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# Lecture 18: Forward Sequent Calculus

The sequent calculus presented so far was motivated by search for natural deductions, starting from a goal sequent towards the initial sequents. The advantage of such a method is that it is immediately goal-directed. The disdvantage is the disjunctive non-determinism which in practice forces a backtracking implementation.

We now develop a sequent calculus appropriate for forward reasoning, from the initial sequents to the goal sequent. This sequent calculus in itself is not yet useful for theorem proving, because the space of all possible proofs is too large. However, in a second step we take advantage of the subformula property for cut-free derivations to specialize the rules of the sequent calculus to the goal sequent, thereby obaining the basis for the inverse method.

• Handout 12 on Forward Sequent Calculus (also availabe in PDF format).

### Key Concepts

• Forward sequent calculus
• Weakening
• Contraction