Geometry of Interaction VI: a blueprintfor Transcendental Syntax

Jean-Yves GirardInstitut de Mathématiques de Luminy, UMR 6206 – CNRS

163, Avenue de Luminy, Case 907, F-13288 Marseille Cedex 09

girard@iml.univ-mrs.fr

January 1, 2013

À Corrado, sans qui le λ-calculne serait pas ce qu’il est.

Et donc, la logique non plus.

AbstractThe paper makes use of a finitary version of Geometry of Interaction

to fix most pending problems in propositional proof-nets. The method-ological guidelines are those of transcendental syntax.

1 Transcendental SyntaxTranscendental syntax starts where semantics fails: foundational issues1.

1.1 Splendor. . .Semantics is very valuable tool: nothing is better than a good model-theoreticargument. Typically the independence of Euclide’s postulat is explained inthe most convincing ways by non-euclidian models: the sphere or the one-sheethyperboloid. I myself made several crucial uses of semantics: linear logic literallyjumped out of coherent spaces.

This use of semantics is, unfortunately, accidental. There are dozens ofsystems of arithmetic, all of them different; but since there is only one decentmodel for arithmetic — styled standard for this reason —, semantics cannot beused to separate them. This shortage of models, known as incompleteness, castsa doubt as to the foundational relevance of semantics.

1.2 . . . and misery of semanticsFoundationnaly speaking, the semantic credo is the idea that logic is aboutA something else B, a reality exterior and prior to language. This realism cantake two aspects: pleonasmatic and structural.

1Work partially supported by the ANR project ANR-2010-BLAN-021301 LOGOI.

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1.2.1 The pleonasm

The pleonasm styles the A something else B as truth or meta. The language issupposed to imitate this preexistent model: when we write A ⇒ A, we meanA if A is true, then A is true B; justifying A ⇒ A by B ⇒ B (with B:= A A istrue B) is dubious: indeed, this is but a metaphysical gimmick condoning ourprejudices, a reality imitating the language, not the other way around2!

1.2.2 Structuration

Coming from the BHK3 interpretation, functional (or category-theoretic) in-terpretations are not pleonasmatic. For instance, A ⇒ A is interpreted bythe identity morphism. This category-theoretic structure is most precious; theproblem is that its is foundationally misleading. A typical example: the BHKinterpretation of Π0

1 formulas is trivial, conveys no information4 and this cannotbe fixed within the category-theoretic universe which forces the identification ofall proofs in this case. Now to claim that there is nothing inside the proof of Π0

1

statements like the quadratic reprocity law, Fermat’s last theorem, etc. is pre-posterous. In other terms, the category-theoretic universe is not primitive; bythe way, categories are heavily formatted5; when analysing the format (secondA light B, infra), we clearly step beyond categories.

In the theory of proof-nets, categories have been a nuisance: typically, theycompel us to identify the proof of `A1

1 & A21, . . . , A

1n & A2

n, B1 ⊗ . . . ⊗ Bn ob-tained from proofs of A1

i , Bi and A2i , Bi (i = 1, . . . , n) with the 2n A slices B

`Aσ(1)1 , . . . , Aσ(n), B1 ⊗ . . . ⊗ Bn, neglecting the monstrous difference between

the two proofs, 2n vs. 2n.

1.2.3 Subjectivism

The only argument for realism is the fear of subjectivism; this is the case withirrelevant subjective distinctions, styled A intensional B, e.g., a left-handed vs.right-handed cups. But, subjectivism can take the perverse disguise of objec-tivism, i.e., the denial of the part taken by the subject in the construction ofthe object: this leads to absurdities like Ptolemaic astronomy which never re-alised that the A backwards motion B of planets was indeed the movement ofthe Earth, i.e., of the observer. The Ptolemaic epicycles survive in the modernKripke models where cognitive distinctions are handled by means of ad hoc par-allel universes. And what to say about the A objective B treatment of databaseswhich erases all punctations (begin,end) on the basis that they are semanti-cally meaningless, thus making the reckoning of the end-of-file a particular caseof the undecidable halting problem?

Logic is indeed subjective, since it deals with reasoning: the subject is apriori involved in a proof. The question is to shed some light on this complexinvolvement. Once removed the irrelevant bureaucratic details, the subject oc-curs in three ways: in an implicit handling of answers, in the choice of a formatfor questions, in producing a reasonable form of certainty.

2Dixit Voltaire: On prétend que Dieu a fait l’homme à son image, mais l’homme le lui abien rendu.

3Brouwer-Heyting-Kolmogoroff, ∼ 1930.4The unique BHK proof of ∀nA is the constant fonction f(n) := true.5Witness the etymology of A morphism B.

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1.3 The three lights

Semantics being discarded, one has to develop a specific methodology; in theabsence of any realistic reference, three basic questions corresponding to thefundamental tensions of logic: implicit vs. explicit, formatted vs. unformatted,apodictic vs. epidictic will enlighten our way.

1.3.1 First light: what is an answer?

Proofs are always implicit, i.e., yield indirect answers. Asking logic to be ex-plicit is like asking a bank to milk cows or print newspapers. Logical coherencecorresponds to the fact that proofs can — eventually — be A explicited B. Thisprocess is known as cut-elimination: proofs, as cheques on the possible, canbe A cashed B. Various negative theorems, mainly Turing’s undecidability of thehalting problem (1936), ensure that this A explicitation B is extremely hazardous,in particular that the implicit cannot be a priori reduced to the explicit.

The first task of transcendental syntax is to find a decent space where im-plicit and explicit coexist, where a hazardous process of explicitation can takeplace. This space, this process should be syntax-free, i.e., use none of theusual wedding-cake recipes of traditional proof-theory: in other terms, syn-thetic. Usual rewriting systems are not synthetic enough; among them, thebest approximation remains pure λ-calculus; various π-calculi are everythingbut synthetic, since they end into cookbooks. By the way, category-theorywhich equates the two terms t (implicit) and u (explicit) of a rewriting t 7→ u— through commutative diagrams of which one side is more commutative thanthe other — fails in the sole light of A answers B.

1.3.2 Second light: what is a question?

The hazardous character of explicitation means that, without adequate hypothe-ses, it may fail. Choosing the adequate restrictions amounts at formatting logic,i.e., discriminating which questions are legitimate. The problem with formats isthat they are a necessary nuisance: typically, the original, unformated, induc-tion axiom of Peano6 is inconsistent; but any consistent arithmetic is incomplete.By the way, Richard’s paradox (1905) A the smallest number not definable inless than twenty words B shows that definable must be formatted, but also thatthis formatting will be unsatisfactory.

The question of the format is usually evacuated by restricting to such andsuch system enjoying cut-elimination: inside the usual wedding-cake approach toproof-theory, the formatting is far more hidden than explicitation. For instance,category theory takes it for granted: morphisms preserve the form, the format,but which one?

In linear logic, the format occurs in proof-nets: some structure claims to bea proof, but we have forgotten in which order the rules have been performed,hence the question of sequentialisation. The ancestor of proof-nets is Herbrand’stheorem: in a prenex form the existentials should be given as functionsy = t[x1, . . . , xn] of the universals. Assuming we forgot the step-by-step con-struction of t, Herbrand replaces x with f(y) in the case of a formula ∃y∀x; ifx actually occurs in t, then we get a cycle (failed unification) y = t[f(y)].

6Induction on all A properties B.

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The choice of the appropriate questions is thus operated through a deon-tic dialogue whose purpose is not to determine truth, but what makes sense.Just like those judicial discussions A Objection, your Honor! Objection sus-tained/overruled B. In other terms, refutation has been replaced with recusal.This deontic standpoint has been central in Geometry of Interaction (GoI).

Proof-nets are based on connectedness and acyclicity, which correspond tovery old intuitions on logic, completely inaccessible to truth, i.e., to semantics.For instance, it is common to speak of a A vicious circle B; but a vicious circle isneither true nor false, it is just vicious. In proof-nets, a vicious circle becomesa cycle, i.e., a deontic mistake. There are legitimate objections against materialimplication, on the grounds that the premise should contribute to the conclusion;semantics can’t help: false⇒ true := false would equate A⇒ B withB ⇒ A. In proof-nets, this translates as A a proof should be connected B.

1.3.3 Third light: what conveys certainty?

As we already observed, the ultimate blow against subjectivism is to reckonthat logic is subjective. This means that we can actually write and manipulateproofs and that we believe in them. In terms of answers, this means that thesyntax-free space should be combinatorial in a certain way. This is is why Ieventually decided to replace the von Neumann algebras of GoI [Gir11a] withsorts of graphs based upon unification7.

As to questions, GoI explains logic in an existentialist way, through a notionof deontic test, a.k.a. ordeal. These ordeals are homegeneous to proofs of thenegation. GoI thus relies upon a duality between proofs of A and proofs of ∼A,thus vindicating Hegel’s contradictory foundations8.

The subjective character of logic forces the set of ordeals for A to be finite. Agabarit9, i.e., a finite set of ordeals, surely conveys absolute, apodictic, certainty.However, the second incompleteness theorem says that proofs are not quiteapodictic, that there is room for a reasonable — albeit limited — doubt. Thisrelative certainty, I style it epidictic. Oustside first order logic, as soon assecond order quantifiers (indeed, existential ones) are used, we lose absolutecertainty: this is the case for Π0

1 formulas. We shall see in a forthcoming paperhow epidictic certainty is still conveyed by gabarits. The difference with thefirst order case being that the loss of the subformula property casts a reasonabledoubt10 as the relevance of the gabarits.

1.4 Analytic vs. synthetic, a priori vs. a posteriori

Coming back to the pleonasm, observe that it is hardly more than a translationof the language into another one. And, by the way, this cannot be avoided:language can only be explained by language. What is wrong with the pleonasmis that this other language is extraterritorial ; an honest approach should explainthe language by itself, by its own properties. To avoid possible circularities inthe explanation, we shall look for an approach synthetic a posteriori.

7A technique invented by Herbrand, decidedly a milestone in logic!8On the base of negation as refutation, it stumbles onconsistency, but it works well with

negation as recusal.9A sort of mould.

10Reasonable, but not too serious!

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Kant styled mathematics synthetic a priori. This means that mathematicaldefinitions take place in a general framework which cannot be put into question.Traditional logic is analytic a priori, like a cookbook: it takes the form of listof cases A if this rule is like this, do that B. It works. . . but don’t ask why!

Our main concern will thus be to make logic synthetic, i.e., to accommodateour three lights (answers, questions, certainty) independently of any syntacticalcommitment. This means finding a space in which all normalisations take place(answers); this includes the general notion of typing (questions) and axiomatics(certainty), which should result from a general paradigm in which these notionscan be discussed. To sum up, we want logic to be synthetic a posteriori.

A posteriori means that language is not primitive, that it is but a conve-nience, the name for certain remarkable artifacts in our synthetic universe. Thefact that we can individualise, give names, formulate rules. . . will be made pos-sible by various structural results, typically the Church-Rosser property, themother of all category-theoretic interpretations.

Synthetic a posteriori, this is the only way in which we can speak of theconditions of possibility of language, i.e., of transcendental syntax. The wholeprogram of GoI was about those conditions of possibility. Since there havebeen many fluctuations in the program, it would be preposterous to style theseconditions necessary ; so let us style them sufficient. Which does not forbid usfrom seeking necessary conditions; we should only be aware that, necessary ornot, we shall never be sure of their necessity.

1.5 This paper

1.5.1 Logical scope

I shall restrict myself to first order propositional linear logic.

Why linear? The expressive powers of classical, intuitionistic or linear logicsare, roughly speaking, the same. But linear logic allows for subtler distinctionswhich are sometimes blurred in the other logics: typically, if we want to under-stand conjunction from a deontic standpoint, we must distinguish between itsadditive and multiplicative versions; one corresponds to &, the other to ⊗; theworse is reached when one side uses ⊗ while the other uses &.

Γ`A,∆ Γ`B,∆

Γ`A ∧B,∆

Γ, A`∆

Γ, A ∧B `∆

Γ, B `∆

Γ, A ∧B `∆

Γ`A,∆ Γ′ `B,∆′

Γ,Γ′ `A ∧B,∆,∆′Γ, A,B `∆

Γ, A ∧B `∆

Why propositional? Because predicate calculus, is very delicate to explain;indeed, first order equality remains terra incognita.

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Why first order? This is because the epidictic part of logic involves a quan-tification over gabarits; something which requires, to start with, some familiaritywith gabarits. The real task of this paper is thus the definite version of gabarits.First order is apodictic, a world of irrefragable certainty, which takes its sourcein the subformula property.

1.5.2 Proof-nets

The idea is that logic should be completely understood in terms of interactionbetween graphs, thus expelling any sort of syntactical decorations. In otherterms, we shall develop a notion of proof-net mainly as a short hand for certainA graphs B. Everything should be contained in the A graphs B, in particular, stillfollowing our three lights:

Explicitation: should be a matter of A graph contraction B, normalisation; itshould be Church-Rosser: two cuts normalise like a single cut.

Formatting: find the right gabarits associated to logical formulas.

Certainty: through adequate sequentialisation results.

1.5.3 Technical details

The original proof-net criterion was formulated in terms of trips, translatedas permutations, thus leading to operators of functional analysis. The Danos-Regnier criterion [DR89], formulated in terms of graphs, was considered moremanageable, but not quite the right thing. The third light (certainty) tells usthat we should stay finitary, thus work in a unification algebra, which generalisespermutations, i.e., the A trip B approach.

Tethers, i.e., formulas linked to axioms, have been usually rejected fromproof-nets for category-theoretic reasons, on the grounds of A irrelevance B. Thisobjection can be overruled by reckoning that irrelevant data are often (e.g., forΠ0

1 formulas) the only way to convey certainty. Tethers will thus be acceptedor rejected depending whether or not one can normalise them according thegeneral, synthetic, pattern. Tethered axioms being links of arity ≥ 2, we mustgive up the original criterion — trips — in favour of graphs, the only decentapproach to n-ary links. The common generalisation of the unification algebraand graphs yields a sort of A thick graphs B

11: galaxies in which an alternativeGoI, closer to the Danos-Regnier version of proof-nets, can be developed.

In order to take care of the second light (questions), it is necessary to find thesynthetic version of the correctness condition for proof-nets. Correctness shouldbe stated independently from the proof, which is quite untractable in the caseof the contraction rule: we cannot foresee and locate the various premises ofthis n-ary rule. This problem can be overcome using an idea of de Naurois andMogbil [dNM11]: the synthetic approach now embodies virtual switches.

The paper leaves many questions open. Typically, it hardly goes beyondthe definition of proof-nets and their gabarits. So, see this paper as a sort ofblueprint.

11A Graphes épais B to use the terminology of Seiller [Sei12].

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2 Stars and galaxiesHere we define the space in which answers dwell.

2.1 UnificationConsider a term language with infinitely many functional symbols of any arity.An equation t = u between terms can be solved by means of substitutions, i.e.,values for all variables, even those not occurring in t, u: x1 = θ1, x2 = θ2, . . .;if θ is the substitution xi = θi, then tθ := t[xi/θi] denotes the result of thereplacement of the xi with the θi. We say that t, u are unifiable when tθ = uθfor some unifier θ. The point is that substitutions do compose, hence Herbrand’sresult:

Theorem 1There is a mother of all unifiers for t, u, i.e., a substitution θ0 such that anyunifier θ for t, u can be uniquely written θ0θ′.

Due to possible repetitions of variables, the size of this principal unifier cangrow exponentially; by presenting it as a product of substitutions, one can keepits size linear in the inputs.

2.2 The unification algebra2.2.1 Flows

A flow is an expression t↼ t′ where t, t′ are terms with quite the same variables.These common variables are internal to the flow, in other terms bound. Inparticular, when combining two flows, one must always rename the variables soas to make them distinct. Composition between t↼ t′ and u↼u′ is obtainedby matching t′ and u: matching is the particular case of unification where theterms have no variable in common, what is the case when the variables of t′, uhave been made distinct. If θ is the principal unifier, we define compositionby (t↼ t′)(u↼u′) := tθ↼u′θ. Composition is thus a partial operation; if weformally add an empty flow 0 to take care of a possible failure of the matching;(t↼ t′)(u↼u′) := 0, composition becomes associative, with neutral elementI := x↼x.

If T is the set of closed terms and if c is a constant, then t↼ t′ inducesa partial bijection [t↼ t′] between two subsets of T : [t↼ t′]t0 is defined when(t↼ t′)(t0↼c) 6= 0, in case it writes [t↼ t′]t0↼c; the condition A quite thesame variables B ensures that [t↼ t′]t0 is closed and that [t↼ t′] is injective.Any flow u↼u is idempotent; its associated function is the identity of a subset[u] ⊂ T . More generally, [t↼ t′] is a bijection bijection between [t′] and [t].

2.2.2 The convolution algebra

One can introduce the convolution algebra of the monoid, i.e., the set of finiteformal sums

∑λiφi where the φi are flows and the λi are complex coefficients,

the improper flow 0 being identified with the empty sum. This algebra actson the Hilbert space `2(T ) by means of (t↼ t′)(

∑i λiti) :=

∑i λi[t↼ t′]ti. The

involution (∑λiti↼t′i)

∗ :=∑λ̄it′i↼ti is implemented by the usual adjunction.

The idempotents t↼ t correspond to the projections on the subpaces `2([t]) and

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t↼ t′ induces a partial isometry of source `2([t′]) and target `2([t]). The earlyversions of GoI [Gir89, Gir95] did associate to proofs finite sums of flows. Thesesums were indeed partial isometries; u =

∑ti↼t′i is partial isometry (i.e.,

uu∗u = u) if the targets ti are pairwise A disjoint B, not unifiable, idem forthe t′i. Indeed, the operators of GoI are even partial symmetries, i.e., hermitianpartial isometries (u = u3 = u∗): typically the identity axioms t↼ t′ + t′↼t(t, t′ disjoint).

2.3 A few examples

The unification algebra enables one to internalise many algebraic constructions.

2.3.1 Matrixes

If I is a finite set of closed terms, the I × I matrix (λij) can be naturallyrepresented by

∑ij λij i↼ j.

2.3.2 Direct sums

The flows P := p(x)↼x,Q := q(x)↼x induce an isometric embedding of`2(T ) ⊕ `2(T ) in `2(T ): x ⊕ y 7→ [P ]x ⊕ [Q]y. The isometricity comes fromP ∗P = Q∗Q = I, P ∗Q = Q∗P = 0. The embedding is not surjective: thiswould require PP ∗ + QQ∗ = I, in other terms that every term matches eitherp(x) or q(x).

P and Q have been heavily used in the early GoI, in particular for multi-plicatives — and, modulo tensorisation with I, for contraction. They enableone to change the size of matrices in a flexible way. Usually, the only possibilityis to divide the size, typically Mmn(C) ' Mm(Mn(C)) replaces a mn × mnmatrix with a m×m matrix whose entries are n×n matrices, i.e., blocks of sizen × n. Thanks to P,Q, one can replace a 3 × 3 matrix with a 2 × 2 one (withfour A blocks B of sizes 2× 2, 2× 1, 1× 2, 1× 1).

2.3.3 Tensor products

The tensor product of two flows makes use of a binary function A · B and isdefined by (t↼ t′) ⊗ (u↼u′) := t · u↼t′ · u′; the variables of the two flowsmust first be made distinct. This corresponds to an internalisation of the tensorproduct, which plays an essential role in the handling of exponentials, i.e., ofrepetition. The flow T := (x · y) · z↼x · (y · z) is such thatT ∗((t↼ t′)⊗ (u↼u′))⊗ (v↼v′)T = (t↼ t′)⊗ ((u↼u′)⊗ (v↼v′)), thus com-pensating the want of associativity of the internal tensor.

2.3.4 Crown products

In the same style as T , the flowσ := x1 ·(x2 ·(. . . (xn−1 ·xn) . . .))↼xσ(1) ·(xσ(2) ·(. . . (xσ(n−1) ·xσ(n)) . . .)) inducesa permutation of the constituants af a n-ary tensor. In [Gir12], crown productshave been used to develop a LOGSPACE GoI.

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2.4 Stars and galaxies2.4.1 Stars

A star Jt1, . . . , tn+1K consists in n + 1 terms; these terms — the rays of thestar — should have exactly the same variables and be pairwise disjoint. Starswill be used to represent the links of proof-nets.

Stars generalise the unification algebra; thus, the axiom link t↼ t′ + t′↼tbecomes Jt, t′K. The possibility of handling a n-ary link as a single element is ofcrucial importance in the additive case. However, since our objects are no longeroperators, there are some difficulties in defining the analogue of composition.For this we shall use coloured stars.

We select pairs of complementary colours, e.g., (green, magenta) togetherwith the neutral colour black; a coloured star is a star in which each ray hasbeen given a colour: typically, Jt, u, v, wK. Pairwise disjointness is required forthe sole rays of the same colour. Coloured stars are not yet another notion,just a shorthand: indeed, consider three unary functions g,m, b and replaceJt, u, v, wK with Jg(t), g(u), b(v),m(w)K; the translation explains why pairwisedisjointness is not required between rays of different colours.

2.4.2 Galaxies

A galaxy is a finite set of coloured stars. Cut-free proofs will be represented byblack galaxies, whereas the cut-rule will make use of complementary colours.

In order to normalise a galaxy, we first form its diagrams. By this I meanany tree (in the topological acception) obtained by attaching N + 1 stars ofthe galaxy by means of N vertices. By a vertex, I mean a pair t = u ofrays of complementary colours. All diagrams can be obtained in the followingincremental way, starting with a single star: we select a free coloured ray in thediagram, i.e., a coloured ray not yet used in a vertex and form a vertex withany ray of the complementary colour occurring in the galaxy, thus aggregatinga new star to the diagram. Since a star may be used several times in a diagram,a galaxy is likely to generate infinitely many diagrams.

The unification of a diagram consists in unifying its vertices, so that tθ = uθis an actual equality. Most unifications will fail; we are basically concerned withcorrect diagrams. A correct diagram is a diagram in which unification has beensuccessfully performed.

2.4.3 Normalisation

In usual GoI, the cut-rule is handled by a partial symmetry σ; the normal formof the proof (u, σ) is given by:

(I − σ2)u(I − σu)−1(I − σ2)

Here σ corresponds to the swapping of complementary colours: σ exchangesgreen and magenta and A kills B black. Under reasonable hypotheses (nilpo-tency), u(I−σu)−1 can be written as a finite sum u+uσu+uσuσu+ . . ., whichcorresponds to the plugging of u with itself through complementary colours.The two I − σ2 correspond to the restriction to the A black stars B.

Summing up, we obtain the following conditions (strong normalisation),which generalise the nilpotency of σu:

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1. There are only finitely many correct diagrams. In other terms, for anappropriate N , all diagrams of size N + 1 fail; this finite N accounts forstrong normalisation.

2. Among the correct diagrams, none of them is closed, i.e., with a free ray.The simplest closed diagram is made of two unary stars: {JtK, JtK} withthe vertex t = t. This condition is thus reminiscent of consistency.

3. In a correct diagram, free rays of complementary colours are disjoint12.The simplest diagram thus excluded consists of a single binary star: {Jt, uK},with t, u not disjoint. This last condition is by far the most important.

The normal form is obtained by collecting the correct diagrams whose free raysare black. And to replace them with their residual star, i.e., the star whose raysare its free rays.

The condition of isometricity of GoI translates as: the rays of the samecolour of the galaxy are pairwise disjoint. It is easy to show (using condition 3.above) that the normal form of an A isometric galaxy B is still A isometric B.

3 Proof-nets, vehicles and gabarits

3.1 Multiplicative case3.1.1 Proof-nets

We assume some familiarity with multiplicative proof-nets, see, e.g., [Gir11b].It remains to get rid of syntactical decorations as much as possible so as todescribe proof-nets in a purely locative way13: in order to represent a proofof `A,B,C unary functions pA, pB , pC will be used to distinguish between thevarious locations available in the sequent; I could as well use p1, p2, p3, but thiswould compel me into a systematic renumbering of the indices.

3.1.2 Vehicles: cut-free case

To each proof π we associate its vehicle, i.e., a galaxy π•; this galaxy is black inthe cut-free case.

Identity axiom: if π is the axiom `A,∼A, then π• := {JpA(x), p∼A(x)K}.

`-rule: if the proof π of `Γ, A`B has been obtained from a proof ν of `Γ, A,B,then π• is defined as ν• in which pA and pB are now defined symbols:pA(x) := pA⊗B(l · x), pB(x) := pA⊗B(r · x). l, r are distinct closed termsand · is a binary function chosen once for all.

⊗-rule: if the proof π of `Γ, A⊗B has been obtained from proofs ν of `Γ, Aand µ of `B,∆, then π• := ν• ∪ µ•, with pA, pB defined bypA(x) := pA⊗B(l · x), pB(x) := pA⊗B(r · x).

The vehicle is thus a galaxy of axiom-links, seen as stars. The rules `,⊗ havebeen used to relocate these links. For instance, the axiom JpA(x), p∼A(x)Krelocates as JpA`(∼A⊗B)(l · x), pA`(∼A⊗B)(r · (l · x))K.

12This is stronger than A non unifiable B.13Only the locations matter: Locus Solum.

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3.1.3 Vehicles: general case

In presence of cuts, additional coloured functions will be needed. The standardsolution consists in using a pair of complementary colours, typically pB , p∼Band pB , p∼B . The interpretation π• now looks as a union V ∪ C: V (in blackand green) is the vehicle proper, C — its feedback — is easily identified as themagenta part of the vehicle.

Cut rule: if the proof π of `Γ has been obtained from proofs ν of `Γ, A and µof `∼A,∆, then π• := ν•∪µ•∪{JpA(x), p∼A(x)K}; furthermore, in ν•∪µ•,pA, p∼A have been painted green: pA(t) 7→ pA(t), p∼A(t) 7→ p∼A(t).

It may be useful to use a specific pair of colours for each cut. Then a vehichlewith two cuts may be written V ∪ C ∪ C. As to normalisation, there are threepossibilities:

1. Identify green = blue, magenta = yellow and normalise.

2. Normalise the cuts blue/yellow, then the residual cuts green/magenta.

3. Normalise the cuts green/magenta, then the residual cuts blue/yellow

The Church-Rosser property will equate (in any possible sense) these threepossibilities. From this, it will be possible to show the compositionality of cut,hence to develop various functional, i.e., category-theoretic, interpretations.

The Church-Rosser property also says that one pair of colours is enough, atleast for theoretical considerations.

3.1.4 Gabarits (I)

We must now make sense of the lower part of the proof-net, the one dealingwith the `,⊗ and Cut links. The main problem is to give a precise definitionof the switching discipline leading to the correctness condition. Indeed, to eachswitch, we shall associate an ordeal, i.e., a coloured galaxy. The finite set of allthese ordeals is called the gabarit.

We already defined the unary functions pA(x) for each formula and subfor-mula of the proof-net. We now introduce qA(x) := pA(g · x), where g is yetanother constant. The replacement of pA with qA in the context of gabarits isdue to the fact that pA⊗B(x) is not disjoint from pA(x) := pA(l · x), whereasqA⊗B(x) and qA(x) are disjoint. In other terms, the qA provide disjoint locationsfor the formulas occurring in the lower part of the proof-net.

Given a proof-net of conclusions Γ, a switch L/R of its `-links induces anordeal, namely the coloured galaxy made of the following stars:

⊗: if A⊗B 6∈ Γ, JqA⊗B(x), qA(x), qB(x)K; if A⊗B ∈ Γ, JqA⊗B(x), qA(x), qB(x)K.

`L: if A`B 6∈ Γ, JqA`B(x), qA(x)K and JqB(x)K; if A`B ∈ Γ, JqA`B(x), qA(x)Kand JqB(x)K. In terms of graphs, JqB(x)K A terminates B all JqB(t)K.

`R: if A`B 6∈ Γ, JqA`B(x), qB(x)K and JqB(x)K; if A`B ∈ Γ, JqA`B(x), qB(x)Kand JqA(x)K.

Cut: JqA(x), q∼A(x)K.

11

Let Vq := V[g·x], and let Vq be Vq painted green; let us select an ordeal S. Thenthe galaxy Vq∪S strongly normalises and its normal form is {JqA(x) ; A ∈ ΓK}.This is the correctness criterion for proof-nets, obviously necessary, indeed suf-ficient.

3.1.5 Gabarits (II)

Let us turn our attention towards an exotic multiplicative, namely the A linearaffine B implication A _ B. A _ B yields a purely multiplicative second-orderreduction of additives: A⊕B := ∀X((A−◦X) _ ((B −◦X) _ X))14.

Indeed, A _ B is an intuitionistic implication without reuse of premises;this is why it interests us. The associated disjunction A n B := ∼A _ B isproblematic in terms of gabarits. Indeed, the n-link:

[A] B

AnB

is problematic: the premise A (written [A] for this reason) might be absent,hence the switch A L B is hazardous, since it may destroy everything in thatcase. On the other hand, we cannot content ourselves with the sole A R B, hencethe idea of a virtual switch, i.e., a way of replacing the missing switch withsomething else.

Virtual switches are inspired from the proof by de Naurois and Mogbil[dNM11] of the NL complexity of multiplicative proof-nets; improving the ideaof contractibility introduced by Danos [Dan90], the authors show that it isenough to switch ` on one side, e.g., always A R B. In our case, the virtualswitch remains weaker than the actual switching A L B of the `15; this is whyI made up this marginal implication as an example where virtual switches domatter.

A virtual switch is a set Jt;u1, . . . , unK (or Jt;u1, . . . , unK), with a distin-guished root (t or t). u1, . . . , un must be pairwise disjoint; each variable occuringin t (or t) must still occur in the ui.

Our notion of ordeal is modified as follows: besides its stars, the ordealcontains a galaxy of virtual switches. Typically, in the case of A n B, besidesJqAnB(x), qB(x)K (or JqAnB(x), qB(x)K) and JqA(x)K we shall add the virtualswitch JqAnB(x); qA(x)K (or JqAnB(x); qA(x)K).

Some condition as to the various virtual switches must be asked: considerthe unique correct diagram in Vq ∪ S, and let us unify it, so as to get a galaxyG. For each virtual switch Jt;u1, . . . , unK (or Jt;u1, . . . , unK, say Jt;u1, . . . , unK),consider all rays obtained by unification from some ui; since uiθ = uiθ

′ impliestθ = tθ′, each such ray A comes from B a specific instanciation of t, its A root B.We require that:

1. If uiθ ∈ G, then its root tθ occurs in G.

2. uθ is A upwards connected B to tθ, i.e., the connection does not transitthrough the vertex t1θ1 = tθ. In the case of a virtual switch Jt;u1, . . . , unK,this condition is empty.

14Instead of ∀X((A−◦X)⇒ ((B −◦X)⇒ X)).15The actual presence of the premise A is dubious; what is not the case in [dNM11].

12

For each tθ (or tθ, say tθ) in G), we can consider the set Gtθ of all rays standing inbetween tθ and some uiθ′ with root tθ (i.e., s.t. tθ = tθ′) including extremities;Gtθ is this a sort of tree, rooted in tθ. We then require that:

3. If Gt1θ1 ∩ Gt2θ2 6= ∅, then either Gt1θ1 ⊂ Gt2θ2 or Gt2θ2 ⊂ Gt1θ1 .

Virtual switches are well-adapted to weakening, since they cope with uncer-tainty as to the presence of a specific premise. Moreover, since ui may containvariables not in t, there is no limitation as to the number of uiθ′ rooted in agiven tθ: the extra variables thus account for contraction.

3.1.6 Normalisation

The normalisation of a proof-net is indeed the normalisation of the union V ∪ Cof its vehicle and its cuts, namely the galaxyC := {JpB(x), p∼B(x)K}, B ranging over all cut formulas of the proof-net.

3.2 Exponentials3.2.1 The problem

The technology of virtual switches allows one to solve the major pending ques-tions related to exponentials. However, we are still unable to cope with expres-sions like ?A ⊗ ?B (dually, !A ` !B). Indeed, we shall restrict to intuitionisticimplication: A⇒ B := !A−◦B, i.e., limit the use of exponentials to the config-urations ?A`B and !A⊗B, where B does not begin with ?, !.

This restriction can be implemented by saying that ?A, !A are sort of pre-formulas than can only be used in sequents. Sequents will assume the forms:`Γ, ?∆ (Γ 6= ∅) or `Γ, ?∆, !A. A more ambitious setting would distinguishthree polarities, namely standard, perennial, co-perennial, so as to allow a lessspartan socialisation for exponentials, typically !A ⊕ !B16, together with theappropriate notion of proof-net in the spirit of Laurent’s polarised nets [Lau03].In this paper, we content ourselves with a solution a minima.

What is definitely missing is the neutral element 1, treated as a formula:!A⊗1 would then be a standard formula and behave like the usual exponential.1 is definable in second order logic as ∀X(X−◦X); hence if we are not too pickywith the subformula property, i.e., if we are only epidictic, we shall gain accessto the standard exponential.

3.2.2 Proof-nets

The rules of sequent calculus are the usual ones. As to proof-nets, the rules ofnatural deduction:

[A]····B

A⇒ B

····A

····A⇒ B

B

conveniently topsy-turvied, become the two exponential links, A ?` B and A !⊗ B:16A ∨B := !A⊕ !B is the linear translation of intuitionistic implication.

13

[A] B?`

?A`BA B

!⊗!A⊗B

As usual, [A] denotes an unspecified number of A occurrences B of A.

3.2.3 Vehicles (I)

pA, pB are defined from p?A`B (or p!A⊗B) in the familiar way. The noveltyis that pA(x) will be replaced with pA(x · y); the extra variable y copes withrepetitions, like the tensorisation with I in the first GoI [Gir89].

⊗-rule: if the proof π of ` ? Γ, !A ⊗ B,∆ has been obtained from proofs ν of` ? Γ, A and µ of `B,∆, then we modify ν• into ν◦:pA(t) 7→ pA(t·y), pB(t·u) 7→ pB(t·(u·y)) (B ∈ Γ), pC(t) 7→ pC(t · y). Thendefine π• := ν◦ ∪ µ•, with pA(x) := p!A⊗B(l · x), pB(x) := p!A⊗B(r · x).

Weakening: if the proof π of `Γ, ?A has been obtained from the proof ν of`Γ, then π• := ν•.

Derelection: if the proof π of `Γ, ?A has been obtained from the proof ν of`Γ, A, then π• is obtained from ν• by means of the replacementpA(t) 7→ pA(t · d), where d is a constant.

Contraction: if the proof π of `Γ, ?A has been obtained from the proof ν of`Γ, ?A′, ?A”, then π• is obtained from ν• by means of the replacementpA′(t · u) 7→ pA(t · (l · u)), pA”(t · u) 7→ pA(t · (r · u)). Contraction thusmerges various A occurrences B of ?A, kept disjoint by adequate choicesof the components u in the pA(x · u). Due to the ⊗-rule, u may involvevariables y1, . . . , yn, where n is the nesting of ?A in terms of boxes.

?`: if the proof π of `Γ, ?A`B has been obtained from the proof ν of `Γ, ?A,B,define π• := ν•, with pA(x) := p?A`B(l · x), pB(x) := p?A`B(r · x).

3.2.4 Vehicles (II)

We should first have a synthetic look at the various pA occuring in a proof-net.In the multiplicative case, the pA were defined unary functions:pA(x) := pA⊗B(l · x), pB(x) := pA⊗B(r · x), etc.

Formulas in a net now receive a depth measuring their exponential nesting :if !A⊗B (or ?A`B) is of depth n, then A is of depth n+ 1. The conclusionsof a net, including the cuts, are at depth 0.

A formula A of depth n induces a n+1-ary defined function pA(x; yn, . . . , y1);the yi stand for possible duplicating instructions occuring in the various depths,the deepmost being y1. To the conclusions C ∈ Γ we associate unary func-tions pC(x), the same with cut formulas. Given any C and its associatedpC(x; yn, . . . , y1, we define functions for the immediate subformulas of C.

C = A⊗B /A`B : pA(x; yn, . . . , y1) := pC(l · x; yn, . . . , y1) andpB(x; yn, . . . , y1) := pC(r · x; yn, . . . , y1).

C = !A⊗B / ?A`B : pA(x; yn+1, yn, . . . , y1) := pC(l · (x · yn+1); yn, . . . , y1)and pB(x; yn, . . . , y1) := pC(r · x; yn, . . . , y1).

14

The pA represent the subformulas of the conclusions, including cut formulas.Now, due to contraction, several A occurrences B of the same A will occur. Thevehicle thus appears as as made of stars JpA(x; tm, . . . , t1), pB(x;un, . . . , u1)K:they correspond to axiom links whose precise location is given by the tj , uj ,following the analytical description given above. Of course, those pA standingabove a cut must be painted green.

3.2.5 Gabarits

We define qA(x; yn, . . . , y1) := pA(g · x; yn, . . . , y1); the qA are pairwise disjoint.The ordeal consists in the following stars, depending upon a switching L/R

of the links ` and !⊗; the switch A R B of !⊗ severs the implicit exponential boxabove A which will thus be destroyed.

⊗: if A⊗B 6∈ Γ, JqA⊗B(x; yn, . . . , y1), qA(x; yn, . . . , y1), qB(x; yn, . . . , y1)K;if A⊗B ∈ Γ, JqA⊗B(x), qA(x), qB(x)K.

`L: ifA`B 6∈ Γ, JqA`B(x; yn, . . . , y1), qA(x; yn, . . . , y1)K and JqB(x; yn, . . . , y1)K;if A`B ∈ Γ, JqA`B(x), qA(x)K and JqB(x)K.

`R: ifA`B 6∈ Γ, JqA`B(x; yn, . . . , y1), qB(x; yn, . . . , y1)K and JqA(x; yn, . . . , y1)K;if A`B ∈ Γ, JqA`B(x), qB(x)K and JqA(x)K.

Cut: JqA(x), q∼A(x)K (cuts are at depth 0).

!⊗L: if !A⊗B 6∈ Γ, Jq!A⊗B(x; yn, . . . , y1), qA(x;x, yn, . . . , y1), qB(x; yn, . . . , y1)K;if !A⊗B ∈ Γ, Jq!A⊗B(x), qA(x;x), qB(x)K.

!⊗R: if !A⊗B 6∈ Γ, Jq!A⊗B(x; yn, . . . , y1), qB(x; yn, . . . , y1)K;if !A ⊗ B ∈ Γ, Jq!A⊗B(x), qB(x)K. Notice the disparition of qA, whichamounts at a physical destruction of everything linked to A (its A box B).

?`: if ?A`B 6∈ Γ, Jq?A`B(x; yn, . . . , y1), qB(x; yn, . . . , y1)K and JqA(x; yn+1, yn, . . . , y1)K;if ?A`B ∈ Γ, Jq?A`B(x), qB(x)K and JqA(x; y1)K.

A ?` B involves no switching: it corresponds to a A ` B switched on its only re-liable premise, B. By the way, there is no reasonable way to link the conclusionwith the premises [A]. On the other hand, JqA(x, ; yn+1, yn, . . . , y1)K A termi-nates B all [A] independently of their number and their locations. Due to theabsence of right switches, virtual switches must be used:

The virtual switches are thus the following:

• JqA`B(x; yn, . . . , y1); qB(x; yn, . . . , y1)K (or JpA`B(x), qB(x)K).

• Jq?A(x; yn, . . . , y1; qA(x; yn+1, yn, . . . , y1)K or Jq?A(x); qA(x, y1)K.

By the way, observe the loss of linearity involved with exponentials: the leftswitch for !⊗ involves the repetition (= contraction) of the variable x inqA(x;x, yn, . . . , y1), the first one so far; the virtual (= left) switch for ?` involvesthe disparition (= weakening) of the variable yn+1 of qA(x; yn+1, yn, . . . , y1).

15

3.2.6 Open questions

Here the questions, starting with the stability of correctness under normalisa-tion, sequentialisation, are less obvious than in the multiplicative case.

What about light logics? The only intuition so far comes from [Gir11a]which makes use of the hyperfinite factor, an algebra with no A atoms B, henceno dereliction. Without coming back to vN algebras, is it possible to makereasonable restrictions on the rays so as to achieve similar phenomenons, e.g.,the refusal of dereliction for A want of atoms B?

3.3 Additives3.3.1 The problem

The delicate question is that of superposition of contexts: the &-rule of sequentcalculus calls for a mingling of a proof of `Γ, A and a proof of Γ, B. The originalA solution B in terms of boxes is a typical example of cooking recipe, for onemust devise various commutative conversions for cuts involving a A side door Bof the box. Much better is my solution [Gir96] in terms of weighted proof-nets,whose counterpart is the dialectal GoI of [Gir95]. Hughes and van Glabbeek[HvG03] proposed an alternative solution based on slices; the exponential sizeof such proofs is problematic (see section 1.2.2), but the real problem is thattheir criterion is too extraterritorial in the light of A questions B.

I will propose here a non dialectal approach: no boxes, no weights, no dialect.Indeed, each axiom link occurring in `Γ, A (resp. `Γ, B) will be tethered toA (resp. B), so that we shall be able to tell the original A slices B. The onlydrawback is that normalisation will produce aberrant axiom links tethered toboth A and B. A sort of garbage collection must thus be designed to erase thoselinks.

3.3.2 Tethers

The tethers [ and ] are purely locative variants of ? and !. Sequents will nowassume one of the forms: `Γ, ?∆, [Λ (Γ 6= ∅), `Γ, ?∆, [Λ, !A or ` [Λ, ]A

`A,∼A, [C1, . . . , [Ck ` ]A, [C1, . . . , [Ck

The [Bi replace the weights of familiar additive proof-nets; ]A is used to weightthe premise A of A⊕B. The tethers [B will usually be repeted many times, withdistinct locations; such repetitions are created by the normalisation procedure.

The only structural rule for tethers is contraction:

`Γ, [A, [A(C)

`Γ, [A

Moreover, in binary rules, the tethered contexts must be the same:

`Γ, [Λ, A ∆, [Λ, B

`Γ,∆, [Λ, A⊗B

`Γ, [Λ, A ∆, [Λ, B

`Γ,∆, [Λ, !A⊗B

The sequent rules for additives now write as:

16

` [Λ,]A `Γ, [Λ,A

`Γ, [Λ,A⊕B` [Λ,]B `Γ, [Λ,B

`Γ, [Λ,A⊕B

`Γ, [Λ,[A,A `Γ, [Λ,[B,B

`Γ, [Λ,A&B

3.3.3 Proof-nets

Besides the axioms, the additive links have four premises:

A ∼A [C1 . . . [Ck A [C1 . . . [Ck

]A A ]B B

A⊕B[A A [B B

A&B

However, in the ⊕-link, it is possible to use only two premises, ]A,A or ]B,B.

3.3.4 Vehicles, analytic version

Identity axiom: interpreted by {JpA(x), pA(∼x), p[C1(x · d), . . . , p[Ck(x · d)K}.

Tether axiom: interpreted by {Jp]A(x; y), p[C1(x; y), . . . , p[Ck(x; y)K}.

&-rule: if the proof π of `Γ, [Λ, A & B has been obtained from proofs ν of`Γ, [A,A and µ of `Γ, [Λ, [B,B, then π• := ν• ∪ µ•, with:pA(x) := p[A(x) := pA&B(l′ · x), pA&B(l · x),p[B(x) := pA&B(r′ · x), pB(x) := pA&B(r · x).

⊕L-rule: if the proof π of `Γ, [C1, . . . , [Ck, A ⊕ B has been obtained fromproofs ν of ` [Λ, ]A and µ of `Γ, [Λ, A and, then π• := ν• ∪ µ•, with:p]A(x) := pA⊕B(l′ · x), pA(x) := pA⊕B(l · x).

⊕R-rule: symmetrical.

As to tethered axioms, the choice p[Ci(x ·d) is quite arbitrary; p[Ci

(ti ·ui) woulddo the same, provided ti · ui has exactly one variable, x. In particular, throughnormalisation, the tethered axioms will assume this general form.

3.3.5 Gabarits

A switching L/R of all with &-links must be selected, as well as a switchingLL/LR/RL/RR of the ⊕-links; in that case we content ourselves To each ordealwe add the following stars:

⊕LL: if A⊕B 6∈ Γ, JqA⊕B(x; yn, . . . , y1), q]A(x;x, yn, . . . , y1), qA(x; yn, . . . , y1)Kand JqA⊕B(x; yn, . . . , y1), q]B(x;x, yn, . . . , y1), qB(x; yn, . . . , y1)K;if A⊕B ∈ Γ, JqA⊕B(x), q]A(x;x), qA(x)K and JqA⊕B(x), q]B(x;x), qB(x)K.

⊕LR: if A⊕B 6∈ Γ, JqA⊕B(x; yn, . . . , y1), q]A(x;x, yn, . . . , y1), qA(x; yn, . . . , y1)Kand JqA⊕B(x; yn, . . . , y1), qB(x; yn, . . . , y1)K;if A⊕B ∈ Γ, JqA⊕B(x), q]A(x;x), qA(x)K and JqA⊕B(x), qB(x)K.

17

⊕RL: if A⊕B 6∈ Γ, JqA⊕B(x; yn, . . . , y1), qA(x; yn, . . . , y1)Kand JqA⊕B(x; yn, . . . , y1), q]B(x;x, yn, . . . , y1), qB(x; yn, . . . , y1)K;if A⊕B ∈ Γ, JqA⊕B(x), qA(x)K and JqA⊕B(x), q]B(x;x), qB(x)K.

⊕RR: if A⊕B 6∈ Γ, JqA⊕B(x; yn, . . . , y1), qA(x; yn, . . . , y1)Kand JqA⊕B(x; yn, . . . , y1), qB(x; yn, . . . , y1)K;if A⊕B ∈ Γ, JqA⊕B(x), qA(x)K and JqA⊕B(x), qB(x)K.

&L: if A&B 6∈ Γ, {qA&B(x; yn, . . . , y1), qA(x; yn, . . . , y1)}and {q[A(x; yn+1, yn, . . . , y1)}; ifA&B ∈ Γ, {qA&B(x), qA(x)} and {q[A(x; y1)}.

&R: if A&B 6∈ Γ, {qA&B(x; yn, . . . , y1), qB(x; yn, . . . , y1)}and {q[B(x; yn+1, yn, . . . , y1)}; ifA&B ∈ Γ, {qA&B(x), qB(x)} and {q[B(x; y1)}.

Together with the virtual switches {qA&B(x; yn, . . . , y1); q[A(x; yn+1, yn, . . . , y1)}(or {qA&B(x); q[A(x; y1)}) in the case of &L, {qA&B(x; yn, . . . , y1); q[B(x; yn+1, yn, . . . , y1)}(or {qA&B(x); q[B(x; y1)}) in the case of &R.

3.3.6 Open questions

Additives modify the general pattern in several ways:

• We are no longer isometric. Typically the vehicles involved in the &-ruleinvolve a real superposition of the context. In the same way, the ordealsof the ⊕ superpose two choices.

• When normalising, this superposition will enable one to mingle the variousslices, with a completely hazardous result.

A theory of normalisation, far more interesting than the sole sequentialisation,should be developed in that case. A central notion would be that of incarnation:a link in the vehicle is incarnated when it occurs in some correct diagram w.r.t.some ordeal. Links involving both tethers [A and [B of A & B are thus notincarnated. The problem therefore reduces at finding the way of restricting nor-malisation to the incarnation of the normal form, without making it a cookingrecipe: they have been so far avoided!

3.4 Additive neutrals3.4.1 The problem

The additive neutral >>> is extremely delicate to treat, but only because of thecategory-theoretic diktats which force identifications to the point that one can-not even write its rules. However, there is an easy solution, namely use thesecond order definition >>> := ∃XX. Second order definitions should be avoidedwhen possible, but this is not a dogma, only a matter of common sense: secondorder quantification may hide cuts. For instance, replace the cut:

`Γ, A `∼A,∆

`Γ,∆

with

18

`Γ, A `∼A,∆

`Γ, A⊗∼A,∆

`Γ,∃X(X ⊗∼X),∆

So that the formula ∃X(X ⊗∼X) A swallows B the cut. This is why the secondorder translation ⊥⊥⊥ := ∃X(X ⊗ ∼X),1 := ∀X(X ⇒ X) of the multiplicativeneutrals is fishy.

This absorbtion is made possible because X occurs both positively and neg-atively in ∃X(X ⊗∼X). But this is not the case for ∃XX. I thus propose:

`Γ, A

`Γ,>>>

3.4.2 Proof-nets

There is not much to object to the link:

A

>>>

except a local loss of the subformula property.This local loss is in turn easy to fix: replace A (a formula) with the gabarit

for A, now seen as part of the vehicle. A gabarit being a combinatoric, logic-freeartifact whose only aim is to connect>>> in a nice way with its context, there is nolonger any mention of the formula A! The bad news is that the gabarit expellsus from the category-theoretic paradise in which >>>, as a terminal element musthave a unique proof: indeed, the category-theoretic quotient basically consistsin forgetting this gabarit.

What we propose is not a paradise — i.e., a sort a perfect jail —, it is aspace of freedom in which certain foundational discussions may be reopened,typically the aporia of Π0

1 formulas. Taking the gabarits into account, a proofof ∀n>>> can become extremely complex! But this is the topic of another paper.

NON SI NON LA

19

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