Derivability from Relational Logic of Charles Sanders Peirce to Essential Laws of Quantum Mechanics

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Charles Sanders Peirce made important contributions in logic, where he invented and elaborated novel system of logical syntax and fundamental logical concepts. The starting point is the binary relation SiRSj between the two ‘individual terms’ (subjects) Sj and Si. In a short hand notation we represent this relation by Rij. Relations may be composed: whenever we have relations of the form Rij, Rjl, a third transitive relation Ril emerges following the rule

RijRkl = δjkRil —– (1)

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce considered the individual as the aggregate of all its relations

Si = ∑j Rij —– (2)

The individual Si thus defined is an eigenstate of the Rii relation

RiiSi = Si —– (3)

The relations Rii are idempotent

R2ii = Rii —– (4)

and they span the identity

i Rii = 1 —– (5)

The Peircean logical structure bears resemblance to category theory. In categories the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category consists of objects A, B, C,… and arrows (morphisms) f, g, h,… . Each arrow f is assigned an object A as domain and an object B as codomain, indicated by writing f : A → B. If g is an arrow g : B → C with domain B, the codomain of f, then f and g can be “composed” to give an arrow gof : A → C. The composition obeys the associative law ho(gof) = (hog)of. For each object A there is an arrow 1A : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, Rij stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ Si is Rii.

Rij may receive multiple interpretations: as a transition from the j state to the i state, as a measurement process that rejects all impinging systems except those in the state j and permits only systems in the state i to emerge from the apparatus, as a transformation replacing the j state by the i state. We proceed to a representation of Rij

Rij = |ri⟩⟨rj| —– (6)

where state ⟨ri | is the dual of the state |ri⟩ and they obey the orthonormal condition

⟨ri |rj⟩ = δij —– (7)

It is immediately seen that our representation satisfies the composition rule equation (1). The completeness, equation (5), takes the form

n|ri⟩⟨ri|=1 —– (8)

All relations remain satisfied if we replace the state |ri⟩ by |ξi⟩ where

i⟩ = 1/√N ∑n |ri⟩⟨rn| —– (9)

with N the number of states. Thus we verify Peirce’s suggestion, equation (2), and the state |ri⟩ is derived as the sum of all its interactions with the other states. Rij acts as a projection, transferring from one r state to another r state

Rij |rk⟩ = δjk |ri⟩ —– (10)

We may think also of another property characterizing our states and define a corresponding operator

Qij = |qi⟩⟨qj | —– (11)

with

Qij |qk⟩ = δjk |qi⟩ —– (12)

and

n |qi⟩⟨qi| = 1 —– (13)

Successive measurements of the q-ness and r-ness of the states is provided by the operator

RijQkl = |ri⟩⟨rj |qk⟩⟨ql | = ⟨rj |qk⟩ Sil —– (14)

with

Sil = |ri⟩⟨ql | —– (15)

Considering the matrix elements of an operator A as Anm = ⟨rn |A |rm⟩ we find for the trace

Tr(Sil) = ∑n ⟨rn |Sil |rn⟩ = ⟨ql |ri⟩ —– (16)

From the above relation we deduce

Tr(Rij) = δij —– (17)

Any operator can be expressed as a linear superposition of the Rij

A = ∑i,j AijRij —– (18)

with

Aij =Tr(ARji) —– (19)

The individual states could be redefined

|ri⟩ → ei |ri⟩ —– (20)

|qi⟩ → ei |qi⟩ —– (21)

without affecting the corresponding composition laws. However the overlap number ⟨ri |qj⟩ changes and therefore we need an invariant formulation for the transition |ri⟩ → |qj⟩. This is provided by the trace of the closed operation RiiQjjRii

Tr(RiiQjjRii) ≡ p(qj, ri) = |⟨ri |qj⟩|2 —– (22)

The completeness relation, equation (13), guarantees that p(qj, ri) may assume the role of a probability since

j p(qj, ri) = 1 —– (23)

We discover that starting from the relational logic of Peirce we obtain all the essential laws of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic measurement.

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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