Old logic and contemporary logic A new logic of ideas ?

Introduction Method Reasoning Judgement Conception

Old logic and contemporary logicA new logic of ideas ?

Arnaud Valence

[email protected]

Rome, June 6 2013


Introduction

Introduction : (very) old logic


Neo-mohist logic

Reference points

Mo Zi (master Mo, 479 à 392 av. JC) is a disciple of ConfuciusThe disciples of Mo Zi are grouped under the name of “Mohistschool” or “Logician school”This school is a variation on the confucian Zhengmingtradition, i.e. the tradition of rectification of names.The mohist philosophy is collected in 53 booksThese books deal with very general thoughts, or platitudes,devoid of logical contentBut there are exceptions : books 32-37, wrote by GongsunLong


Neo-mohist logic

Gongsun Long’s reasoning

Examples of Gongsun Long’s reasoning :

(All men are mortal) M ∈ Mor

Socrates is mortal. Socrates is a man. S ∈ M

Because he is a man. Therefore Socrates is mortal. S ∈ Mor

A thief is a man. A (man) killer is a killer MK ∈ K

But killing a thief A thief killer is a man killer. TK ∈ MK

is not killing a man. Therefore a thief killer is a killer TK ∈ K

Conclusion : Gongsun Long’s reasoning is not syllogism. It is aboutrepresentation.


Greek logic

Aristotle’s Organon

The Categories deal with classification. 10 categories :substance, quantity, quality, relation, place, time,situation, condition, action, and passion.

On Interpretation deals with proposition and judgement, andwith relations between affirmative, negative,universal, and particular propositions.

The Prior Analytics deal with validity within syllogistic method.The Posterior Analytics deal with soundness within

demonstrative syllogisms (scientific knowledge).The Topics deal with non demonstrative syllogisms (art of

dialectic).The Sophistical Refutations deal with false syllogisms (sophisms

or paralogisms).


Conclusion

Two ways to consider logic

Two ancient schools of logicThe chinese school focusing on representation, i.e. oncorrectness of predicates.The greek school stressing on reasoning, i.e. on validity ofinferences (but don’t forget Cratylus!).

In fact, both components are déjà là in the Organon, at least ingerm.

Reasoning (On Interpretation, Analytics), better developedthan in Plato.Representation, even if it is more incomplete foundationalissues (Categories, Topics).


Conclusion

Two ways to consider logic

Aristotle’s Organon propose a canonical division of logic into fourparts.

1 Categories How to categorize predicables?

2 On Interpretation How to connect predicables?

3 Analytics How to lead a valid (deductive) inference?

4 Topics How to construct a valid argument regardless of the truth ofthe premises?


Conclusion

Contemporary logic

ProposalFragments of contemporary logic are particular constructivereconsiderations of each of these four canonical parts.

1 Conception What is a proposition?2 Judgement What is a judgement on a proposition?3 Reasoning What is a valid (deductive) inference?4 Method What part of the axiomatico-deductive method in the

decision problem?

Let’s start with a top-down survey.


Method

Method


Example

Proof of parity of 4 without rule of calculus.

` ∀x∀y, x = y ⇒ s(x) = s(y)

` ∀x, x = 0⇒ s(x) = 1

` (1× 0) + 0 = 0⇒ s((1× 0) + 0) = 1

` ∀x, x × 0 = 0

` 1× 0 = 0

` (1× 0) + 0 = 0

` s((1× 0) + 0) = 1

` ∀x∀y∀z, x = y ⇒ x = z ⇒ y = z

` ∀x∀y, x = y ⇒ x = 1⇒ y = 1

` ∀x, x = (1× 0) + 1⇒ x = 1⇒ (1× 0) + 1 = 1

` s((1× 0) + 0) = (1× 0) + 1⇒ s((1× 0) + 0) = 1⇒ (1× 0) + 1 = 1

` ∀x∀y, x × 0 = 0

` ∀x, s(x + 0) = x + 1

` s((1× 0) + 0) = (1× 0) + 1

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` s((1× 0) + 0) = 1⇒ (1× 0) + 1 = 1

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` (1× 0) + 1 = 1

` ∀x∀y∀z, x = y ⇒ x = z ⇒ y = z

` ∀x∀y, x = y ⇒ x = 1⇒ y = 1

` ∀x, x = 1× 1⇒ x = 1⇒ 1× 1 = 1

` (1× 0) + 1 = 1× 1⇒ (1× 0) + 1 = 1⇒ 1× 1 = 1

` ∀x∀y, (x × y) + x = x × s(y)

` ∀x, (x × 0) + x = x × 1

` (1× 0) + 1 = 1× 1

` (1× 0) + 1 = 1⇒ 1× 1 = 1

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` 1× 1 = 1

` (1× 1) + 0 = 1

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.` (2× 1) + 1 = 3

` ∀x∀y, x = y ⇒ s(x) = s(y)

` ∀x, x = 3⇒ s(x) = 4

` (2× 1) + 1 = 3⇒ s((2× 1) + 1) = 4

` s((2× 1) + 1) = 4

` ∀x∀y∀z, x = y ⇒ x = z ⇒ y = z

` ∀x∀y, x = y ⇒ x = 4⇒ y = 4

` ∀x, x = (2× 1) + 2⇒ x = 4⇒ (2× 1) + 2 = 4

` s((2× 1) + 1) = (2× 1) + 2⇒ s((2× 1) + 1) = 4⇒ (2× 1) + 2 = 4

` ∀x∀y, s(x + y) = x + s(y)

` ∀x, s(x + 1) = x + 2

` s((2× 1) + 1) = (2× 1) + 2

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` s((2× 1) + 1) = 4⇒ (2× 1) + 2 = 4

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` (2× 1) + 2 = 4

` ∀x∀y∀z, x = y ⇒ x = z ⇒ y = z

` ∀x∀y, x = y ⇒ x = 4⇒ y = 4

` x = 2× 2⇒ x = 4⇒ 2× 2 = 4

` (2× 1) + 2 = 2× 2⇒ (2× 1) + 2 = 4⇒ 2× 2 = 4

` ∀x∀y, (x × y) + x = x × s(y)

` ∀x, (x × 1) + x = x × 2

` (2× 1) + 2 = 2× 2

` (2× 1) + 2 = 4⇒ 2× 2 = 4

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` 2× 2 = 4` ∃x, x × 2 = 4

` ∃x∃y, x × y = 4

` C(4)


Example

Proof of parity of 4 with several rewriting rules.

Rewriting rules :s(x + y)→ x + s(y)x × s(y)→ (x × y) + xx + 0→ xx × 0→ 0x = x → >

` ∀x , x = x

` 2 = 2` 0 + 2 = 2

` (2× 0) + 2 = 2` 2× 1 = 2

` (2× 1) + 0 = 2` s((2× 1 + 0) = 3` (2× 1) + 1 = 3` s((2× 1) + 1) = 4` (2× 1) + 2 = 4` 2× 2 = 4` ∃x , x × 2 = 4` ∃x∃y , x × y = 4

` C (4)


Example

Proof of parity of 4 with a single rewriting rule.

Rewriting rule :(a congruence relationship)s(x + y)→ x + s(y)andx × s(y)→ (x × y) + xandx + 0→ xandx × 0→ 0andx = x → >

` >` 2× 2 = 4` ∃x , x × 2 = 4` ∃x∃y , x × y = 4

` C (4)


Proof and certificate

The notion of certificate

The notion of proof can be generalized by the notion ofcertificate.There are different kinds of certificate :

Some (proofs) are purely axiomatic : certification is entirely onthe sender’s side.Some skip the purely axiomatic steps : certification is entirelyon the receiver’s side.Some are intermediary : certification is shared between senderand receiver.

What’s the point? Take a look at mathematics in action.



Mathematics in action

1976 Appel and Haken prove the four color theorem: the firsttheorem ever proved using a computer.

This result was considered as an exception. Is it really sure thatthere is no “manually” proof?Yes, anyway, it is not an exception ! From the 1980’s we haveseveral other theorems of the same kind:

1989 Proof of the non-existence of finite projective planes of order10

1995 Proof of the double bubble theorem.1998 Proof of the Hales theorem.



Conclusion

Some mathematical problems needs calculus to be proved :complicated problems (four colors theorem)...... as well as simple problems, more surprisingly (double bubbletheorem).

But are the simple ones so simple ?Not really, often, the size of proofs is explained by a long casestudy.Are long case proofs authentic explanations?In short, what is an explanation ? How complex a proof is anexplanation ?


Reasoning

Reasoning


The notion of cut-elimination


Ancient logic Birth of syllogistics, which is then improved by themedieval scholastics.

Modern logic Port-Royal reconstruction: syllogistics is replaced inthe background of a larger idea : “the premises mustcontain the conclusion”. This striking intuition iscompleted by Gentzen’s cut-elimination theorem in1935.

Contemporary logic Girard reconstruction: the cut-elimination asa feedback equation, i.e. an algebraic relationshipbetween permutations (2005).



Extensions of the notion of cut-elimination

Model-theoretic approaches In1959 Rasiowa and Sikorskiprovide a first semantic proof of cut-eliminationtheorem. In 1967, Takahashi provides a semanticproof of cut-elimination theorem for high order logic.

Mixed rewriting/axiomatic approaches Dowek, Hardin andKirchner provide a cut-elimination theorem in modulodeduction. Works in progress towards model-theoreticapproaches modulo rewriting rules.


Conclusion

Reasoning, method and judgement

sequents

structures

cut-free sequents

tables

sequents modulo structures modulo

cut-freesequentsmodulo

tables modulo

soundness

completeness

synt

actic

cut-

elim

inat

ion

stron

gco

mpleten

ess

com

plet

enes

sof

tabl

es

soundness of tables

Method

Reasoning

judgement


Conclusion


sequents

structures

cut-free sequents

tables

sequents modulo

structures modulo


tables modulo

soundness

completeness

synt

actic

cut-

elim

inat

ion

stron

gco

mpleten

ess

com

plet

enes

sof

tabl

es

soundness of tables

Method

Reasoning

judgement


Conclusion


sequents structures

cut-free sequents tables

sequents modulo structures modulo


tables modulo

soundness

completeness

synt

actic

cut-

elim

inat

ion

stron

gco

mpleten

ess

com

plet

enes

sof

tabl

es

soundness of tables

Method

Reasoning

judgement


Conclusion

What kind of reasoning

In a way, a reasoning is a constructive method.So what? How to reason ?Should we accept the rule of the excluded middle? Could it bemore constructive?Is it a matter of taste between classical logic and intuitionisticlogic?Is the witness part of logic?


Judgement

Judgement


Historical reference points

Modern logic

In 1662, Port Royal renews the notions of comprehension andextension. For the first time comprehension seems to have anintensional meaning.In 1790, Kant introduces the notion of judgement (Urteil)instead of proposition (Satz), but the power of judgement isthrown out in a extrinsic (transcendental) logic.In 1892, Frege introduces the notions of sense (Sinn) andreference (Bedeutung): key-notions to conceptualize intensionand extension.In 1930, Wittgenstein sheds a new light on intensionalconception of logical consequence (Marion), stressing ondeduction rules rather than on axioms. A few years later,Gentzen introduces natural deduction.In the 1930’s, Heyting and Kolmogorov introduce(intuitionistic) proof-theoretic semantics (BHK interpretation).


Historical reference points

Contemporary logic

60’s Curry–Howard (or proofs-as-programs) correspondence:propositions are seen as types, proofs as terms.

70’s Martin-Löf introduces intuitionistic type theory, an extensionof the Curry–Howard correspondence where types depend onterms. At the same time Girard introduces system F, anextension of the Curry–Howard correspondence where termsdepend on types.

80’s Coquand introduces calculus of constructions, includingpolymorphism, type theory and types depending on types.


From proposition to judgement


Frege: premises are not formulas (A,B ,...) but propositions(` A, ` B ,...).Martin-Löf: ok, but then, the choice of propositions dependson the form of judgement.

` A prop ` B prop ` A true` A ∨ B true

To know a judgement is the same as to possess a proof of it, and toknow a judgement of the particular form “A is true” is the same asto know how, or be able, to verify the proposition A. [...] Knowledgeof a judgement of [this] form is knowledge how, whereas knowledgeof a judgement of the form “A is a proposition” is knowledge [...]what to do, simply. Here I am introducing knowledge what as acounterpart of Ryle’s knowledge how. So the difference betweenthese two kinds of knowledge is the difference between knowledgewhat to do and knowledge how to do it. (Martin-Löf 1980)



Another order of priorities

Frege’s conceptual order of priorities:proposition → assertion → logic rule.

Martin-Löf’s conceptual order of priorities:form of judgement → judgement → evident judgement → logic rule.

logic rule︷︸︸︷evident judgement︷︸︸︷

I know that

judgement︷︸︸︷A

form of judgement︷︸︸︷is ............

evident judgement︷︸︸︷I know that

judgement︷︸︸︷B

form of judgement︷︸︸︷is ............

I know that A and B are ............


Conclusion

Conclusion

A “full-blooded” theory of meaning (Dummett).A close connection between constructive mathematics andprogramming (Martin-Löf).A normalization property richer than the (semantic)cut-elimination property.But how to choose types? Were does they come from?Example: the term t = λx .λy .((xλz .0)F )((yλz ′.1)F ) whereF = λf .λg .gf is not typable in system F.How to get the interesting programs without going through thead hoc description of such and such type of formal system?Girard’s answer: Forgetting syntax to understand it.


Conception

Conception


Proofs as permutations

Plugging permutations

Ax1 Ax2 Ax3

⊗1 ⊗2

`1 `2

(3

Figure : Proofnet of formula ` (A` B)⊗ C ( A` (B ⊗ C )


Proofs as permutations

The long-trip criterion

Ax1 Ax2

⊗ `

`

Figure : Long-trip in the proof structure A⊗ A ( A⊗ A


A new referential

A new referential

We have the following change of perspective :

topological space → functions algebras → operators algebras

ou (via the GNS construction)

Hilbert spaces → functions algebras → von Neumann algebras

Transcendantal syntaxHow can we recover formulas (topological spaces or Hilbert spaces)from interactions (C*-algebras or von Neumann algebras)?


A new referential

A new referential

Let A a unital C ∗-algebra, and a ∈ A. The map P 7→ P(a) whereP(a) = λ01 + λ1a + λ2a

2 + · · ·+ λnan is a unital homomorphism.

The spectrum σ(a) of a is the set of characters λ such that:

σ(a) = {λ ∈ C|(a− λ1) is not invertible}

The polynomial function P(a) can generalized (for continuousfunctions) by the Stone–Weierstrass theorem:

Spectral theoremσ(C (a)) = C (σ(a))


A story of categories

A equivalence of categories

Gelfand’s transformationLet Spec(a) the set of algebra morphisms of A and C (X ) a unitalC ∗-algebra, then

Spec(C (X )) = C (Spec(X ))

X Y

Spec(C (X )) Spec(C (Y ))

f

εX εY

Spec(C (f ))

(Hausdorff) topological spaces

C ∗-algebras

f (x) = εX (f )f ◦ ε = ε ◦ Spec(C (f ))


Conclusion

Open questions

Dependant types They are embedded in transcendental syntax1.0.

Πa ∈ A B(a) := {f | ∀a ∈ A, [f ]a ∈ B(a)}

Σa ∈ A B(a) := {a⊗ b | a ∈ A, b ∈ B(a)}⊥⊥

but not in transcendental syntax 2.0.Normativity (deontic level) Internalization of typing. A massive

program !

Old logic and contemporary logic A new logic of ideas ?

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