Logic Selection

In logic, there are no morals. Everyone is at liberty to build up his own logic, i.e. his own form of language, as he wishes.

This chapter formalizes the logic and absence-policy dimensions of context, defining Anchor A15 (Logic Selection): each view carries a quadruple (signature, constraints, proof logic, absence policy), and coherence across views requires reconciling both the inference rules and the interpretation of missing data. The three-valued logic adapter is introduced as a first-class reconciliation mechanism for CWA/OWA mismatches. T5 (Negation/Absence) receives a precise policy diagnosis in the constructed example; other failures may arise from the model, records or implementation. The formal treatment here extends the context structure of Chapter 10 and corresponds to the narrative discussion of evidentiary gaps in Vol I, Chapter 6 ("Evidence without Custody").

The Problem with "Not"

Consider a hypothetical hospital merging two patient databases. Database A lists allergies explicitly: if a patient has a penicillin allergy, the record says so. If the allergy field is empty, the system treats it as unknown. Database B uses a standardized allergy field: if the field is empty, the patient has no known allergies.

After the merge, patients with missing allergy data from Database A are flagged as "no allergies." A physician prescribes penicillin. The patient has an allergic reaction. The system did exactly what it was told to do.

Stipulate that both databases were internally consistent. The integration still contains a software or specification error: two systems with different interpretations of absence were merged without reconciliation.

The same pattern appears everywhere:

• Inventory: "not in stock" may mean "confirmed out" or "we haven't checked yet."
• Legal eligibility: "not prohibited" may mean "permitted" or "not yet ruled on."
• Product safety: an empty toxicity field may be intended to encode a completed negative test or no test at all. Its emptiness does not establish which occurred, and a negative test has a scope.

The common thread is negation. What does "not p" mean? The answer depends on two independent choices the system has made, often implicitly—the proof logic governing valid inferences, and the absence policy governing how missing data is interpreted.

Two Independent Axes

The first axis is proof logic. Different logics have different inference rules.

Classical logic accepts the law of excluded middle: for any proposition p, either p or ¬p holds. It also accepts double negation elimination: ¬¬p implies p. Semantically, propositions are bivalent: every proposition is either true or false, even if the system does not know which.

Intuitionistic logic does not accept either as an unrestricted principle¹. The law of excluded middle is not assumed; p ∨ ¬p is not a theorem. Double negation elimination fails: ¬¬p means "assuming ¬p leads to contradiction," but that does not prove p is true. Negation is asymmetric: proving ¬p requires a refutation procedure, not merely the absence of a proof for p.

A complete, consistent theory can decide each sentence in its language; an arbitrary classical theory need not. Completeness of a proof calculus means that semantic consequences are derivable, not that an unsuccessful search proves a sentence false. Intuitionistic logic likewise does not identify lack of proof with a third truth value.

Classical logic itself does not license "not provable implies false." That inference requires an additional commitment, an epistemic policy layered on top of the proof system. This policy is independent of the proof logic.

The second axis is absence policy.

Closed-World Assumption (CWA): A declared closure rule licenses a negative conclusion for the relevant ground atom when the required positive derivability test fails². This is more than absence of an explicit assertion: rules may derive a positive claim. A completed database query over a specified finite relation supplies one useful setting; an unfinished proof search does not.

Open-World Assumption (OWA): Absence of p does not itself establish ¬p³. If neither polarity is derivable, the question remains undetermined. A knowledge base can nevertheless derive p from other facts or explicitly establish its negation.

You can mix these independently. Classical logic with OWA is coherent: you accept excluded middle as a logical principle but treat missing data as unknown. Intuitionistic logic with CWA is also coherent, though unusual: you reject excluded middle but treat absence as negation for operational purposes.

Classical description logics can use open-world semantics; knowledge graphs are not generally intuitionistic systems. The choices must be identified in the particular implementation. The conflation to avoid: "classical logic says absence implies negation." It does not. CWA says that.

CWA and OWA in Practice

Finite relational databases can evaluate ordinary relational queries over their recorded domain. Their decidability comes from the query language and evaluation setting; CWA alone does not make an arbitrary logic decidable. If you ask "which products are in stock?" the system returns all rows where in_stock = true. Rows with false or NULL in that field are not returned by this filter, for different reasons. This works well when the database is the authoritative source and completeness is guaranteed.

A knowledge graph can adopt OWA when combining sources that make no collective completeness claim. If you ask "is product X sustainable?" and the graph has no sustainability data for X, the answer is "unknown," not "no." This works well when the knowledge base is expected to grow and no single source is authoritative.

The merge problem emerges when these systems meet:

• Database (CWA): "product X not listed" means X does not exist in our catalog.
• Knowledge graph (OWA): "product X not in graph" means we have no information about X.
• Merge: If we treat the combined data under CWA, products not in the database are treated as nonexistent, even if the knowledge graph is merely silent about them. If we treat it under OWA, the database's authoritative "not in catalog" becomes "unknown status," which may be operationally useless.

Without explicit absence policy annotation, the system silently picks one interpretation. The merge succeeds. The downstream queries are wrong. This is T5.

A15: Logic Selection

This definition extends Chapter 10's context structure. Ctx(U) is the full specification of a view; Th(U) is the set of sentences derivable in that view. The separation of L and P makes the two degrees of freedom explicit.

The constructed T5 example is a P mismatch with the same underlying proof logic. The databases often share classical logic but differ on absence policy. The fix is not to change the logic; it is to reconcile the policies.

The Logic Adapter

When views with different absence policies overlap, we need a reconciliation mechanism. One useful approach is to translate both sides into a common status space while retaining provenance and policy scope.

The 3-valued adapter: Map all claims into true/false/unknown.

Definitions:

• unknown = neither side established under the declared procedure and coverage; an unfinished search retains that limitation
• CWA-derived = falsity arising from absence, not from explicit negative evidence

Source	Mapping
OWA view, p asserted true	→ true
OWA view, p asserted false	→ false
OWA view, neither p nor its negation established	→ unknown
CWA view, p asserted true	→ true
CWA view, p asserted false	→ false
CWA view, p absent from the completed positive extension covered by its completeness declaration	→ false (tagged: CWA-derived)

Comparison and reconciliation:

• true / true and false / false agree as status values; their evidence may still differ.
• unknown / unknown agrees only as a report that neither side has established the proposition.
• true / false conflicts when both concern the same proposition, scope and date.
• true / unknown supplies one positive assertion and one silent source. The positive assertion does not become reliable merely because it is unopposed.
• false (CWA-derived) / unknown juxtaposes a negative derived under a completeness policy with a source that does not decide the question.

The last two pairs are not contradictions. A receiving institution may adopt the supported assertion, continue inquiry, or retain the two source reports. It must examine the evidence and the scope of any completeness assumption. An adapter cannot decide that a register is complete by tagging it CWA.

These are proposed reconciliation policies, not the exact equality condition of A13. If the adapter selects one value from unequal status reports, it has changed the reports or the comparison; it has not glued the original unequal family under strict equality. A mathematical gluing claim requires the presheaf and restrictions that make its resulting family compatible.

The adapter should record what it mapped, what it retained and which conclusion the chosen rule permits. An unresolved policy question is not automatically a cohomological obstruction.

Worked Example: Allergen Data

In a second constructed example, two data sources describe product X.

Source A (Inventory Database, CWA):

• Ctx(A) = (Σ, I, classical, CWA)
• contains_allergens(X) not in database
• CWA interpretation: X does not contain allergens

Source B (Supplier Knowledge Graph, OWA):

• Ctx(B) = (Σ, I, classical, OWA)
• contains_allergens(X) not in graph
• OWA interpretation: allergen status of X is unknown

Overlap via logic adapter:

Source A	Source B	Mapped A	Mapped B	Result
contains_allergens(X) = true	contains_allergens(X) = true	true	true	Agreement
(absent)	(absent)	false	unknown	Different grounds; examine CWA scope
(absent)	= true	false	true	Real conflict
= true	(absent)	true	unknown	A asserts; B supplies no contrary finding

Resolution for policy conflict:

1. Adopt A's policy: X does not contain allergens. This is dangerous for medical applications. The absence from A's database may reflect data entry lag, not safety certification.

2. Adopt B's policy: Allergen status is unknown. This may block operational workflows that require a definite answer. If A’s completeness claim is inadequate for the proposed use, its negative has not been established for that use.

3. Record disagreement: "Source A claims ¬contains_allergens (CWA-derived from absence). Source B has no information. Policy conflict on overlap."

Keeping both source reports preserves the grounds for further inquiry. The receiving institution must establish whether A’s completeness assumption supports the use it proposes. It may obtain adequate evidence elsewhere; it may also have to refrain from treating allergen absence as established. Merely assigning the consumer a risk choice does not supply the missing evidence.

The preserved reports let the recipient examine the premise on which the negative depends. The merge has made that inquiry possible; it has not completed it.

Beyond Two-Valued

Classical and intuitionistic are not the only proof logics. Three-valued logics (Kleene⁴, Łukasiewicz⁵) make "unknown" a first-class truth value. Negation of unknown is unknown (strong Kleene), which captures the behavior we want for absent data.

Paraconsistent logics tolerate contradictions without explosion⁶. In classical logic, a contradiction (p ∧ ¬p) implies everything (ex falso quodlibet). This is catastrophic when merging inconsistent sources. Paraconsistent logics allow local contradictions to be quarantined: the merge can proceed, flagging the contradiction, without the entire system becoming trivially true.

Default logic and non-monotonic reasoning allow conclusions to be retracted when new information arrives⁷. This is useful for CWA-style assumptions that should be overridden by explicit evidence. If the database assumes "no allergen data means no allergens" but later receives explicit allergen information, the default is defeated.

The parameter space is not binary. Each view occupies a point in (L, P) space. The system must annotate which point each view occupies, or the merge will silently pick one.

The Missing Premise in T5

T5 (Negation/Absence) was: System treats absence as negation, e.g., "no allergy listed" → "no allergy."

T5 in A15 terms: The source view has $P_{source} = \text{OWA}$. The target view has $P_{target} = \text{CWA}$. The system translated without policy reconciliation. The CWA interpretation was applied to OWA data.

The fix: Require Ctx(U) = (Σ, I, L_U, P_U) for every view. When translating claims across views, check both L and P compatibility. If $P_{source} \neq P_{target}$ and the claim involves absence, apply the logic adapter or record disagreement.

In this example, the policy mismatch must be repaired wherever the output is produced. Explicit (L, P) annotation helps locate it; the integration must actually honor that annotation. A better model could help recover or interpret evidence, but could not make the missing premise true by producing a more confident answer.

The honest system response:

Query: "Does product X contain allergens?"

System: "Source A (CWA): no allergen data, interpreted as no allergens. Source B (OWA): no allergen data, interpreted as unknown. Policy conflict on overlap. Recommend: treat as unknown for safety-critical decisions."

This response reports different grounds without confusing a silent source with a refutation. Whether A’s negative is usable depends on the completeness claim it relies on.

Consequence

The source’s silence can support a negative only through the completeness claim governing it. That claim remains available for scrutiny after an adapter has produced an answer. Preserving the original report and the rule used to interpret it lets a later recipient ask a different question without mistaking the derived value for another observation.

The next chapter follows substitution itself: when two representations have been related, which properties can a receiving operation actually carry across? A shared status value has not yet supplied that transport.