Baseswhether

Baseswhether is a term used in some discussions of formal logic and knowledge representation to describe the capacity of a base, a foundational set of assumptions or axioms, to determine the truth value of a given statement. In this usage, a base B is said to decide a statement φ if the truth of φ is fixed by B.

The notion can be treated in two closely related ways. Syntactically, a statement φ is BW-decided by B if either φ is provable from B (B ⊢ φ) or its negation is provable from B (B ⊢ ¬φ). Semantically, φ is BW-decided by B if φ holds in every model of B (B ⊨ φ) or ¬φ holds in every model of B (B ⊨ ¬φ). The syntactic and semantic perspectives align in complete systems, but they can diverge in incomplete or non-standard bases.

Baseswhether is not a standard term in formal literature; it appears mainly as a descriptive label in expository or hypothetical discussions about how different bases influence what can be concluded or decided within a theory. It is related to, but distinct from, concepts such as entailment, decidability, and independence, and it can be used to frame questions about knowledge bases and belief revision.

Example: in propositional logic, let B = {p, p → q}. From B, one can derive q, so B decides q (B ⊢ q). However, B does not decide r unless r or ¬r is derivable from B. This illustrates how a base can fix the truth value of some statements while leaving others undecided.

See also: logic, model theory, entailment, decidability, knowledge bases.