11slides(2Gentzen)

11slides(2Gentzen) - Chapter 11(Part 2 Gentzen Sequent...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 11 (Part 2) Gentzen Sequent Calculus GL The proof system GL for the classical propo- sitional logic is a version of the original Gentzen (1934) systems LK . A constructive proof of the completeness the- orem for the system GL is very similar to the proof of the completeness theorem for the system RS . Expressions of the system like in the original Gentzen system LK are Gentzen sequents . Hence we use also a name Gentzen sequent calculus. 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Language of GL: L = L {∪ , , , ¬ , } . We add a new symbol to the alphabet: -→ . It is called a Gentzen arrow. The sequents are built out of finite sequences (empty included) of formulas, i.e. elements of F * , and the additional sign -→ . We denote, as in the RS system, the finite sequences of formulas by Greek capital let- ters Γ , Δ , Σ, with indices if necessary. Sequent definition: a sequent is the expres- sion Γ -→ Δ , where Γ , Δ ∈ F * .
Background image of page 2
Meaning of sequents Intuitively, we interpret a sequent A 1 ,...,A n -→ B 1 ,...,B m , where n,m 1 as a formula ( A 1 ... A n ) ( B 1 ... B m ) . The sequent: A 1 ,...,A n -→ (where n 1) means that A 1 ... A n yields a contra- diction. The sequent -→ B 1 ,...,B m (where m 1) means | = ( B 1 ... B m ). The empty sequent: -→ means a contra- diction. 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Γ, Δ, we de- note by σ Γ any conjunction of all formulas of Γ, and by δ Δ any disjunction of all formulas of Δ. The intuitive semantics
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.

Page1 / 17

11slides(2Gentzen) - Chapter 11(Part 2 Gentzen Sequent...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online