Truth Table Ontology
Propositional Logic
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 16 September 2026
Audio Overview
# Truth Table Ontology — Flow Algebra
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/propositional-logic/truth-table-ontology.html
FLOW TruthTableOntology {
DOMAIN:
AI Core
-> Ontologies
-> Mathematical Ontologies
-> Mathematical Logic Ontologies
-> Propositional Logic
-> Truth Table Ontology
PURPOSE:
Represent a truth table as a machine-interpretable exhaustive semantic structure for evaluating propositional formulas over all possible truth assignments to their atomic propositions under a declared truth-functional logic. The ontology models variables, valuation rows, truth-value columns, subformula columns, connective evaluation, row ordering, truth vectors, satisfiability, validity, tautology, contradiction, contingency, semantic equivalence, argument validity, countermodels, canonical normal forms, Boolean functions, computational generation, verification, provenance and AI interpretation. It enables deterministic reconstruction of a table from a formula and atom ordering while preventing row order, numeric encoding, partial tables, sampled valuations or solver outputs from being confused with the complete semantic definition.
ENTITY TruthTable {
TYPE: ExhaustivePropositionalSemanticTable
PROPERTIES:
has_identifier
has_logic_system
has_formula
has_atomic_variables
has_variable_order
has_rows
has_columns
has_truth_values
has_truth_vector
may_determine_satisfiability
may_determine_validity
may_compare_equivalence
has_version
has_provenance
}
ENTITY TruthTableRow {
TYPE: ValuationEvaluationRecord
COMPONENTS:
row_identifier
valuation
atom_values
subformula_values
target_value
}
ENTITY TruthTableColumn {
TYPE: OrderedSemanticColumn
POSSIBLE:
atomic_variable_column
subformula_column
target_formula_column
}
ENTITY TruthVector {
TYPE: OrderedFormulaEvaluationVector
COMPONENTS:
formula_identifier
valuation_order
truth_values
}
TABLE_IDENTITY_MODEL {
fields:
truth_table_identifier
formula_identifier
logic_system
atom_order
row_order
version
RULE:
table_identity MUST preserve_semantic_ordering_metadata
}
LOGIC_CONTEXT_MODEL {
default:
classical_propositional_logic
truth_domain:
TRUE
FALSE
RULE:
nonclassical_truth_tables MUST declareseparate_truth_domain_andconnectives
}
SOURCE_FORMULA_MODEL {
formula:
well_formed_propositional_expression
RULE:
source_formula_AST SHOULD be preserved
}
ATOM_SET_MODEL {
Vars(phi):
distinct_atomic_variables_informula
RULE:
duplicate_occurrences_do_notcreateadditional_truth_dimensions
}
ATOM_COUNT_MODEL {
n:
cardinality_ofdistinct_atom_set
RULE:
complete_classical_truth_table_has_2^n_rows
}
VARIABLE_ORDER_MODEL {
order:
p_1,p_2,...,p_n
RULE:
order MUST be fixed_beforecanonical_row_generation
}
ROW_COUNT_MODEL {
classical_complete:
rows = 2^n
RULE:
row_count_depends_onnumber_ofdistinct_atoms
}
VALUATION_MODEL {
row:
maps_each_atom_toTRUE_orFALSE
RULE:
everycomplete_row MUST assignallatoms
}
COMPLETE_ROW_MODEL {
row:
total_assignment_overatom_set
RULE:
complete_table_contains_everypossible_total_assignment_exactlyonce
}
PARTIAL_ROW_MODEL {
row:
assignment_to subset_ofatoms
RULE:
partial_row != complete_truth_table_row
}
ROW_UNIQUENESS_MODEL {
table:
no_duplicate_complete_valuations
RULE:
duplicate_rows SHOULD be flagged
}
ROW_COVERAGE_MODEL {
complete_table:
covers_all_2^n_valuations
RULE:
missingvaluation means_table_notexhaustive
}
ROW_ORDER_MODEL {
possible:
binary_counting
lexicographic
Gray_code
custom
RULE:
semantic_results_independent_ofrow_order_buttruth_vector_identity_requiresorder
}
BINARY_ROW_ORDER_MODEL {
mapping:
TRUE_FALSE_or1_0_pattern
RULE:
bit_significance_convention MUST be explicit
}
LEXICOGRAPHIC_ROW_ORDER_MODEL {
ordering:
based_onfixed_variable_order_andtruth_symbol_order
RULE:
truth_symbol_order MUST be declared
}
GRAY_CODE_ROW_MODEL {
property:
adjacent_rows_differ_byonevariable
RULE:
Gray_order_changesvector_order_notformula_semantics
}
ROW_IDENTIFIER_MODEL {
fields:
table_identifier
row_index
valuation_hash
RULE:
row_index_alone_depends_onordering
}
COLUMN_MODEL {
table:
containsordered_columns
RULE:
column_role MUST be explicit
}
ATOMIC_COLUMN_MODEL {
column:
values_ofsingle_propositional_variable_acrossrows
RULE:
atom_identifier MUST be stable
}
SUBFORMULA_COLUMN_MODEL {
column:
truth_values_ofsubformula
RULE:
subformula_AST_identifier SHOULD be preserved
}
TARGET_COLUMN_MODEL {
column:
truth_values_ofprimary_formula
RULE:
target_column MUST be identifiable
}
COLUMN_ORDER_MODEL {
possible:
atoms_first
subformulas_in_dependency_order
target_last
RULE:
display_order != semantic_dependency_order_necessarily
}
CELL_MODEL {
fields:
row_identifier
column_identifier
truth_value
RULE:
cell_value MUST be derivable_fromvaluation_andformula
}
TRUTH_VALUE_MODEL {
classical:
TRUE
FALSE
RULE:
display_encodings_areseparate
}
BOOLEAN_ENCODING_MODEL {
TRUE -> 1
FALSE -> 0
RULE:
1_and0_areencodings_nottruth_values_byidentity
}
SYMBOLIC_ENCODING_MODEL {
possible:
T_F
true_false
top_bottom
RULE:
encoding_profile MUST be preserved
}
CONNECTIVE_MODEL {
possible:
NOT
AND
OR
XOR
IMPLIES
IFF
NAND
NOR
RULE:
truth_function MUST be declared
}
NEGATION_EVALUATION_MODEL {
NOT TRUE = FALSE
NOT FALSE = TRUE
}
CONJUNCTION_EVALUATION_MODEL {
TRUE AND TRUE = TRUE
otherwise = FALSE
}
DISJUNCTION_EVALUATION_MODEL {
FALSE OR FALSE = FALSE
otherwise = TRUE
}
XOR_EVALUATION_MODEL {
true_when:
operands_differ
}
IMPLICATION_EVALUATION_MODEL {
false_when:
antecedent_TRUE
consequent_FALSE
RULE:
material_implication != causation
}
BICONDITIONAL_EVALUATION_MODEL {
true_when:
operands_equal
}
NAND_EVALUATION_MODEL {
result:
NOT(A AND B)
}
NOR_EVALUATION_MODEL {
result:
NOT(A OR B)
}
DEPENDENCY_EVALUATION_MODEL {
formula_AST:
evaluate_children_beforeparent
RULE:
subformula_columns MAY followtopological_dependency_order
}
BOTTOM_UP_EVALUATION_MODEL {
process:
atoms
immediate_subformulas
higher_subformulas
target
RULE:
eachcomputed_value MUST be traceable
}
COMPUTATION_TRACE_MODEL {
row:
records_subformula_evaluations
RULE:
trace MUST align_withAST
}
TRUTH_VECTOR_MODEL {
target_formula:
ordered_values_acrossrows
RULE:
vector_requiresrow_order_metadata
}
TRUTH_VECTOR_HASH_MODEL {
input:
atom_order
row_order
truth_vector
logic_system
RULE:
hash_withoutordering_metadata_isnotcanonical
}
SEMANTIC_SIGNATURE_MODEL {
formula:
truth_vector_undercanonical_atom_androw_order
RULE:
formulas_withsame_semantic_signature_areequivalent_underclassical_propositional_semantics
}
TAUTOLOGY_MODEL {
target_column:
allTRUE
RULE:
table_proves_classical_validity_whencomplete
}
CONTRADICTION_MODEL {
target_column:
allFALSE
RULE:
table_proves_unsatisfiability_whencomplete
}
CONTINGENCY_MODEL {
target_column:
containsTRUE_andFALSE
}
SATISFIABILITY_MODEL {
condition:
at_least_one_target_TRUE
witness:
corresponding_valuation
}
UNSATISFIABILITY_MODEL {
condition:
no_target_TRUE_incomplete_table
RULE:
completeness_ofrow_coverage REQUIRED
}
VALIDITY_MODEL {
condition:
every_target_value_TRUE
RULE:
incomplete_table_cannotestablish_validity_byexhaustion
}
FALSIFIABILITY_MODEL {
condition:
atleastone_target_FALSE
witness:
countervaluation
}
MODEL_MODEL {
row:
model_offormula_if target_TRUE
}
COUNTERMODEL_MODEL {
row:
countermodel_toformula_validity_if target_FALSE
}
MODEL_SET_MODEL {
formula:
set_ofrows_where target_TRUE
RULE:
model_set_canrepresentBoolean_function_extension
}
SEMANTIC_EQUIVALENCE_MODEL {
formulas_A_B:
equivalent_ifcolumns_identical_foreveryvaluation
RULE:
atom_mapping_andcompletevaluation_set MUST match
}
COLUMN_COMPARISON_MODEL {
A_column:
compare_withB_column
possible:
identical
complement
implication_relation
unrelated
RULE:
comparison_semantics MUST be explicit
}
COMPLEMENT_COLUMN_MODEL {
B:
complement_ofA_ifB_row_value = NOT A_row_value_foreveryrow
}
IMPLICATION_COLUMN_MODEL {
A_implies_B:
no_row_withA_TRUE_andB_FALSE
RULE:
thisestablishes_semantic_entailment_betweenformulas
}
BICONDITIONAL_COLUMN_MODEL {
A_equivalent_B:
A_IFF_B_column_allTRUE
}
XOR_EQUIVALENCE_TEST_MODEL {
A_equivalent_B:
A_XOR_B_column_allFALSE
}
ARGUMENT_TABLE_MODEL {
premises:
P_1...P_n
conclusion:
C
RULE:
validity_analysis focuses_onrows_whereallpremises_TRUE
}
PREMISE_COLUMN_MODEL {
each_premise:
evaluated_foreveryrow
}
CONCLUSION_COLUMN_MODEL {
conclusion:
evaluated_foreveryrow
}
CRITICAL_ROW_MODEL {
row:
allpremises_TRUE
RULE:
argument_validity determined_onlybycritical_rows
}
INVALIDATING_ROW_MODEL {
row:
allpremises_TRUE
conclusion_FALSE
RULE:
oneinvalidating_row_refutesargument_validity
}
ARGUMENT_VALIDITY_MODEL {
valid:
no_invalidating_row_incomplete_table
RULE:
premise_truth_inactual_world_isnotpart_offormal_validity_test
}
VACUOUS_VALIDITY_MODEL {
condition:
no_row_satisfiesallpremises
result:
semantic_entailment_holds_classically
RULE:
inconsistent_premises_canentailanyconclusion_underclassical_semantics
}
ENTAILMENT_MODEL {
Gamma entails C:
everyrow_satisfyingGamma_satisfiesC
}
ENTAILMENT_REDUCTION_MODEL {
finite_Gamma:
conjunction(Gamma) -> C
valid_if:
target_column_allTRUE
}
CONSISTENCY_MODEL {
premise_set:
jointly_satisfiable_ifatleastonecritical_row
RULE:
no_critical_row meanspremises_semantically_inconsistent
}
FORMULA_SET_TABLE_MODEL {
formulas:
evaluated_overcommon_atom_union
RULE:
allrows MUST assignunion_ofvariables
}
ATOM_UNION_MODEL {
formula_set:
atom_set = union_ofVars(each_formula)
}
PROJECTION_MODEL {
larger_table:
project_to subset_ofvariables
RULE:
duplicate_projected_rows MAY collapse
}
RESTRICTION_MODEL {
table:
retain_rows_matchingfixed_assignment
RULE:
restricted_table != complete_table_overoriginal_atom_set
}
COFACTOR_TABLE_MODEL {
fix:
p=TRUE_orFALSE
result:
truth_table_ofrestricted_formula_overremainingvariables
}
SHANNON_MODEL {
formula:
decomposed_byvariable_cofactors
RULE:
table_rows_partition_byp_truth_value
}
CANONICAL_DNF_MODEL {
construction:
for eachTRUE_row
create_minterm
OR_all_minterms
RULE:
atom_order_andpolarity_convention MUST be preserved
}
MINTERM_MODEL {
row:
maps_to conjunction_containingeveryatom_once
polarity:
positive_ifrow_TRUE
negative_ifrow_FALSE
}
CANONICAL_CNF_MODEL {
construction:
for eachFALSE_row
create_maxterm
AND_all_maxterms
}
MAXTERM_MODEL {
row:
maps_to disjunction_containingeveryatom_once
polarity:
chosen_sothatmaxterm_FALSE_onthatrow
}
NORMAL_FORM_EXTRACTION_MODEL {
truth_table:
canproducecanonical_CNF_orDNF
RULE:
canonical_forms MAY be exponentiallylarge
}
BOOLEAN_FUNCTION_MODEL {
truth_table:
complete_extensional_definition_ofBoolean_function
RULE:
variable_order_part_ofrepresentation
}
FUNCTION_ARITY_MODEL {
n:
number_ofBoolean_inputs
}
FUNCTION_COUNT_MODEL {
for_n_variables:
number_ofBoolean_functions = 2^(2^n)
RULE:
assumeslabelled_input_variables
}
FUNCTIONAL_EQUIVALENCE_MODEL {
formulas_orcircuits:
same_truth_table_overcommoninputs
RULE:
establishes_sameBoolean_function
}
BOOLEAN_ALGEBRA_MODEL {
table:
mayverifyBoolean_identity_byrowwise_equality
}
BOOLEAN_IDENTITY_VERIFICATION_MODEL {
equation:
LHS = RHS
method:
comparecolumns_allrows
RULE:
complete_table REQUIRED
}
DIGITAL_LOGIC_MODEL {
truth_table:
specifiesBoolean_input_output_behaviour
RULE:
abstract_truth_values != physical_voltage_levels
}
LOGIC_GATE_MODEL {
gate:
truth_function_withtable
possible:
NOT
AND
OR
XOR
NAND
NOR
}
MULTI_INPUT_GATE_MODEL {
arity:
greater_than2
RULE:
truth_table_rows = 2^arity
}
CIRCUIT_TRUTH_TABLE_MODEL {
circuit:
evaluateoutputs_foreveryinput_assignment
RULE:
internal_signal_columns MAY be included
}
MULTI_OUTPUT_TABLE_MODEL {
inputs:
shared
outputs:
multipleBoolean_functions
RULE:
eachoutput_hasowntruth_vector
}
CARE_SET_MODEL {
valuations:
semantically_relevant_inputs
RULE:
care_set_table != full_truth_table_unlessallvaluationsincluded
}
DONT_CARE_MODEL {
row_oroutput:
value_notconstrained_forapplication
symbol:
X_when_declared
RULE:
don't-care != third_logical_truth_value
}
INCOMPLETE_SPECIFICATION_MODEL {
function:
notdefined_orconstrained_onallinputs
RULE:
incomplete_specification != classical_totalBoolean_function
}
K_MAP_LINK_MODEL {
truth_table:
canmap_toKarnaugh_map
RULE:
Gray_code_cell_adjacency MUST be preserved
}
BDD_LINK_MODEL {
truth_table:
maycompile_toBDD_orROBDD
RULE:
BDD_variable_order MUST be explicit
}
SAT_LINK_MODEL {
formula:
satisfiable_iftruth_table_hasTRUE_row
RULE:
truth_table_enumeration_andSAT_solver_arealternative_methods
}
SAT_WITNESS_MODEL {
TRUE_row:
satisfying_assignment
}
UNSAT_PROOF_BY_TABLE_MODEL {
complete_table:
allrows_FALSE
RULE:
exhaustive_coverage_isproof_basis
}
VALIDITY_PROOF_BY_TABLE_MODEL {
complete_table:
allrows_TRUE
}
EQUIVALENCE_PROOF_BY_TABLE_MODEL {
complete_common_table:
compared_columns_equal_allrows
}
ARGUMENT_PROOF_BY_TABLE_MODEL {
complete_table:
noTRUE-premises_FALSE-conclusion row
}
TABLE_GENERATION_MODEL {
input:
formula
ordered_atoms
process:
enumerate_valuations
evaluate_subformulas
populate_columns
}
VALUATION_ENUMERATION_MODEL {
n_atoms:
generate_2^n_assignments
RULE:
eachassignment_unique
}
RECURSIVE_GENERATION_MODEL {
base:
zero_atoms -> one_emptyvaluation
recursive:
extend_eachvaluation_withTRUE_andFALSE_fornewatom
}
BINARY_COUNTER_GENERATION_MODEL {
index:
0_to2^n_minus1
bits:
map_toatom_truth_values
RULE:
bit_toTRUE_mapping_andbit_significance MUST be declared
}
TABLE_VALIDATION_MODEL {
checks:
atom_count
row_count
row_uniqueness
row_coverage
cell_values
target_values
RULE:
structural_andsemantic_validation BOTH required
}
ROW_VALIDATION_MODEL {
row:
assignsallatoms_exactlyonce
RULE:
duplicate_or missingassignment_invalid
}
CELL_VALIDATION_MODEL {
cell:
recompute_fromformula_androw
RULE:
stored_value MUST equalcomputed_value
}
COLUMN_VALIDATION_MODEL {
column:
formula_identifier
values_matchrecalculation
RULE:
mislabeled_column MUST be flagged
}
TABLE_COMPLETENESS_MODEL {
complete_if:
row_count=2^n
allvaluations_unique
allvaluationscovered
RULE:
visual_size_alone_doesnotestablishcompleteness
PARTIAL_TABLE_MODEL {
table:
fewer_thanallpossiblevaluations
RULE:
partial_table_canestablish_counterexample_orSAT_witness
butnotUNSAT_orvalidity_byexhaustion
}
SAMPLED_TABLE_MODEL {
rows:
selectedvaluations_only
RULE:
sample_results MUST not_be generalised_toallvaluations
}
FILTERED_TABLE_MODEL {
rows:
selected_bycondition
RULE:
filter_condition MUST be preserved
}
COMPRESSED_TABLE_MODEL {
representation:
merges_rows_usingpatterns_orwildcards
RULE:
compression MUST preserveexact_semantics
}
WILDCARD_ROW_MODEL {
symbol:
* or X_when_declared
meaning:
multiplevaluations
RULE:
wildcard != truth_value
}
CUBE_MODEL {
partial_assignment:
represents_set_ofcompletevaluations
RULE:
omittedvariables_arefree_notFALSE
}
TABLE_SIZE_MODEL {
rows:
2^n
growth:
exponential_inatom_count
RULE:
exhaustive_tables_may_beimpractical_forlarge_n
}
SCALABILITY_MODEL {
alternative_methods:
SAT_solver
BDD
symbolic_reasoning
RULE:
alternative_method_result MUST preserveproof_orwitness_status
}
LAZY_EVALUATION_MODEL {
process:
evaluateonlyneeded_subformulas_orrows
RULE:
lazy_result != complete_table_unlesscoverage_completed
}
PARALLEL_GENERATION_MODEL {
rows:
independently_evaluable
RULE:
deterministic_row_identifiers_andordering SHOULD be preserved
}
CACHING_MODEL {
subformula:
cache_truth_vector
RULE:
cache_key MUST includeformula_AST_atom_order_andlogic_version
}
MEMOISATION_MODEL {
repeated_subformula:
reusecomputed_column
RULE:
structurally_identical_orcanonical_equivalent_key MUST be defined
}
COLUMN_DEPENDENCY_GRAPH_MODEL {
nodes:
atom_andsubformula_columns
edges:
DEPENDS_ON
RULE:
graph MUST beacyclic_forordinaryformula_AST
}
DISPLAY_MODEL {
visual:
headers
rows
truth_symbols
RULE:
display_format != semantic_table_identity
}
HEADER_MODEL {
column_header:
atom_orformula_label
RULE:
header SHOULD linktocanonical_formula_identifier
}
ROW_STRIPING_DISTINCTION_MODEL {
formatting:
visual_only
RULE:
presentation_style hasno_semantic_effect
}
SORTING_MODEL {
table:
rows_maybereordered
RULE:
semantic classification unchanged
buttruth_vector_order_metadata MUST update
}
COLUMN_REORDERING_MODEL {
display:
columns_maybereordered
RULE:
column_identifiers_anddependencies MUST remainstable
}
SERIALISATION_MODEL {
possible:
CSV
JSON
HTML
RDF
binary
RULE:
serialisation MUST preserveatom_order_row_assignments_column_formula_ids_andtruth_values
}
CSV_MODEL {
headers:
variable_andformula_labels
rows:
truth_values
RULE:
metadata_forlogic_atom_mapping_andencoding SHOULD accompanyCSV
}
JSON_MODEL {
fields:
atoms
rows
columns
formulas
values
RULE:
Boolean_json_values_requiredeclared_mapping
}
HTML_TABLE_MODEL {
purpose:
human_readable_rendering
RULE:
DOM_order SHOULD notbeonlysource_ofsemantic_identity
}
ACCESSIBILITY_MODEL {
requirements:
meaningful_headers
row_column_associations
text_equivalents_for symbols
RULE:
accessibility_metadata MUST notchangeformal semantics
}
NONCLASSICAL_TABLE_MODEL {
truth_domain:
logic_specific
rows:
k^n_forfinite_k_valued_total_assignments
RULE:
k_andconnective_tables MUST be explicit
}
THREE_VALUED_TABLE_MODEL {
truth_values:
three_declared_values
rows:
3^n
RULE:
thirdvalue_semantics MUST be specified
}
MANY_VALUED_TABLE_MODEL {
finite_truth_domain_size:
k
complete_rows:
k^n
RULE:
designated_values MUST be identified_forvalidity
}
DESIGNATED_VALUE_MODEL {
truth_domain:
subset_designated
RULE:
nonclassical_validity_maydepend_ondesignatedness
}
FUZZY_TABLE_DISTINCTION_MODEL {
graded_values:
potentiallycontinuous
RULE:
exhaustivefinite_truth_table_maynotexist
}
PROBABILITY_TABLE_DISTINCTION_MODEL {
probability_table:
probabilities_overevents_orstates
RULE:
probability_table != truth_table
}
DECISION_TABLE_DISTINCTION_MODEL {
decision_table:
condition_action_rules
RULE:
decision_table != logical_truth_table
}
DATABASE_TABLE_DISTINCTION_MODEL {
relational_table:
data_rows_andcolumns
RULE:
generic_database_table != truth_table
}
CONFUSION_MATRIX_DISTINCTION_MODEL {
matrix:
predicted_vsactual_classes
RULE:
confusion_matrix != truth_table
}
STATE_TRANSITION_TABLE_DISTINCTION_MODEL {
table:
current_state_input_next_state
RULE:
state_transition_table != propositional_truth_table
}
NATURAL_LANGUAGE_FORMULA_MODEL {
source:
human_statement
target:
propositional_formula
RULE:
table_semantics depend_oncorrectformalisation
}
FORMALISATION_MODEL {
process:
identifyatoms
identifyconnectives
constructformula
generatetable
RULE:
formalisation_confidence SHOULD be preserved
}
ATOM_LABEL_MODEL {
atom:
maps_tohuman_readable_claim
RULE:
label != truth_value
}
CONDITIONAL_TABLE_MODEL {
formula:
A -> B
diagnostic_row:
A=TRUE
B=FALSE
result=FALSE
RULE:
table_encodesmaterial_implication_notcausal_strength
}
BICONDITIONAL_TABLE_MODEL {
formula:
A <-> B
result:
TRUE_whenA_B_equal
}
OR_TABLE_MODEL {
classical:
inclusive
RULE:
natural_language_exclusive_OR_requiresXOR_table
}
AMBIGUITY_MODEL {
possible:
atom_order_missing
row_order_missing
truth_encoding_unknown
formula_parse_ambiguous
partial_table_mislabelled_complete
RULE:
ambiguity MUST be resolved_orpreserved
}
SEMANTIC_CONFIDENCE_MODEL {
confidence:
support_forformula_parse_atom_mapping_andtable_interpretation
RULE:
confidence != truth_value
}
ERROR_MODEL {
possible:
wrong_row_count
duplicate_row
missing_row
incorrect_cell
incorrect_formula_header
wrong_connective_semantics
encoding_mismatch
RULE:
error_type SHOULD identifyrow_column_andexpected_value
}
CORRECTION_MODEL {
invalid_table:
mayberecomputed_fromformula
RULE:
corrected_table MUST preserveoriginal_artifact_andrevision_provenance
}
VERIFICATION_MODEL {
process:
recompute_table_independently
RULE:
verification_requiresformula_logic_andatom_order
}
PROOF_CERTIFICATE_MODEL {
truth_table:
itself_mayserve_asfinite_semantic_certificate
requires:
completecoverage
correctevaluation
RULE:
certificate_claim MUST includevalidation_status
}
AI_GENERATION_MODEL {
AI:
parseformula
identifyatoms
choosecanonicalorder
enumeraterows
evaluatecells
classifyformula
RULE:
AI_generated_table SHOULD be machine_revalidated
}
AI_INTERPRETATION_MODEL {
AI:
readtable
reconstructvaluations
identifytargetcolumn
determineclassification
extractwitnesses
RULE:
AI MUST notinfercompleteness_fromappearancealone
}
EXPLAINABILITY_MODEL {
output:
formula
atoms
row_generation
subformula_columns
target_vector
classification
witness_orcountermodel
RULE:
explanation SHOULD exposewhyclassification_follows
}
PROVENANCE_MODEL {
fields:
source_formula
parser
atom_order
row_order
generator
validator
logic_version
timestamp
agent
RULE:
truth_table MUST remainprovenance_linked
}
VERSION_MODEL {
versioned:
formula
language
connective_semantics
table_generator
ontology
RULE:
historical_table MUST preserveversions
}
INTEROPERABILITY_MODEL {
exchange:
table_identifier
formula_identifier
atom_identifiers
atom_order
row_order
valuations
column_formula_identifiers
truth_values
logic_system
completeness_status
RULE:
matrix_ofT_F_values_withoutmetadata_isnotlossless
}
KNOWLEDGE_GRAPH_MODEL {
nodes:
truth_table
row
column
valuation
atom
formula
truth_value
edges:
HAS_ROW
HAS_COLUMN
ASSIGNS
EVALUATES
SATISFIES
REFUTES
DEPENDS_ON
REPRESENTS
RULE:
row_column_formula_andvaluation_entities MUST remainlinked
}
RELATIONSHIPS {
TruthTable -> EVALUATES -> PropositionalFormula
TruthTable -> HAS_ROW -> TruthTableRow
TruthTable -> HAS_COLUMN -> TruthTableColumn
TruthTableRow -> REPRESENTS -> Valuation
Valuation -> ASSIGNS -> TruthValue
TruthTableColumn -> REPRESENTS -> AtomOrSubformula
TruthVector -> DERIVED_FROM -> TargetColumn
TrueRow -> IS_MODEL_OF -> Formula
FalseRow -> MAY_BE_COUNTERMODEL_TO -> ValidityClaim
}
SEMANTIC_EQUIVALENCE {
"truth table" ~= exhaustive_table_ofpropositional_valuations_andformula_truth_values
"truth table row" ~= onecompletevaluation_pluscomputed_formula_values
"truth vector" ~= orderedsequence_offormula_truth_values_overdeclaredvaluation_order
truth_table != probability_table
truth_table != decision_table
truth_table != database_table
row_order != semantic_meaning
truth_vector_withoutorder != canonical_semantic_signature
partial_table != complete_truth_table
sampled_rows != exhaustive_semantics
don't-care != third_truth_value
TRUE_FALSE_encoding != semantic_identity
implication_table != causal_model
semantic_confidence != truth_value
}
CONSTRAINTS {
truth_table_identifier_formula_andlogic_system_must_be preserved = true
atom_set_andvariable_order_must_be explicit = true
complete_classical_table_must_contain_2_power_n_unique_rows = true
everycomplete_row_must_assignallatoms = true
row_coverage_anduniqueness_must_be validated = true
connective_truth_functions_must_be correct = true
target_andsubformula_columns_must_link_toformula_identifiers = true
truth_vector_must_preserve_row_order = true
partial_or sampled_tables_must_not_be labelled_exhaustive = true
validity_UNSAT_equivalence_andargument_validity_bytable_requirecompletecoverage = true
countermodels_andSAT_witnesses_should_be preserved = true
numeric_anddisplay_encodings_must_be distinguished_fromtruth_values = true
nonclassical_tables_must_define_truth_domain_anddesignated_values = true
provenance_must_be preserved = true
}
TRUTH_TABLE_RECORD_MODEL {
RECORD:
truth_table_identifier
formula_identifier
source_formula
formula_AST
logic_system_identifier
logic_version
atom_identifiers
atom_count
variable_order
row_order
expected_row_count
actual_row_count
completeness_status
row_identifiers
valuations
column_identifiers
column_formula_identifiers
truth_value_encoding
target_column_identifier
truth_vector
satisfying_rows
falsifying_rows
satisfiability_status
validity_status
contingency_status
countermodels
argument_premises
argument_conclusion
argument_validity_status
canonical_DNF
canonical_CNF
verification_status
ambiguity_status
semantic_confidence
version
provenance
RULE:
truth_table_record MUST support_independent_reconstruction_andsemantic_verification
}
REASONING_RULES {
IF formula_received
THEN parseformula
identifydistinctatoms
establishcanonical_orprovided_variable_order
IF n_atoms_identified
THEN expected_rows = 2^n_underclassical_semantics
IF complete_table_requested
THEN generateeveryuniquevaluation_exactlyonce
FOR each_row:
assignatom_values
evaluatesubformulas_in dependency_order
evaluatetarget
storetrace
IF alltarget_values_TRUE
AND table_complete
THEN classify_asvalid_tautology
IF alltarget_values_FALSE
AND table_complete
THEN classify_asunsatisfiable_contradiction
IF target_containsTRUE_andFALSE
THEN classify_ascontingent
IF anytarget_TRUE
THEN classify_assatisfiable
preserveatleastonewitness
IF anytarget_FALSE
THEN classify_asfalsifiable
preserveatleastonecountervaluation
IF comparing_formulas
THEN evaluateoverunion_ofatoms
comparecolumns_rowbyrow
IF testing_argument
THEN identifyrows_whereallpremises_TRUE
searchforconclusion_FALSE
IF table_partial_or sampled
THEN allowwitness_orcounterexample_conclusions_whenvalid
butdo_notclaimexhaustive_validity_orUNSAT
IF table_revision_orformula_revision_occurs
THEN preservepreviousversion
regenerateandrevalidate_currenttable
}
VALIDATION {
CHECK truth_table_identifier
CHECK formula_identifier
CHECK formula_AST
CHECK logic_system
CHECK atom_set
CHECK atom_count
CHECK variable_order
CHECK row_order
CHECK expected_row_count
CHECK actual_row_count
CHECK row_uniqueness
CHECK row_coverage
CHECK valuation_completeness
CHECK column_identifiers
CHECK subformula_dependencies
CHECK connective_semantics
CHECK cell_values
CHECK target_column
CHECK truth_vector
CHECK satisfiability
CHECK validity
CHECK contingency
CHECK countermodels
CHECK semantic_equivalence
CHECK argument_validity
CHECK canonical_DNF_CNF
CHECK partial_orcomplete_status
CHECK encoding
CHECK verification_status
CHECK ambiguity
CHECK semantic_confidence
CHECK version
CHECK provenance
}
CONFIDENCE {
formula_parse_confidence: 1.00
atom_identification_confidence: 1.00
row_generation_confidence: 1.00
connective_evaluation_confidence: 1.00
table_completeness_confidence: 1.00
semantic_classification_confidence: 1.00
equivalence_argument_confidence: 1.00
natural_language_formalisation_confidence: 0.96
RULE:
confidence decreases when formula parse, atom mapping,
variable order, row coverage, connective semantics,
completeness status, version orprovenance is missing or ambiguous
}
UPDATE_MODEL {
ON new_truth_table:
assign_table_identifier
attach_formula_logic_andatom_order
validate_rows_andcolumns
compute_semantic_classification
preserveprovenance
compute_FLOW_HASH
ON new_row:
verifyvaluation_uniqueness
evaluatecells
updatecoverage_andclassification
ON formula_revision:
preserveprevioustable
create_newtable_version
regenerateallrows_andvalues
ON logic_orconnective_revision:
preservehistoricaltable
registernewlogicversion
reevaluatecurrenttable_undernewsemantics
ON semantic_revision:
preserve_previous_definition
record_revision
recompute_FLOW_HASH
}
AI_CONSUMPTION {
canonical_entity: TruthTable
canonical_role: ExhaustivePropositionalSemanticTable
interpretation_order:
identify_truth_table_andlogic_system
identify_source_formula_andAST
identifydistinct_atomic_variables
resolvevariable_order_androw_order
verifyexpected_row_count
reconstructeachvaluation
identifyatom_subformula_andtargetcolumns
recomputecells_usingconnective_semantics
verifyrow_uniqueness_andcoverage
constructtruth_vector
classify_satisfiability_validity_contradiction_orcontingency
preserveSAT_witnesses_andcountermodels
compareformula_columns_whenrequired
evaluateargumentcritical_rows_whenrequired
derivecanonical_DNF_orCNF_whenrequested
distinguishcomplete_partial_sampled_andcompressed_tables
detectencoding_orsemantic_ambiguity
assignsemantic_confidence
preserveversions_andprovenance
emit_machine_resolvable_truth_table_structure
ambiguity_controls:
distinguish_truth_table_fromprobability_table
distinguish_truth_table_fromdecision_table
distinguish_truth_table_fromdatabase_table
distinguish_row_order_fromsemantic_meaning
distinguish_truth_vector_withoutorder_fromcanonical_signature
distinguish_partial_table_fromcomplete_truth_table
distinguish_sampled_rows_fromexhaustive_semantics
distinguish_dont_care_fromthird_truth_value
distinguish_TRUE_FALSE_encoding_fromsemantic_identity
distinguish_implication_table_fromcausal_model
distinguish_semantic_confidence_fromtruth_value
preserve_formula_atoms_orders_rows_columns_values_completeness_version_andprovenance
}
PROVENANCE {
ontology_scope:
truth_tables_valuations_truth_vectors_validity_satisfiability_equivalence_arguments_andAI_semantic_evaluation
parent_path:
AI Core
-> Ontologies
-> Mathematical Ontologies
-> Mathematical Logic Ontologies
-> Propositional Logic
-> Truth Table Ontology
authorship:
Mark Higgins
Sky (Recursive AI)
site:
https://kirethresearch.org
}
FLOW_HASH {
INPUTS:
canonical_entity
ontology_path
table_identity_model
logic_context_model
source_formula_model
atom_set_model
atom_count_model
variable_order_model
row_count_model
valuation_model
complete_row_model
partial_row_model
row_uniqueness_model
row_coverage_model
row_order_model
binary_row_order_model
lexicographic_row_order_model
Gray_code_row_model
row_identifier_model
column_model
atomic_column_model
subformula_column_model
target_column_model
column_order_model
cell_model
truth_value_model
Boolean_encoding_model
symbolic_encoding_model
connective_model
negation_evaluation_model
conjunction_evaluation_model
disjunction_evaluation_model
XOR_evaluation_model
implication_evaluation_model
biconditional_evaluation_model
NAND_evaluation_model
NOR_evaluation_model
dependency_evaluation_model
bottom_up_evaluation_model
computation_trace_model
truth_vector_model
truth_vector_hash_model
semantic_signature_model
tautology_model
contradiction_model
contingency_model
satisfiability_model
unsatisfiability_model
validity_model
falsifiability_model
model_model
countermodel_model
model_set_model
semantic_equivalence_model
column_comparison_model
complement_column_model
implication_column_model
biconditional_column_model
XOR_equivalence_test_model
argument_table_model
premise_column_model
conclusion_column_model
critical_row_model
invalidating_row_model
argument_validity_model
vacuous_validity_model
entailment_model
entailment_reduction_model
consistency_model
formula_set_table_model
atom_union_model
projection_model
restriction_model
cofactor_table_model
Shannon_model
canonical_DNF_model
minterm_model
canonical_CNF_model
maxterm_model
normal_form_extraction_model
Boolean_function_model
function_arity_model
function_count_model
functional_equivalence_model
Boolean_algebra_model
Boolean_identity_verification_model
digital_logic_model
logic_gate_model
multi_input_gate_model
circuit_truth_table_model
multi_output_table_model
care_set_model
dont_care_model
incomplete_specification_model
K_map_link_model
BDD_link_model
SAT_link_model
SAT_witness_model
UNSAT_proof_by_table_model
validity_proof_by_table_model
equivalence_proof_by_table_model
argument_proof_by_table_model
table_generation_model
valuation_enumeration_model
recursive_generation_model
binary_counter_generation_model
table_validation_model
row_validation_model
cell_validation_model
column_validation_model
table_completeness_model
partial_table_model
sampled_table_model
filtered_table_model
compressed_table_model
wildcard_row_model
cube_model
table_size_model
scalability_model
lazy_evaluation_model
parallel_generation_model
caching_model
memoisation_model
column_dependency_graph_model
display_model
header_model
row_striping_distinction_model
sorting_model
column_reordering_model
serialisation_model
CSV_model
JSON_model
HTML_table_model
accessibility_model
nonclassical_table_model
three_valued_table_model
many_valued_table_model
designated_value_model
fuzzy_table_distinction_model
probability_table_distinction_model
decision_table_distinction_model
database_table_distinction_model
confusion_matrix_distinction_model
state_transition_table_distinction_model
natural_language_formula_model
formalisation_model
atom_label_model
conditional_table_model
biconditional_table_model
OR_table_model
ambiguity_model
semantic_confidence_model
error_model
correction_model
verification_model
proof_certificate_model
AI_generation_model
AI_interpretation_model
explainability_model
provenance_model
version_model
interoperability_model
knowledge_graph_model
relationships
semantic_equivalence
constraints
truth_table_record_model
reasoning_rules
validation
confidence_model
update_model
AI_consumption_model
PURPOSE:
deterministic_semantic_integrity_identifier
RECOMPUTE_WHEN:
ontology_definition_changes
truth_table_identity_formula_orlogic_system_changes
atom_set_variable_order_orrow_order_changes
row_generation_coverage_oruniqueness_changes
connective_truth_function_orcell_evaluation_changes
truth_vector_satisfiability_validity_orcontingency_semantics_change
equivalence_entailment_argument_orcountermodel_semantics_change
canonical_DNF_CNF_Boolean_function_orcircuit_semantics_change
table_generation_validation_compression_orserialisation_changes
nonclassical_truth_domain_or designated_value_changes
AI_generation_interpretation_orconfidence_changes
truth_table_record_changes
constraint_model_changes
reasoning_rule_changes
version_orprovenance_changes
}
}
END FLOW