Introduction Method Reasoning Judgement Conception
Old logic and contemporary logicA new logic of ideas ?
Arnaud Valence
Rome, June 6 2013
Introduction Method Reasoning Judgement Conception
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
Introduction Method Reasoning Judgement Conception
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.
Introduction Method Reasoning Judgement Conception
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).
Introduction Method Reasoning Judgement Conception
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).
Introduction Method Reasoning Judgement Conception
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?
Introduction Method Reasoning Judgement Conception
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.
Introduction Method Reasoning Judgement Conception
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)
Introduction Method Reasoning Judgement Conception
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)
Introduction Method Reasoning Judgement Conception
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)
Introduction Method Reasoning Judgement Conception
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)
Introduction Method Reasoning Judgement Conception
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)
Introduction Method Reasoning Judgement Conception
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.
Introduction Method Reasoning Judgement Conception
Proof and certificate
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.
Introduction Method Reasoning Judgement Conception
Proof and certificate
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 ?
Introduction Method Reasoning Judgement Conception
The notion of cut-elimination
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).
Introduction Method Reasoning Judgement Conception
The notion of cut-elimination
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.
Introduction Method Reasoning Judgement Conception
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
Introduction Method Reasoning Judgement Conception
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
Introduction Method Reasoning Judgement Conception
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
Introduction Method Reasoning Judgement Conception
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?
Introduction Method Reasoning Judgement Conception
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).
Introduction Method Reasoning Judgement Conception
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.
Introduction Method Reasoning Judgement Conception
From proposition to judgement
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)
Introduction Method Reasoning Judgement Conception
From proposition to judgement
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 ............
Introduction Method Reasoning Judgement Conception
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.
Introduction Method Reasoning Judgement Conception
Proofs as permutations
Plugging permutations
Ax1 Ax2 Ax3
⊗1 ⊗2
`1 `2
(3
Figure : Proofnet of formula ` (A` B)⊗ C ( A` (B ⊗ C )
Introduction Method Reasoning Judgement Conception
Proofs as permutations
The long-trip criterion
Ax1 Ax2
⊗ `
`
Figure : Long-trip in the proof structure A⊗ A ( A⊗ A
Introduction Method Reasoning Judgement Conception
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)?
Introduction Method Reasoning Judgement Conception
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))
Introduction Method Reasoning Judgement Conception
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 ))
Introduction Method Reasoning Judgement Conception
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 !