Intuitionistic Logic

(1)

Intuitionistic Logic

Akim Demaille akim@lrde.epita.fr

EPITA — École Pour l’Informatique et les Techniques Avancées

June 10, 2016

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Intuitionistic Logic

1 Constructivity

2 Intuitionistic Logic

A. Demaille Intuitionistic Logic 2 / 34

(3)

Constructivity

1 Constructivity

2 Intuitionistic Logic

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Constructivity

Classical logic does not build truth it discovers a preexisting truth

Classical logic assumes facts are either true or false

` A ∨ ¬A Excluded middle, tertium non datur

(5)

Constructivity

Classical logic does not build truth it discovers a preexisting truth

Classical logic assumes facts are either true or false

` A ∨ ¬A Excluded middle, tertium non datur

(6)

Constructivity

Classical logic does not build truth it discovers a preexisting truth

Classical logic assumes facts are either true or false

` A ∨ ¬A Excluded middle, tertium non datur

(7)

Constructivity

Classical logic does not build truth it discovers a preexisting truth

Classical logic assumes facts are either true or false

` A ∨ ¬A Excluded middle, tertium non datur

(8)

Excluded Middle

XM A ∨ ¬A

· ·

¬¬A ·

¬¬

A [¬A] ·

· · B

[¬A] ·

· ·

¬B

Contradiction A

A ` A

` ¬A, A ` ¬

`r ∨

` A ∨ ¬A, A

`l ∨

` A ∨ ¬A, A ∨ ¬A

`C

` A ∨ ¬A

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Reductio ad Absurdum

In Real Life

A You should respect C’s belief, for all beliefs are of equal validity and cannot be denied.

B What about D’s belief? (Where D believes something that is considered to be wrong by most people, such as nazism or the world being flat)

A I agree it is right to deny D’s belief.

B If it is right to deny D’s belief, it is not true that no belief

can be denied. Therefore, I can deny C’s belief if I can give

reasons that suggest it too is incorrect.

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Reductio ad Absurdum

In Real Life

A You should respect C’s belief, for all beliefs are of equal validity and cannot be denied.

B

¹

I deny that belief of yours and believe it to be invalid.

2

According to your statement, this belief of mine (1) is valid, like all other beliefs.

3

However, your statement also contradicts and invalidates mine, being the exact opposite of it.

4

The conclusions of 2 and 3 are incompatible and contradictory, so your statement is logically absurd.

(11)

Reductio ad Absurdum

Mathematics: The Smallest Positive Rational

There is no smallest positive rational.

1

Suppose there exists one such rational r

2

r/2 is rational and positive

3

r/2 < r

4

Contradiction

(12)

Reductio ad Absurdum

Mathematics: The Smallest Positive Rational

There is no smallest positive rational.

1

Suppose there exists one such rational r

2

r/2 is rational and positive

3

r/2 < r

4

Contradiction

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Reductio ad Absurdum

Mathematics: The Smallest Positive Rational

There is no smallest positive rational.

1

Suppose there exists one such rational r

2

r/2 is rational and positive

3

r/2 < r

4

Contradiction

(14)

Reductio ad Absurdum

Mathematics: The Smallest Positive Rational

There is no smallest positive rational.

1

Suppose there exists one such rational r

2

r/2 is rational and positive

3

r/2 < r

4

Contradiction

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Reductio ad Absurdum

Mathematics: The Smallest Positive Rational

There is no smallest positive rational.

1

Suppose there exists one such rational r

2

r/2 is rational and positive

3

r/2 < r

4

Contradiction

(16)

Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

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Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

(18)

Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

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Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

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Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

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Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

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Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

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Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

(24)

Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

(25)

Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

(26)

Reductio ad Absurdum

Mathematics: √

2 is irrational

√ 2 is irrational.

1

Assume √

2 is rational: ∃a, b integers st. a/b = √ 2

2

a, b can be taken coprime

3

∴ a ² /b ² = 2 and a ² = 2b ²

4

∴ a ² is even (a ² = 2b ² )

5

∴ a is even

6

Because a is even, ∃k st. a = 2k.

7

We insert the last equation of (3) in (6): 2b ² = (2k) ² is equivalent to 2b ² = 4k ² is equivalent to b ² = 2k ² .

8

Since 2k ² is even, b ² is even, hence, b is even

9

By (5) and (8) a, b are even

10

Contradicts 2

(27)

Constructivity

Mathematics: Rationality and Power

There are irrational positive numbers a, b such that a ^b is rational.

1

√ 2 is known to be irrational

2

Consider √ 2

√ 2 :

1

If it is rational, take a = b = √ 2

2

Otherwise, take a = √ 2

√2

, b = √

2, a

^b

= 2

But it is not known which numbers.

We proved A ∨ B, but neither A nor B .

(28)

Constructivity

Mathematics: Rationality and Power

There are irrational positive numbers a, b such that a ^b is rational.

1

√ 2 is known to be irrational

2

Consider √ 2

√ 2 :

1

If it is rational, take a = b = √ 2

2

Otherwise, take a = √ 2

√2

, b = √

2, a

^b

= 2

But it is not known which numbers. We proved A ∨ B, but neither A nor B .

(29)

Constructivity

Mathematics: Rationality and Power

There are irrational positive numbers a, b such that a ^b is rational.

1

√ 2 is known to be irrational

2

Consider √ 2

√ 2 :

1

If it is rational, take a = b = √ 2

2

Otherwise, take a = √ 2

√2

, b = √

2, a

^b

= 2

But it is not known which numbers.

We proved A ∨ B, but neither A nor B .

(30)

Constructivity

Mathematics: Rationality and Power

There are irrational positive numbers a, b such that a ^b is rational.

1

√ 2 is known to be irrational

2

Consider √ 2

√ 2 :

1

If it is rational, take a = b = √ 2

2

Otherwise, take a = √ 2

√2

, b = √

2, a

^b

= 2

But it is not known which numbers. We proved A ∨ B, but neither A nor B .

(31)

Constructivity

Mathematics: Rationality and Power

There are irrational positive numbers a, b such that a ^b is rational.

1

√ 2 is known to be irrational

2

Consider √ 2

√ 2 :

1

If it is rational, take a = b = √ 2

2

Otherwise, take a = √ 2

√2

, b = √

2, a

^b

= 2

But it is not known which numbers.

We proved A ∨ B, but neither A nor B .

(32)

Constructivity

Mathematics: Rationality and Power

There are irrational positive numbers a, b such that a ^b is rational.

1

√ 2 is known to be irrational

2

Consider √ 2

√ 2 :

1

If it is rational, take a = b = √ 2

2

Otherwise, take a = √ 2

√2

, b = √

2, a

^b

= 2 But it is not known which numbers.

We proved A ∨ B, but neither A nor B .

(33)

Constructivity

Mathematics: Rationality and Power

There are irrational positive numbers a, b such that a ^b is rational.

1

√ 2 is known to be irrational

2

Consider √ 2

√ 2 :

1

If it is rational, take a = b = √ 2

2

Otherwise, take a = √ 2

√2

, b = √

2, a

^b

= 2 But it is not known which numbers.

We proved A ∨ B, but neither A nor B.

(34)

Constructivity

Mathematics: Unknown Numbers

Let σ be the number defined below. Its value is unknown, but it is rational.

For each decimal digit of π, write 3. Stop if the sequence 0123456789 is found.

1

If 0123456789 occurs in π, then σ = 0, 3 . . . 3 = ¹⁰ _3.10

^k

⁻¹

_k

2

If it does not, σ = 0, 3 . . . = 1/3

We proved ∃x.P (x), but know no t : P (t).

(35)

Constructivity

Mathematics: Unknown Numbers

Let σ be the number defined below. Its value is unknown, but it is rational.

For each decimal digit of π, write 3. Stop if the sequence 0123456789 is found.

1

If 0123456789 occurs in π, then σ = 0, 3 . . . 3 = ¹⁰ _3.10

^k

⁻¹

_k

2

If it does not, σ = 0, 3 . . . = 1/3

We proved ∃x.P (x), but know no t : P (t).

(36)

Constructivity

Mathematics: Unknown Numbers

Let σ be the number defined below. Its value is unknown, but it is rational.

For each decimal digit of π, write 3. Stop if the sequence 0123456789 is found.

1

If 0123456789 occurs in π, then σ = 0, 3 . . . 3 = ¹⁰ _3.10

^k

⁻¹

_k

2

If it does not, σ = 0, 3 . . . = 1/3

We proved ∃x.P (x), but know no t : P (t).

(37)

Constructivity

Mathematics: Unknown Numbers

Let σ be the number defined below. Its value is unknown, but it is rational.

For each decimal digit of π, write 3. Stop if the sequence 0123456789 is found.

1

If 0123456789 occurs in π, then σ = 0, 3 . . . 3 = ¹⁰ _3.10

^k

⁻¹

_k

2

If it does not, σ = 0, 3 . . . = 1/3

We proved ∃x.P (x), but know no t : P (t).

(38)

Constructivity

Disjunction Property

If A ∨ B is provable, then either A or B is provable, and reading the proof tells which one.

Existence Property

If ∃x · A(x) is provable, then reading the proof allows to exhibit a witness t (i.e., such that A(t)).

(39)

Constructivity

Disjunction Property

If A ∨ B is provable, then either A or B is provable, and reading the proof tells which one.

Existence Property

If ∃x · A(x) is provable, then reading the proof allows to exhibit a witness t

(i.e., such that A(t)).

(40)

Intuitionistic Logic

1 Constructivity

2 Intuitionistic Logic

NJ: Intuitionistic Natural Deduction LJ: Intuitionistic Sequent Calculus

(41)

Intuitionistic Logic

Luitzen Egbertus Jan Brouwer (1881–1966)

(42)

Intuitionistic Logic

Classical logic focuses on truth (hence truth values) Intuitionistic logic focuses on provability (hence proofs) A is true if it is provable

The excluded middle is. . . excluded

A ` A

` ¬A, A ` ¬

` r∨

` A ∨ ¬A, A

` l ∨

` A ∨ ¬A, A ∨ ¬A

` C

` A ∨ ¬A

(43)

Intuitionistic Logic

Classical logic focuses on truth (hence truth values) Intuitionistic logic focuses on provability (hence proofs) A is true if it is provable

The excluded middle is. . . excluded

A ` A

` ¬A, A ` ¬

` r∨

` A ∨ ¬A, A

` l ∨

` A ∨ ¬A, A ∨ ¬A

` C

` A ∨ ¬A

(44)

Intuitionistic Logic

Classical logic focuses on truth (hence truth values) Intuitionistic logic focuses on provability (hence proofs) A is true if it is provable

The excluded middle is. . . excluded

A ` A

` ¬A, A ` ¬

` r∨

` A ∨ ¬A, A

` l ∨

` A ∨ ¬A, A ∨ ¬A

` C

` A ∨ ¬A

(45)

Intuitionistic Logic

Classical logic focuses on truth (hence truth values) Intuitionistic logic focuses on provability (hence proofs) A is true if it is provable

The excluded middle is. . . excluded

A ` A

` ¬A, A ` ¬

` r∨

` A ∨ ¬A, A

` l ∨

` A ∨ ¬A, A ∨ ¬A

` C

` A ∨ ¬A

(46)

Intuitionistic Logic

Classical logic focuses on truth (hence truth values) Intuitionistic logic focuses on provability (hence proofs) A is true if it is provable

The excluded middle is. . . excluded

A ` A

` ¬A, A ` ¬

` r∨

` A ∨ ¬A, A

` l ∨

` A ∨ ¬A, A ∨ ¬A

` C

` A ∨ ¬A

(47)

Intuitionistic Logic

Classical logic focuses on truth (hence truth values) Intuitionistic logic focuses on provability (hence proofs) A is true if it is provable

The excluded middle is. . . excluded

A ` A

` ¬A, A ` ¬

` r∨

` A ∨ ¬A, A

` l ∨

` A ∨ ¬A, A ∨ ¬A

` C

` A ∨ ¬A

(48)

NJ: Intuitionistic Natural Deduction

1 Constructivity

2 Intuitionistic Logic

NJ: Intuitionistic Natural Deduction LJ: Intuitionistic Sequent Calculus

(49)

Intuitionistic Natural Deduction

Natural deduction supports very well intuistionistic logic.

In fact, classical logic does not fit well in natural deduction.

[A] ·

· ·

⊥

¬A ¬I

· ·

· A

· ·

¬A ·

⊥ ¬E

XM A ∨ ¬A

· ·

¬¬A ·

¬¬

A

[¬A] ·

· · B

[¬A] ·

· ·

¬B

Contradiction

A

(50)

Intuitionistic Natural Deduction

Natural deduction supports very well intuistionistic logic.

In fact, classical logic does not fit well in natural deduction.

[A] ·

· ·

⊥

¬A ¬I

· ·

· A

· ·

¬A ·

⊥ ¬E

XM A ∨ ¬A

· ·

¬¬A ·

¬¬

A

[¬A] ·

· · B

[¬A] ·

· ·

¬B

Contradiction A

(51)

Intuitionistic Natural Deduction

Natural deduction supports very well intuistionistic logic.

In fact, classical logic does not fit well in natural deduction.

[A] ·

· ·

⊥

¬A ¬I

· ·

· A

· ·

¬A ·

⊥ ¬E

XM A ∨ ¬A

· ·

¬¬A ·

¬¬

A

[¬A] ·

· · B

[¬A] ·

· ·

¬B

Contradiction

A

(52)

Intuitionistic Natural Deduction

Natural deduction supports very well intuistionistic logic.

In fact, classical logic does not fit well in natural deduction.

[A] ·

· ·

⊥

¬A ¬I

· ·

· A

· ·

¬A ·

⊥ ¬E

XM A ∨ ¬A

· ·

¬¬A ·

¬¬

A

[¬A] ·

· · B

[¬A] ·

· ·

¬B

Contradiction A

(53)

Intuitionistic Natural Deduction

Natural deduction supports very well intuistionistic logic.

In fact, classical logic does not fit well in natural deduction.

[A] ·

· ·

⊥

¬A ¬I

· ·

· A

· ·

¬A ·

⊥ ¬E

XM A ∨ ¬A

· ·

¬¬A ·

¬¬ A

[¬A] ·

· · B

[¬A] ·

· ·

¬B

Contradiction

A

(54)

Intuitionistic Negation

[A] ·

· ·

⊥ ¬I

¬A

· · A ·

· ·

¬A ·

¬E

⊥ [A] ·

· ·

B ⇒I A ⇒ B

· ·

· A

· ·

· A ⇒ B

⇒E B

So define ¬A := A ⇒ ⊥.

(55)

Intuitionistic Negation

[A] ·

· ·

⊥ ¬I

¬A

· · A ·

· ·

¬A ·

¬E

⊥ [A] ·

· ·

B ⇒I A ⇒ B

· ·

· A

· ·

· A ⇒ B

⇒E B

So define ¬A := A ⇒ ⊥.

(56)

Prove A ` ¬¬A

A [A ⇒ ⊥] ¹

⊥ ⇒E

⇒I ₁ (A ⇒ ⊥) ⇒ ⊥

(57)

Prove A ` ¬¬A

A [A ⇒ ⊥] ¹

⊥ ⇒E

⇒I ₁

(A ⇒ ⊥) ⇒ ⊥

(58)

Prove ¬¬¬A ` ¬A

[A] ² [A ⇒ ⊥] ¹

⇒E

⊥ ⇒I ₁

(A ⇒ ⊥) ⇒ ⊥ ((A ⇒ ⊥) ⇒ ⊥) ⇒ ⊥

⇒E

⊥ ⇒I ₂ A ⇒ ⊥

(59)

Prove ¬¬¬A ` ¬A

[A] ² [A ⇒ ⊥] ¹

⇒E

⊥ ⇒I ₁

(A ⇒ ⊥) ⇒ ⊥ ((A ⇒ ⊥) ⇒ ⊥) ⇒ ⊥

⇒E

⊥ ⇒I ₂

A ⇒ ⊥

(60)

Intuitionistic Natural Deduction

[A] ·

· ·

B ⇒I A ⇒ B

· · A ·

· · A ⇒ · B

⇒E B

· ·

⊥ ·

⊥E A A B

∧I A ∧ B

A ∧ B

∧l E A

A ∧ B

∧r E B

· ·

· A ∨lI A ∨ B

· ·

· B ∨rI A ∨ B

· ·

· A ∨ B

[A] ·

· · C

[B ]

· ·

· C ∨E

C

(61)

LJ: Intuitionistic Sequent Calculus

1 Constructivity

2 Intuitionistic Logic

NJ: Intuitionistic Natural Deduction

LJ: Intuitionistic Sequent Calculus

(62)

LJ — Intuitionistic Sequent Calculus

LJ — Gentzen 1934

Logistischer intuitionistischer Kalkül

(63)

LK: Identity Group

Id A ` A

Γ ` A, ∆ Γ ⁰ , A ` ∆ ⁰

Cut

Γ, Γ ⁰ ` ∆,∆ ⁰

(64)

LK: Identity Group

Id A ` A

Γ ` A, ∆ Γ ⁰ , A ` ∆ ⁰ Cut Γ, Γ ⁰ ` ∆,∆ ⁰

(65)

LJ: Identity Group

Id A ` A

Γ ` A

, ∆

Γ ⁰ , A ` B Cut Γ, Γ ⁰ `

∆,

B

(66)

LK: Structural Group

Γ ` ∆

`X Γ ` τ (∆)

Γ ` ∆ X ` σ(Γ) ` ∆ Γ ` ∆

`W Γ ` A, ∆

Γ ` ∆ W`

Γ, A ` ∆ Γ ` A, A, ∆

` C Γ ` A, ∆

Γ, A, A ` ∆ C ` Γ, A ` ∆

(67)

LK: Structural Group

Γ ` ∆

`X Γ ` τ (∆)

Γ ` ∆ X ` σ(Γ) ` ∆ Γ ` ∆

`W Γ ` A, ∆

Γ ` ∆ W`

Γ, A ` ∆ Γ ` A, A, ∆

` C Γ ` A, ∆

Γ, A, A ` ∆

C `

Γ, A ` ∆

(68)

LJ: Structural Group

Γ ` ∆

` X Γ ` τ (∆)

Γ ` B X ` σ(Γ) ` B

Γ ` ∆

` W Γ ` A, ∆

Γ ` B W ` Γ, A ` B

Γ ` A, A, ∆

` C Γ ` A, ∆

Γ, A, A ` B C`

Γ, A ` B

(69)

LK: Logical Group: Negation

Γ, A ` ∆

` ¬ Γ ` ¬A, ∆

Γ ` A, ∆

¬ `

Γ, ¬A ` ∆

(70)

LK: Logical Group: Negation

Γ, A ` ∆

` ¬ Γ ` ¬A, ∆

Γ ` A, ∆

¬ ` Γ, ¬A ` ∆

(71)

LJ: Logical Group: Negation

Γ, A ` ∆

` ¬ Γ ` ¬A, ∆

Γ ` A, ∆

¬ `

Γ, ¬A ` ∆

(72)

LK: Logical Group: Conjunction

Γ ` A, ∆ Γ ` B , ∆

` ∧ Γ ` A ∧ B, ∆

Γ, A ` ∆ l ∧ ` Γ, A ∧ B ` ∆

Γ, B ` ∆ r ∧ ` Γ, A ∧ B ` ∆

(73)

LK: Logical Group: Conjunction

Γ ` A, ∆ Γ ` B , ∆

` ∧ Γ ` A ∧ B, ∆

Γ, A ` ∆ l ∧ ` Γ, A ∧ B ` ∆

Γ, B ` ∆

r ∧ `

Γ, A ∧ B ` ∆

(74)

LJ: Logical Group: Conjunction

Γ ` A

, ∆

Γ ` B

, ∆

` ∧ Γ ` A ∧ B

, ∆

Γ, A ` C

l∧ ` Γ, A ∧ B ` C

Γ, B ` C

r∧ ` Γ, A ∧ B ` C

(75)

LK: Logical Group: Disjunction

Γ ` A, ∆

` l∨

Γ ` A ∨ B, ∆ Γ ` B , ∆

` r∨

Γ ` A ∨ B, ∆

Γ, A ` ∆ Γ, B ` ∆

∨ `

Γ, A ∨ B ` ∆

(76)

LK: Logical Group: Disjunction

Γ ` A, ∆

` l∨

Γ ` A ∨ B, ∆ Γ ` B , ∆

` r∨

Γ ` A ∨ B, ∆

Γ, A ` ∆ Γ, B ` ∆

∨ ` Γ, A ∨ B ` ∆

(77)

LJ: Logical Group: Disjunction

Γ ` A

, ∆

` l∨

Γ ` A ∨ B

, ∆

Γ ` B

, ∆

` r∨

Γ ` A ∨ B

, ∆

Γ, A ` C Γ, B ` C

∨ `

Γ, A ∨ B ` C

(78)

LK: Logical Group: Implication

Γ ` A, ∆ Γ ⁰ , B ` ∆ ⁰

⇒`

Γ, Γ ⁰ , A ⇒ B ` ∆,∆ ⁰

Γ, A ` B, ∆

`⇒

Γ ` A ⇒ B, ∆

(79)

LK: Logical Group: Implication

Γ ` A, ∆ Γ ⁰ , B ` ∆ ⁰

⇒`

Γ, Γ ⁰ , A ⇒ B ` ∆,∆ ⁰

Γ, A ` B, ∆

`⇒

Γ ` A ⇒ B, ∆

(80)

LJ: Logical Group: Implication

Γ ` A

, ∆

Γ ⁰ , B ` B ⁰

⇒`

Γ, Γ ⁰ , A ⇒ B `

∆,

B ⁰

Γ, A ` B

, ∆

`⇒

Γ ` A ⇒ B

, ∆

(81)

LJ

Id A ` A

Γ ` A Γ ⁰ , A ` B Cut Γ, Γ ⁰ ` B

Γ ` B X ` σ(Γ) ` B

Γ ` B W`

Γ, A ` B

Γ, A, A ` B C ` Γ, A ` B Γ ` A Γ ` B

` ∧ Γ ` A ∧ B

Γ, A ` C

l ∧ ` Γ, A ∧ B ` C

Γ, B ` C

r∧ ` Γ, A ∧ B ` C Γ ` A

` l ∨ Γ ` A ∨ B

Γ ` B

` r∨

Γ ` A ∨ B

Γ, A ` C Γ, B ` C

∨ ` Γ, A ∨ B ` C

Γ ` A Γ ⁰ , B ` C

⇒`

Γ, Γ ⁰ , A ⇒ B ` C

Γ, A ` B

`⇒

Γ ` A ⇒ B

(82)

Prove A ` ¬¬A

A − ` A + ⊥ ₋ ` ⊥ ₊

⇒` A − , A + ⇒ ⊥ ₋ ` ⊥ ₊

`⇒ A − ` (A + ⇒ ⊥ ₋ ) ⇒ ⊥ ₊

(83)

Prove A ` ¬¬A

A ` A ⊥ ` ⊥

⇒`

A, A ⇒ ⊥ ` ⊥

`⇒

A ` (A ⇒ ⊥) ⇒ ⊥

(84)

Prove A ` ¬¬A

A − ` A + ⊥ ₋ ` ⊥ ₊

⇒`

A − , A + ⇒ ⊥ ₋ ` ⊥ ₊

`⇒

A − ` (A + ⇒ ⊥ ₋ ) ⇒ ⊥ ₊

(85)

Prove ¬¬¬A ` ¬A

A − ` A ₊ ⊥ ₋ ` ⊥ ₊

⇒` A − , A ₊ ⇒ ⊥ ₋ ` ⊥ ₊

`⇒

A − ` (A ₊ ⇒ ⊥ − ) ⇒ ⊥ ₊ ⊥ ⁰ ₋ ` ⊥ ⁰ ₊

⇒` A − , ((A + ⇒ ⊥ − ) ⇒ ⊥ ₊ ) ⇒ ⊥ ⁰ ₋ ` ⊥ ⁰ ₊

`⇒

((A + ⇒ ⊥ − ) ⇒ ⊥ ₊ ) ⇒ ⊥ ⁰ ₋ ` A − ⇒ ⊥ ⁰ ₊

Therefore, in intuistionistic logic ¬¬¬A ≡ ¬A, but ¬¬A 6≡ A.

(86)

Prove ¬¬¬A ` ¬A

A ` A ⊥ ` ⊥

⇒`

A, A ⇒ ⊥ ` ⊥

`⇒

A ` (A ⇒ ⊥) ⇒ ⊥ ⊥ ⁰ ` ⊥ ⁰

⇒`

A, ((A ⇒ ⊥) ⇒ ⊥) ⇒ ⊥ ⁰ ` ⊥ ⁰

`⇒

((A ⇒ ⊥) ⇒ ⊥) ⇒ ⊥ ⁰ ` A ⇒ ⊥ ⁰

Therefore, in intuistionistic logic ¬¬¬A ≡ ¬A, but ¬¬A 6≡ A.

(87)

Prove ¬¬¬A ` ¬A

A − ` A ₊ ⊥ ₋ ` ⊥ ₊

⇒`

A − , A ₊ ⇒ ⊥ ₋ ` ⊥ ₊

`⇒

A − ` (A ₊ ⇒ ⊥ − ) ⇒ ⊥ ₊ ⊥ ⁰ ₋ ` ⊥ ⁰ ₊

⇒`

A − , ((A + ⇒ ⊥ − ) ⇒ ⊥ ₊ ) ⇒ ⊥ ⁰ ₋ ` ⊥ ⁰ ₊

`⇒

((A + ⇒ ⊥ ₋ ) ⇒ ⊥ ₊ ) ⇒ ⊥ ⁰ ₋ ` A − ⇒ ⊥ ⁰ ₊

Therefore, in intuistionistic logic ¬¬¬A ≡ ¬A, but ¬¬A 6≡ A.

(88)

Bibliography I

van Atten, M. (2009).

Stanford Encyclopedia of Philosophy, chapter The Development of Intuitionistic Logic.

The Metaphysics Research Lab.