Independence of premise¶
A constructive-logical principle allowing an existential witness to be moved outside an implication when the premise does not contain that witness variable.
Core Idea¶
In a standard form, from phi implies there exists x theta(x) one infers there exists x such that phi implies theta(x), with an inhabited domain and x not free in phi; validity depends on the logic and formula restrictions. Because the antecedent carries no information about the witness variable, the principle asserts that one witness can be chosen independently of whether or how the premise is established. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Independence of premise belongs to proof theory and is useful where the analyst can specify the typed proof theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the formal theory and logic, inhabited-domain assumption and distinguished term, formulas phi and theta, variable-freeness condition, premise and conclusion schemata, syntactic restrictions and classical or intuitionistic validity status are explicit. The scope is broad within that domain but bounded by the need for the formal theory and logic, inhabited-domain assumption and distinguished term, formulas phi and theta, variable-freeness condition, premise and conclusion schemata, syntactic restrictions and classical or intuitionistic validity status are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the formal theory and logic, inhabited-domain assumption and distinguished term, formulas phi and theta, variable-freeness condition, premise and conclusion schemata, syntactic restrictions and classical or intuitionistic validity status are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Independence of premise. Independence of premise compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed proof theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the formal theory and logic, inhabited-domain assumption and distinguished term, formulas phi and theta, variable-freeness condition, premise and conclusion schemata, syntactic restrictions and classical or intuitionistic validity status are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of proof theory because they reuse the typed proof theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, Because the antecedent carries no information about the witness variable, the principle asserts that one witness can be chosen independently of whether or how the premise is established., and type the carrier, state every parameter and convention in the definition, test that the formal theory and logic, inhabited-domain assumption and distinguished term, formulas phi and theta, variable-freeness condition, premise and conclusion schemata, syntactic restrictions and classical or intuitionistic validity status are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Independence of premise Domain-specific
Parents (1) — more general patterns this builds on
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Independence of premise is a kind of Deductive Reasoning Prime
The proposed strict upward parent is
prime:deductive_reasoning.
Hierarchy path (1) — routes to 1 parentless root
- Independence of premise → Deductive Reasoning
Neighborhood in Abstraction Space¶
Independence of premise sits in a crowded region of the domain-specific corpus (7th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Formal Logic & Type Theory (34 abstractions)
Nearest neighbors
- Negation introduction — 0.93
- Type theory — 0.93
- Proof-theoretic semantics — 0.93
- Constructive dilemma — 0.93
- Identity type — 0.93
Computed from structural-signature embeddings · 2026-09-08