Reality Calculus (RC) — A Unified Formal Language

Introduction

Overview

Reality Calculus (RC) is a unified formal language for representing and reasoning about reality.

It provides a common framework for expressing things, properties, relations, existence, causation, time, modality, perception, experience, and belief.

A central principle is that different kinds of claims should not be treated as interchangeable. What exists is not necessarily what is perceived; what is perceived is not necessarily what is experienced; and what is experienced is not necessarily what is believed.

This allows the reality being represented to be distinguished from the representations used to describe, perceive, experience, or believe it.

RC is a formal language rather than a theory of reality by itself. Its expressions acquire specific inferential force through definitions, axioms, theories, and explicit bridge principles.

RC-Core

Core Vocabulary

RC	Meaning
`x`	Thing or entity
`A(x)`	x has property A
`ex x(...)`	There exists an x satisfying...
`all x(...)`	For every x...
`an`	And
`or`	Or
`no`	Not
`imp`	Implies
`iff`	If and only if
`if`	Introduces a conditional
`el`	Else; otherwise; the alternative consequent when the condition is not satisfied
`th`	Therefore; thus; logical conclusion from preceding premises
`eq`	Identical
`neq`	Distinct
`R(x,y)`	Relation between x and y
`ca(x,y)`	x causes y
`do(x,A)`	x performs action A
`will A`	A will occur or be performed
`A imp will do(I,B)`	If A happens, I will do B
`rep(x,y)`	x represents y
`map(x,y)`	x maps or models y
`dep(x,y)`	x depicts y
`sim(x,y)`	x simulates y
`simul(x,y)`	x is a simulacrum of y
`pr A`	A was previously true
`nu A`	A is true now
`fu A`	A will subsequently be true
`A at t`	A at a particular time
`po A`	A is possible
`nec A`	A is necessary
`can A`	A is within capacity
`may A`	A is permitted
`mus A`	A is required
`wou A`	A would be the case
`cou A`	A could be the case
`sho A`	A should be the case

Formal Syntax

Syntax

Atomic properties

Red(x)
Alive(x)
Human(x)

These express properties of things.

Combined propositions

Red(x) an Round(x)

Red(x) or Blue(x)

no Red(x)

Implication

Human(x) imp Mortal(x)

The operator imp expresses a conditional logical relationship: if the antecedent holds, the consequent follows according to the stated rule.

Conditional alternatives

if A th B el C

This means: if A, then B; otherwise, C.

The el operator means else or otherwise. It introduces the alternative consequent when the condition is not satisfied.

The conditional form can be understood in terms of two implications:

A imp B
no A imp C

el is a conditional alternative operator. It does not replace imp; rather, imp expresses the underlying implication while el provides a compact way to express what occurs otherwise.

Biconditional

Human(x) iff Mortal(x)

Logical conclusion

Human(s)
Human(x) imp Mortal(x)
th Mortal(s)

The operator th marks the conclusion drawn from the preceding premises or derivation. It corresponds to “therefore,” “thus,” or “hence” in ordinary language.

th is an inferential operator. It does not itself state a property or relation in the world; it marks the transition from premises to a logical conclusion.

Quantification

Existence

ex x(
    Human(x)
)

There exists at least one x such that x is human.

Multiple conditions

ex x(
    Human(x) an Mortal(x)
)

Two distinct things

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

Universal statements

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

Nested existence

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

Relations & Causation

Relations

Loves(x,y)
Parent(x,y)
Near(x,y)

Relations connect two or more things.

Causation

ca(Fire(x), Smoke(x))

Causation is represented separately from logical implication.

Temporal causation

ca(
    Event(x) at t₁,
    Event(y) at t₂
)

Causation is not the same as implication. An implication is a logical relationship between propositions; causation represents a causal relationship between events or states.

Action Extension

Action & Conditional Action

RC can represent actions performed by agents and can distinguish an action itself from propositions about what is possible, permitted, required, or expected.

Action

do(x,A)

x performs action A.

Future occurrence or performance

will A

A will occur or be performed in the future.

Conditional future action

A imp will do(I,B)

If A happens, I will do B.

Conditional action with an alternative

if A th will do(I,B) el will do(I,C)

If A happens, I will do B; otherwise, I will do C.

The el clause represents the alternative action associated with the negation of the stated condition. At the underlying conditional level, this corresponds to:

A imp will do(I,B)
no A imp will do(I,C)

Conditional modal actions

RC	Meaning
`A imp should do(I,B)`	If A happens, I should do B.
`A imp must do(I,B)`	If A happens, I must do B.
`A imp may do(I,B)`	If A happens, I may do B.
`A imp can do(I,B)`	If A happens, I can do B.

Examples

do(a,Open(door))

will Rain()

Rain() imp will do(a,Close(window))

Alarm(a) imp should do(a,Leave())

Emergency(a) imp must do(a,Stop())

Invitation(a) imp may do(a,Enter())

ToolAvailable(a) imp can do(a,Repair(x))

if Rain() th will do(a,Close(window))
el will do(a,Open(window))

The conditional operator imp connects a condition with a consequent. The if ... th ... el ... construction provides a compact conditional form with an explicit alternative. Action expressions specify what an agent does, while will, should, must, may, and can qualify the consequent in different ways. These distinctions should not be collapsed.

Temporal Extension

Time

RC	Meaning
`pr A`	A was previously true
`nu A`	A is true now
`fu A`	A will subsequently be true
`A at t`	A at a particular time
`t₁ < t₂`	t₁ precedes t₂
`before A B`	A occurs before B
`after A B`	A occurs after B
`during A B`	A occurs during B
`until A B`	A continues until B
`since A B`	A has held or occurred since B
`while A B`	A and B overlap or occur concurrently

Temporal relations

RC can also express temporal relations among events, states, and processes. These relations are distinct from the temporal modes below.

before(A,B)
after(A,B)
during(A,B)
until(A,B)
since(A,B)
while(A,B)

while(A,B) expresses temporal overlap or concurrency: A occurs while B occurs. It does not by itself specify which event begins first, which ends first, or whether their durations are equal.

while(
    do(a,Work(x)),
    Event(y)
)

before(Event(a),Event(b))

during(Event(a),Process(b))

until(
    State(a),
    Event(b)
)

since(
    State(a),
    Event(b)
)

Temporal modes

RC can distinguish different modes in which temporal phenomena may be represented. These modes do not replace pr, nu, fu, or at. Instead, they qualify the character or perspective of temporal representation.

RC	Term	Meaning
`ch A`	Chronos	A considered in measured, sequential, ordered, or clock-based time.
`ka A`	Kairos	A considered in significant, opportune, timely, or qualitatively appropriate time.
`ai A`	Aion	A considered across an age, era, enduring period, or extended temporal horizon.
`am A`	Anamnesis	A considered as remembered, recollected, or retrieved from temporal memory.

Chronos

ch Event(x)

The event is considered in measured or sequential time.

ch(
    Event(x) at t₁ an
    Event(y) at t₂ an
    t₁ < t₂
)

The events are considered according to an ordered temporal sequence.

ch(
    ca(
        Fire(x) at t₁,
        Smoke(x) at t₂
    ) an
    t₁ < t₂
)

The causal sequence is represented in measured temporal order.

Kairos

ka Event(x)

The event is considered in terms of its significant, opportune, or appropriate time.

ka(
    Opportunity(x) an
    Action(y)
)

The action is considered in relation to a significant or opportune temporal circumstance.

ka(
    Action(a) an
    Opportunity(a) an
    may Action(a)
)

The action is considered as timely or appropriate within a particular opportunity, while permission is represented separately by may.

Aion

ai Era(x)

The entity or state is considered across an extended temporal horizon such as an age or era.

ai(
    Civilization(x) at t₁ an
    Civilization(x) at t₂ an
    t₁ < t₂
)

The civilization is considered across an extended temporal span rather than as an isolated moment.

ai(
    Civilization(x) an
    Stable(x)
)

The state of the civilization is considered from the perspective of enduring or extended historical time.

Anamnesis

am Event(x)

The event is considered as remembered or recollected in time.

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

A remembered event is related to the earlier event that occurred at a particular time.

am(
    Perceives(a,x) at t₁ an
    Experiences(a,Red(x)) at t₁ an
    Believes(a,Red(x)) at t₂
)

The original perception and experience occurred earlier, while the later belief concerns what was previously experienced.

Combining temporal modes

ch(
    Event(x) at t₁ an
    Event(y) at t₂ an
    t₁ < t₂
)

ka(
    Event(y) an
    Opportunity(y)
)

ai(
    Civilization(x) at t₁ an
    Civilization(x) at t₂
)

am(
    Remembers(a,Event(x))
)

These expressions can describe the same broader reality from different temporal perspectives: sequential time, significant time, extended historical time, and remembered time.

Temporal mode with causation

ch(
    ca(
        Event(x) at t₁,
        Event(y) at t₂
    ) an
    t₁ < t₂
)

Chronos emphasizes the ordered temporal sequence of the causal events.

Temporal mode with modality

ka(
    po(
        fu Action(x)
    )
)

A future action is considered possible within a significant or opportune temporal context.

Temporal mode with perception and belief

am(
    Perceives(a,x) at t₁ an
    Experiences(a,Red(x)) at t₁ an
    Believes(a,Red(x)) at t₂
)

The remembered perspective concerns an earlier perception and experience and a later belief about that experience.

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)

The temporal operators and temporal modes are an extension layer. Their precise model-theoretic semantics remain an area for further formal development. The modes are intended to distinguish temporal perspectives rather than collapse all temporal representation into a single notion of time.

Modal Extension

Modality

RC	Meaning
`po A`	A is possible
`nec A`	A is necessary
`can A`	A is within capacity
`may A`	A is permitted
`mus A`	A is required
`wou A`	A would be the case
`cou A`	A could be the case
`sho A`	A should be the case

Possibility

po Open(x)

Necessity

nec Mortal(x)

Capacity

can Move(x)

Permission

may Enter(x)

Requirement

mus Stop(x)

Would

wou Help(x,y)

Could

cou Win(x)

Should

sho Help(x,y)

Possibility, capacity, permission, obligation, counterfactuality, and normative recommendation are distinct modal concepts and should not be collapsed into a single operator.

Representation & Simulation

Representation, Mapping, Depiction & Simulation

RC distinguishes reality from the representational structures used to describe, model, depict, simulate, or stand in for it. These relations extend RC's existing distinctions among ontology, perception, experience, and belief.

RC	Meaning
`rep(x,y)`	x represents y; the broadest representational relation.
`map(x,y)`	x maps or models y.
`dep(x,y)`	x depicts y.
`sim(x,y)`	x simulates y.
`simul(x,y)`	x is a simulacrum of y.

Representation

rep(Map(x),Territory(y))

The first entity represents the second.

Mapping or modeling

map(Model(x),System(y))

A model maps or models a system.

Depiction

dep(Image(x),Person(y))

An image depicts a person.

Simulation

sim(Simulation(x),Process(y))

A simulation simulates a process.

Simulacrum

simul(Representation(x),Object(y))

The representation is treated as a simulacrum of the object.

Combined representation and belief

rep(Image(x),Object(y)) an
Perceives(a,Image(x)) an
Experiences(a,Red(y)) an
Believes(a,Red(y))

A representation can be perceived and experienced by an agent, and the agent can form beliefs concerning what it represents. These levels remain distinct: the representation is not thereby identical with the represented object.

Representation, mapping, depiction, simulation, and simulacrum are related but distinct relations. In particular, sim(x,y) expresses a simulating relation, whereas simul(x,y) expresses that x is a simulacrum of y.

Ontology

Ontology concerns what exists and what things are.

Object existence

ex x(
    Object(x)
)

Object with properties

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

The existence of an object is represented independently from later claims about perception or belief concerning that object.

Phenomenology

Perception & Experience

Perception

Perceives(p,x)

Person p perceives object x.

Experience

Experiences(p,A(x))

Person p experiences the property or state A(x).

Perception does not guarantee reality

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

This does not by itself entail that the object is actually red.

Epistemology

Experience & Belief

Belief

Believes(p,A(x))

Person p believes that A(x).

False belief

Believes(p,Red(x)) an
no Red(x)

The person believes that x is red, while x is not actually red.

Perceptual error

Perceives(p,x) an
Experiences(p,Red(x)) an
no Red(x)

Belief about causation

Believes(p,ca(A,B))

Belief does not entail truth. Perception does not entail the perceived property. Experience does not entail ontology.

Inference

Inference is the process of deriving a conclusion from one or more premises according to an inferential rule. RC uses th to explicitly mark the conclusion of an inference.

The conclusion operator

A
B
th C

Here A and B are premises and C is the conclusion. The operator th means “therefore,” “thus,” or “hence.”

The operator th marks an inferential step. It should not be confused with imp: imp expresses an implication between propositions, while th marks the conclusion actually derived from preceding premises.

Socrates is mortal

Human(s)
Human(x) imp Mortal(x)
th Mortal(s)

The first line states that Socrates is human. The second gives the rule that humans are mortal. The conclusion follows by applying the rule to Socrates.

Universal syllogistic form

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

Human(s)

th Mortal(s)

The universal rule applies to every human. Since Socrates is human, the rule yields the conclusion that Socrates is mortal.

Modus Ponens

A imp B
A
th B

If A implies B, and A is true, then B follows.

Modus Tollens

A imp B
no B
th no A

If A implies B, and B is false, then A must be false.

Disjunctive Syllogism

A or B
no A
th B

If A or B is the case, and A is not the case, then B follows.

Hypothetical Syllogism

A imp B
B imp C
th A imp C

If A implies B and B implies C, then A implies C.

Conditional alternative

if A th B el C

This expresses a two-way conditional structure: B follows when A holds, while C is the alternative when A does not hold.

Its underlying conditional form is:

A imp B
no A imp C

Inference is not the same as implication

A imp B

This states an implication: if A, then B.

A imp B
A
th B

This performs an inference: given the premises, B is concluded.

RC therefore distinguishes the logical relationship expressed by imp, the conditional alternative expressed by el, and the inferential operation marked by th.

Rules matter

RC distinguishes between representing facts and deriving conclusions from them. A conclusion requires an explicit inferential rule or theory connecting the premises to the conclusion.

A representation such as Believes(p,A(x)) does not entail A(x). The fact that something is believed is distinct from the truth of what is believed.

Worked Examples

01 · Things and properties

Red(x)
Round(x)
Heavy(x)

These describe properties of a thing.

02 · Existence

ex x(
    Human(x)
)

There exists a human.

03 · Two distinct philosophers

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

There are at least two distinct philosophers.

04 · Relation

Parent(x,y)

x stands in the parent relation to y.

05 · Complete syllogism

Human(s)
Human(x) imp Mortal(x)
th Mortal(s)

Socrates is mortal.

06 · Modus Ponens

Rain() imp Wet(Ground)
Rain()
th Wet(Ground)

If rain implies that the ground is wet, and it is raining, the ground is concluded to be wet.

07 · Modus Tollens

Rain() imp Wet(Ground)
no Wet(Ground)
th no Rain()

If rain implies a wet ground, and the ground is not wet, then rain is ruled out by modus tollens.

08 · Disjunctive Syllogism

Rain() or Snow()
no Rain()
th Snow()

If it is either raining or snowing, and it is not raining, snowing follows.

09 · Hypothetical Syllogism

Rain() imp Wet(Ground)
Wet(Ground) imp Slippery(Ground)
th Rain() imp Slippery(Ground)

The two implications combine into a chain: rain implies wetness, wetness implies slipperiness, therefore rain implies slipperiness.

10 · Conditional alternative

if Rain()
th Wet(Ground)
el Dry(Ground)

If it rains, the ground is wet; otherwise, the ground is dry.

At the underlying conditional level, this corresponds to:

Rain() imp Wet(Ground)
no Rain() imp Dry(Ground)

11 · Conditional action with alternative

if Rain()
th will do(a,Close(window))
el will do(a,Open(window))

If it rains, the agent will close the window; otherwise, the agent will open it.

12 · Temporal change

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

The object is red at an earlier time and blue at a later time.

13 · While — temporal overlap

while(
    Event(a),
    Event(b)
)

Event a occurs while event b occurs; the two events temporally overlap.

14 · Action and conditional action

do(a,Open(door))

Rain() imp will do(a,Close(window))

Alarm(a) imp must do(a,Leave())

Invitation(a) imp may do(a,Enter())

The examples distinguish an action, a future action, and conditional actions qualified by requirement or permission.

15 · Representation and simulation

rep(Map(x),Territory(y))
map(Model(x),System(y))
dep(Image(x),Person(y))
sim(Simulation(x),Process(y))
simul(Representation(x),Object(y))

The expressions distinguish representation, modeling, depiction, simulation, and simulacrum.

16 · Combined action and temporal overlap

while(
    do(a,Observe(x)),
    Event(y)
)

Observation(a,x) imp will do(a,Record(x))

An action can be situated within a temporal overlap and can occur conditionally after another event or state.

17 · Possible future

po(
    fu Open(x)
)

It is possible that x will subsequently be open.

18 · Chronos — measured temporal sequence

ch(
    Event(x) at t₁ an
    Event(y) at t₂ an
    t₁ < t₂
)

The events are considered in measured and sequential temporal order.

19 · Chronos and causation

ch(
    ca(
        Fire(x) at t₁,
        Smoke(x) at t₂
    ) an
    t₁ < t₂
)

The causal relationship is represented together with its ordered temporal sequence.

20 · Kairos — significant or opportune time

ka(
    Opportunity(x) an
    Action(y)
)

The action is considered in relation to a significant or opportune temporal circumstance.

21 · Kairos and modality

ka(
    po(
        fu Action(x)
    )
)

A future action is considered as a possibility within a significant or opportune temporal context.

22 · Aion — extended temporal horizon

ai(
    Civilization(x) at t₁ an
    Civilization(x) at t₂ an
    t₁ < t₂
)

The civilization is considered across an extended historical period rather than as an isolated moment.

23 · Aion and enduring states

ai(
    Civilization(x) an
    Stable(x)
)

The state of the civilization is considered from the perspective of an enduring temporal horizon.

24 · Anamnesis — remembered time

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

A remembered event is related to the earlier event that occurred at a particular time.

25 · Anamnesis with perception, experience and belief

am(
    Perceives(a,x) at t₁ an
    Experiences(a,Red(x)) at t₁ an
    Believes(a,Red(x)) at t₂
)

The remembered perspective connects an earlier perception and experience with a later belief.

26 · Four temporal perspectives

ch(
    Event(x) at t₁ an
    Event(y) at t₂ an
    t₁ < t₂
)

ka(
    Event(y) an
    Opportunity(y)
)

ai(
    Civilization(x) at t₁ an
    Civilization(x) at t₂
)

am(
    Remembers(a,Event(x))
)

The expressions distinguish four temporal perspectives: sequential time, significant time, extended historical time, and remembered time.

27 · Temporal modes with reality and perception

ex x(
    Object(x) an
    Red(x) at t₁ an

    ch(
        Perceives(a,x) at t₂
    ) an

    ka(
        Opportunity(a) at t₂
    ) an

    ai(
        Object(x) at t₁ an
        Object(x) at t₃
    ) an

    am(
        Believes(a,Red(x)) at t₄
    ) an

    t₁ < t₂ an
    t₂ < t₃ an
    t₃ < t₄ an

    rep(Image(x),Object(x)) an
    sim(Simulation(x),Process(y)) an

    Event(z) imp will do(a,Record(x))
)

A single object is represented across several temporal perspectives: its chronological existence and perception, the significance of the moment, its persistence across an extended period, and the later recollection or belief concerning it.

28 · Temporal modes with causation and modality

ch(
    ca(
        Event(x) at t₁,
        Event(y) at t₂
    ) an
    t₁ < t₂
)

ka(
    may Action(a)
)

ai(
    State(x) at t₁ an
    State(x) at t₃
)

am(
    Remembers(a,Event(x))
)

po(
    fu Event(z)
)

The example combines chronological causation, significant timing, extended duration, recollection, and future possibility within one RC representation.

29 · Reality, perception, experience and belief

ex x(
    Object(x) an
    Red(x) an
    Perceives(a,x) an
    Experiences(a,Red(x)) an
    Believes(a,Red(x))
)

An object exists, is red, is perceived by Alice, is experienced by Alice as red, and is believed by Alice to be red.

30 · Perceptual error

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

Alice perceives an object and experiences it as red, believes it is red, but it is not actually red.

Integrated Example

Master Example

An object exists and is red at an earlier time. Alice later perceives it and experiences it as red. The moment of perception is represented chronologically, while also being treated as a significant opportunity. The object persists across an extended temporal horizon and later changes state. Alice subsequently remembers the earlier event and continues to believe that the object was red.

ex x(
    Object(x) an

    Red(x) at t₁ an

    ch(
        Perceives(a,x) at t₂ an
        Experiences(a,Red(x)) at t₂ an
        t₁ < t₂
    ) an

    ka(
        Opportunity(a) at t₂ an
        may Observe(a,x)
    ) an

    ai(
        Object(x) at t₁ an
        Object(x) at t₃ an
        t₁ < t₃
    ) an

    Blue(x) at t₃ an
    no Red(x) at t₃ an

    am(
        Remembers(a,Perceives(a,x)) at t₄ an
        Believes(a,Red(x)) at t₄
    ) an

    t₂ < t₃ an
    t₃ < t₄
)

Flow

01 · Object exists

Object(x)

02 · Object is initially red

Red(x) at t₁

03 · Chronos

ch(Perceives(a,x) at t₂)

04 · Alice experiences redness

Experiences(a,Red(x)) at t₂

05 · Kairos

ka(Opportunity(a) at t₂)

06 · Permission

may Observe(a,x)

07 · Aion

ai(Object(x) at t₁ an Object(x) at t₃)

08 · Object changes

Blue(x) at t₃

09 · Redness ends

no Red(x) at t₃

10 · Anamnesis

am(Remembers(a,Perceives(a,x)) at t₄)

11 · Later belief

Believes(a,Red(x)) at t₄

12 · Representation

rep(Image(x),Object(x))

13 · Simulation

sim(Simulation(x),Process(y))

14 · Conditional action

Event(z) imp will do(a,Record(x))

15 · Conditional alternative

if Event(z) th B el C

16 · Temporal order

t₁ < t₂ < t₃ < t₄

What the example demonstrates

The master example shows how RC can represent several different aspects of one reality without collapsing them into one another.

Ontology

The object exists independently of Alice's perception or belief.

Chronos

The events are ordered within sequential temporal structure.

Kairos

The moment is represented as significant or opportune rather than merely as a point on a clock.

Aion

The object is considered across an extended temporal horizon.

Anamnesis

A later mental state refers back to an earlier event.

Phenomenology

Alice perceives and experiences the object.

Epistemology

Alice forms and retains a belief about the object.

Action

RC can represent actions performed by agents and conditional actions that follow specified circumstances.

Conditional alternative

el represents the otherwise branch of a conditional construction.

Representation

RC distinguishes what represents, models, depicts, simulates, or stands as a simulacrum for something else.

Temporal overlap

while represents concurrency or temporal overlap without collapsing the events into one.

Inference

th explicitly marks a logical conclusion derived from preceding premises.

Temporal change

The object is red at one time and blue at a later time.

The temporal modes do not assert that time itself has four independent substances or dimensions. They provide distinct representational perspectives through which temporally situated propositions can be expressed. Their exact formal semantics remain subject to further development.

Reference

Cheat Sheet

RC	Meaning
`x`	Thing
`A(x)`	x is A
`ex x(...)`	There exists an x
`all x(...)`	For every x
`an`	And
`or`	Or
`no`	Not
`imp`	Implies
`iff`	If and only if
`if`	Introduces a conditional
`el`	Else; otherwise; alternative conditional branch
`th`	Therefore; thus; logical conclusion
`eq`	Identical
`neq`	Distinct
`R(x,y)`	Relation
`ca(x,y)`	Causes
`do(x,A)`	x performs action A
`will A`	A will occur or be performed
`A imp will do(I,B)`	If A happens, I will do B
`if A th B el C`	If A, then B; otherwise C
`A imp should/must/may/can do(I,B)`	Conditional normative, required, permitted, or capacity-based action
`rep(x,y)`	x represents y
`map(x,y)`	x maps/models y
`dep(x,y)`	x depicts y
`sim(x,y)`	x simulates y
`simul(x,y)`	x is a simulacrum of y
`pr A`	A was previously true
`nu A`	A is true now
`fu A`	A will subsequently be true
`A at t`	A at a particular time
`t₁ < t₂`	t₁ precedes t₂
`before A B`	A occurs before B
`after A B`	A occurs after B
`during A B`	A occurs during B
`until A B`	A continues until B
`since A B`	A has held or occurred since B
`while A B`	A and B overlap or occur concurrently
`ch A`	Chronos — measured, sequential, or ordered time
`ka A`	Kairos — significant, opportune, or qualitatively appropriate time
`ai A`	Aion — age, era, enduring period, or extended temporal horizon
`am A`	Anamnesis — remembered or recollected time
`po A`	A is possible
`nec A`	A is necessary
`can A`	A is within capacity
`may A`	A is permitted
`mus A`	A is required
`wou A`	A would be the case
`cou A`	A could be the case
`sho A`	A should be the case
`Object(x)`	x is an object
`Perceives(p,x)`	p perceives x
`Experiences(p,A(x))`	p experiences A(x)
`Believes(p,A(x))`	p believes A(x)

Inference at a Glance

Rule	Form	Conclusion
Modus Ponens	`A imp B A th B`	If A implies B and A holds, conclude B.
Modus Tollens	`A imp B no B th no A`	If A implies B and B does not hold, conclude no A.
Disjunctive Syllogism	`A or B no A th B`	If A or B and not A, conclude B.
Hypothetical Syllogism	`A imp B B imp C th A imp C`	If A implies B and B implies C, conclude A implies C.
Conditional Alternative	`if A th B el C`	If A, then B; otherwise C.

Action & Representation at a Glance

Construct	Meaning	Example
`do(x,A)`	x performs A	`do(a,Open(door))`
`will A`	A will occur	`will Rain()`
`A imp will do(I,B)`	If A happens, I will do B	`Rain() imp will do(a,Close(window))`
`if A th B el C`	If A, then B; otherwise C	`if Rain() th Wet(Ground) el Dry(Ground)`
`rep(x,y)`	x represents y	`rep(Image(x),Object(y))`
`map(x,y)`	x models y	`map(Model(x),System(y))`
`dep(x,y)`	x depicts y	`dep(Image(x),Person(y))`
`sim(x,y)`	x simulates y	`sim(Simulation(x),Process(y))`
`simul(x,y)`	x is a simulacrum of y	`simul(Representation(x),Object(y))`

Temporal Modes at a Glance

Mode	Question it emphasizes	Example
`ch`	When does something occur in sequential or measured time?	`ch(Event(x) at t₁)`
`ka`	When is something significant, timely, or opportune?	`ka(Opportunity(x))`
`ai`	How does something exist across an extended age or era?	`ai(Civilization(x))`
`am`	How is an earlier event represented through recollection?	`am(Remembers(a,Event(x)))`

Reality Calculus is designed to keep distinct what exists, what is true, what is perceived, what is experienced, what is believed, what is possible, what is necessary, what causes what, what follows logically from what, how something is situated in time, how agents act, how conditional alternatives are expressed, and how things can represent, model, depict, simulate, or stand as simulacra of other things.