Sequent Calculus: A Formal System for Logical Deduction

Overview

Sequent calculus is a formal system used in mathematical logic to represent and derive logical entailments. Unlike other proof systems, it represents deductions as sequences of formulas, known as sequents. This approach offers a different perspective on proof theory, focusing on the structural manipulation of formulas.

Key Concepts

The fundamental unit in sequent calculus is the sequent, typically written as $\Gamma \implies \Delta$. Here, $\Gamma$ and $\Delta$ are finite sets (or multisets) of formulas. The sequent represents the idea that the conjunction of formulas in $\Gamma$ logically entails the disjunction of formulas in $\Delta$.

• Axioms: Basic sequents that are assumed to be true.
• Rules of Inference: Transformations that allow deriving new sequents from existing ones. These rules are divided into:
• Left Rules: Introduce logical connectives into the antecedent ($\Gamma$).
• Right Rules: Introduce logical connectives into the succedent ($\Delta$).
• Structural Rules: Manipulate the formulas within the antecedent or succedent (e.g., weakening, contraction, exchange).

Deep Dive into Rules

The power of sequent calculus lies in its precise rules. For instance, the rule for conjunction introduction in the antecedent (AND-Left) might look like:

A, B, \Gamma \implies \Delta
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(A \land B), \Gamma \implies \Delta

And for conjunction introduction in the succedent (AND-Right):

\Gamma \implies A, \Delta \quad \Gamma \implies B, \Delta
----------------------------
\Gamma \implies (A \land B), \Delta

Structural rules are crucial for managing the formulas and ensuring the system’s properties. For example, weakening allows adding formulas, while contraction allows removing duplicates.

Applications

Sequent calculus is a foundational tool in proof theory and has significant applications in:

• Automated theorem proving
• Logic programming
• Formal verification of software and hardware
• Investigating the properties of logical systems (e.g., completeness, decidability)

Challenges & Misconceptions

A common misconception is that sequent calculus is overly complex. While it has a rich set of rules, its emphasis on structural manipulation can simplify proofs and analyses. Another challenge is choosing the right variant of sequent calculus (e.g., Gentzen’s LJ for intuitionistic logic, LK for classical logic) for a specific problem.

FAQs

What is a sequent?

A sequent is a statement of the form $\Gamma \implies \Delta$, representing a logical entailment.

How does it differ from natural deduction?

Natural deduction focuses on introducing and eliminating logical connectives within a proof, mimicking human reasoning. Sequent calculus uses sequences and structural rules to manipulate formulas, offering a more uniform derivation process.

Is sequent calculus decidable?

Decidability depends on the specific logic and the sequent calculus formulation. Many propositional and first-order theories formulated within sequent calculus are decidable.