A closer look at the completeness of first order intuitionistic logic w.r.t. (pre)sheaves of classical models

A closer look at the

completeness of first order intuitionistic logic w.r.t. (pre)sheaves of classical models

Christian Retor ´e (LIRMM, Universit ´e de Montpellier) with Jacques van de Wiele (Paris) 1987,

Ivano CIardelli (M ¨unchen) 2010-2011, David Th ´eret (Montpellier) 2016-2019,

...

GDR IM — GT LHC & SCALP ENS Lyon 18 oct. 2019

Remarks

Motivation: natural language semantics, models of FO S4.

A beautiful subject — but not my main research area.

Quite difficult to know what has exactly been achieved on this question.

The presentation of Kripke-Joyal forcing and counter exam- ple is from a lecture by Jacques Van de Wiele in 1987.

Special dedication to Laurent Regnier who attended as well.

The direct completeness proof is essentially due to Ivano Ciardelli (in TACL 2011 cf. reference at the end).

It has been re-worked and extended with David Th ´eret since 2016.

Thanks to Alexis Saurin for suggesting a talk at this meeting.

Thanks to the Topos & Logic group (Abdelkader Gouaich, Jean Malgoire, Nicolas Saby, David Theret) of the Institut Montpelli ´erain Alexander Grothendieck

A Logic?

formulas

proofs ! interpretations

A.1. Formulas, proofs and models

Formulas of a given first-order logical language, sayL can be true (or not) in a givenL- structure.

An L- structure is simply a set M with an interpretation of constants inM and interpretation ofn-ary predicates asn-ary relations on M, n-ary functions symbols as n-ary functions fromMⁿ toM etc.

Soundness: what is provable is true in anyL-structure.

Completeness: what is true in everyL-structure is provable.

A.2. Morphisms of L -structures

LetMu andMv be twoL structures over the same language

— interpretation in u or in v of function symbols (e.g. ϑ) and predicates (e.g. R) are denoted with a subscriptu or v (e.g. R_u ϑu: interpretations inM_u and R_v ϑv: interpretations inM_v).

A map ρuv from |Mu| to |Mv| is said to be a morphism of L-structures when:

• For anyk-ary functionϑ symbol ofL:

∀c₁, ...,c_k∈ |Mu|

ρuv(ϑu(c₁, ...,c_k)) =ϑv(ρuv(c₁), ...,ρuv(c_k))

• For anyn-ary predicateR ofL:

∀c₁, ...,cn∈ |Mu|

if(c₁, ...,cn)∈Ru then(ρuv(c₁), ...,ρuv(c_k))∈Rv

A.3. Presheaf semantics: models

Apresheaf modelMforL is a presheaf of first-orderL−structures over a Grothendieck site(C,)(or a topological space viewed

as a poset for inclusion):

• for any objectu anL structureMu

• for any arrow f :v ,→u a morphism (cf. supra) of L structuresM(f):M_u →M_v

satisfying the following extra conditions.

SeparatenessFor any elementsa,bofM_u,

ifthere is a coveru{f_i :u_i ,→u|i ∈I}such that for alli∈I we haveM(f_i)(a) =M(f_i)(b),

thena=b.

Local character of atomsFor anyn−ary relation symbolR, for any tuple(a₁, ... ,a_n)fromM_u

if there is a cover u{f_i :u_i ,→u|i ∈I} such that ∀i ∈I one has(M(f_i)(a₁), ... ,M(f_i)(a_n))∈R_ui,

then(a₁, ... ,a_n)∈R_u.

A.4. Presheaf semantics: Kripke-Joyal forc- ing — 1/4 assignments

Given a presheaf modelM, and some openu, we inductively define for any formulaF ofL the relation uF (”meaning”:

F is true atu).

Assignment A usual, in order to define uF, we need an assignmentν inM_u the free variables ofF, and this is written uνF withν= [z₁7→c₁;· · ·;z_p7→c_p]where thez_i are the free variables inF andc_i ∈ |M_u|.

As we shall see, u ν F can be defined from v ν⁰ F⁰ with f :v ,→ u and with F⁰ having free variables among those of F (plus possibly one free variable in the ∃ and ∀ cases, but its assignment will be defined when dealing with quanti- fiers). If ν = [z₁7→c₁;· · ·;z_p 7→c_p] we naturally define ν⁰ by ν⁰= [z₁7→M(f)(c₁);· · ·;z_p7→M(f)(c_p)]whereM(f)is the re- strictionM(f):|M_u| → |M_v|.

A.5. Presheaf semantics: Kripke-Joyal forc- ing — 2/4 atoms and conjunction

• uν R(t₁, ... ,t_n)iff([t₁]_ν, ... ,[t_n]_ν)∈R_u.

• uν t₁=t₂iff [t₁]_ν = [t₂]_ν.

• u ν ⊥ iff u = /0 It is so, because the empty covering is a covering (with 0 open) of the empty open. Hence, because of the locality condition on atoms, the empty open forces all atomic formulas including⊥.

• uν ϕ∧ψ iffuν ϕ anduν ψ.

A.6. Presheaf semantics: Kripke-Joyal forc- ing — 3/4 disjunction and existential

• u ν ϕ∨ψ iff there exists a covering family {f_i :u_i ,→ u|i ∈I} such that for any i ∈I we have u_i νi ϕ or u_i νi ψ.

Alternatively,uνϕ∨ψ there exist two opensu₁,u₂with u₁∪u₂=u u_i ϕ andu₂ψ.

• uν ∃xϕ iff there exists a covering family{f_i :u_i ,→u|i∈ I}and elementsa_i∈ |M_ui|fori∈I such thatu_i νi∪[x7→ai]

ϕ for any index i.

A.7. Presheaf semantics: Kripke-Joyal forc- ing — 4/4 implication and universal

• uν ϕ→ψ iff for allf :v ,→u, ifv νv ϕ thenv νv ψ.

• uν ¬ϕ iff for allf :v ,→u, withv 6= /0, v 6νv ϕ. This is obtain from /0⊥and→cases because¬ϕ=ϕ→ ⊥.

• uν ∀xϕiff for allf :v ,→u and alla∈M_v,vνv∪[x7→a]ϕ.

A.8. Validity

A formula F is said to be valid in a topological model in a presheaf model over a topological space(X,O(X))or a pre- topology whenever

X F

i.e. F is true at the global section.

B An example

B.1. Language

Let us consider the language of ring theory:

• two constants 0,1

• two binary functions+,·

• equality as the only predicate

B.2. The (pre)sheaf of L -structures

A presheaf model over the topological space R for this lan- guage is defined by|M_u|=C(u,R) the continuous functions fromutoRwith0_u(x) =0and1_u(x) =1for allx ∈u+_u point- wise addition (f +_ug)(x) =f(x) +g(x),·_u pointwise multipli- cation(f ·ug)(x) =f(x)·g(x).

The restriction ρuv :|M_u| → |M_v| morphism, whenv ,→u is defined by: ∀f ∈C(u,R)∀x ∈v ρuv(f)(x) =f(x).

ρuv is a morphism, because:

• ρuv(0_u) =0_v,

• ρ_u

v(1_u) =1_v

• (”=” is the only predicate) ∀f,g ∈ |M_u|=C(u,R)if f =g inC(u,R)thenρ_u

v(f) =ρ_u

v(g)inC(v,R).

B.3. Locality and separateness conditions

Locality condition for atoms:

”=” is the only predicate so we just have to check that, given two elementsa andbofM_u

ifthere is a coveru{f_i :u_i ,→u|i ∈I}such that∀i∈I we have(ρ_u

ui(a)) =ρ_u

ui(b))in|M_ui|,

then a=b in |M_u|. This is true, because two functions that are equal on each open of a covering ofu are equal onu.

Separateness is exactly the locality condition for our unique predicate, i.e. the ”=” predicate, which is interpreted as ”=”.

Remark: This presheaf is a sheaf: given a coveru_i ofRand a family f_i ∈ C(u_i,R) such that any two f_j and f_k agree on u_j=u_k for allj,k there exists a uniquef inC(R,R)such that f

ui =f_i.

B.4. A remark on C (U , R ) 1/3

Given any non empty open subsetU ⊂Rthere exist - an open subsetV =]a,b[⊂U

- and a continuous function`:V 7→R such thatV 6[x7→`](x=0)∨ ¬(x =0)with`:

`: ]a,b[ 7→ R

x 7→ 0 ifx 6(a+b)/2 x 7→ x−(a+b)/2 ifx >(a+b/2)

B.5. A remark on C (U , R ) 2/3

]a,b[6[x7→`](x=0∨ ¬(x =0)).

We proceed by contradiction (the meta logic is classical).

Let us assume that]a,b[[x7→`](x =0∨ ¬(x =0)).

Then there exists open setsu₁,u₂st. u₁∪u₂=]a,b[, such that:

• u₁[x7→`u1]x =0i.e. ∀x<sub>1</sub>∈u<sub>1</sub> `(x₁) =0

• u₂[x7→`u2]¬(x=0)i.e. ∀v<sub>2</sub>⊂u<sub>2</sub>,v<sub>2</sub>6= /0 v<sub>2</sub>6[x7→`u2]`=0 i.e. ` never is constantly 0 on a (non empty) sub open v₂ ofu₂.

B.6. A remark on C (U , R ) 2/3

This is impossible because(a+b)/2must be inu₁ or inu₂.

• If(a+b)/2∈u₁then`should be constantly0on a neigh- bourhood of (a+b)/2, but it is false on the right side of (a+b)/2.

• If (a+b)/2∈u₂ then` should never be constantly0 on any sub open of u<sub>2</sub> but if (a+b)/2∈u<sub>2</sub> there are sub opens inu<sub>2</sub> on the left side of (a+b)/2 where` is con- stantly0.

B.7. A classically valid but intuitionistically non valid formula

C(R,R)validates¬∀x (x=0)∨ ¬(x =0)(*).

Indeed, according to Kripke-JoyalR¬∀x (x=0)∨ ¬(x=0) means that for every non empty open u ⊂R, u 6∀x (x = 0)∨ ¬(x=0).

Butu∀x (x=0)∨ ¬(x=0)means that for every openv ⊂u and for everyf ∈C(v,R)v [x7→f](x =0)∨ ¬(x =0).

We precisely established supra (with`) thatu6∀x (x =0)∨ ¬(x =0).

ButC(R,R)validates∀x ¬¬((x =0)∨ ¬(x=0))(**)

— because` ¬¬(C∨ ¬C)is provable for allC.

However in classical logic (*) is the negation of (**) !!!

C Completeness

C.1. Statements

Soundness:F intuitionistically provable⇒F true at any open of any topological interpretation.

Completeness: F true at a global section any topological in- terpretation⇒F intuitionistically provable.

Two lemmas:

• (functoriality) ifF[c₁, ...,c_n]true atU

thenF[c₁^V, ...,c_n^V]true at any openV ⊂U.

• (locality) The locality condition for atomic formula (cf.

above) extends to any formula: if (u_i) covers u, for all i u_xk7→c^ui

k F thenuxk7→c F.

C.2. Proof of soundness

Induction on the proof height, looking at every possible last rule, e.g. in natural deduction. Below: ∨e case.

Θ`(A∨B) A, Γ`C B, ∆`C

∨e

Θ, Γ, ∆`C

We have to show thatUΘ, Γ, ∆thenU C.

If U Θ by induction hypothesis, U A∨B. Hence, there exists a covering(U_i)such that for everyi U_i AorU_i B.

If U_i A, because U Γ we have U_i Γ (functor property), and by induction hypothesis (proof ofA, Γ`C)U_i C. Similarly, ifU_i B, thenU_i C.

So for alli U_i C and by locality lemmaUC.

C.3. Canonical model construction:

the underlying site

For a direct proof, we consider this particular ”syntactic” sheaf model.

Canonical site:

• Category: we take the Lindenbaum-Tarski algebraL – Objects: classes of provably equivalent formulasϕ.

– Arrows: ϕ≤ψ ⇐⇒ ϕ`ψ

• Grothendieck topology: ϕ{ψ_i}_i∈I whenever

∀χ

ϕ`χ iff (∀i ∈I ψi `χ)

Think of the last line asϕ=^W_iψ_i

(incorrect, because FOL formulae are finite!)

C.4. Properties of this site

The proposed site is actually a site i.e. it enjoys the three properties.

1. ϕ{ϕ};

2. ifψ `ϕ andϕ{ϕ_i|i ∈I}thenψ{ψ∧ϕ_i|i ∈I};

3. ifϕ{ϕ_i|i ∈I}

and if for eachi ∈I,ϕ_i{ψ_i,k|k∈Ki}, thenϕ{ψ_i,k|i∈I,k∈Ki}.

C.5. Canonical model construction: the presheaf

• Putt ≡_ϕ t⁰ in caseϕ `t=t⁰.

• Denote by t^ϕ the equivalence class oft modulo≡_ϕ.

Canonical presheaf:

• ModelM_ϕ:

1. Universe|M_ϕ|:

set of equivalence classest^ϕ of closed terms;

2. Function symbols: f_ϕ(~t^ϕ) =f(~t)^ϕ;

3. Relation symbols:~t^ϕ ∈R_ϕ ⇐⇒ ϕ `R(~t).

• Restriction. Ift^ψ ∈M_ψ andϕ≤ψ, putt^ψϕ=t^ϕ.

C.6. The canonical presheaf is well defined

The canonical presheaf is separated. If two elements have the same restrictions on each part of a cover, then they are equal.

The interpretation of atomic formulas is local. If an atomic formula holds on each part of a cover ofU then it holds on U.

(Ivano Ciardelli claims that the glueing of compatible ele- ments may not exists, i disagree, I think the canonical (sep- arated) presheaf is actually a sheaf).

C.7. Method for the proof of completeness

∀ψ

∀ϕ [if ϕψ then ϕ `ψ] or without much additional effort

∀ψ

∀ϕ[ϕ ψ iff ϕ`ψ] By induction on the formulaψ.

What is fun is thatsoundnessmainly usesintroductionrules whilecompletenessmainly useseliminationrules.

The method and the construction can be parametrised by a contextΓfor obtaining what is calledstrong completeness: if in every interpretationuΓentails uX for any openuthen (iff)Γ`X.

The quotient on formulas is not really needed.

Having equality is not mandatory but pleasant.

C.8. Sketch of completeness proof

Truth Lemma 1. For any formula ϕ and sentenceψ, ϕψ ⇐⇒ ϕ`ψ

Proof By induction on ψ. The two directions of each induc- tive step amount to the introduction and elimination rules for the given logical constant.

Let us look at the case of the existential quantifier.

C.9. Completeness ∃ direction ⇒

• Supposeϕ∃xψ(x).

• There is a family {ϕ_i|i ∈ I} and elements t_i^ϕi ∈M_ϕi such thatϕ_i `_[x7→tϕi

i ]ψ(x)for alli ∈I.

• Since[t] =t^ϕi for closed t atϕi, this isϕi ψ(t_i).

• By induction hypothesis amounts toϕi `ψ(t_i).

• By rule(∃i), for anyi ∈I we haveϕ_i ` ∃xψ(x).

• Since ϕ{ϕ_i|,i ∈I}, by the meaning of we have ϕ` ∃xψ(x).

C.10. Completeness ∃ direction ⇐

• Supposeϕ` ∃xψ(x).

• We must provide a covering ofϕ and local witnesses.

• For any constantc, defineϕc =ϕ∧ψ(c).

• Sinceϕc `ψ(c), by induction hypothesisϕc ψ(c).

• Since [c] =c^ϕc atϕ_c, also ϕ_c `_[x7→cϕc]ψ(x), i.e. the ele- mentc^ϕc is a witness for the existential atϕ_c.

• It remains to be seen thatϕ{ϕ_c|c a constant}.

C.11. Completeness ∃ direction ⇐, continued

• Suppose ξ is derivable from ϕ∧ψ(c) for any constant c.

• Letc^∗ be a constant that occurs neither inϕ nor inξ.

• In particular, ϕ∧ψ(c^∗)`ξ, that is, ϕ,ψ(c<sup>∗</sup>)`ξ.

• But sincec^∗occurs neither inϕ nor inξ, by the rule(∃e) we have ϕ,∃xψ(x)`ξ.

• Thus by the assumptionϕ` ∃xψ(x)we also haveϕ`ξ.

• This shows thatϕ{ϕ_c|c a constant}.

• Hence we concludeϕ ∃xψ(x).

C.12. State of the art: hard to tell

Before 1995 : survey by Makkay and Reyes in 1995.

After 1995, other work in particular by Awodey.

Direct completeness via canonical presheaf: Ivano Ciardelli.

A Canonical Model for Presheaf Semantics. Talk at Topology, Algebra and Categories in Logic (TACL) 2011, Jul 2011, Mar-

seille, France. 2011.HAL Id: inria-00618862 https://hal.inria.fr/inria- 00618862

Ongoing work with David Th ´eret in Montpellier.

C.13. Future work

Connection toΩsets of Dana Scott (roughly speaking, if i un- derstand properly: one classical model, but the truth value of P(a) varies in a Heyting algebra, like Boolean valued mod- els) ?

Can we construct a canonical sheaf and not just separated presheaf e.g. with the sheaf completion method that basi- cally simply formally adds the missing global sections? Or by imposing some additional locality condition on terms and equality?

(Pre)sheaves are particular kinds of Kripke models, conversely can any Kripke model be viewed as a pre(sheaf) with the or- der topology?

Is it possible to do so with a standard topology (instead of a pretopology / Grothendieck topology)?

Does it applies to first order S4?

Thank you for your attention.