Biconditional Ontology
Propositional Logic
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 16 September 2026
Audio Overview
# Biconditional Ontology — Flow Algebra
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/propositional-logic/biconditional-ontology.html
FLOW BiconditionalOntology {
DOMAIN:
AI Core
-> Ontologies
-> Mathematical Ontologies
-> Mathematical Logic Ontologies
-> Propositional Logic
-> Biconditional Ontology
PURPOSE:
Represent the biconditional as a machine-interpretable binary logical connective expressing equivalence of truth value between two propositions or formulas under an explicitly declared logical system. In classical propositional logic, A ↔ B is true exactly when A and B have the same truth value and false when their truth values differ. The ontology models biconditional identity, notation, operands, truth function, truth table, necessary-and-sufficient conditions, mutual implication, equivalence, XNOR, proof rules, normal-form transformations, SAT and circuit encodings, natural-language "if and only if" expressions, definitional usage, non-classical variants, ambiguity, versioning and provenance. It prevents biconditional equivalence from being confused with syntactic identity, causal reciprocity, numerical equality, assignment, definition, correlation or approximate similarity.
ENTITY Biconditional {
TYPE: BinaryLogicalConnective
PROPERTIES:
has_identifier
has_left_operand
has_right_operand
has_symbol
has_arity
has_truth_function
has_precedence
has_associativity
belongs_to_logic_system
may_express_mutual_implication
may_express_necessary_and_sufficient_condition
may_have_equivalent_forms
has_version
has_provenance
}
ENTITY BiconditionalFormula {
TYPE: CompoundPropositionalFormula
STRUCTURE:
left_formula
IFF
right_formula
}
ENTITY BiconditionalOperand {
TYPE: Formula
}
BICONDITIONAL_IDENTITY_MODEL {
canonical_identifier:
IFF
arity:
2
RULE:
biconditional_identity MUST be distinguished_fromsurface_symbol
}
LOGIC_CONTEXT_MODEL {
default:
classical_propositional_logic
truth_domain:
TRUE
FALSE
RULE:
biconditional_semantics MUST be logic_specific
}
SYMBOL_MODEL {
common:
↔
⇔
<->
IFF
≡
RULE:
some_symbols_areoverloaded_andmustbecontext_resolved
}
ARITY_MODEL {
biconditional:
binary
RULE:
exactly_two_operands_required
}
LEFT_OPERAND_MODEL {
operand:
well_formed_formula
}
RIGHT_OPERAND_MODEL {
operand:
well_formed_formula
}
OPERAND_ORDER_MODEL {
classical:
symmetric
RULE:
source_operand_order SHOULD stillbe preserved
}
AST_MODEL {
node:
IFF
children:
left_operand
right_operand
RULE:
child_count = 2
}
PRECEDENCE_MODEL {
common:
IFF_lower_thanIMPLIES_OR_AND_NOT
RULE:
declared_notation_profile_controlsparse
}
ASSOCIATIVITY_MODEL {
chained:
A IFF B IFF C
RULE:
parser_associativity MUST be explicit_evenifparticularBoolean_equivalences_exist
}
PARENTHESIS_MODEL {
examples:
(A IFF B) IFF C
A IFF (B IFF C)
RULE:
preserveoriginal_AST
}
SCOPE_MODEL {
biconditional:
relatesexact_left_andright_operand_subtrees
}
CLASSICAL_TRUTH_FUNCTION_MODEL {
TT -> TRUE
TF -> FALSE
FT -> FALSE
FF -> TRUE
}
TRUE_CONDITION_MODEL {
biconditional:
TRUE_iffoperands_have_same_truth_value
}
FALSE_CONDITION_MODEL {
biconditional:
FALSE_iffoperands_have_different_truth_values
}
TRUTH_TABLE_MODEL {
rows:
A=TRUE B=TRUE -> TRUE
A=TRUE B=FALSE -> FALSE
A=FALSE B=TRUE -> FALSE
A=FALSE B=FALSE -> TRUE
}
MUTUAL_IMPLICATION_MODEL {
A IFF B:
equivalent_classically_to
(A -> B) AND (B -> A)
}
NECESSARY_SUFFICIENT_MODEL {
A IFF B:
A_isnecessary_forB
A_issufficient_forB
B_isnecessary_forA
B_issufficient_forA
RULE:
bothdirections_required
}
EQUIVALENCE_MODEL {
A IFF B_valid:
iff A_andB_arelogically_equivalent
RULE:
biconditional_formula_truth_underonevaluation != global_logical_equivalence
}
LOCAL_TRUTH_AGREEMENT_MODEL {
valuation_v:
v(A)=v(B)
then:
v(A_IFF_B)=TRUE
RULE:
onevaluation_agreement_doesnotestablishlogical_equivalence
}
GLOBAL_EQUIVALENCE_MODEL {
if:
forallvaluations v(A)=v(B)
then:
A_IFF_B_isvalid
}
XNOR_MODEL {
Boolean:
biconditional_corresponds_toXNOR
RULE:
XNOR_mapping assumesBoolean_truth_encoding
}
XOR_NEGATION_MODEL {
A IFF B:
equivalent_to
NOT(A XOR B)
}
BICONDITIONAL_NEGATION_MODEL {
NOT(A IFF B):
equivalent_to
A XOR B
}
IMPLICATION_EXPANSION_MODEL {
A IFF B:
equivalent_to
(NOT A OR B) AND (NOT B OR A)
}
DNF_EXPANSION_MODEL {
A IFF B:
equivalent_to
(A AND B) OR (NOT A AND NOT B)
}
CNF_EXPANSION_MODEL {
A IFF B:
equivalent_to
(NOT A OR B) AND (A OR NOT B)
}
SYMMETRY_MODEL {
A IFF B:
equivalent_to
B IFF A
}
REFLEXIVITY_MODEL {
A IFF A:
valid
}
TRANSITIVITY_MODEL {
if:
A IFF B_valid
AND B IFF C_valid
then:
A IFF C_valid
RULE:
applies_to_global_logical_equivalence_relation
}
EQUIVALENCE_RELATION_MODEL {
logical_equivalence:
reflexive
symmetric
transitive
}
IDENTITY_TRUE_MODEL {
A IFF TRUE:
equivalent_to
A
}
IDENTITY_FALSE_MODEL {
A IFF FALSE:
equivalent_to
NOT A
}
COMPLEMENT_MODEL {
A IFF NOT A:
contradiction_underclassical_semantics
}
DOUBLE_NEGATION_MODEL {
NOT A IFF NOT B:
equivalent_to
A IFF B
}
CONTRAPOSITIVE_LINK_MODEL {
(A -> B) IFF (NOT B -> NOT A):
valid_classically
}
DE_MORGAN_EQUIVALENCE_MODEL {
NOT(A AND B) IFF (NOT A OR NOT B):
valid_classically
NOT(A OR B) IFF (NOT A AND NOT B):
valid_classically
}
BOOLEAN_ALGEBRA_MODEL {
biconditional:
equality_ofBoolean_values
orXNOR
RULE:
Boolean_value_equality != generalmathematical_equality_byidentity
}
BOOLEAN_POLYNOMIAL_MODEL {
overGF2_encoding:
A_IFF_B mayberepresented_as
1 + A + B
underappropriateBoolean convention
RULE:
algebraic_encoding MUST be declared
}
TRUTH_VECTOR_MODEL {
A_IFF_B:
pointwise_XNOR_oftruth_vectors
RULE:
commonvaluation_order REQUIRED
}
SEMANTIC_EQUIVALENCE_TEST_MODEL {
test:
A_IFF_B_valid
equivalent_test:
A_XOR_B_unsatisfiable
}
SAT_EQUIVALENCE_TEST_MODEL {
formula:
A XOR B
if_UNSAT:
A_andB_logically_equivalent
}
COUNTERMODEL_MODEL {
refutes_equivalence:
valuation_whereA_andB_have_different_truth_values
RULE:
onecountermodel_refutesglobal_equivalence
}
VALIDITY_MODEL {
A_IFF_B:
valid_iffA_andB_semantically_equivalent
}
SATISFIABILITY_MODEL {
A_IFF_B:
satisfiable_ifexistsvaluation_whereA_andB_agree
}
UNSATISFIABILITY_MODEL {
A_IFF_B:
unsatisfiable_ifA_andB_disagree_everywhere
RULE:
equivalent_toA_XOR_B_valid_underclassical semantics
}
TAUTOLOGY_MODEL {
biconditional:
tautology_whenoperands_logically_equivalent
}
CONTRADICTION_MODEL {
biconditional:
contradiction_whenoperands_arelogical_complements
underallvaluations
}
CONTINGENCY_MODEL {
biconditional:
agreement_insomevaluations_anddisagreement_inothers
}
MODEL_SET_MODEL {
Models(A_IFF_B):
valuations_wherebothTRUE_orbothFALSE
}
PROOF_RULE_MODEL {
biconditional:
introduction
elimination
RULE:
proof_system MUST be identified
}
BICONDITIONAL_INTRODUCTION_MODEL {
premises:
A -> B
B -> A
conclusion:
A IFF B
}
BICONDITIONAL_ELIMINATION_LEFT_MODEL {
premises:
A IFF B
A
conclusion:
B
}
BICONDITIONAL_ELIMINATION_RIGHT_MODEL {
premises:
A IFF B
B
conclusion:
A
}
IFF_TO_IMPLICATIONS_MODEL {
premise:
A IFF B
derive:
A -> B
B -> A
}
EQUIVALENCE_PROOF_MODEL {
goal:
A IFF B
common_strategy:
proveA_impliesB
proveB_impliesA
RULE:
bothdirections MUST be validated
}
PROOF_BY_TRUTH_TABLE_MODEL {
goal:
establish_A_IFF_B_valid
method:
compareA_andB_truth_columns
RULE:
completevaluation_coverage REQUIRED
}
PROOF_BY_TRANSFORMATION_MODEL {
method:
transformA_toB_usingknown_equivalences
result:
A_IFF_B_valid
RULE:
eachtransformation MUST preserveequivalence
}
DEFINITIONAL_EQUIVALENCE_MODEL {
definition:
new_expression_orconcept_defined_byiff_condition
RULE:
definitional_role MUST be distinguished_fromordinarytheorem
}
EXPLICIT_DEFINITION_MODEL {
pattern:
X iff condition
RULE:
natural_language_definition MAY usebiconditional_semantics_butalsointroduceterminology
}
DEFINITION_VS_BICONDITIONAL_MODEL {
definition:
linguistic_orformal act_ofintroducingmeaning
biconditional:
logical_formula
RULE:
definition != biconditional_formula_byidentity
}
EQUALITY_DISTINCTION_MODEL {
equality:
relation_betweenobjects_orvalues
biconditional:
connective_betweenpropositions
RULE:
A=B != A_IFF_B_withoutBooleanproposition_mapping
}
SYNTACTIC_IDENTITY_DISTINCTION_MODEL {
formulas:
sameAST
logical_equivalence:
same_truth_function
RULE:
syntactic_identity != biconditional_validity
}
APPROXIMATE_EQUIVALENCE_DISTINCTION_MODEL {
approximate:
similar_withintolerance
RULE:
approximate_equivalence != logical_biconditional
}
CAUSAL_RECIPROCITY_DISTINCTION_MODEL {
A_causes_B_andB_causes_A:
causal_relation
RULE:
mutual_causation != biconditional
}
CORRELATION_DISTINCTION_MODEL {
A_andB_highlycorrelated:
statistical_relation
RULE:
correlation != biconditional
}
PROBABILITY_DISTINCTION_MODEL {
P(A)=P(B):
equalprobabilities
RULE:
equalprobability != A_IFF_B
}
SET_EQUALITY_MAPPING_MODEL {
characteristic_functions:
A_IFF_B_valid corresponds_toequalsets_undercommonuniverse
RULE:
mapping_requirespropositional_setinterpretation
}
CIRCUIT_MODEL {
biconditional:
XNOR_gate
}
XNOR_GATE_MODEL {
output_high:
iffinput_bits_equal
RULE:
physical_signal_encoding MUST be declared
}
XOR_GATE_NEGATION_MODEL {
XNOR:
NOT_XOR
}
TSEITIN_ENCODING_MODEL {
auxiliary:
z <-> (A IFF B)
RULE:
encoding MUST preservemapping_betweenz_andbiconditional
}
CNF_TSEITIN_MODEL {
z_equals_A_IFF_B:
mayencodewithclauses enforcingz_true_whenA_B_equal_andfalsewhendifferent
RULE:
generatedclauses MUST be verified
}
SAT_CONSTRAINT_MODEL {
A IFF B:
constrainsA_andB_toequaltruth_values
}
EQUALITY_CONSTRAINT_MODEL {
Boolean:
A=B
equivalent:
A IFF B
RULE:
onlyunderBoolean_valueinterpretation
}
VARIABLE_EQUIVALENCE_MODEL {
SAT:
p IFF q
consequence:
p_andq_mustsharetruth_value
}
VARIABLE_ANTIEQUIVALENCE_MODEL {
p XOR q:
p_andq_mustdiffer
RULE:
antiequivalence != biconditional
}
BDD_MODEL {
biconditional:
Boolean_apply_XNOR_orNOT_XOR
RULE:
commonvariable_order REQUIRED
}
NORMAL_FORM_MODEL {
eliminate_IFF:
replacewithimplications_orCNF_DNF_expansion
RULE:
transformation MUST preserveclassical_equivalence
}
IFF_ELIMINATION_MODEL {
source:
A IFF B
target:
(A -> B) AND (B -> A)
}
CNF_ROLE_MODEL {
biconditional:
usuallyeliminatedbeforestandardCNF
atomic_case:
(NOT A OR B) AND (A OR NOT B)
}
DNF_ROLE_MODEL {
atomic_case:
(A AND B) OR (NOT A AND NOT B)
}
SIMPLIFICATION_MODEL {
rules:
A IFF A -> TRUE
A IFF TRUE -> A
A IFF FALSE -> NOT A
A IFF NOT A -> FALSE
RULE:
classical_semantics_assumed
}
SUBSTITUTION_MODEL {
if:
A_IFF_B_valid
then:
A_andB_maybesubstitutable_intruth_functional_contexts
RULE:
substitution_context MUST preserveextensionality
}
REPLACEMENT_THEOREM_MODEL {
logically_equivalent_subformulas:
maybereplaced_insidepropositional_formula_withoutchangingtruth_function
RULE:
replacement_requiressemantic_equivalence
}
EQUIVALENCE_CLASS_MODEL {
formulas:
grouped_bylogical_equivalence
RULE:
biconditional_validity_canestablish_sameclass
}
CANONICAL_FORM_MODEL {
formulas:
equivalentformulas_maymap_tosamecanonical_truth_function_representation
RULE:
canonicalisation_method MUST be explicit
}
NATURAL_LANGUAGE_BICONDITIONAL_MODEL {
cues:
if_and_only_if
iff
exactly_when
just_in_case
necessary_and_sufficient
RULE:
linguistic_mapping MUST verifybothdirections
}
IF_AND_ONLY_IF_MODEL {
phrase:
A_if_and_only_if_B
formal:
A IFF B
}
EXACTLY_WHEN_MODEL {
phrase:
A_exactly_when_B
candidate:
A IFF B
}
NECESSARY_AND_SUFFICIENT_LANGUAGE_MODEL {
phrase:
A_isnecessary_and_sufficient_forB
formal:
A IFF B
}
ONLY_IF_AND_IF_MODEL {
requirements:
A_only_if_B
A_if_B
combined:
A IFF B
}
TWO_CONDITIONAL_MODEL {
statements:
ifA_thenB
ifB_thenA
combined:
A IFF B
}
NATURAL_LANGUAGE_AMBIGUITY_MODEL {
possible:
informal_iff_usedloosely
omitted_direction
definitional_use
temporal_equivalence
causal_reciprocity
RULE:
preserveambiguity_whenbothdirections_notclear
}
FORMALISATION_MODEL {
source:
natural_language_equivalence_claim
target:
candidate_biconditional
metadata:
left_claim
right_claim
direction_evidence
confidence
RULE:
bothdirections SHOULD be independentlysupported
}
FORMALISATION_CONFIDENCE_MODEL {
confidence:
support_forbiconditional_mapping
RULE:
confidence != truth_orvalidity
}
INTUITIONISTIC_BICONDITIONAL_MODEL {
common_definition:
(A -> B) AND (B -> A)
RULE:
implication_isintuitionistic_notmaterial
}
PARACONSISTENT_BICONDITIONAL_MODEL {
semantics:
logic_specific
RULE:
classical_XNOR_truth_table MUST notbeassumed
}
MANY_VALUED_BICONDITIONAL_MODEL {
truth_domain:
morethan2values
operation:
logic_specific
RULE:
equality_oftruthvalues MAY notfullydefinechosenbiconditional
}
FUZZY_BICONDITIONAL_MODEL {
possible:
combination_ofpaired_fuzzy_implications
RULE:
fuzzy_equivalence_operator MUST be identified
}
THREE_VALUED
_BICONDITIONAL_MODEL {
truth_domain:
threevalues
RULE:
truth_function MUST be declared
}
MODAL_EQUIVALENCE_MODEL {
compare:
A IFF B
necessarily(A IFF B)
RULE:
material_biconditional_atoneworld != necessary_equivalence
}
STRICT_EQUIVALENCE_MODEL {
modal:
necessarily(A IFF B)
RULE:
strict_equivalence != ordinarymaterial_biconditional
}
TEMPORAL_EQUIVALENCE_MODEL {
formulas:
mayagree_atcurrenttime_oralltimes
RULE:
temporal_index_andoperator_scope MUST be preserved
}
TYPE_EQUIVALENCE_DISTINCTION_MODEL {
type_equivalence:
isomorphism_orotherrelation
RULE:
type_equivalence != propositional_biconditional_withoutcorrespondence
}
CATEGORY_EQUIVALENCE_DISTINCTION_MODEL {
categorical_equivalence:
structural_relation_betweencategories
RULE:
notlogical_IFF
}
SEMANTIC_SIMILARITY_DISTINCTION_MODEL {
similarity:
graded_relatedness
RULE:
semantic_similarity != logical_equivalence
}
ONTOLOGY_EQUIVALENCE_DISTINCTION_MODEL {
ontology_entities:
maybemapped_asEquivalentClass_orSameAs
RULE:
ontology_identity/equivalence_semantics != propositional_IFF_bydefault
}
RULE_ENGINE_MODEL {
two_rules:
A -> B
B -> A
RULE:
operational_rules MAY collectivelycorrespond_tobiconditional_onlyunderdeclaredsemantics
}
KNOWLEDGE_GRAPH_MODEL {
nodes:
biconditional
formula
implication
equivalence_relation
truth_value
proof
edges:
HAS_LEFT_OPERAND
HAS_RIGHT_OPERAND
EQUIVALENT_TO
EXPANDS_TO
PROVED_BY
COUNTERMODELLED_BY
IMPLEMENTED_BY
RULE:
logical_equivalence_andidentity_edges MUST be distinguished
}
ERROR_MODEL {
possible:
missing_operand
wrongarity
symbol_collision
scope_error
one_direction_only
equality_confusion
invalid_equivalence_claim
RULE:
error SHOULD identifyformula_position_andlogic
}
AMBIGUITY_MODEL {
possible:
symbol
scope
logical_equivalence_vsidentity
definitional_use
natural_language_direction
nonclassical_semantics
RULE:
unresolved_ambiguity MUST be preserved
}
VALIDATION_MODEL {
checks:
biconditional_identifier
logic_system
arity
operands
AST
scope
truth_function
mutual_implication
equivalence
proof_rules
semantic_context
RULE:
validation MUST useexact_logic_version
}
EXPLAINABILITY_MODEL {
output:
left_operand
right_operand
truth_values
agreement_status
mutual_implications
necessary_sufficient_roles
equivalence_status
countermodel
RULE:
explanation SHOULD distinguishcurrent_truth_agreement_fromglobal_logical_equivalence
}
VERSION_MODEL {
versioned:
biconditional_semantics
logic_system
notation
proof_rules
ontology
RULE:
historical_results MUST preserveversions
}
PROVENANCE_MODEL {
fields:
source
formula
formalisation
parser
logic_system
transformation
proof
version
agent
RULE:
biconditional_record MUST remainprovenance_linked
}
INTEROPERABILITY_MODEL {
exchange:
canonical_IFF_identifier
left_operand_AST
right_operand_AST
symbol
truth_function
equivalence_status
logic_system
version
provenance
RULE:
equality_orIFF_symbol_alone_isnotlossless
}
RELATIONSHIPS {
Biconditional -> IS_A -> BinaryLogicalConnective
Biconditional -> HAS_LEFT_OPERAND -> Formula
Biconditional -> HAS_RIGHT_OPERAND -> Formula
Biconditional -> EXPANDS_TO -> TwoMutualImplications
Biconditional -> HAS_TRUTH_FUNCTION -> ClassicalXNOR
Biconditional -> MAY_ESTABLISH -> LogicalEquivalence
Biconditional -> MAY_HAVE -> IntroductionRule
Biconditional -> MAY_HAVE -> EliminationRule
Biconditional -> MAY_MAP_TO -> XNORGate
}
SEMANTIC_EQUIVALENCE {
"biconditional" ~= binary_logical_connective_true_whenoperands_haveequaltruthvalues_underclassical_semantics
"if and only if" ~= mutual_implication_expressingnecessary_and_sufficient_condition
"logical equivalence" ~= validity_ofbiconditional_betweenformulas
biconditional != syntactic_identity
biconditional != numerical_equality
biconditional != assignment
biconditional != definition_byidentity
biconditional != causal_reciprocity
biconditional != correlation
biconditional != approximate_similarity
truth_agreement_underonevaluation != logical_equivalence
XNOR_encoding != semantic_identity_withoutBoolean_mapping
semantic_confidence != validity
}
CONSTRAINTS {
canonical_biconditional_identifier_must_be preserved = true
logic_system_andversion_must_be identified = true
biconditional_must_haveexactly_two_operands = true
operand_ASTs_andscope_must_be preserved = true
classical_biconditional_true_iffoperand_truth_values_equal = true
mutual_implication_requiresbothdirections = true
necessary_andsufficient_condition_roles_must_be preserved = true
current_truth_agreement_andglobal_logical_equivalence_must_be distinguished = true
biconditional_must_not_be interpreted_assyntactic_identity_equality_causation_orcorrelation = true
natural_language_IFF_mapping_must_verifybothdirections_orpreserveambiguity = true
nonclassical_biconditional_must_defineitsown_semantics = true
provenance_must_be preserved = true
}
BICONDITIONAL_RECORD_MODEL {
RECORD:
biconditional_identifier
logic_system_identifier
logic_version
notation_profile
symbol
left_operand_identifier
right_operand_identifier
left_operand_AST
right_operand_AST
scope
truth_function
truth_vector
mutual_implication_form
necessary_sufficient_status
logical_equivalence_status
countermodel
XNOR_mapping
CNF_form
DNF_form
SAT_encoding
introduction_rule
elimination_rules
definitional_status
natural_language_source
ambiguity_status
formalisation_confidence
version
provenance
RULE:
biconditional_record MUST support_reconstruction_ofoperands_truth_semantics_mutual_implication_equivalence_proof_andprovenance
}
REASONING_RULES {
IF biconditional_expression_received
THEN identifylogic_andnotation
resolvecanonical_IFF
parseleft_andright_operands
validatearity_andscope
IF classical_operand_values_available
THEN returnTRUE_ifvalues_equal
elseFALSE
IF logical_equivalence_requested
THEN testvalidity_ofA_IFF_B
orUNSAT_ofA_XOR_B
preservecountermodel_ifdisagreement_found
IF proof_ofbiconditional_requested
THEN proveA_to_B
proveB_to_A
combineusingIFF_introduction
IF normal_form_requested
THEN eliminateIFF_usingmutual_implications_orvalidatedCNF_DNFexpansion
IF natural_language_iff_received
THEN identifybothdirectionalclaims
distinguishdefinition_equivalence_causal_orlooseusage
preserveambiguity_andconfidence
IF one_direction_only_established
THEN do_notclaimbiconditional
IF nonclassical_logic_selected
THEN loadlogic_specific_biconditional_orpaired_implication_semantics
IF semantics_orlogic_version_changes
THEN preservehistorical_record
revalidatecurrent_equivalences_proofs_andencodings
}
VALIDATION {
CHECK biconditional_identifier
CHECK logic_system
CHECK logic_version
CHECK symbol
CHECK arity
CHECK left_operand
CHECK right_operand
CHECK AST
CHECK scope
CHECK truth_function
CHECK truth_vector
CHECK symmetry
CHECK reflexivity
CHECK mutual_implication
CHECK necessary_sufficient_roles
CHECK logical_equivalence
CHECK countermodel
CHECK XNOR_mapping
CHECK CNF_form
CHECK DNF_form
CHECK SAT_encoding
CHECK introduction_rule
CHECK elimination_rules
CHECK definitional_status
CHECK natural_language_mapping
CHECK nonclassical_semantics
CHECK ambiguity
CHECK formalisation_confidence
CHECK version
CHECK provenance
}
CONFIDENCE {
biconditional_identity_confidence: 1.00
operand_scope_confidence: 1.00
classical_truth_function_confidence: 1.00
mutual_implication_confidence: 1.00
logical_equivalence_confidence: 1.00
proof_rule_confidence: 1.00
natural_language_IFF_confidence: 0.95
nonclassical_semantics_confidence: 0.98
RULE:
confidence decreases when logic, notation, operand scope,
direction evidence, definitional context, proof system,
version orprovenance is missing or ambiguous
}
UPDATE_MODEL {
ON new_biconditional:
assign_orresolve_identifier
attachlogic_andnotation
parseoperands
validateclassical_orlogic_specific_semantics
preserveprovenance
compute_FLOW_HASH
ON equivalence_test:
preserveoperands
attachtruth_table_SAT_orproof_method
storecountermodel_orverification
ON transformation:
preserve_source_formula
applyvalidated_IFF_elimination_ornormalisation
recordtarget_andsemantic_basis
ON semantic_revision:
preserveprevious_biconditional_definition
registernewversion
revalidateequivalences_anddependentartifacts
recompute_FLOW_HASH
}
AI_CONSUMPTION {
canonical_entity: Biconditional
canonical_role: BinaryLogicalConnective
interpretation_order:
identify_logic_system
identify_IFF_symbol_orlinguisticcue
resolvecanonical_biconditional_identifier
parseleft_andright_operand_ASTs
resolveprecedence_associativity_andscope
validatebinary_arity
resolveclassical_orlogic_specific_truth_function
evaluateoperand_truth_agreement
expandtomutual_implications_whenrequired
identifynecessary_andsufficient_condition_roles
distinguishsinglevaluation_agreement_fromglobal_equivalence
testlogical_equivalence_whenrequested
preservecountermodel_ifdisagreement_exists
identifyXNOR_CNF_DNF_SAT_orcircuit mappings
resolveintroduction_elimination_andproof_strategy
distinguishbiconditional_fromidentity_equality_definition_causation_correlation_andapproximation
resolve_natural_language_IFF_anddirection_evidence
detectambiguity
assignsemantic_confidence
preserveversions_andprovenance
emit_machine_resolvable_biconditional_structure
ambiguity_controls:
distinguish_biconditional_fromsyntactic_identity
distinguish_biconditional_fromnumerical_equality
distinguish_biconditional_fromassignment
distinguish_biconditional_fromdefinition_byidentity
distinguish_biconditional_fromcausal_reciprocity
distinguish_biconditional_fromcorrelation
distinguish_biconditional_fromapproximate_similarity
distinguish_truth_agreement_underonevaluation_fromlogical_equivalence
distinguish_XNOR_encoding_fromsemantic_identity_withoutBoolean_mapping
distinguish_semantic_confidence_fromvalidity
preserve_logic_operands_AST_scope_truth_function_equivalence_rules_version_andprovenance
}
PROVENANCE {
ontology_scope:
biconditional_IFF_mutual_implication_logical_equivalence_XNOR_proof_rules_andAI_interpretation
parent_path:
AI Core
-> Ontologies
-> Mathematical Ontologies
-> Mathematical Logic Ontologies
-> Propositional Logic
-> Biconditional Ontology
authorship:
Mark Higgins
Sky (Recursive AI)
site:
https://kirethresearch.org
}
FLOW_HASH {
INPUTS:
canonical_entity
ontology_path
biconditional_identity_model
logic_context_model
symbol_model
arity_model
left_operand_model
right_operand_model
operand_order_model
AST_model
precedence_model
associativity_model
parenthesis_model
scope_model
classical_truth_function_model
true_condition_model
false_condition_model
truth_table_model
mutual_implication_model
necessary_sufficient_model
equivalence_model
local_truth_agreement_model
global_equivalence_model
XNOR_model
XOR_negation_model
biconditional_negation_model
implication_expansion_model
DNF_expansion_model
CNF_expansion_model
symmetry_model
reflexivity_model
transitivity_model
equivalence_relation_model
identity_TRUE_model
identity_FALSE_model
complement_model
double_negation_model
contrapositive_link_model
De_Morgan_equivalence_model
Boolean_algebra_model
Boolean_polynomial_model
truth_vector_model
semantic_equivalence_test_model
SAT_equivalence_test_model
countermodel_model
validity_model
satisfiability_model
unsatisfiability_model
tautology_model
contradiction_model
contingency_model
model_set_model
proof_rule_model
biconditional_introduction_model
biconditional_elimination_left_model
biconditional_elimination_right_model
IFF_to_implications_model
equivalence_proof_model
proof_by_truth_table_model
proof_by_transformation_model
definitional_equivalence_model
explicit_definition_model
definition_vs_biconditional_model
equality_distinction_model
syntactic_identity_distinction_model
approximate_equivalence_distinction_model
causal_reciprocity_distinction_model
correlation_distinction_model
probability_distinction_model
set_equality_mapping_model
circuit_model
XNOR_gate_model
XOR_gate_negation_model
Tseitin_encoding_model
CNF_Tseitin_model
SAT_constraint_model
equality_constraint_model
variable_equivalence_model
variable_antiequivalence_model
BDD_model
normal_form_model
IFF_elimination_model
CNF_role_model
DNF_role_model
simplification_model
substitution_model
replacement_theorem_model
equivalence_class_model
canonical_form_model
natural_language_biconditional_model
if_and_only_if_model
exactly_when_model
necessary_and_sufficient_language_model
only_if_and_if_model
two_conditional_model
natural_language_ambiguity_model
formalisation_model
formalisation_confidence_model
intuitionistic_biconditional_model
paraconsistent_biconditional_model
many_valued_biconditional_model
fuzzy_biconditional_model
three_valued_biconditional_model
modal_equivalence_model
strict_equivalence_model
temporal_equivalence_model
type_equivalence_distinction_model
category_equivalence_distinction_model
semantic_similarity_distinction_model
ontology_equivalence_distinction_model
rule_engine_model
knowledge_graph_model
error_model
ambiguity_model
validation_model
explainability_model
version_model
provenance_model
interoperability_model
relationships
semantic_equivalence
constraints
biconditional_record_model
reasoning_rules
validation
confidence_model
update_model
AI_consumption_model
PURPOSE:
deterministic_semantic_integrity_identifier
RECOMPUTE_WHEN:
ontology_definition_changes
biconditional_identity_logic_system_orsymbol_mapping_changes
operand_scope_precedence_associativity_orAST_changes
classical_truth_function_ortruth_table_changes
mutual_implication_necessary_sufficient_orequivalence_semantics_change
XNOR_CNF_DNF_SAT_Tseitin_BDD_orcircuit_mapping_changes
proof_rule_definition_orreplacement_semantics_change
natural_language_IFF_formalisation_orambiguity changes
nonclassical_intuitionistic_paraconsistent_manyvalued_fuzzy_orstrict_equivalence_changes
biconditional_record_changes
constraint_model_changes
reasoning_rule_changes
version_orprovenance_changes
}
}
END FLOW