Gentzen Calculus

Gentzen Calculus

Proof Systems / Logical Calculi

Mateus Gomes Chetan Arora Christian Kayser SVHOL2011

Gentzen Calculus

Consists of:

Gentzen Calculus

Consists of:

One Axiom

Gentzen Calculus

Consists of:

One Axiom

Multiple Rules

Gentzen Calculus

Consists of:

One Axiom

Multiple Rules

Weakening (W)

Gentzen Calculus

Consists of:

One Axiom

Multiple Rules

Weakening (W)

Conjunction (^I)

Conjunction (^E) Conjunction (^E )

Gentzen Calculus

Multiple Rules

Disjunction (vI^l) Disjunction (vI^r)

Disjunction (vE)

Gentzen Calculus

Multiple Rules

Disjunction (vI^l) Disjunction (vI^r)

Disjunction (vE)

Truth Values (FalseE)

Gentzen Calculus

Multiple Rules

Disjunction (vI^l) Disjunction (vI^r)

Disjunction (vE)

Truth Values (FalseE)

Negation (¬I) Negation (¬E)

Gentzen Calculus

Multiple Rules

Implication (→I )

Implication (→E )

Modus Ponens

Gentzen Calculus

Multiple Rules

Implication (→I )

Implication (→E )

Modus Ponens

Until here: Ready for Exercise 1.a

Gentzen Calculus

Exercise 1.a - Warmup

Gentzen Calculus

Exercise 1.a - Warmup

→I

Gentzen Calculus

Exercise 1.a - Warmup

→I

Conjunction (^El) Conjunction (^ER)

Implication (→E )

Modus Ponens

Disjunction (vE)

?

possible choices

Gentzen Calculus

Exercise 1.a - Warmup

→I

vE

Axiom

Axiom:

vE

Gentzen Calculus

Exercise 1.a - Warmup

→I

vE

Axiom

Axiom Axiom Axiom Axiom

Axiom:

→E →E

vE

Gentzen Calculus

Exercise 1.a - Warmup

→I

vE

Axiom

Axiom Axiom Axiom Axiom

! Proven !

Axiom:

→E →E

vE

Gentzen Calculus

Exercise 1.a

Gentzen Calculus

Exercise 1.a

Gentzen Calculus

Exercise 1.a

^I Left side

Gentzen Calculus

Exercise 1.a

^I VE

Axiom

Left side

Gentzen Calculus

Exercise 1.a

^I VE

Axiom

W W

Left side

Axiom

Gentzen Calculus

Exercise 1.a

^I VE

Axiom

W W

Left side

vI^l Axiom

vI^R

Gentzen Calculus

Exercise 1.a

^I VE

Axiom

W W

Left side

vI^l Axiom

vI^R Axiom

^E^l

Gentzen Calculus

Exercise 1.a

^I Right side

Gentzen Calculus

Exercise 1.a

^I Right side

Axiom

VE

Gentzen Calculus

Exercise 1.a

^I Right side

Axiom W W

Gentzen Calculus

Exercise 1.a

^I Right side

Axiom W W

vI^l Axiom

vI^R

Gentzen Calculus

Exercise 1.a

^I Right side

Axiom W W

vI^l Axiom

vI^R

^E^R Axiom

^I

Gentzen Calculus

Exercise 1.a

All Axioms All Axioms

^I

Gentzen Calculus

Exercise 1.a

All Axioms All Axioms

! Proven !

Gentzen Calculus

Exercise 1.b Additional Rules All-quantor

Existence-quantor

Gentzen Calculus

Exercise 1.b Additional Rules All-quantor

Existence-quantor

Until here: Ready for Exercise 1.b

Gentzen Calculus

Exercise 1.b

Gentzen Calculus

Exercise 1.b

→I

Gentzen Calculus

Exercise 1.b

→I

! Proven !

Gentzen Calculus

Exercise 1.b

→I

! Proven !

Axiom

Gentzen Calculus

Exercise 1.b

→I

! Proven !

Axiom

Gentzen Calculus

Exercise 1.b

→I

! Proven !

Axiom

Gentzen Calculus

Exercise 1.b

→I Axiom

W Axiom

Gentzen Calculus

Exercise 1.b

→I

! Proven !

Axiom

W Axiom

Gentzen Calculus

Thanks for listening....