Natural Deduction

(1)

Natural Deduction

Akim [email protected]

EPITA — École Pour l’Informatique et les Techniques Avancées

June 10, 2016

(2)

Define “logic” [fuc, ]

(3)

Natural Deduction

1 Logical Formalisms

2 Natural Deduction

3 Additional Material

A. Demaille Natural Deduction 3 / 49

(4)

Preamble

The following slides are implicitly dedicated to classicallogic.

(5)

Logical Formalisms

1 Logical Formalisms Syntax

Proof Types Proof Systems

2 Natural Deduction

(6)

Syntax

2 Natural Deduction

(7)

Terminal Symbols

Propositional Calculus

Constants a,b,c, . . . Propositional Variables A,B,C, . . .

Unary Connective ¬ Binary Connectives ∧,∨,⇒

Punctuation (,),[,].

(8)

Terminal Symbols

(9)

Terminal Symbols

(10)

Terminal Symbols

(11)

Terminal Symbols

(12)

Terminal Symbols

Predicate calculus

Individual Variables x,y,z, . . .

Functions f,g,h, . . ., with a fixed arity Predicates P,Q,R, . . ., with a fixed arity Quantifiers ∀,∃

Punctuation ·.

(13)

Terminal Symbols

Predicate calculus

Punctuation ·.

(14)

Terminal Symbols

Predicate calculus

Punctuation ·.

(15)

Terminal Symbols

Predicate calculus

Punctuation ·.

(16)

Terminal Symbols

Predicate calculus

Punctuation ·.

(17)

Propositional Formulas

hformulai ::= hpropositional variablei

| ¬hformulai

| hformulai ∧ hformulai

| hformulai ∨ hformulai

| hformulai ⇒ hformulai

(18)

Terms

htermi ::= hconstanti

| hfunctioni(htermi, . . .)

With the proper arity.

(19)

First Order Formulas

hformulai ::= hpropositional variablei

| ¬hformulai

| hformulai ∧ hformulai

| hformulai ∨ hformulai

| hformulai ⇒ hformulai

| hpredicatei(htermi, . . .)

| ∀hindividual variablei · hformulai

| ∃hindividual variablei · hformulai

With the proper arity.

(20)

Syntactic Conventions

Associativity ∧,∨ are left-associative (unimportant)

⇒ is right-associative (very important) Precedence (increasing)

1 ∀,∃

2 ⇒

3 ∨

4 ∧

5 ¬

(21)

Syntactic Conventions

Associativity ∧,∨ are left-associative (unimportant)

⇒ is right-associative (very important) Precedence (increasing)

1 ∀,∃

2 ⇒

3 ∨

4 ∧

5 ¬

(22)

Free Variables

FV(X) = ∅

FV(P(x1,x2,· · · ,xn)) = {x₁,x2,· · ·,xn} FV(¬A) = FV(A)

FV(A∨B) = FV(A)∪FV(B) FV(A∧B) = FV(A)∪FV(B) FV(A⇒B) = FV(A)∪FV(B)

FV(∀x·A) = FV(A)− {x}

FV(∃x·A) = FV(A)− {x}

(23)

Proof Types

2 Natural Deduction

(24)

Different Proof Types

(25)

(26)

(27)

(28)

(29)

Proof Systems

2 Natural Deduction

(30)

Proof Systems

Hilbertian Systems Natural Deduction Sequent Calculus

Natural Deduction in Sequent Calculus

(31)

Proof Systems

(32)

Proof Systems

(33)

Proof Systems

(34)

Axioms

Axioms are formulas that are considered true a priori

∀x·x+0=x Axiom schemesuse meta-variables

(that range over a specific domain)

X+Y =Y +X

Axiom schemes are used when quantifiers are not welcome

SXYZ → XZ(YZ) KXY → X

Axiom schemes are used when quantifiers do not apply

(35)

Axioms

X+Y =Y +X

Axiom schemes are used when quantifiers are not welcome

SXYZ → XZ(YZ) KXY → X

A∨ ¬A

(36)

Axioms

X+Y =Y +X

Axiom schemes are used when quantifiers are not welcome SXYZ → XZ(YZ)

KXY → X

(37)

Axioms

X+Y =Y +X

Axiom schemes are used when quantifiers are not welcome SXYZ → XZ(YZ)

KXY → X

Axiom schemes are used when quantifiers do not apply A∨ ¬A

(38)

Inference Rules

H1 H2 · · · Hn

Rule name C

Axiom name A

(39)

Logical Formalisms

David Hilbert (1862–1943)

(40)

Hilbertian System

A single inference rule: themodus ponens A A⇒B

modus ponens B

Many axioms to define the connectives

A⇒B ⇒A∧B A∧B ⇒A A∧B ⇒B A⇒A∨B B ⇒A∨B

A∨B⇒ (A⇒C)⇒(B ⇒C)⇒C

A⇒B⇒A (A⇒(B ⇒C))⇒(A⇒B)⇒A⇒C

⇒

A∨ ¬A A⇒ ¬A⇒B

(41)

Hilbertian System

modus ponens B

A∨B⇒ (A⇒C)⇒(B ⇒C)⇒C

A⇒B⇒A (A⇒(B ⇒C))⇒(A⇒B)⇒A⇒C

⇒

A∨ ¬A A⇒ ¬A⇒B

(42)

Hilbertian System

modus ponens B

A∨B⇒ (A⇒C)⇒(B ⇒C)⇒C

A⇒B⇒A (A⇒(B ⇒C))⇒(A⇒B)⇒A⇒C

⇒A∨ ¬A A⇒¬A⇒B

(43)

Hilbertian System: Prove A ⇒ A

(A⇒((A⇒A)⇒A))⇒(A⇒A⇒A)⇒A⇒A A⇒(A⇒A)⇒A

(A⇒A⇒A)⇒A⇒A A⇒A⇒A

A⇒A

(44)

Hilbertian System: Prove A ⇒ A

(A⇒((A⇒A)⇒A))⇒(A⇒A⇒A)⇒A⇒A A⇒(A⇒A)⇒A

(A⇒A⇒A)⇒A⇒A A⇒A⇒A

A⇒A

(45)

Natural Deduction

2 Natural Deduction Syntax

Normalization

(46)

Syntax

Normalization

(47)

Deduction

Adeduction is a tree whose root (A) is theconclusionand whose

active

leafs (Γ) is the set ofhypotheses.

Γ·

·· A Any formula Ais a valid hypothesis.

Proof (Demonstration)

Aproof is a deduction without hypotheses.

(48)

Deduction

Adeduction is a tree whose root (A) is theconclusionand whoseactive leafs (Γ) is the set ofhypotheses.

Γ·

·· A Any formula Ais a valid hypothesis.

Proof (Demonstration)

Aproof is a deduction without hypotheses.

(49)

Deductions

What’s this?

A

A deduction of Aunder the hypothesis A.

(50)

Deductions

What’s this?

A A deduction of Aunder the hypothesisA.

(51)

Implication

[A]·

··

B ⇒I A⇒B

··

· A

··

· A⇒B

⇒E B

Deduction theorem, and Modus Ponens.

Note the connection with (left) contraction: any number ofA (including 0) is discharged.

(52)

Implication

[A]·

··

B ⇒I A⇒B

··

· A

··

· A⇒B

⇒E B

(53)

Implication

[A]·

··

B ⇒I A⇒B

··

· A

··

· A⇒B

⇒E B

(54)

Implication

[A]·

··

B ⇒I A⇒B

··

· A

··

· A⇒B

⇒E B

(55)

Proving A ⇒ A in Natural Deduction

[A] ⇒I A⇒A

(56)

Proving A ⇒ A in Natural Deduction

[A] ⇒I A⇒A

(57)

Conjunction

··

· A

··

· B ∧I A∧B

··

· A∧B

∧lE A

··

· A∧B

∧rE B

(58)

Conjunction

··

· A

··

· B ∧I A∧B

··

· A∧B

∧lE A

··

· A∧B

∧rE B

(59)

Universal Quantification

··

· A[y/x]

∀I y 6∈FV(hyp(A))

∀x·A

··

∀x··A

∀E A[t/x]

[A] ∀I

∀x·A

⇒I A⇒ ∀x·A

[A] ⇒I

A⇒A

∀I

∀x·(A⇒A)

(60)

Universal Quantification

··

· A[y/x]

∀x·A

··

∀x··A

∀E A[t/x]

[A] ∀I

∀x·A

⇒I A⇒ ∀x·A

[A] ⇒I

A⇒A

∀I

∀x·(A⇒A)

(61)

Universal Quantification

··

· A[y/x]

∀x·A

··

∀x··A

∀E A[t/x]

[A] ∀I

∀x·A

⇒I A⇒ ∀x·A

[A] ⇒I A⇒A

∀I

∀x·(A⇒A)

(62)

Absurd

··

⊥·

⊥E A

(63)

Disjunction

··

· A ∨lI A∨B

··

· B ∨rI A∨B

··

· A∨B

[A]·

·· C

[B]

··

· C ∨E C

(64)

Existential Quantification

··

· A[t/x]

∃I

∃x·A

··

∃x··A

[A[y/x]]

··

·

B ∃E y 6∈FV(B,hyp(B)) B

For elimination, y6∈hyp(B), i.e., not in the hypotheses other than the dischargedA.

(65)

Negation

[A]·

··

⊥

¬I

¬A

·· A·

··

¬A·

⊥ ¬E

Plus one of these equivalent formulations of the fact that

classical

negation is involutive.

XM A∨ ¬A

··

¬¬A·

¬¬ A

[¬A]·

·· B

[¬A]·

··

¬B

Contradiction A

(66)

Negation

[A]·

··

⊥

¬I

¬A

·· A·

··

¬A·

⊥ ¬E

Plus one of these equivalent formulations of the fact that classicalnegation is involutive.

XM A∨ ¬A

··

¬¬A·

¬¬

A

[¬A]·

·· B

[¬A]·

··

¬B

Contradiction A

(67)

Normalization

(68)

Cut

Cut: Introduction of a connective followed by its elimination.

·· A·

·· B·

∧I A∧B

∧lE A·

··

(69)

Normalization

Thenormalization process eliminates the cuts.

··

· A

··

· B ∧I A∧B

∧lE A

;

··

· A·

··

(70)

Normalizing Conjunctions

·· A·

·· B·

∧I A∧B

∧lE A·

··

;

··

· A·

··

· A

··

· B ∧I A∧B

∧rE ;

··

· B·

··

(71)

Normalizing Implications

··

· A

[A]·

··

B ⇒I A⇒B

⇒E B·

··

;

··

· A·

·· B·

··

(72)

Normalizing Universal Quantifiers

··

· A ∀I

∀x·A

∀E A[t/x]

··

·

;

··

· A[t/x]

··

·

x must not be free in the hypotheses, otherwise the reduction would change them.

(73)

Normalizing Disjunction

·· A·

∨lI A∨B

[A]·

·· C

[B]

··

· C ∨E C·

··

;

··

· A·

·· C·

··

· B ∨rI A∨B

[A]·

·· C

[B]

··

· C ∨E C·

··

;

··

· B·

·· C·

··

(74)

Additional Material

2 Natural Deduction

(75)

Logicians in a Bar

Three logicians walk into a bar,

and the bartender asks “Would you all like a drink?”

The first one says, “Maybe.”

The second one says, “Maybe.”

The third one says, “Yes.”

(76)

Edukera

(77)

Edukera

(78)

Edukera

http://edukera.appspot.com

8245EEC

(79)

Bibliography Notes

[Girard et al., 1989]

A short (160p.) book addressing all the concerns of this course, and more (especially Linear Logic). Easy and pleasant to read. Now available for free.

[Girard, 2004]

A much more comprehensive book focusing on logic and its connections with computer science. In French.

(80)

Bibliography I

Fuck theory — Experiments in visceral philosophy.

http://fucktheory.tumblr.com/post/12492878134/

dictionary-of-philosophy-logic.

Girard, J.-Y. (2004).

Cours de Logique, Rome, Automne 2004.

http://logica.uniroma3.it/uif/corso/.

Girard, J.-Y., Lafont, Y., and Taylor, P. (1989).

Proofs and Types.

Cambridge University Press.

http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html.