A syntactic object of the form Γ ⇒ Δ (where Γ and Δ are, typically, finite multisets or sequences of formulas) expressing that the conjunction of formulas in Γ entails the disjunction of formulas in Δ within a sequent-style proof system.

Principle

A sequent internalizes entailment as a single formula-like token so that inference rules operate locally on antecedent and succedent components, enabling systematic manipulation of entailment structure.

Demonstration

A sequent example: {A, A→B} ⇒ {B}. Interpreted in a classical sequent calculus, the rules permit transforming premises into the succedent; this single sequent stands for the claim that A and A→B together entail B.

Misapplication

Reading a sequent Γ ⇒ Δ as simply the material implication (∧Γ)→(∨Δ) without respecting structural rules or context, or treating sequents as semantic conditionals in contexts where the calculus has restricted structural rules (e.g., intuitionistic sequents).

Consequence

Using sequents makes structural phenomena (weakening, contraction, exchange, cut) explicit; the sequent representation supports meta-results such as cut-elimination and systematic proof search strategies.

Reversal

The inversion is to swap antecedent and succedent (Δ ⇒ Γ) or to convert sequent judgments back into formula-level theorems in a Hilbert-style system; such reversals highlight asymmetries (e.g., single-conclusion intuitionistic sequents).

Boundary

Applies within sequent calculi and related syntactic frameworks; it is a formal device and not itself a semantic valuation or model; sequents presuppose a choice about whether Γ/Δ are sets, multisets or sequences and about allowed structural rules.

Semantic Tension

Tension exists between the sequent as a syntactic entailment token and alternative representations of entailment (Hilbert proofs, natural deduction, tableau): all express consequence but distribute logical work differently across rules and contexts.

Synthesis

A sequent is a concentrated syntactic representation of entailment, Γ ⇒ Δ, designed so that local inference rules manipulate antecedent and succedent components, enabling structural analysis and algorithmic proof transformations.