Reality Calculus (RC)

What is Reality Calculus?

Reality Calculus (RC) is a proposed unified formal language for representing and reasoning about reality across ordinary properties and relations, causation, time, modality, ontology, perception, experience, and belief.

RC is designed as a common representational architecture rather than a claim that one notation should replace every existing logical system. Its central ambition is to let different kinds of claims be expressed consistently while preserving distinctions between them.

A simple RC statement

Human(s)

This represents the proposition that s is human. RC then allows that proposition to be combined with existence, relations, time, modality, perception, experience, belief and inference without requiring those categories to be collapsed into one another.

A combined statement

ex x(
    Human(x) an
    Philosopher(x) an
    Perceives(a,x)
)

RC-Core

RC-Core contains the basic vocabulary for constructing ordinary logical statements.

Properties

Human(s)
Mortal(s)
Wise(s)

an

Human(s)
an
Mortal(s)
an
Wise(s)

Negation

no Mortal(s)

Negation can also apply to complex formulas:

no(
    Human(s)
    an
    Mortal(s)
)

Disjunction

Human(s)
or
Animal(s)

Disjunction and conjunction are logical operators. They should not automatically be interpreted as claims about physical alternatives or causal possibilities.

Syntax

RC uses predicates to attribute properties to things and relation symbols to connect terms.

Atomic formulas

Human(x)
Loves(x,y)
Between(x,y,z)

x eq y
x neq y

Predicate arity matters. Human(x) is unary, Loves(x,y) binary, and Between(x,y,z) ternary.

Connectives

A an B
A or B
no A
A imp B
A iff B

Schematic grammar

Term ::= Variable | Constant | Function(Term,...)

Atomic ::= Predicate(Term,...) | Term eq Term | Term neq Term

Formula ::= Atomic
         | no Formula
         | Formula an Formula
         | Formula or Formula
         | Formula imp Formula
         | Formula iff Formula
         | ex Variable(Formula)
         | all Variable(Formula)

Nested structure

ex x(
    Human(x) an
    ex y(
        Philosopher(y) an
        Teaches(y,x)
    )
)

This demonstrates that RC formulas can contain formulas inside quantified scopes, allowing complex relational structures to be represented explicitly.

Identity inside a structure

ex x ex y(
    Human(x) an
    Human(y) an
    x neq y an
    Knows(x,y)
)

Quantification

Existential quantification

ex x(
    Human(x)
)

There exists at least one human.

Existence with multiple properties

ex x(
    Human(x) an
    Philosopher(x) an
    Wise(x)
)

At least two distinct philosophers

ex x ex y(
    Philosopher(x) an
    Philosopher(y) an
    x neq y
)

Universal statements

all x(
    Human(x) imp Mortal(x)
)

Everything that is human is mortal.

Nested existence

ex x(
    Philosopher(x) an
    ex y(
        Student(y) an
        Teaches(x,y)
    )
)

Nested universal and existential quantification

all x(
    Human(x)
    imp
    ex y(
        Parent(y,x)
    )
)

This represents the general form: every human has at least one parent.

Universal relation

all x all y(
    Human(x) an
    Human(y) an
    x neq y
    imp
    CanInteract(x,y)
)

Unique existence pattern

ex x(
    King(x)
    an
    all y(
        King(y)
        imp
        y eq x
    )
)

This expresses the structure of there being a king and every king being identical with that same entity.

Relations and Causation

Relations

Teaches(s,p)
Admires(p,s)

Argument order matters unless a theory establishes symmetry. Loves(s,p) is not automatically equivalent to Loves(p,s).

Three-place relations

Between(a,b,c)
Gives(a,b,c)
LocatedBetween(x,y,z)

Relations can connect more than two entities where the relevant domain requires it.

Causation

ca(x,y)

ca(x,y) represents that x causes y. It is deliberately distinct from implication.

ca(x,y) ≠ x imp y

RC does not automatically infer that an effect is necessary, future, or temporally later merely because a causal relation is represented.

Causal chain

ca(a,b)
an
ca(b,c)

th

causal_chain(a,b,c)

The final expression would require a theory defining causal_chain; the th operator marks the conclusion rather than making the conclusion true by notation alone.

Temporal causation

Earthquake(e) at t₁
an
Collapse(c) at t₂
an
ca(e,c)
an
t₁ < t₂

Causation and implication

Rain(r)
imp
WetGround(g)

ca(r,g)

The first line is a conditional relationship. The second represents a causal relation. A theory may connect them, but RC keeps them formally distinct.

Time

Temporal change

Alive(s) at t₁
an
no Alive(s) at t₂
an
t₁ < t₂

Historical states

pr Student(s)
nu Philosopher(s)
fu Remembered(s)

Before and after

Event(a) at t₁
an
Event(b) at t₂
an
t₁ < t₂

Temporal sequence

Event(a) at t₁
an
Event(b) at t₂
an
Event(c) at t₃
an
t₁ < t₂
an
t₂ < t₃

Time and causation

ca(a,b)
an
A(a) at t₁
an
B(b) at t₂
an
t₁ < t₂

Time and modality

fu(
    Possible(Event(e))
)

The temporal layer can be combined with modality, causation, perception, experience and belief. These combinations do not automatically determine their own semantics and therefore remain subject to the relevant theory.

Chronos, Kairos, Aion and Anamnesis

RC can distinguish several ways in which temporal structure may be represented. These proposed temporal modes do not replace ordinary temporal ordering such as t₁ < t₂; instead, they describe different forms or modes of temporal significance.

Chronos

ch represents temporal order, sequence or measured temporal progression.

ch(
    Event(a) at t₁
    an
    Event(b) at t₂
    an
    t₁ < t₂
)

Chronos and change

ch(
    Child(s) at t₁
    an
    Adult(s) at t₂
    an
    t₁ < t₂
)

Kairos

ka represents temporal significance, opportunity, decision or an appropriate/decisive moment.

ka(
    Opportunity(o)
    an
    Available(o) at t
)

Kairos and action

ka(
    DecisionPoint(d) at t
)
an
do(a,Choose(a,d))

The temporal mode identifies the event as significant or opportune; the action relation represents what the agent does.

Aion

ai represents an extended temporal span such as an age, era or epoch.

ai(
    Empire(e)
    an
    Exists(e) at t₁
    an
    Exists(e) at t₂
    an
    t₁ < t₂
)

Aion and historical duration

ai(
    Civilization(c)
    an
    Begins(c) at t₁
    an
    Ends(c) at t₂
)

Anamnesis

am represents recollection or remembered temporal content.

am(
    Remembers(a,e)
    an
    Event(e) at t₁
)

Anamnesis and belief

am(
    Remembers(a,e)
    an
    Event(e) at t₁
)
an
Believes(a,Event(e) at t₁)

Chronos + Kairos

ch(
    Event(e) at t
)
an
ka(
    Significant(e)
)

This distinguishes the occurrence of an event in measured temporal order from the claim that the moment is significant or decisive.

Aion + Chronos

ai(
    Era(e)
)
an
ch(
    Begins(e) at t₁
    an
    Ends(e) at t₂
    an
    t₁ < t₂
)

Anamnesis + Chronos

am(
    Remembers(a,e)
    an
    Event(e) at t₁
)
an
ch(
    Remembrance(a) at t₂
)
an
t₁ < t₂

All four temporal modes

ch(
    Event(e) at t₁
    an
    Event(e) at t₂
    an
    t₁ < t₂
)
an
ka(
    Significant(e)
)
an
ai(
    HistoricalEra(h)
)
an
am(
    Remembers(a,e)
)

The four modes can therefore be combined while retaining their conceptual distinctions: chronological order, temporal significance, extended temporal span and recollected temporal content.

Modality

Possibility and necessity

po(Alive(s))

nec(
    all x(
        Human(x)
        imp
        Mortal(x)
    )
)

Time + modality

po(
    fu Victory(s)
)

po(
    pr Event(x)
)

po(
    Alive(s) at t₁
)

Operator order matters. po(fu A) and fu(po A) need not have the same meaning.

Capacity, permission and obligation

can Run(s)
may Leave(s)
mus Leave(s)
sho Study(s)

Nested modality

po(
    nec(
        Human(s)
        imp
        Mortal(s)
    )
)

Temporal mode + modality

ka(
    po(
        Opportunity(o)
    )
)

This can represent a possible significant opportunity, subject to the precise semantics of the temporal and modal extensions.

Modality + causation

po(
    ca(a,b)
)

This represents a modal claim about a causal relation rather than asserting that the causal relation necessarily holds.

Ontology

RC can represent claims about what kinds of things exist while keeping ontological vocabulary distinct from perception and belief.

ex x(
    Object(x) an
    Red(x)
)

Different ontological categories

ex x(
    Object(x)
)

ex y(
    Event(y)
)

ex z(
    Abstract(z)
)

Object before perception

Object(x) at t₁
an
Perceives(a,x) at t₂
an
t₁ < t₂

The object is represented as existing at an earlier time than Alice's later perception of it.

Ontology and properties

ex x(
    Physical(x)
    an
    Spatial(x)
    an
    Mass(x)
)

Ontology does not imply perception

Object(x)
an
Exists(x)
an
no Perceives(a,x)

The represented existence of an object does not by itself entail that an agent perceives it.

Perception

Perceives(p,x)

This represents a relation between a perceiving subject and an object. It does not by itself determine the ontology of the object.

Ontology ≠ perception

Tree(x)
an
no Perceives(a,x)

A tree can be represented as a tree while Alice does not perceive it. Conversely, perception does not automatically prove the perceived thing exists as represented.

Experience

Experiences(a,Red(x))

This represents Alice experiencing x as red, or an experience represented by Red(x). It is distinct from the proposition that x actually is red.

Perception and experience

Perceives(a,x)
an
Experiences(a,Red(x))

The first expression represents a perceptual relation. The second represents the associated experiential content.

Perceptual disagreement with reality

Blue(x)
an
no Red(x)
an
Perceives(a,x)
an
Experiences(a,Red(x))

Experience and Belief

Belief

Believes(a,Wise(s))

Alice believes that Socrates is wise. This does not entail Wise(s).

False belief

no Wise(s)
an
Believes(a,Wise(s))

Perceptual error

Object(x)
an
Blue(x)
an
no Red(x)
an
Perceives(a,x)
an
Experiences(a,Red(x))
an
Believes(a,Red(x))

The object is blue and not red; Alice perceives it; she experiences it as red; and she believes it is red.

Belief about causation

no ca(x,y)
an
Believes(a,ca(x,y))

Belief about the future

Believes(a,fu Event(e))
an
no fu Event(e)

This illustrates that a belief about a future event can be represented separately from whether the future event actually occurs.

Experience without truth

Experiences(a,Red(x))
an
no Red(x)

The experiential content and the external property are represented as distinct claims.

Inference

RC notation does not create inference merely by being written. Inference depends on the theory and proof rules adopted.

The inference operator: th

th means therefore or hence. It marks a conclusion derived from preceding premises or rules.

Socrates: the first complete inference

all x(
    Human(x)
    imp
    Mortal(x)
)

Human(s)

th

Mortal(s)

This uses universal instantiation and modus ponens.

Extended syllogism

all x(
    Philosopher(x)
    imp
    Human(x)
)

all x(
    Human(x)
    imp
    Mortal(x)
)

Philosopher(s)

th

Human(s)

th

Mortal(s)

Conjunction elimination

Human(s)
an
Mortal(s)

th

Human(s)

Modus ponens

Human(s)
an
Human(s)
imp
Mortal(s)

th

Mortal(s)

Identity substitution

x eq y
an
Red(x)

th

Red(y)

This represents the general substitution pattern, subject to the identity rules adopted by the formal system.

Basic inference rules

• Conjunction elimination
• Modus ponens
• Identity substitution
• Universal instantiation
• Existential introduction
• Existential elimination with appropriate witness restrictions
• Classical double negation

Inference is not implication

A imp B

represents a conditional within the language.

A

th

B

represents a derivation in which B is concluded from A together with whatever rule or premise licenses that inference.

Worked Examples

The examples progress from things and properties to integrated representations.

s

Human(s)
Philosopher(s)

ex x(
    Human(x)
    an
    Philosopher(x)
)

ex x ex y(
    Philosopher(x)
    an
    Philosopher(y)
    an
    x neq y
)

ex x(
    Philosopher(x)
    an
    ex y(
        Student(y)
        an
        Teaches(x,y)
    )
)

all x(
    Human(x)
    imp
    Mortal(x)
)

Human(s)

th

Mortal(s)

Child(s) at t₁
an
Adult(s) at t₂
an
t₁ < t₂

po(
    fu Victory(s)
)

ex x(
    Tree(x)
    an
    Green(x)
    an
    Perceives(a,x)
    an
    Experiences(a,Green(x))
    an
    Believes(a,Green(x))
)

ex x(
    Tree(x)
    an
    Blue(x)
    an
    no Green(x)
    an
    Perceives(a,x)
    an
    Experiences(a,Green(x))
    an
    Believes(a,Green(x))
)

ex x(
    Human(x)
    an
    ex y(
        Parent(y,x)
        an
        ex z(
            Human(z)
            an
            Knows(y,z)
        )
    )
)

Earthquake(e) at t₁
an
Collapse(c) at t₂
an
ca(e,c)
an
t₁ < t₂

ch(
    Event(e) at t₁
    an
    Event(e) at t₂
    an
    t₁ < t₂
)

ka(
    Opportunity(o)
    an
    Available(o) at t
)

ai(
    HistoricalEra(e)
    an
    Begins(e) at t₁
    an
    Ends(e) at t₂
)

am(
    Remembers(a,e)
    an
    Event(e) at t₁
)

do(a,OpenDoor)

OpenDoor
imp
do(a,EnterRoom)

rep(map,x)

map(map,x)

dep(image,x)

sim(model,system)

simul(model,system)

Blue(x)
an
no Red(x)
an
Believes(a,Red(x))

Object(x)
an
Red(x) at t₁
an
Perceives(a,x) at t₂
an
Experiences(a,Red(x)) at t₂
an
Believes(a,Red(x)) at t₂
an
Blue(x) at t₃
an
no Red(x) at t₃
an
t₁ < t₂
an
t₂ < t₃

The RC Master Example

A single example can combine objecthood, properties, time, perception, experience, belief and change.

A red object existed before Alice perceived it. Alice perceived it, experienced it as red, and believed it was red. The object later became blue.

ex x(
    Object(x)
    an
    Red(x) at t₁
    an
    Perceives(a,x) at t₂
    an
    Experiences(a,Red(x)) at t₂
    an
    Believes(a,Red(x)) at t₂
    an
    Blue(x) at t₃
    an
    no Red(x) at t₃
    an
    t₁ < t₂
    an
    t₂ < t₃
)

OBJECT EXISTS
      ↓
OBJECT IS RED
      ↓
ALICE PERCEIVES OBJECT
      ↓
ALICE EXPERIENCES REDNESS
      ↓
ALICE BELIEVES OBJECT IS RED
      ↓
OBJECT BECOMES BLUE

Master example with Chronos and Anamnesis

ex x(
    Object(x)
    an
    Red(x) at t₁
    an
    Perceives(a,x) at t₂
    an
    Experiences(a,Red(x)) at t₂
    an
    Believes(a,Red(x)) at t₂
    an
    Blue(x) at t₃
    an
    no Red(x) at t₃
    an
    ch(
        t₁ < t₂
        an
        t₂ < t₃
    )
    an
    am(
        Remembers(a,x)
    )
)

This demonstrates how temporal mode operators can be added without eliminating the underlying temporal relations.

Cheat Sheet

Extensions

RC is designed as an extensible framework. Extensions add specialised vocabulary and semantics while inheriting the underlying RC architecture and preserving distinctions between kinds of claim.

Possible model structure

M = <D,I,T,W,E,C,P,A,V,R,...>

D is a domain, I an interpretation, T temporal structure and W modal structure. Additional components may represent epistemic, causal, probabilistic, phenomenological, agency, value, information, mathematical, process, system, measurement and other extension-specific structures.

Agency and Action

Agency and Action RC extends RC with a formal vocabulary for agents, actions, intentions, goals, capacities, permissions, obligations and conditional actions.

Action

do(a,OpenDoor)

Future action

will do(a,OpenDoor)

Conditional action

A
imp
do(x,B)

This represents: if A occurs, x performs B. The action relation is distinct from the truth-functional conditional itself.

Conditional action modalities

A
imp
should do(x,B)

A
imp
must do(x,B)

A
imp
may do(x,B)

A
imp
can do(x,B)

Conditional future action

A
imp
will do(x,B)

Action and intention

Intends(x,A)
an
will do(x,A)

Whether intention entails future performance is a theory-dependent question and is not built into the representation itself.

Representation and Simulation

Representation RC distinguishes a thing from what it represents, models, depicts or simulates. These relations are especially important when moving between reality, models, images, symbols, simulations and beliefs.

Representation

rep(image,tree)

Mapping or modelling

map(model,system)

Depiction

dep(portrait,person)

Simulation

sim(simulation,climate_system)

Simulacrum

simul(copy,original)

Representation is not identity

rep(image,tree)
an
image neq tree

Simulation and belief

sim(model,system)
an
Believes(a,modelRepresentsSystem(model,system))

The representation relation can therefore be connected to epistemic claims without collapsing the representation into the thing represented.

Integrated representation

ex x(
    Object(x)
    an
    rep(image,x)
    an
    dep(image,x)
    an
    Perceives(a,image)
    an
    Believes(a,Represents(image,x))
)

Epistemic RC

Epistemic RC formalises claims about knowledge, belief, evidence, doubt, justification and information available to an agent.

Believes(x,p)
Knows(x,p)
Evidence(e,p)
Justified(x,p)
Doubts(x,p)

Belief and truth

Believes(x,p)
an
no p

This represents a false belief without collapsing belief into truth.

Evidence

Evidence(e,p)
an
Justifies(e,x,p)

th

Justified(x,p)

Knowledge and belief

Knows(x,p)
an
Believes(x,p)

A theory may define knowledge as involving belief together with additional conditions, but RC does not force one definition merely by using the predicate Knows.

Probability and Uncertainty

Probabilistic RC provides a way to represent numerical uncertainty without identifying probability with modality.

Prob(p) eq 0.7

Prob(p | q) eq 0.8

Probability can also be represented as changing with evidence or time.

Evidence(e,p) at t1
Prob(p | e) eq 0.8

Probability and possibility

po(p)
an
Prob(p) eq 0.01

Value and Normativity

Value RC provides a general vocabulary for representing evaluative and normative relations without building a particular ethical theory into RC itself.

Normative conditional

Action(a)
imp
sho(
    Good(Action(a))
)

This represents a normative relationship as a formula; a theory would be required to establish whether the conditional is valid.

This extension can support formal representation in ethics, aesthetics, decision theory and other value-related domains.

Information and Language

Information and Language RC formalises messages, meaning, reference, description, communication and symbolic representation.

Information(i)
Message(m)
Means(m,p)
RefersTo(w,x)
Describes(d,x)
Communicates(x,y,p)
Receives(y,p)

Communication through time

Communicates(x,y,p) at t1
Receives(y,p) at t2

Representation and language

Message(m)
an
rep(m,p)
an
Means(m,p)

This makes it possible to distinguish a proposition, its representation, its transmission and its reception.

Mathematical RC

Mathematical RC provides a common RC vocabulary for representing mathematical objects and quantitative relations.

Number(n)
Set(S)
Member(x,S)
Function(f)
Maps(f,x,y)
Mass(x) eq 5kg

Mathematical relation

Number(x)
an
Number(y)
an
x < y

The purpose is representational: mathematical structures can be incorporated into RC without confusing mathematical relations with empirical or causal relations.

Process and Systems RC

Process and Systems RC represents processes, components, transformations, dependencies and structured systems.

Process(p)
System(s)
Part(x,s)
Component(x,s)
DependsOn(x,y)
Transforms(p,x,y)

Process through time

Begins(p) at t1
Continues(p) at t2
Ends(p) at t3

Process and causation

Transforms(p,x,y)
an
ca(x,y)

This extension is useful for physical, biological, computational and social processes.

Identity and Persistence

Identity and Persistence RC develops RC's existing equality and distinctness machinery to represent identity through change and time.

x eq y at t1
an
x eq y at t2

Persists(x,t1,t2)

A thing may change properties while remaining numerically identical. Persistence therefore remains distinct from simple property continuity.

Identity through change

Red(x) at t1
an
Blue(x) at t2
an
x eq x at t1
an
x eq x at t2

Distinct things

x neq y
an
Red(x)
an
Blue(y)

Counterfactual RC

Counterfactual RC represents claims about what would have occurred under alternative conditions.

A cf B

Counterfactual dependence is distinct from ordinary implication and from causation.

Counterfactual action

A cf do(x,B)

This can represent a claim about what an agent would do under an alternative condition.

Measurement and Observation

Measurement RC distinguishes observation, measurement, recorded results and conscious perception.

Observes(x,y)
Measurement(m)
Measures(m,y)
Result(m,r)

Measurement is not perception

Measures(m,y)
neq
Perceives(x,y)

A physical measurement need not itself be a conscious experience. Conversely, a perception need not constitute a scientific measurement.

Measurement and time

Measures(m,y) at t1
an
Result(m,r) at t2
an
t1 < t2

Emergence

Emergence RC provides a formal way to represent claims that a property, entity or process emerges from a system.

System(S)
an
Property(P)
an
Emerges(P,S)

Emerges(P,S) represents the claim that P emerges from S; it does not by itself establish why P emerges, whether the emergence is causal, or which theory of emergence is correct.

Emergence through time

System(S)
an
no Property(P) at t1
an
Property(P) at t2
an
Emerges(P,S)
an
t1 < t2

Theology Calculus (TC)

Theology Calculus (TC) is a proposed domain extension of Reality Calculus for representing and reasoning about theological entities, properties, relations, doctrines, revelation, Scripture, religious experience, and theological arguments.

TC inherits the general RC machinery and adds theological vocabulary and structures without making the language itself commit to a particular theology.

God(G)
an
Divine(G)
an
Eternal(G)
an
Good(G)

Human(Christ)

all x(
    Human(x)
    imp
    Mortal(x)
)

th

Mortal(Christ)

Quantum Reality Calculus (QRC)

Quantum Reality Calculus (QRC) is a proposed domain extension of Reality Calculus for representing and reasoning about quantum-theoretical systems, states, observables, measurements, amplitudes, probabilities, superposition, entanglement, operators, uncertainty, and quantum state evolution.

QRC is a formal representational extension of RC, not a new interpretation of quantum mechanics. Particular physical theories or interpretations can be represented by definitions, axioms, models, and inference rules.

Core vocabulary

Quantum states

QuantumState(ψ)

QuantumSystem(S)
an
hasState(S,ψ)

Superposition

Superposition is not ordinary classical disjunction. In RC, A or B expresses logical disjunction; in QRC, a superposition is represented as a structure involving a quantum state and amplitudes.

QuantumState(ψ)
an
superposition(ψ,A,B)
an
amplitude(ψ,A,α)
an
amplitude(ψ,B,β)

Probability and amplitude

probability(ψ,r,p)

amplitude(ψ,r,α)

QRC distinguishes numerical probability from amplitude and from RC's qualitative modal operators such as possible and necessary.

Measurement

Measurement(M)
an
measures(M,S,O)
an
result(M,r)

Entanglement and causation

entangled(S1,S2)

Entanglement and causal dependence are distinct relations. QRC does not infer Causes(S1,S2) merely from entangled(S1,S2).

Observables and non-commutativity

Observable(X)
an
Observable(P)
an
no commutes(X,P)

This can represent the relevant structural fact for theories involving non-commuting observables and uncertainty relations, while leaving the precise mathematical semantics to the associated QRC theory.

Quantum evolution

QuantumState(ψ1)
an
QuantumState(ψ2)
an
evolves(ψ1,ψ2,t1,t2)

State update

Measurement(M)
an
updates(M,ψ,ψ')

The update relation does not by itself commit QRC to a particular interpretation of quantum mechanics. Whether an update is interpreted as physical collapse, conditional state assignment, information update, or another theoretical construct belongs to the associated theory.

Key distinctions

QRC architecture

Reality Calculus (RC)
        ↓
Quantum Reality Calculus (QRC)
        ↓
Quantum-theoretical vocabulary
        ↓
Definitions + axioms + models + inference rules

QRC therefore provides an RC-compatible representational layer for quantum theory while keeping the underlying mathematical and interpretive commitments explicit.

Limitations and Future Formal Work

Version 1 is a formal proposal for a unified logical language and extensible semantic architecture. It does not claim that every extension already has complete semantics or proof theory.

Outstanding research

1. Complete BNF grammar
2. Fixed precedence and associativity
3. Exact temporal semantics
4. Formal semantics for Chronos, Kairos, Aion and Anamnesis
5. Exact modal semantics
6. Causal semantics
7. Epistemic semantics
8. Phenomenological semantics
9. Counterfactual semantics
10. Representation and simulation semantics
11. Complete proof theory
12. Soundness and completeness results
13. Consistency, decidability and complexity analysis
14. Comparison with established logical systems

Recommended development sequence

RC Version 1
     ↓
Complete RC-Core grammar
     ↓
Formal RC-Core semantics
     ↓
RC-Core proof system
     ↓
Soundness / completeness
     ↓
Temporal RC
     ↓
Chronos / Kairos / Aion / Anamnesis
     ↓
Modal RC
     ↓
Epistemic RC
     ↓
Phenomenological RC
     ↓
Causal RC
     ↓
Counterfactual RC
     ↓
Agency & Action RC
     ↓
Representation & Simulation RC
     ↓
Probabilistic RC
     ↓
Information & Language RC
     ↓
Mathematical RC
     ↓
Process & Systems RC
     ↓
Identity & Persistence RC
     ↓
Measurement RC
     ↓
Integrated RC system

A Common Language for Different Layers of Reality

RC's central value lies in providing a common formal architecture in which different categories of claim can be represented without collapsing them into one another.

The same language can move from Red(x) to an integrated representation involving existence, causation, temporal change, modality, perception and belief while preserving the distinctions between those layers.

The expanded RC progression

THING
  ↓
PROPERTY
  ↓
EXISTENCE
  ↓
RELATION
  ↓
RULE
  ↓
INFERENCE
  ↓
TIME
  ↓
CHRONOS
  ↓
KAIROS
  ↓
AION
  ↓
ANAMNESIS
  ↓
MODALITY
  ↓
ONTOLOGY
  ↓
PERCEPTION
  ↓
EXPERIENCE
  ↓
BELIEF
  ↓
KNOWLEDGE
  ↓
ACTION
  ↓
CAUSATION
  ↓
COUNTERFACTUAL
  ↓
REPRESENTATION
  ↓
SIMULATION
  ↓
INFORMATION
  ↓
PROBABILITY
  ↓
VALUE
  ↓
PROCESS
  ↓
SYSTEM