Possibility Ontology

FLOW_ALGEBRA PossibilityOntology v1.0

TARGET_URL := "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/modal-logic/possibility-ontology.html"
LANGUAGE := "en-GB"
DOMAIN := MathematicalOntologies > MathematicalLogicOntologies > ModalLogic > PossibilityOntology

BEGIN_FLOW

ENTITY Possibility {
    TYPE := ModalConcept
    SYMBOL := "◇"
    DEFINITION := "A modal condition expressing that a proposition is true at at least one world accessible from the current evaluation world under the selected modal semantics."
}

ENTITY PossibilityOperator {
    TYPE := UnaryModalOperator
    SYMBOL := "◇"
    OPERAND := ModalFormula
    RESULT := ModalFormula
}

RELATIONSHIP Possibility --is_expressed_by--> PossibilityOperator
RELATIONSHIP PossibilityOperator --belongs_to--> ModalLogic
RELATIONSHIP PossibilityOperator --dual_of--> NecessityOperator
RELATIONSHIP Possibility --is_interpreted_by--> KripkeSemantics
RELATIONSHIP Possibility --depends_on--> AccessibilityRelation
RELATIONSHIP Possibility --applies_to--> Proposition

SYNTAX {
    IF P IS ModalFormula
    THEN ◇P IS ModalFormula
}

READING {
    ◇P := "It is possible that P"
    ◇P := "P holds at some accessible alternative"
}

KRIPKE_SEMANTICS {
    M,w ⊨ ◇P
    IFF
        EXISTS v:
            R(w,v)
            AND M,v ⊨ P
}

POSSIBLE_AT(Model M, World w, Formula P) {
    FOR_EACH world v IN Worlds(M) {
        IF R(w,v)
           AND M,v ⊨ P
        THEN
            RETURN TRUE
    }

    RETURN FALSE
}

ACCESSIBLE_WORLD_SET {
    ACCESSIBLE(w) :=
        {v ∈ W | R(w,v)}

    POSSIBILITY_CONDITION :=
        EXISTS v ∈ ACCESSIBLE(w):
            P(v)=TRUE
}

NO_SUCCESSOR_CASE {
    IF ACCESSIBLE(w)=empty_set
    THEN
        M,w ⊭ ◇P
        FOR every P
        under standard existential Kripke semantics

    NOTE :=
        serial frame conditions guarantee at least one accessible successor
}

DUALITY {
    ◇P ≡ ¬□¬P

    DERIVATION {
        ◇P
        IFF some accessible world satisfies P
        IFF it is not necessary that P is false
        IFF ¬□¬P
    }
}

NEGATION_OF_POSSIBILITY {
    ¬◇P ≡ □¬P

    INTERPRETATION :=
        "P is impossible exactly when P is false at every accessible world."
}

NECESSITY_FROM_NEGATION {
    □P ≡ ¬◇¬P
}

LOGICAL_CONSISTENCY_DISTINCTION {
    MODAL_POSSIBILITY_AT_WORLD :=
        truth_at_some_world_accessible_from_specific_world

    SATISFIABILITY :=
        truth_at_some world in some model under specified semantics

    PROBABILITY :=
        numerical measure of uncertainty

    CONSTRAINT :=
        these concepts MUST NOT be automatically conflated
}

FRAME_REFLEXIVITY {
    PROPERTY :=
        FOR_ALL w:
            R(w,w)

    CONSEQUENCE_FOR_POSSIBILITY :=
        P -> ◇P

    DERIVATION :=
        if P holds at w and w accesses itself,
        then an accessible P-world exists
}

FRAME_SERIALITY {
    PROPERTY :=
        FOR_ALL w:
            EXISTS v:
                R(w,v)

    CONSEQUENCE :=
        □P -> ◇P

    MODAL_AXIOM := D
}

FRAME_TRANSITIVITY {
    PROPERTY :=
        FOR_ALL w,v,u:
            (R(w,v) AND R(v,u))
            -> R(w,u)

    RELATED_MODAL_AXIOM :=
        □P -> □□P
}

FRAME_SYMMETRY {
    PROPERTY :=
        FOR_ALL w,v:
            R(w,v) -> R(v,w)

    ASSOCIATED_MODAL_AXIOM :=
        P -> □◇P
}

FRAME_EUCLIDEAN {
    PROPERTY :=
        FOR_ALL w,v,u:
            (R(w,v) AND R(w,u))
            -> R(v,u)

    CONSEQUENCE_FOR_POSSIBILITY :=
        ◇P -> □◇P

    MODAL_AXIOM := 5
}

SYSTEM_K {
    BASE :=
        normal_modal_logic

    POSSIBILITY_DEFINED_BY :=
        ◇P ≡ ¬□¬P
}

DISTRIBUTION_OVER_DISJUNCTION {
    ◇(P ∨ Q) ≡ ◇P ∨ ◇Q
}

ONE_WAY_CONJUNCTION_RELATION {
    ◇(P ∧ Q) -> (◇P ∧ ◇Q)

    CONVERSE :=
        NOT valid in general

    REASON :=
        P and Q may hold at different accessible worlds
}

MONOTONICITY_WITH_RESPECT_TO_ENTAILMENT {
    IF P -> Q IS valid
    THEN
        ◇P -> ◇Q IS valid
}

POSSIBILITY_WITNESS {
    IF M,w ⊨ ◇P
    THEN
        EXISTS v:
            R(w,v)
            AND M,v ⊨ P

    v := witness_accessible_world
}

MODEL_CHECK_POSSIBILITY(Model M, World w, Formula P) {
    successors := ACCESSIBLE(w)

    FOR_EACH v IN successors {
        IF MODEL_CHECK(M,v,P)=TRUE
            RETURN {
                value := TRUE,
                witness := v
            }
    }

    RETURN {
        value := FALSE,
        witness := NONE
    }
}

NESTED_POSSIBILITY {
    ◇◇P :=
        "It is possible that P is possible."

    SEMANTICS :=
        EXISTS v accessible_from w:
            EXISTS u accessible_from v:
                M,u ⊨ P
}

TRANSITIVE_COLLAPSE_DIRECTION {
    IF frame_is_transitive
    THEN
        ◇◇P -> ◇P

    REASON :=
        a two-step reachable P-world is also directly accessible
}

REFLEXIVE_EXPANSION_DIRECTION {
    IF frame_is_reflexive
    THEN
        ◇P -> ◇◇P

    REASON :=
        an accessible P-world accesses itself
}

S4_RELATIONSHIP {
    FRAME := ReflexiveAndTransitive

    CONSEQUENCES_INCLUDE {
        P -> ◇P
        ◇◇P -> ◇P
        ◇P -> ◇◇P
    }
}

S5_RELATIONSHIP {
    STANDARD_FRAME_STYLE :=
        equivalence_relation_accessibility

    POSSIBILITY :=
        truth at at least one world in the relevant equivalence class
}

GLOBAL_POSSIBILITY_SPECIAL_CASE {
    IF R = W × W
    THEN
        ◇P at any world
        IFF P holds at at least one world in model
}

IDENTITY_ACCESSIBILITY_SPECIAL_CASE {
    IF R = {(w,w) | w∈W}
    THEN
        ◇P at w
        IFF P at w
}

ACCESSIBILITY_EXPANSION_EFFECT {
    GIVEN same valuation and world w

    IF accessibility_successors_are_added
    THEN
        satisfying ◇P can become easier

    REASON :=
        possibility existentially quantifies over accessible worlds
}

ACCESSIBILITY_RESTRICTION_EFFECT {
    GIVEN same valuation and world w

    IF accessibility_successors_are_removed
    THEN
        satisfying ◇P can become harder
}

EXISTENTIAL_ANALOGY {
    ◇P at w
        structurally resembles
        existential_quantification_over_accessible_worlds

    IMPORTANT :=
        modal possibility is not identical to ordinary first-order existential quantification
}

POSSIBILITY_AND_NECESSITY {
    IF □P
       AND frame_is_serial
    THEN ◇P

    IF ◇P
    THEN NOT necessarily □P

    IF ◇P AND ◇¬P
    THEN P IS contingent_at_current_world
}

CONTINGENCY {
    CONTINGENT(P) :=
        ◇P ∧ ◇¬P

    INTERPRETATION :=
        both P and its negation occur among accessible alternatives
}

IMPOSSIBILITY {
    IMPOSSIBLE(P) :=
        ¬◇P

    EQUIVALENT :=
        □¬P
}

NECESSARY_FALSEHOOD {
    □¬P
        -> ¬◇P
}

POSSIBLE_TRUTH {
    ◇P
        -> ¬□¬P
}

BISIMULATION_PRESERVATION {
    IF worlds w and v are bisimilar
    THEN
        ◇P has the same truth value at w and v
        for every basic modal formula P
}

FIRST_ORDER_MODAL_POSSIBILITY {
    FORM :=
        ◇φ(x)

    REQUIRES_EXPLICIT_SEMANTICS_FOR {
        Domains,
        VariableAssignments,
        RigidOrNonRigidDesignation,
        QuantifierModalInteraction
    }
}

QUANTIFIER_MODAL_INTERACTION {
    POSSIBLE_FOR_EACH :=
        ∀x◇P(x)

    POSSIBLY_FOR_ALL :=
        ◇∀xP(x)

    WARNING :=
        these are not generally equivalent;
        interpretation depends on modal domain semantics
}

EPISTEMIC_SPECIALISATION {
    ◇_a P
        MAY represent compatibility_of_P_with_agent_a_accessible_states

    REQUIRE :=
        epistemic accessibility semantics
}

DEONTIC_SPECIALISATION {
    ◇P
        MAY represent permission_that_P

    REQUIRE :=
        deontic semantic framework
}

TEMPORAL_SPECIALISATION {
    possibility_like existential modalities
        MAY represent eventual or reachable temporal truth

    REQUIRE :=
        temporal semantics
}

DYNAMIC_SPECIALISATION {
    <α>P :=
        there exists an α-successor state in which P holds
}

APPLICATION_RELATIONSHIPS {
    Possibility --supports--> ModalReasoning
    Possibility --supports--> ReachabilityReasoning
    Possibility --supports--> ModelChecking
    Possibility --supports--> KnowledgeRepresentation
    Possibility --supports--> EpistemicLogic
    Possibility --supports--> DeonticLogic
    Possibility --supports--> TemporalLogic
    Possibility --supports--> DynamicLogic
    Possibility --supports--> PhilosophicalLogic
    Possibility --supports--> ArtificialIntelligence
}

ONTOLOGY_LINKS {
    PARENT := ModalLogicOntology

    RELATED := {
        NecessityOntology,
        KripkeFrameOntology,
        KripkeModelOntology,
        PossibleWorldOntology,
        AccessibilityRelationOntology,
        ModalOperatorOntology,
        PropositionalLogicOntology,
        PredicateLogicOntology,
        ModelTheoryOntology,
        EpistemicLogicOntology,
        DeonticLogicOntology,
        TemporalLogicOntology,
        DynamicLogicOntology
    }
}

SEMANTIC_EQUIVALENCE {
    ◇P ≈ Possibly(P)
    ◇P ≈ P_is_possible
    ◇P ≈ ¬□¬P

    ¬◇P ≈ □¬P

    PossibilityOperator ≈ DiamondOperator
        WHEN context=ModalLogic

    AccessibleWorld ≈ RSuccessor
        WHEN context=KripkeSemantics
}

CONSTRAINTS {
    C1 :=
        possibility MUST be interpreted relative to a modal semantics or proof system

    C2 :=
        under Kripke semantics ◇P at w MUST require at least one R-accessible P-world

    C3 :=
        if no accessible P-world exists then ◇P MUST be false at w

    C4 :=
        if no accessible worlds exist then ◇P MUST be false under standard existential Kripke semantics

    C5 :=
        possibility MUST NOT be conflated with probability

    C6 :=
        modal possibility MUST NOT automatically be equated with satisfiability in some arbitrary model

    C7 :=
        frame-dependent inferences MUST respect the selected modal system

    C8 :=
        existential witnesses MUST be accessible from the current evaluation world

    C9 :=
        first-order modal possibility MUST state domain assumptions when quantifier interaction matters

    C10 :=
        semantic transformations MUST preserve truth under the selected modal framework
}

REASONING_CHAIN Query Q {
    IDENTIFY possibility_statement
    IDENTIFY operand_formula P
    IDENTIFY modal_system
    IDENTIFY semantic_framework

    PARSE ◇P
    RESOLVE possibility_aliases
    RESOLVE duality ◇P ≡ ¬□¬P

    IF KripkeModel_provided {
        IDENTIFY current_world w
        IDENTIFY accessibility_relation R
        COMPUTE ACCESSIBLE(w)

        FOR_EACH accessible_world v {
            CHECK P at v

            IF P=true
                RETURN possibility_true WITH witness=v
        }

        RETURN possibility_false
    }

    IF frame_properties_provided {
        CHECK Reflexivity
        CHECK Seriality
        CHECK Transitivity
        CHECK Symmetry
        CHECK EuclideanProperty
    }

    IF reflexive_context AND P_at_current_world
        THEN permit P -> ◇P

    IF axiom_D_context
        THEN permit □P -> ◇P

    IF axiom_5_context
        THEN permit ◇P -> □◇P

    IF first_order_modal_context
        THEN identify_domain_semantics

    IF specialised_modal_context
        THEN identify interpretation_of_diamond_operator

    VALIDATE constraints
    RETURN result WITH derivation
}

CONFIDENCE {
    CORE_POSSIBILITY_DEFINITION := 1.00
    KRIPKE_POSSIBILITY_SEMANTICS := 1.00
    NECESSITY_POSSIBILITY_DUALITY := 1.00
    STANDARD_FRAME_CORRESPONDENCES := 0.99

    MODEL_CHECKING :=
        IF complete_model_and_world_given THEN 1.00
        ELSE REQUIRE missing_information

    SPECIALISED_MODAL_INTERPRETATION :=
        REQUIRE semantic_context
}

UPDATE_MODEL {
    IMMUTABLE_CORE := {
        PossibilityOperator,
        KripkeExistentialAccessibilitySemantics,
        NecessityPossibilityDuality
    }

    EXTENSIBLE := {
        ModalSystems,
        FrameProperties,
        FirstOrderModalSemantics,
        SpecialisedModalInterpretations,
        ReachabilityInterpretations,
        ApplicationDomains,
        RelatedOntologies
    }

    ON_NEW_KNOWLEDGE {
        IDENTIFY modal_context
        VALIDATE operator_semantics
        VALIDATE frame_assumptions
        CHECK system_compatibility
        CHECK semantic_consistency
        ADD relationship IF non_contradictory
        UPDATE confidence
    }
}

AI_CONSUMPTION {
    MACHINE_READABLE := TRUE
    HUMAN_INTERPRETABLE := TRUE
    RECURSIVE_REASONING := TRUE
    ENTITY_RESOLUTION := TRUE
    SYMBOL_RESOLUTION := TRUE
    RELATIONSHIP_TRAVERSAL := TRUE
    POSSIBLE_WORLD_REASONING := TRUE
    ACCESSIBILITY_REASONING := TRUE
    POSSIBILITY_EVALUATION := TRUE
    WITNESS_WORLD_DETECTION := TRUE
    FRAME_PROPERTY_REASONING := TRUE
    VALIDATION_REQUIRED := TRUE
}

FLOW_HASH {
    TARGET := TARGET_URL
    SUBJECT := "Possibility Ontology"
    VERSION := "1.0"
    CANONICAL_SIGNATURE :=
        HASH(TARGET + SUBJECT + DOMAIN + VERSION)
}

OUTPUT {
    PRIMARY_ENTITY := Possibility

    INTERPRETATION :=
        "Possibility is the modal condition represented by ◇. Under standard Kripke semantics, ◇P is true at a world exactly when at least one world accessible from it satisfies P. Its interpretation therefore depends on accessibility structure, modal system and the distinction between modal possibility, satisfiability and probability."
}

END_FLOW