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 S_{i}RS_{j} between the two ‘individual terms’ (subjects) S_{j} and S_{i}. In a short hand notation we represent this relation by R_{ij}. Relations may be composed: whenever we have relations of the form R_{ij}, R_{jl}, a third transitive relation R_{il} emerges following the rule

R_{ij}R_{kl} = δ_{jk}R_{il} —– (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

S_{i} = ∑_{j} R_{ij} —– (2)

The individual S_{i} thus defined is an eigenstate of the R_{ii} relation

R_{ii}S_{i} = S_{i} —– (3)

The relations R_{ii} are idempotent

R^{2}_{ii} = R_{ii} —– (4)

and they span the identity

∑_{i} R_{ii} = 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 1_{A} : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, R_{ij} stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ S_{i} is R_{ii}.

R_{ij} 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 R_{ij}

R_{ij} = |r_{i}⟩⟨r_{j}| —– (6)

where state ⟨r_{i} | is the dual of the state |r_{i}⟩ and they obey the orthonormal condition

⟨r_{i} |r_{j}⟩ = δ_{ij} —– (7)

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

∑_{n}|r_{i}⟩⟨r_{i}|=1 —– (8)

All relations remain satisfied if we replace the state |r_{i}⟩ by |ξ_{i}⟩ where

|ξ_{i}⟩ = 1/√N ∑_{n} |r_{i}⟩⟨r_{n}| —– (9)

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

R_{ij} |r_{k}⟩ = δ_{jk} |r_{i}⟩ —– (10)

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

Q_{ij} = |q_{i}⟩⟨q_{j} | —– (11)

with

Q_{ij} |q_{k}⟩ = δ_{jk} |q_{i}⟩ —– (12)

and

∑_{n} |q_{i}⟩⟨q_{i}| = 1 —– (13)

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

R_{ij}Q_{kl} = |r_{i}⟩⟨r_{j} |q_{k}⟩⟨q_{l} | = ⟨r_{j} |q_{k}⟩ S_{il} —– (14)

with

S_{il} = |r_{i}⟩⟨q_{l} | —– (15)

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

Tr(S_{il}) = ∑_{n} ⟨r_{n} |S_{il} |r_{n}⟩ = ⟨q_{l} |r_{i}⟩ —– (16)

From the above relation we deduce

Tr(R_{ij}) = δ_{ij} —– (17)

Any operator can be expressed as a linear superposition of the R_{ij}

A = ∑_{i,j} A_{ij}R_{ij} —– (18)

with

A_{ij} =Tr(AR_{ji}) —– (19)

The individual states could be redefined

|r_{i}⟩ → e^{iφi} |r_{i}⟩ —– (20)

|q_{i}⟩ → e^{iθi} |q_{i}⟩ —– (21)

without affecting the corresponding composition laws. However the overlap number ⟨r_{i} |q_{j}⟩ changes and therefore we need an invariant formulation for the transition |r_{i}⟩ → |q_{j}⟩. This is provided by the trace of the closed operation R_{ii}Q_{jj}R_{ii}

Tr(R_{ii}Q_{jj}R_{ii}) ≡ p(q_{j}, r_{i}) = |⟨r_{i} |q_{j}⟩|^{2} —– (22)

The completeness relation, equation (13), guarantees that p(q_{j}, r_{i}) may assume the role of a probability since

∑_{j} p(q_{j}, r_{i}) = 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.