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

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.

# Wittgenstein’s Form is the Possibility of Structure

For given two arbitrary objects x and y they can be understood as arguments for a basic ontological connection which, in turn, is either positive or negative. A priori there exist just four cases: the case of positive connection – MP, the case of negative connection – MI, the case that connection is both positive and negative, hence incoherent, denoted – MPI, and the most popular in combinatorial ontology the case of mutual neutrality – N( , ). The first case is taken here to be fundamental.

Explication for σ

Now we can offer the following, rather natural explication for a powerful, nearly omnipotent, synthesizer: y is synthetizable from x iff it is be made possible from x:

σ(x) = {y : MP(x,y)}

Notice that the above explication connects the second approach (operator one) with the third (internal) approach to a general theory of analysis and synthesis.

Quoting one of the most mysterious theses of Wittgenstein’s Tractatus:

(2.033) Form is the possibility of structure.

Ask now what the possibility means? It has been pointed out by Frank Ramsey in his famous review of the Tractatus that it cannot be read as a logical modality (i. e., form cannot be treated as an alternative structure), for this reading would immediately make Tractatus inconsistent.

But, rather ‘Form of x is what makes the structure of y possible’.

Formalization: MP(Form(x), Str(y)), hence – through suitable generalization – MP(x, y).

Wittgensteinian and Leibnizian clues make the nature of MP more clear: form of x is determined by its substance, whereas structurality of y means that y is a complex built up in such and such way. Using syntactical categorization of Lésniewski and Ajdukiewicz we obtain therefore that MP has the category of quantifier: s/n, s – which, as is easy to see, is of higher order and deeply modal.

Therefore M P is a modal quantifier, characterized after Wittgenstein’s clue by

MP(x, y) ↔ MP(S(x), y)

# Leibniz’s Compossibility and Compatibility

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!).