Leibniz’s Compossibility and Compatibility

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Leibniz believed in discovering a suitable logical calculus of concepts enabling its user to solve any rational question. Assuming that it is done he was in power to sketch the full ontological system – from monads and qualities to the real world.

Thus let some logical calculus of concepts (names?, predicates?) be given. Cn is its connected consequence operator, whereas – for any x – Th(x) is the Cn-theory generated by x.

Leibniz defined modal concepts by the following metalogical conditions:

M(x) :↔ ⊥ ∉ Th(x)

x is possible (its theory is consistent)

L(x) :↔ ⊥ ∈ Th(¬x)

x is necessary (its negation is impossible)

C(x,y) :↔ ⊥ ∉ Cn(Th(x) ∪ Th(y))

x and y are compossible (their common theory is consistent).

Immediately we obtain Leibnizian ”soundness” conditions:

C(x, y) ↔ C(y, x) Compossibility relation is symmetric.

M(x) ↔ C(x, x) Possibility means self-compossibility.

C(x, y) → M(x)∧M(y) Compossibility implies possibility.

When can the above implication be reversed?

Onto\logical construction

Observe that in the framework of combination ontology we have already defined M(x) in a way respecting M(x) ↔ C(x, x).

On the other hand, between MP( , ) and C( , ) there is another relation, more fundamental than compossibility. It is so-called compatibility relation. Indeed, putting

CP(x, y) :↔ MP(x, y) ∧ MP(y, x) – for compatibility, and C(x,y) :↔ M(x) ∧ M(y) ∧ CP(x,y) – for compossibility

we obtain a manageable compossibility relation obeying the above Leibniz’s ”soundness” conditions.

Wholes are combinations of compossible collections, whereas possible worlds are obtained by maximalization of wholes.

Observe that we start with one basic ontological making: MP(x, y) – modality more fundamental than Leibnizian compossibility, for it is definable in two steps. Observe also that the above construction can be done for making impossible and to both basic ontological modalities as well (producing quite Hegelian output in this case!).

Conjuncted: Occam’s Razor and Nomological Hypothesis. Thought of the Day 51.1.1

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Conjuncted here, here and here.

A temporally evolving system must possess a sufficiently rich set of symmetries to allow us to infer general laws from a finite set of empirical observations. But what justifies this hypothesis?

This question is central to the entire scientific enterprise. Why are we justified in assuming that scientific laws are the same in different spatial locations, or that they will be the same from one day to the next? Why should replicability of other scientists’ experimental results be considered the norm, rather than a miraculous exception? Why is it normally safe to assume that the outcomes of experiments will be insensitive to irrelevant details? Why, for that matter, are we justified in the inductive generalizations that are ubiquitous in everyday reasoning?

In effect, we are assuming that the scientific phenomena under investigation are invariant under certain symmetries – both temporal and spatial, including translations, rotations, and so on. But where do we get this assumption from? The answer lies in the principle of Occam’s Razor.

Roughly speaking, this principle says that, if two theories are equally consistent with the empirical data, we should prefer the simpler theory:

Occam’s Razor: Given any body of empirical evidence about a temporally evolving system, always assume that the system has the largest possible set of symmetries consistent with that evidence.

Making it more precise, we begin by explaining what it means for a particular symmetry to be “consistent” with a body of empirical evidence. Formally, our total body of evidence can be represented as a subset E of H, i.e., namely the set of all logically possible histories that are not ruled out by that evidence. Note that we cannot assume that our evidence is a subset of Ω; when we scientifically investigate a system, we do not normally know what Ω is. Hence we can only assume that E is a subset of the larger set H of logically possible histories.

Now let ψ be a transformation of H, and suppose that we are testing the hypothesis that ψ is a symmetry of the system. For any positive integer n, let ψn be the transformation obtained by applying ψ repeatedly, n times in a row. For example, if ψ is a rotation about some axis by angle θ, then ψn is the rotation by the angle nθ. For any such transformation ψn, we write ψ–n(E) to denote the inverse image in H of E under ψn. We say that the transformation ψ is consistent with the evidence E if the intersection

E ∩ ψ–1(E) ∩ ψ–2(E) ∩ ψ–3(E) ∩ …

is non-empty. This means that the available evidence (i.e., E) does not falsify the hypothesis that ψ is a symmetry of the system.

For example, suppose we are interested in whether cosmic microwave background radiation is isotropic, i.e., the same in every direction. Suppose we measure a background radiation level of x1 when we point the telescope in direction d1, and a radiation level of x2 when we point it in direction d2. Call these events E1 and E2. Thus, our experimental evidence is summarized by the event E = E1 ∩ E2. Let ψ be a spatial rotation that rotates d1 to d2. Then, focusing for simplicity just on the first two terms of the infinite intersection above,

E ∩ ψ–1(E) = E1 ∩ E2 ∩ ψ–1(E1) ∩ ψ–1(E2).

If x1 = x2, we have E1 = ψ–1(E2), and the expression for E ∩ ψ–1(E) simplifies to E1 ∩ E2 ∩ ψ–1(E1), which has at least a chance of being non-empty, meaning that the evidence has not (yet) falsified isotropy. But if x1 ≠ x2, then E1 and ψ–1(E2) are disjoint. In that case, the intersection E ∩ ψ–1(E) is empty, and the evidence is inconsistent with isotropy. As it happens, we know from recent astronomy that x1 ≠ x2 in some cases, so cosmic microwave background radiation is not isotropic, and ψ is not a symmetry.

Our version of Occam’s Razor now says that we should postulate as symmetries of our system a maximal monoid of transformations consistent with our evidence. Formally, a monoid Ψ of transformations (where each ψ in Ψ is a function from H into itself) is consistent with evidence E if the intersection

ψ∈Ψ ψ–1(E)

is non-empty. This is the generalization of the infinite intersection that appeared in our definition of an individual transformation’s consistency with the evidence. Further, a monoid Ψ that is consistent with E is maximal if no proper superset of Ψ forms a monoid that is also consistent with E.

Occam’s Razor (formal): Given any body E of empirical evidence about a temporally evolving system, always assume that the set of symmetries of the system is a maximal monoid Ψ consistent with E.

What is the significance of this principle? We define Γ to be the set of all symmetries of our temporally evolving system. In practice, we do not know Γ. A monoid Ψ that passes the test of Occam’s Razor, however, can be viewed as our best guess as to what Γ is.

Furthermore, if Ψ is this monoid, and E is our body of evidence, the intersection

ψ∈Ψ ψ–1(E)

can be viewed as our best guess as to what the set of nomologically possible histories is. It consists of all those histories among the logically possible ones that are not ruled out by the postulated symmetry monoid Ψ and the observed evidence E. We thus call this intersection our nomological hypothesis and label it Ω(Ψ,E).

To see that this construction is not completely far-fetched, note that, under certain conditions, our nomological hypothesis does indeed reflect the truth about nomological possibility. If the hypothesized symmetry monoid Ψ is a subset of the true symmetry monoid Γ of our temporally evolving system – i.e., we have postulated some of the right symmetries – then the true set Ω of all nomologically possible histories will be a subset of Ω(Ψ,E). So, our nomological hypothesis will be consistent with the truth and will, at most, be logically weaker than the truth.

Given the hypothesized symmetry monoid Ψ, we can then assume provisionally (i) that any empirical observation we make, corresponding to some event D, can be generalized to a Ψ-invariant law and (ii) that unconditional and conditional probabilities can be estimated from empirical frequency data using a suitable version of the Ergodic Theorem.

Bayesianism in Game Theory. Thought of the Day 24.0

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Bayesianism in game theory can be characterised as the view that it is always possible to define probabilities for anything that is relevant for the players’ decision-making. In addition, it is usually taken to imply that the players use Bayes’ rule for updating their beliefs. If the probabilities are to be always definable, one also has to specify what players’ beliefs are before the play is supposed to begin. The standard assumption is that such prior beliefs are the same for all players. This common prior assumption (CPA) means that the players have the same prior probabilities for all those aspects of the game for which the description of the game itself does not specify different probabilities. Common priors are usually justified with the so called Harsanyi doctrine, according to which all differences in probabilities are to be attributed solely to differences in the experiences that the players have had. Different priors for different players would imply that there are some factors that affect the players’ beliefs even though they have not been explicitly modelled. The CPA is sometimes considered to be equivalent to the Harsanyi doctrine, but there seems to be a difference between them: the Harsanyi doctrine is best viewed as a metaphysical doctrine about the determination of beliefs, and it is hard to see why anybody would be willing to argue against it: if everything that might affect the determination of beliefs is included in the notion of ‘experience’, then it alone does determine the beliefs. The Harsanyi doctrine has some affinity to some convergence theorems in Bayesian statistics: if individuals are fed with similar information indefinitely, their probabilities will ultimately be the same, irrespective of the original priors.

The CPA however is a methodological injunction to include everything that may affect the players’ behaviour in the game: not just everything that motivates the players, but also everything that affects the players’ beliefs should be explicitly modelled by the game: if players had different priors, this would mean that the game structure would not be completely specified because there would be differences in players’ behaviour that are not explained by the model. In a dispute over the status of the CPA, Faruk Gul essentially argues that the CPA does not follow from the Harsanyi doctrine. He does this by distinguishing between two different interpretations of the common prior, the ‘prior view’ and the ‘infinite hierarchy view’. The former is a genuinely dynamic story in which it is assumed that there really is a prior stage in time. The latter framework refers to Mertens and Zamir’s construction in which prior beliefs can be consistently formulated. This framework however, is static in the sense that the players do not have any information on a prior stage, indeed, the ‘priors’ in this framework do not even pin down a player’s priors for his own types. Thus, the existence of a common prior in the latter framework does not have anything to do with the view that differences in beliefs reflect differences in information only.

It is agreed by everyone that for most (real-world) problems there is no prior stage in which the players know each other’s beliefs, let alone that they would be the same. The CPA, if understood as a modelling assumption, is clearly false. Robert Aumann, however, defends the CPA by arguing that whenever there are differences in beliefs, there must have been a prior stage in which the priors were the same, and from which the current beliefs can be derived by conditioning on the differentiating events. If players differ in their present beliefs, they must have received different information at some previous point in time, and they must have processed this information correctly. Based on this assumption, he further argues that players cannot ‘agree to disagree’: if a player knows that his opponents’ beliefs are different from his own, he should revise his beliefs to take the opponents’ information into account. The only case where the CPA would be violated, then, is when players have different beliefs, and have common knowledge about each others’ different beliefs and about each others’ epistemic rationality. Aumann’s argument seems perfectly legitimate if it is taken as a metaphysical one, but we do not see how it could be used as a justification for using the CPA as a modelling assumption in this or that application of game theory and Aumann does not argue that it should.

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