The Tao of Logic - · PDF fileThe Tao of Logic Flavio Zelazek Sapienza University of Rome, Italy ﬂ[email protected] March 30, 2014 Flavio Zelazek (ﬂ[email protected])

The Tao of Logic

Flavio Zelazek

Sapienza University of Rome, Italy

[email protected]

March 30, 2014

Flavio Zelazek ([email protected]) The Tao of Logic March 30, 2014 1 / 37

mailto:[email protected]


Introduction

‘Logic’

Logic deals with what is common to every branch of our cognitiveactivity, and so to every scientific discipline.

[Abrusci(2009)]

Dialectica est ars artium, ad omnium methodorum principia viamhabens.

Petrus Hispanus, Summulae logicales



Introduction

‘Logic’

Logic deals with what is common to every branch of our cognitiveactivity, and so to every scientific discipline.

[Abrusci(2009)]

Dialectica est ars artium, ad omnium methodorum principia viamhabens.

Petrus Hispanus, Summulae logicales



Introduction

‘Duality’

Duality requires the presence of two points of view which arealternative to each other, and from which the objects of our knowledge,the informational contents, can be expressed or considered.

[Abrusci(2009)]



Duality in Early Chinese Philosophy Duality in the I Ching

Outline

1 Duality in Early Chinese PhilosophyDuality in the I ChingDuality in T’ai Chi Ch’uan

2 Duality in LogicDuality as Symmetry in Classical LogicThe Curry-Howard CorrespondenceLinear Negation as Computational Duality




The I Ching

The I Ching (Yijing) [Wilhelm and Baynes(1967)] is composed of:

the Chou I (Zhouyi), the oracular text (Han Dynasty, 2ndcentury BC)the Ten Wings, the commentary, which includes:

the Hsi Tz’u (Xici), or Ta Chuan, the ‘Great Commentary ’




The I Ching







The I Ching







Tao and T’ai Chi

That which let now the dark, now the light appear is Tao.

Ta Chuan, ch. 5

Therefore there is in the Changes the Great Primal Beginning. Thisgenerates the two primary forces. The two primary forces generate thefour images. The four images generate the eight trigrams.

Ta Chuan, ch. 11




Tao and T’ai Chi

That which let now the dark, now the light appear is Tao.

Ta Chuan, ch. 5

Therefore there is in the Changes the Great Primal Beginning. Thisgenerates the two primary forces. The two primary forces generate thefour images. The four images generate the eight trigrams.

Ta Chuan, ch. 11




Chinese ‘cosmogony’

The Great Primal Beginning: T’ai Chi (Taiji)

The two primary forces: Liang I (Liangyi),which dynamically manifest themselves as the ‘dark’ and the ‘light’

in the development of the Tao (Dao); or: Yin–Yang

The four images: Sze Hsiang (Sixiang)

The eight trigrams: Pa Kua (Bagua)































The T’ai ChiTu

The T’ai Chi Tu (Taijitu), that is the ‘Diagram of the T’ai Chi’,symbolizes:

Coexistence of Yin–YangThe cyclical change of one into the otherTheir relation of complementarity




The T’ai ChiTu






The T’ai ChiTu






The T’ai ChiTu





Duality in Early Chinese Philosophy Duality in T’ai Chi Ch’uan

Outline






T’ai Chi Ch’uan

T’ai Chi Ch’uan (Taijiquan), ‘Boxe of the T’ai Chi’: at the same time amartial art, a healthy exercise and a form of meditation (cf. e.g.[Ch’ing(1985)]).




T’ai Chi Classics

The following quotations are from two ‘T’ai Chi Classics’ of the 13thand 17th century AD, both collected in [Liao(1990)].

T’ai Chi is born out of infinity. It is the origin of the positive and thenegative. When T’ai Chi is in motion, the positive and the negativeseparate; when T’ai Chi stops, the positive and negative integrate.




T’ai Chi Classics

Besides clearly separating the positive and negative from one another,you should also clearly locate the substantial and insubstantial. Whenthe entire body is integrated with all parts connected together, itbecomes a vast connection of positive and negative energy units.Each positive and negative unit of energy should be connected toevery other unit and permit no interruption among them.

[. . . ] Your movements should be constantly changing from thesubstantial to the insubstantial. If your left side feels heavy, you shouldmake your left side light. If your right side feels heavy, you shouldmake your right side disappear.




T’ai Chi Classics

[. . . ] You should also follow the T’ai Chi principle of opposites: whenyou move upward, the mind must beware of down; when movingforward, the mind also thinks of moving back; when shifting to the leftside, the mind should simultaneously notice the right side – so that ifthe mind is going up, it is also going down.




T’ai Chi Classics

If your opposite side is hard, change your own side to make it soft.This is called following. If your opponent is moving and you adhere tohim while following in the same direction, it is called sticking. Then youare attached to your opponent: when he moves faster, you also movefaster; when he moves slower, you move slower, thereby matching hismovement.




T’ai Chi Classics

Following the changing situation, you move as is necessary. If you areunable to respond in this way you will become double-weighted. Oftenmartial artists who have practiced for years still cannot move properlyand so cannot follow the flow of their opponent’s movement. This isessentially because they are hindered by their mistake ofdouble-weightedness.

To avoid double-weightedness you should further understand thatpositive and negative must complement each other. [. . . ] The T’ai Chiprinciple is as simple as this: yield yourself and follow the externalforces. [. . . ]




Duality in T’ai Chi philosophy

The Yin–Yang duality in the I Ching and in the T’ai Chi Classics hasthe following aspects:

CoexistenceComplementaritySymmetryInterchangeabilityConstant changeSameness of natureInteraction













































Duality in Logic Duality as Symmetry in Classical Logic

Outline






Duality in logic

Two levels of duality:

The level of formulas: duality between a formula and its negationThe level of proofs: duality between proofs and refutations




Logical calculi

Classical logic has three main kinds of calculi:Hilbert-style calculusSequent calculus (G. Gentzen)Natural deduction (G. Gentzen, D. Prawitz)




Sequents

A sequent has the following form:

A1, . . . ,Aj − B1, . . . ,Bk

Intuitive meaning: ‘A1 ∧ · · · ∧ Aj implies B1 ∨ · · · ∨ Bk ’.




Sequents

A sequent has the following form:

A1, . . . ,Aj − B1, . . . ,Bk

Intuitive meaning: ‘A1 ∧ · · · ∧ Aj implies B1 ∨ · · · ∨ Bk ’.




Rules for negation

Γ − A,∆¬L

Γ,¬A − ∆

Γ,A − ∆¬R

Γ − ¬A,∆




The cut rule

Γ − ∆,A A, Γ′ − ∆′CUT

Γ, Γ′ − ∆,∆′




One-sided sequents

− Θ,A − Θ′,¬ACUT− Θ,Θ′




Non-constructivity

Due to its very symmetry, classical logic is not constructive:cut-elimination is vitiated by phenomena of non-determinism, so proofshave no computational content.



Duality in Logic The Curry-Howard Correspondence

Outline






Intuitionistic sequents

An intuitionistic sequent has only one formula on the right side:

A1, . . . ,Aj − B




λ-calculus

λ-calculus has just two operators:

λ abstraction

( ) application

Computation (i.e. program execution) is performed by the rule ofβ-reduction (or β-conversion):

(λx .t)s ; t [s/x ]




The proofs-as-programs paradigm

NATURAL DEDUCTION TYPED λ-CALCULUS

formulas types

(intuitionistic) proofs λ-terms (programs)

normalization β-conversion

(cf. e.g. [Wadler(2000)])






formulas types









formulas types









formulas types







β-conversion

β-conv:

Γ, x : AD1

t : B ABSλx . t : A→ B

∆D2

s : AAPP

(λx . t)s : B

;

Γ,∆

D1[D2/x ]

t [s/x ] : B




Non-symmetry

Intuitionistic logic, hence λ-calculus, is not symmetrical: there is aninput-output asymmetry.



Duality in Logic Linear Negation as Computational Duality

Outline






The main linear connectives

Decomposition of intuitionistic implication in linear logic[Girard(1987), Girard(1995)]:

A→ B ' !A ( B

Involutiveness of negation:

A = A⊥⊥




Intuitionistic logic

AX

x :

A −

x :

AΓ −

s :

A

x :

A,∆ −

t :

BCUT

Γ,∆ −

t [s/x ] :

B

Γ −

t :

BW

Γ,

x :

A −

t :

BΓ,

x :

A,

y :

A −

t :

BC

Γ,

z :

A −

t [z/x , z/y ] :

B

Γ −

t :

A

x :

B,∆ −

s :

C→ L

z :

A → B, Γ,∆ −

s[zt/x ] :

BΓ,

x :

A −

t :

B→ R

Γ −

λx . t :

A → B

Γ,

x :

A −

t :

BD!

Γ,

x :

!A −

t :

B!Γ −

t :

A!

!Γ −

t :

!A

Figure: LJ→ calculus




Intuitionistic λ-caclulus

AXx : A − x : A

Γ − s : A x : A,∆ − t : BCUT

Γ,∆ − t [s/x ] : B

Γ − t : BW

Γ, x : A − t : BΓ, x : A, y : A − t : B

CΓ, z : A − t [z/x , z/y ] : B

Γ − t : A x : B,∆ − s : C→ L

z : A → B, Γ,∆ − s[zt/x ] : BΓ, x : A − t : B

→ RΓ − λx . t : A → B

Γ, x : A − t : BD!

Γ, x : !A − t : B!Γ − t : A

!!Γ − t : !A

Figure: Term-assignment system for LJ→




Linear λ-calculus

AXx : A − x : A

Γ − s : A x : A,∆ − t : BCUT

Γ,∆ − t [s/x ] : B

Γ − t : BW!

Γ, x : !A − t : BΓ, x : !A, y : !A − t : B

C!Γ, z : !A − t [z/x , z/y ] : B

Γ − t : A x : B,∆ − s : C( L

z : A ( B, Γ,∆ − s[zt/x ] : BΓ, x : A − t : B

( RΓ − λx . t : A ( B

Γ, x : A − t : BD!

Γ, x : !A − t : B!Γ − t : A

!!Γ − t : !A

Figure: Term-assignment system for IMELL




Duality in logic (and computation)

Duality in linear logic resides in the following characteristics:

Symmetry (due to involutive negation)Interchangeability (input-output)Complementarity (at the formula level)Interaction (at the proof level)

Moreover, in Ludics [Girard(1999), Girard(2001)] we have:

Sameness of nature between proofs and tests (refutations):a ‘monist duality’











































Bibliography

Chinese Philosophy

Ch’ing, C. M. (1985).Cheng Tzu’s Thirteen Treatises on T’ai Chi Ch’uan.North Atlantic Books, Berkeley, California.

Liao, W. (1990).T’ai Chi Classics.Shambhala Publications, Boston.

Wilhelm, R. and Baynes, C. F., editors (1967).The I Ching, or Book of Changes, volume 19 of Bollingen series.Princeton University Press, Princeton.



Bibliography

Logic

Abrusci, V. M. (2009).Logica. Lezioni di primo livello.CEDAM, Padova.

Girard, J.-Y. (1987).Linear logic.Theoretical Computer Science, 50, 1–102.

Girard, J.-Y. (1995).Linear logic : its syntax and semantics.In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances inLinear Logic, pages 1–42. Cambridge University Press. Alsoavailable athttp://iml.univ-mrs.fr/˜girard/Articles.html.


http://iml.univ-mrs.fr/~girard/Articles.html


Bibliography

Logic

Girard, J.-Y. (1999).On the meaning of logical rules I : syntax vs. semantics.In U. Berger and H. Schwichtenberg, editors, Computational Logic,NATO series F 165, pages 215–272. Springer-Verlag, Heidelberg.

Girard, J.-Y. (2001).Locus Solum.Mathematical Structures in Computer Science, 11, 301–506.

Wadler, P. (2000).Proofs are Programs: 19th Century Logic and 21st CenturyComputing.Available at http://homepages.inf.ed.ac.uk/wadler/topics/history.html.


http://homepages.inf.ed.ac.uk/wadler/topics/history.html

http://homepages.inf.ed.ac.uk/wadler/topics/history.html


Documents

The Tao of Logic - · PDF fileThe Tao of Logic Flavio Zelazek Sapienza University of Rome, Italy ﬂ[email protected] March 30, 2014 Flavio Zelazek (ﬂ[email protected])