQ/T 0 [see e.g.Young and Freedman (2012), pp. 669-670].If we now assume that such a system exists and that dQ is added to it, we can apply modus ponens -which after all is an instance of implication as inclusion (Section 2, point 8) -and conclude that the system will increase its entropy by dS.The addition of an amount of energy ΔQ -a sum of little dQ's -then leads to an increase ΔS in entropy, which for low temperatures will be large, so that a sufficiently large ΔQ will lead to dissolution of the system's structure.
In such cases, the logical gap between two particulars is closed by assuming that a particular equation applies to a system.Then, an initial situation, which can be viewed as "cause", together with such an equation or constant includes, and so implies an outcome or "effect", in the ways just shown.Implication as inclusion therefore also accounts for these cases.However, said assumption can only be made if the system is isolated.This can never be fully guaranteed: for example, a cosmic light ray, unpredictable in principle [cf.Popper (1991), pp. 57-62], could always knock particles off their trajectory.A rock about to be molten in the sun could, in principle, be saved in some way, even if only by a miracle.Hence, natural law implication is a quasi-implication which approximates implication in the true sense, sometimes extremely closely, but never quite reaches it.

Counterfactual Dependence
In the previous section, it was seen that, just as there is no strict implication from c to e, nor is there one from e to c.However, everyday counterfactual causal statements, such as "if nothing had pulled or pushed the bike, it would not have started moving", still seem to be obviously true.Based on implication as inclusion, it will now be shown that such counterfactuals really are true, and that they are strict, not merely approximative.I will limit myself to deriving these counterfactuals for relatively macroscopic physical objects with classical, not quantum-mechanical, identity conditions [cf.Lowe (2003), p. 78] -say, bacteria, rocks, or galaxies -in order to arrive at a simple model.
First some terminology: the familiar concept of "change" means that an object has at least two different states, one of which is before the other in time.But to account for counterfactuals such as the one just mentioned, we will not need the time order, but only the fact that two different states exist, whatever their order in time.Therefore, an object will be said to "proto-change" if and only if at least two different states of the object exist.To intuit this concept, imagine that you are given a stack of photos which document all states of a given object, such as a football, shuffled in random order.You pick two photos, one where the football has a red spot on it, and one where it is blank.This is an instance of proto-change.A change, then, is a proto-change with a temporal order, just as an ordered set is a set.

The Timeless Counterfactual Dependence in Proto-Change
Now, consider an entity a whose identity criteria are given by the collection [c 1 , . . ., c n ]: any entity x is a if and only if it satisfies each c i .The c i 's can refer, for example, to material constituents, sortal criteria (e.g."is a mammal"), or qualitative properties.Of course, it is notoriously difficult to specify which criteria apply for a given entity.But for what follows, it matters only that some apply -an assumption which must be made in both everyday life and scientific practice, since otherwise it would not be possible to describe the evolution of self-identical objects in spacetime.We can now add to a specifications d 1 , . . ., d m , obtaining [c 1 , . . ., c n , d 1 , . . ., d m ], where the d j 's again refer to intrinsic properties of some sort (e.g."is green") or to constituents (e.g."contains iron").Entity [c 1 , . . ., c n , d 1 , . . ., d m ] satisfies the identity criteria for a, as do, for example, [c 1 , . . ., c n ], [c 1 , . . ., c n , d 3 ], etc.Thus, a comes with a basic modality, that is, a potential to be in different states, where a "state" of a is any x satisfying [c 1 , . . ., c n ], with or without some specification.This basic modality is simply a consequence of a's definition.Given this, it is easy to see that the Leibniz principle holds -indiscernibles are identical -but not its converse.
Consider now the simple case of a physical object a with the following identity criterion: any x is a if and only if it includes s 1 , . . ., s n , where the s i 's are material constituents.Also, it is assumed that the s i 's have low, i.e. non-relativistic, kinetic energies.Suppose now that two states of a exist: the collections [s 1 , . . ., s n ] and [s 1 , . . ., s n , t], for some material constituent t.This is an instance of proto-change.We now obtain the following simple syllogism, where again underlining symbolizes "exists": (from 2, using "→") 4. |[s 1 , . . . ,s n ]| → ¬t.
(using the "blocking-off" operator) (3 and 4, modus tollens) In step ( 4), if we augment the left-hand side by any entity u which does not include t, writing |[s 1 , . . . ,s n ], u|, we likewise get ¬t.Therefore, given (1) and ( 2), there must be a collection T including t, whether it does so properly or improperly.In the latter case, T = [t] = t.T cannot always be included in a, since otherwise, a's state [s 1 , . . ., s n ] could not exist, and must therefore, at least in some of its states, be distinct from a.
Given this, we can now introduce a new implication operator |→ and write: [s 1 , . . . ,s n ] ∧ [s 1 , . . ., s n , t] |→ T: t is included in T ∧ T ≠ a.Using |→ is simply a shorthand which saves us going through the whole argument of the previous paragraph in similar cases.The argument itself, however, relies wholly on implication as inclusion.The operator |→ is, in other words, an instance of implication as inclusion, where the above argument functions as background knowledge.Thus, |→ is a strict implication, not a quasi-implication.Implication via |→ will be called "|implication" (read: "bar implication").Note that the |implication is a counterfactual: the protochange "[s 1 , . . ., s n ] and [s 1 , . . ., s n , t]" cannot occur unless T.
Also, T will be called an "aition" of t. τo ατιoν is the Greek word for "cause", but in its etymological sense, an α , ίτιoν of x is simply something which has x as its part (α , ίσα) [Gemoll and Vretska (2006), "α , ίτιoς"].It is in this latter sense that I use "aition" here and in what follows.In the case considered above, the proto-change of an object depends counterfactually on a suitable aition.
In which other cases is there |implication?To explore this, I will first define: An entity y will be called "within the possibilities of" (henceforth abbreviated as other hand, there are also cases of qualitative proto-change where at least one state is OP the other, so that there is counterfactual dependence.This is true when the difference in quality is due to different material constituents, as for example, when a dish has a bland state and a spicy state. Also, two entities may be OP each other.For example, this is true of [s 1 , . . ., s n , t] and [s 1 , . . ., s n , u] (where t ≠ u), since each contains a constituent which is not in the other.
In sum, for any two non-identical entities x and y, y may be WP x, or vice versa, or both, or neither.The relation "WP" therefore generates a partial order on any set of entities.To illustrate: a complex carbon chain is WP the early Solar System five billion years ago (the former can be obtained from what is in the latter, with no momenta or energy from outside the Solar System required), but not vice versa.The Milky Way and Andromeda galaxies are OP each other (given only one, you cannot obtain the other).A cat rearranging its limbs as it falls in mid-air has different states which are WP each other.There is counterfactual dependence on a suitable aition whenever at least one entity is OP the other.This counterfactual dependence is independent of temporal order: it exists already in instances of proto-change, not only in change.
Having considered some examples from the world of classical objects, what about the wave function  ?As has been noted, it is a disjunction with weighted probabilities: " 1 with probability mod(c 1 ) 2 , or . . .".If we block it off and write | | (i.e."only the system described by  exists"), then -in virtue of Section 2, point 2 -for all k, ¬(| | → ¬ k ).Hence,  k is WP  .In this way, eigenstates of wave functions are WP their wave functions, just as rearranged cat states are WP a cat being dropped out of the window.The relevance of this point can be appreciated by contrasting these cases with that of a "Humean swerve" of a classical particle: Hume notes that a particle cannot just for no reason depart from its line of motion, any more "than it can convert itself into an angel" [(1739), II, 3, 1].Hume is right, because such a swerve would mean that there are two momentum states, so these two states are OP each other.But all too often, this point is misconstrued as excluding all modality in motion, including by Hume himself, who holds that "there are not the least traces of indifference or liberty" (ibid.).The case looks very different given implication as inclusion: on it, neither the famous indeterminism of the collapse of the wave function nor the modality of different spatial rearrangements of macroscopic objects should be viewed as a logically scandalous "violation" of causality, a gratuitous something-from-nothing as in a Humean swerve.In the former two cases, a state is realized which is WP of the system under consideration, but not in the latter case.

Counterfactual Dependence in Time
We can now analyse the change of a physical object A in time: as with proto-change, A has two states, but in addition, there is now a distinction between an earlier and a later state.Also, let "cause of x" have its "naïve", everyday meaning: something "from which" x originates as its "source" (as in Suárez' philosophy).Do changes require causes, just as proto-changes require suitable aitia?From the considerations in Section 4.1, given a change of A there is some a included properly or improperly in A having two states a * and a ∼ such that at least one is OP the other.Let a ∼ be that state.Since we are now assuming a temporal order, we get two cases: 1. a * is before a ∼ , as is the case, for example, when a acquires a constituent c, that is, a ∼ is the state with c, a * the state without it.The existence of the two states depends counterfactually on an aition of c.Given the assumed time order, this aition is one from which a acquires c, and which can therefore be viewed as a cause of a's change.2. a ∼ is before a * , as occurs when a loses c.Then, the aition of c acquires c from a, and cannot be identified with a cause of a's change.However, in this case, there is now a sub-object of a itself, call it s, which acquires some momentum p in its own rest frame, i.e. s recoils as c is lost.But since s acquiresrather than loses -momentum, we can argue just as in case 1: the change from s-without-p to s-with-p depends counterfactually on the existence of an aition of p from which s acquires it, i.e. on a cause of it.
These examples illustrate how the notion of "cause" is linked, via the "from-to" distinction, to our temporal perspective: reverse the direction of time and, at least in many cases, also the cause-effect relation will be reversed.However, it was seen in cases 1 and 2 that the change of a does indeed depend counterfactually on what we would call a "cause" in everyday language.This counterfactual dependence is therefore not a perspectival effect.Thus, we are justified in concluding, at least for the cases considered, that the change of a physical object |implies another object as its cause.

Conclusion
It is now possible to summarize and interpret the results: Implication as inclusion, symbolized by the operator →, has highly intuitive properties and allows syllogistic reasoning.It is "Wittgensteinian" in that you can only get out of propositions what's already inside them.This property makes natural law implication impossible in the strict sense, but also accounts for the fact that some cases do come extremely close to such implication.
In many instances of proto-change, the operator |→, itself reducible to implication as inclusion, allows to infer, strictly, the existence of an object distinct from the proto-changing system, namely of an aition of the difference in momentum, or energy, or material constituents between the two states.That such inference is justified was shown in Section 4.1.What is inferred is a type of object, not a token (e.g."some T which includes t", or "something with sufficient momentum to account for the bike's two momentum states").Thus, any token taken to be necessary can in principle be replaced, but the counterfactual dependence on the type remains true.Counterfactual dependence, in time, of a change on a "cause" as "source" of the change is a special case of the counterfactual dependence of a proto-change on an aition.We therefore don't need an extra operator for causal implication.Rather, causal implications (e.g."if the bike started moving, something must have pushed or pulled it") are instances of |implication viewed in time.Thus, the reason for counterfactual dependence in causation is, I submit, not to be found in global conditions such as laws of nature or other worlds, as has so widely been thought in recent centuries.Rather, it is established by inspecting two entities and finding how they are related in terms of WP and OP.Causal asymmetry is therefore neither primitive, nor a naïve illusion.
There are no free lunches, i.e. no differences in total energy and momentum of a given system -and in the non-relativistic case, no differences in material constituents -without a suitable aition.But genuine modality, in the sense of openness of possibilities, is allowed, also for some isolated systems.Implication as inclusion does not claim to solve the questions concerning the "choice" of the possibility being realized: Why was this eigenstate realized, not some other?Why did the cat turn its right front paw clockwise, not anticlockwise?Rather, implication as inclusion accounts for counterfactual dependence in causation, and in addition shows that modality in nature is not illogical and crazy, like a rock's turning into an angel is.Implication as inclusion, while incompatible with free lunches, is compatible with choice on the menu.

FP:
Any proposition implies only propositions included in it.
conjunctions implied by it (given (1)) is true.One or more of the p i 's may be a disjunction.Since, by assumption, it is true, it includes at least one true proposition.It may of course also include false ones, but in virtue of (2), doesn't imply them.-Afalse proposition is one where at least one of the p i 's is false (the others may be true or false).Hence, any false proposition implies at least one false proposition, and many false propositions also imply true ones.This is because any proposition P can always be written as p 1 ∧ . . .∧ p n , and any S implied by P as p i or as p j ∧ p k ∧ . . . .On the other hand, ¬P = ¬p 1 ∨ . . .∨ ¬p n , and implies only that there is an l such that ¬p l .Hence, ¬P doesn't imply S.
such that each of the p i 's is true.Thus, each of the 2 n − 1