Logical Inference Ontology

FLOW_ALGEBRA LogicalInferenceOntology v1.0
TARGET_URL := "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/logical-reasoning/logical-inference-ontology.html"
LANGUAGE := "en-GB"
DOMAIN := MathematicalOntologies > MathematicalLogicOntologies > LogicalReasoning > LogicalInferenceOntology

BEGIN_FLOW

ENTITY LogicalInference {
 TYPE := FormalReasoningStep
 DEFINITION := "A rule-governed transition from one or more premises to a conclusion within a specified logical system."
}
ENTITY Premise { TYPE := PropositionOrFormula }
ENTITY Conclusion { TYPE := PropositionOrFormula }
ENTITY InferenceRule { TYPE := FormalRuleSchema }

RELATIONSHIP LogicalInference --takes_input--> Premise
RELATIONSHIP LogicalInference --produces--> Conclusion
RELATIONSHIP LogicalInference --instantiates--> InferenceRule
RELATIONSHIP LogicalInference --belongs_to--> LogicalSystem
RELATIONSHIP LogicalInference --supports--> Derivation
RELATIONSHIP LogicalInference --supports--> Proof

INFERENCE_SCHEMA {
 PREMISES := {P1,...,Pn}
 CONCLUSION := C
}

VALID_RULE_APPLICATION {
 GIVEN rule R and instantiation σ
 IF premises match σ(R.premises)
 AND side_conditions_hold
 THEN infer σ(R.conclusion)
}

RULE_SIDE_CONDITION {
 EXAMPLES := {
  FreshVariable,
  FreeForSubstitution,
  EigenvariableCondition,
  TypeCompatibility
 }
}

SEMANTIC_SOUNDNESS {
 Rule R sound IFF
 FOR_ALL models M:
  if all premises true then conclusion true
}

SYNTACTIC_INFERENCE {
 Γ⊢C := C obtainable from Γ through permitted inference steps
}
SEMANTIC_ENTAILMENT {
 Γ⊨C := every model of Γ satisfies C
}
DERIVATION {
 TYPE := SequenceOrTreeOfLogicalInferences
}
PROOF_TREE {
 LEAVES := {Premises,Assumptions,Axioms}
 INTERNAL_NODES := RuleConclusions
 ROOT := FinalConclusion
}

MODUS_PONENS {
 P
 P->Q
 ----
 Q
}
CONJUNCTION_INTRODUCTION {
 P
 Q
 ----
 P∧Q
}
CONJUNCTION_ELIMINATION {
 P∧Q
 ----
 P
}
DISJUNCTION_INTRODUCTION {
 P
 ----
 P∨Q
}
IMPLICATION_INTRODUCTION {
 [P] ... Q
 ----
 P->Q
 REQUIRE discharge_assumption_P
}
NEGATION_INTRODUCTION {
 [P] ... contradiction
 ----
 ¬P
}
EX_FALSO {
 contradiction
 ----
 P
 REQUIRE explosive_logic
}
DOUBLE_NEGATION_ELIMINATION {
 ¬¬P
 ----
 P
 REQUIRE classical_or_compatible_logic
}

UNIVERSAL_ELIMINATION {
 ∀xP(x)
 ----
 P(t)
 REQUIRE t_free_for_x
}
UNIVERSAL_INTRODUCTION {
 P(x)
 ----
 ∀xP(x)
 REQUIRE eigenvariable_or_generality_condition
}
EXISTENTIAL_INTRODUCTION {
 P(t)
 ----
 ∃xP(x)
}
EXISTENTIAL_ELIMINATION {
 ∃xP(x)
 [P(c)] ... Q
 ----
 Q
 REQUIRE freshness_conditions
}

RESOLUTION {
 P∨A
 ¬P∨B
 ----
 A∨B
}
FIRST_ORDER_RESOLUTION {
 C1∨L
 C2∨¬L'
 UNIFY L,L' WITH θ
 ----
 (C1∨C2)θ
}
UNIFICATION {
 INPUT := expressions E1,E2
 OUTPUT := substitution θ
 REQUIRE E1θ=E2θ
}
MOST_GENERAL_UNIFIER {
 ABBREVIATION := MGU
 PROPERTY := every_other_unifier_factors_through_MGU
}
SUBSTITUTION {
 θ := mapping_variables_to_terms
}
CAPTURE_AVOIDANCE {
 substitution MUST avoid unintended_variable_capture
}

SEQUENT {
 FORM := Γ⊢Δ
}
CUT_RULE {
 Γ⊢A
 A,Π⊢Δ
 ----
 Γ,Π⊢Δ
}
STRUCTURAL_RULES {
 MAY_INCLUDE := {Weakening,Contraction,Exchange}
 WARNING := not_available_unrestricted_in_every_logic
}
LINEAR_LOGIC_WARNING {
 weakening_and_contraction are not generally unrestricted
}

NATURAL_DEDUCTION {
 RULE_ORGANISATION := {IntroductionRules,EliminationRules}
}
HILBERT_CALCULUS {
 COMPONENTS := {AxiomSchemas,InferenceRules}
}
TABLEAU_INFERENCE {
 Formulae -> branch_expansion -> open_or_closed_branches
}

FORWARD_CHAINING {
 START := KnownFacts
 APPLY rules whose premises hold
 ADD conclusions
 STOP := fixed_point_or_goal
}
BACKWARD_CHAINING {
 START := Goal
 SELECT rules concluding goal
 REDUCE := Subgoals
 RECURSE until facts_or_premises
}
INFERENCE_ENGINE {
 COMPONENTS := {
  KnowledgeBase,
  RuleSet,
  MatchingStrategy,
  SearchStrategy,
  ProofTrace
 }
}
RULE_MATCHING {
 rule -> match premises -> instantiate -> validate
}
LOOP_DETECTION {
 repeated_reasoning_state -> prune_or_memoise
}
HEURISTIC_SEARCH {
 MAY improve efficiency
 WARNING := failure_to_find_proof_does_not_establish_non_derivability
}

INFERENCE_TRACE {
 RECORD := {
  RuleName,
  PremisesUsed,
  Substitution,
  SideConditions,
  ConclusionDerived
 }
}

VERIFY_INFERENCE_STEP(S,L) {
 IDENTIFY rule R
 VERIFY R belongs_to L
 VERIFY premise_match
 VERIFY substitution
 VERIFY side_conditions
 VERIFY conclusion
 RETURN valid_or_failure_reason
}

VERIFY_DERIVATION(D,L) {
 FOR_EACH step S:
  VERIFY_INFERENCE_STEP(S,L)
  IF invalid RETURN invalid_derivation
 RETURN valid_derivation
}

ADMISSIBLE_RULE {
 adding R does_not_enlarge_theorem_set
 under standard proof_theoretic_definition
}
DERIVABLE_RULE {
 conclusion derivable from premises using existing_rules
}
INVERTIBLE_RULE {
 REQUIRE calculus_specific_definition
}

INFERENCE_VS_IMPLICATION {
 ObjectLanguage := P->Q
 MetaLevel := P⊢Q
 MUST_REMAIN_DISTINCT := TRUE
}
INFERENCE_VS_ENTAILMENT {
 InferenceStep := syntactic_rule_application
 Entailment := semantic_consequence
}
INFERENCE_MODES {
 DeductiveInference := truth_preserving
 InductiveInference := ampliative_support
 AbductiveInference := explanatory_hypothesis_generation
}

NONMONOTONIC_INFERENCE {
 new_information MAY retract prior_conclusion
 REQUIRE nonmonotonic_calculus
}
PARACONSISTENT_INFERENCE {
 contradictions do_not_necessarily_explode
 REQUIRE paraconsistent_logic
}
MODAL_INFERENCE {
 REQUIRE modal_rules_and_semantics
}
TEMPORAL_INFERENCE {
 REQUIRE temporal_rules_and_semantics
}
FUZZY_INFERENCE {
 activation := graded
 REQUIRE fuzzy_inference_semantics
}

PROOF_GRAPH {
 NODES := FormulaOccurrences
 EDGES := InferenceDependencies
}
DEPENDENCY_TRACKING {
 DerivedFormula -> exact_premises_and_assumptions
}
ASSUMPTION_DISCHARGE {
 temporary_assumption removed according_to proof_rule
}
SCOPE_VALIDATION {
 assumptions MUST NOT escape scope unless properly_discharged
}

APPLICATION_RELATIONSHIPS {
 LogicalInference --supports--> DeductiveReasoning
 LogicalInference --supports--> FormalProof
 LogicalInference --supports--> AutomatedTheoremProving
 LogicalInference --supports--> LogicProgramming
 LogicalInference --supports--> ArtificialIntelligence
 LogicalInference --supports--> KnowledgeRepresentation
 LogicalInference --supports--> FormalVerification
 LogicalInference --supports--> ProofTheory
}

ONTOLOGY_LINKS {
 PARENT := LogicalReasoningOntology
 RELATED := {
  DeductiveReasoningOntology,
  InductiveReasoningOntology,
  AbductiveReasoningOntology,
  PropositionalLogicOntology,
  PredicateLogicOntology,
  NaturalDeductionOntology,
  SequentCalculusOntology,
  ResolutionOntology,
  UnificationOntology,
  ProofTheoryOntology,
  EntailmentOntology
 }
}

SEMANTIC_EQUIVALENCE {
 LogicalInference ≈ FormalInference
 InferenceRule ≈ RuleOfInference
 RuleApplication ≈ InferenceStep
 Derivation ≈ SequenceOrTreeOfInferenceSteps
}

CONSTRAINTS {
 C1 := every inference MUST identify its logical system when rule validity depends on it
 C2 := rule premises MUST match selected rule schema
 C3 := all side conditions MUST be satisfied
 C4 := substitution MUST avoid capture and respect typing/sorts
 C5 := syntactic inference MUST NOT be conflated with semantic entailment
 C6 := object-language implication MUST NOT be conflated with meta-level inference
 C7 := structural rules MUST NOT be assumed where restricted
 C8 := failed heuristic search MUST NOT establish non-derivability
 C9 := assumption scopes and discharge MUST be preserved
 C10 := nonclassical inference MUST use declared semantics
 C11 := proof traces MUST record actual dependencies
 C12 := derived formulae MUST remain well-formed
}

REASONING_CHAIN Query Q {
 IDENTIFY logical_system L
 IDENTIFY premises Γ
 IDENTIFY target C
 IDENTIFY available_rules
 PARSE formulae
 CHECK well_formedness

 FOR_EACH candidate_rule {
  MATCH premises
  COMPUTE substitution_if_needed
  CHECK side_conditions
  DERIVE conclusion_if_valid
  RECORD inference_trace
 }

 IF forward_chaining
  THEN iterate_until_goal_or_fixed_point
 IF backward_chaining
  THEN decompose_goal_into_subgoals
 IF resolution_context
  THEN unify_and_resolve
 IF natural_deduction_context
  THEN manage_assumptions_and_discharge
 IF sequent_context
  THEN apply_allowed_sequent_rules
 IF nonclassical_context
  THEN enforce_logic_specific_rules
 IF target_derived
  THEN VERIFY_DERIVATION

 VALIDATE constraints
 RETURN inference_result WITH proof_trace
}

CONFIDENCE {
 LOGICAL_INFERENCE_DEFINITION := 1.00
 RULE_APPLICATION_PATTERN := 1.00
 NATURAL_DEDUCTION_RULES := 1.00
 RESOLUTION_RULE := 1.00
 QUANTIFIER_SIDE_CONDITIONS := 0.99
 SYNTACTIC_SEMANTIC_DISTINCTION := 1.00
 SPECIFIC_INFERENCE :=
  IF logic_rules_premises_and_side_conditions_complete THEN 1.00
  ELSE REQUIRE missing_information
}

UPDATE_MODEL {
 IMMUTABLE_CORE := {
  PremiseConclusionStructure,
  RuleSchema,
  RuleInstantiation,
  SideConditionValidation,
  DerivationDependency
 }
 EXTENSIBLE := {
  InferenceRules,
  ProofCalculi,
  SearchStrategies,
  UnificationMethods,
  NonmonotonicRules,
  ModalTemporalRules,
  AutomatedReasoningMethods,
  ApplicationDomains,
  RelatedOntologies
 }
 ON_NEW_KNOWLEDGE {
  IDENTIFY logical_calculus
  VALIDATE rule_schema
  VALIDATE substitutions
  CHECK side_conditions
  CHECK assumption_scopes
  CHECK semantic_consistency
  ADD relationship IF non_contradictory
  UPDATE confidence
 }
}

AI_CONSUMPTION {
 MACHINE_READABLE := TRUE
 HUMAN_INTERPRETABLE := TRUE
 RECURSIVE_REASONING := TRUE
 ENTITY_RESOLUTION := TRUE
 FORMULA_RESOLUTION := TRUE
 RULE_RESOLUTION := TRUE
 RULE_MATCHING := TRUE
 SUBSTITUTION_REASONING := TRUE
 SIDE_CONDITION_VALIDATION := TRUE
 FORWARD_CHAINING := TRUE
 BACKWARD_CHAINING := TRUE
 RESOLUTION_REASONING := TRUE
 ASSUMPTION_SCOPE_REASONING := TRUE
 PROOF_TRACE_GENERATION := TRUE
 DERIVATION_VALIDATION := TRUE
}

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

OUTPUT {
 PRIMARY_ENTITY := LogicalInference
 INTERPRETATION := "Logical inference is a formal transition from premises to a conclusion by an explicitly defined rule. Valid inference requires correct rule matching, substitution, side-condition handling and assumption scope, while complete derivations compose these individual steps into proof structures."
}

END_FLOW