Algunas cuestiones relevantes desde la visión del alumnado de Primaria

The logic BD2, consisting of formulas valid precisely on frames of depth at most 2, is axiomatized by the axiom p2 ∨(p2 → (¬p1 ∨p1)) which belongs to the class P4 and so the Ciabattoni-Galatos-Terui Theorem does not apply. However, in [25] a cut-free hypersequent calculus which is sound and complete with respect toBD2, is presented. This calculus is obtained by adding the rule

G|Γ1,Γ2 ⇒∆2 G|Γ2, ϕ1⇒ϕ0,∆1

(HBd2) G|Γ1 ⇒∆2 |Γ2 ⇒ϕ1 →ϕ0,∆1

to the multi-succedent hypersequent calculusHJ L0 forInt.

Proposition 5.9. An intuitionistic one-step frame(P1, P0, f)validates the rule(HBd2)

iff every maximal element of a2-chain in P1 is the root of P1.

Proof. A straightforward – but rather involved – application of the basic calculus for one-step correspondence to the inequality

l(x1∧x2, y2) &l(x2∧i(z1), i(z0)∨y1)≤l(x1, y2)⊕l(x2,(i(z1)→i(z0))∨y1),

yields

(j3∧i(i[(j1))≤i(i!(n2))∨n1andj1→n2≤n1andj2 6≤j2∧j3andj3 6≤n1)) =⇒ i≤m,

for all nomimals i,j1,j2,j3 and all co-nominalsm,n1,n2.

By the now standard argument this is equivalent to the statement that

(j3∧i(i[(j1))≤i(i!(n2))∨n1 and j1 →n2 ≤n1 and j3 6≤n1) =⇒ j2 ≤j2∧j3,

for all j1,j2,j3 and alln1,n2. This will translate to the first-order one-step frame con- dition on (P1, P0, f) that for alla1, a2, a3, b1, b2 ∈P1 if

↓a3∩f−1(↓f(↓a1))⊆f−1(P0\↑[f(P1\↑b2)c])∪P1\↑b1

and

P1\↑(↓a1\(P1\↑b2))⊆P1\↑b1 and ↓a3 6⊆P\↑b1,

then

Some basic rewriting will show that this is equivalent to for all a1, a2, a3, b1, b2 ∈P1:

(↓a3∩f−1(↓f(a1))⊆P1\(f−1(↑f(b2))∩ ↑b1)andb2 ≤a1, b1 andb1 ≤a3)) =⇒ a2≤a3,

where we have used the fact thatf(P1\↑b2)c=f(↑b2), as f is surjective, as well as the fact that ↑f(↑b2) =↑f(b2) and ↓f(↓a1) =↓f(a1).

Observe that a2 does not occur in the antecedent of the above implication. Thus we see that the above is equivalent to requiring that for all a1, a2, b1, b2 ∈ P1, such that b2 ≤b1, a2 andb1 ≤a1 if

∀c≤a1 (f(c)≤f(a2) =⇒ b1 6≤c orf(b2)6≤f(c)) (†)

then max{P1}=a1, i.e. that a1 is a root.

We claim that this is equivalent to the condition that every maximal element of a 2-chain is a root. To see this we assume first that a1 ∈P1 is a maximal element of a 2-chain i.e. that there exists an element strictly belowa1. Now assume towards a contradiction that for everya < a1there existsc≤a1such thatf(c)≤f(a) impliesa1≤candf(a)≤f(c). Then we must have that f(a) = f(a1) for all a≤a1 and thus that f(↓a1) = {f(a1)}. So as (P1, P0, f) is conservative it follows that↓a1 ={a1} in direct contradiction with the assumption that a1 had an element strictly below it. Therefore, leta2 < a1 be such that

∀c≤a1(f(c)≤f(a2) =⇒ a1 6≤c orf(a2)6≤f(c)).

We have that (†) is satisfied witha1=b1 anda2 =b2 and hence we must conclude that a1 is a root.

Conversely, suppose that there existsa1, a2, b1, b2 ∈P1 withb2 ≤b1, a2 andb1 ≤a1such that (†) obtains, but a1 is not a root.

Now assume contrary to the desired conclusion that the maximal element of every 2- chain is a root. This entails thata1 =b1 =b2 =a2, but then (†) cannot obtain for these elements in direct contradiction of our initial assumption abouta1, a2, b1 and b2.

We have thus shown that a finite conservative one-step frame (P1, P0, f) validates the rule (HBd2) iff every maximal element of a 2-chain is a root.

Now consider the logicBD3, of formulas valid on frames of depth at most 3, axiomatized by the formula

p3∨(p3 →(p2∨(p2 →(p1∨ ¬p1))). (bd3)

Consider the rule

G|Γ1,Γ3⇒∆3 G|Γ2,Γ3, ϕ1 ⇒ϕ0,∆1 G|Γ2,Γ3 ⇒∆2

(HBd3) G| Γ1⇒∆3 |Γ2,Γ3⇒ϕ1→ϕ0,∆1 |Γ3 ⇒∆2

We prove that the hypersequent calculusHLJ0+ (HBd3) is a calculus for the logicBD3 and moreover that this calculus has the bounded proof property as well as the finite model property.

Unlike in the case for BD2 we will work out the one-step correspondence manually as the algorithmic one-step correspondence becomes a bit to involved to manage.

Proposition 5.10. A one-step frame (P1, P0, f) validates the rule (HBd3) iff P1 does

not contain a 3-chain, whose maximal element is not a root.

Proof. We first note that a step frame (P1, P0, f) validates the rule HBd3 iff for all U1, U2, U3, V1, V2, V3 ∈Do(P1) and allW0, W1∈Do(P0) we have that if

U1∩U3 ⊆V3 and U2∩U3∩f−1(W1)⊆f−1(W0)∪V1 and U2∩U3 ⊆V2,

then

U1⊆V3 or U2∩U3 ⊆P1\↑f−1(W1\W0)∪V1 or U3 ⊆V2.

Evidently this is equivalent to the statement where we restrict the downsetsU1, U2 and U3 to downsets of the form↓x1,↓x2 and ↓x3.

Thus if (P1, P0, f) fails to validate the ruleHBd3 then we have x1, x2, x3 ∈ P1 as well asV1, V2, V3 ∈Do(P1) andW1, W0 ∈Do(P0) such that

↓x1∩ ↓x3⊆V3 and ↓x2∩ ↓x3∩f−1(W1)⊆f−1(W0)∪V1 and ↓x2∩ ↓x3 ⊆V2,

but

↓x16⊆V3 and ↓x2∩ ↓x36⊆P1\↑f−1(W1\W0)∪V1 and ↓x36⊆V2.

It follows that there existsy≤x2, x3 such thaty6∈V1 but for somey0 ≤y we have that f(y0) ∈ W1\W0 whence y0 must belong to ↓x2 ∩ ↓x3∩f−1(W1) and consequently we must have y0 ∈V1. We thus obtain that y0 < y. Now as ↓x2∩ ↓x3 ⊆V2 but ↓x3 6⊆V2, we have thatx36≤x2. So if y=x2 then we have thaty < x3.

We have thus shown thatP1 contains a 3-chainy0 < y < x, where x is eitherx2 orx3. If y < x2 then evidently there exists a 3-chain in P1 whose maximal element is not a root. On the other hand ifx2 =ytheny0< y < x3. We claim thatx3 is not a root. For suppose that x3 is a root. Then in particularx1≤x3 but thenV3⊆ ↓x1∩ ↓x3 =↓x1 in direct contradiction with the assumption that↓x1 6⊆V3.

Conversely, if P1 contains a 3-chain x1 < x2 < x3 such that x3 is not a root then if we let

U1 :=P1, U2 :=↓x2, U3 :=↓x3,

V1 :=f−1(↓f(x1)), V2:=↓x2, V3 :=↓x3,

W0:=∅, W1 :=↓f(x1).

we immediately see that

U1∩U3 ⊆V3 and U2∩U3∩f−1(W1)⊆f−1(W0)∪V1 and U2∩U3 ⊆V2.

However, as x3 is not a root U1 6⊆V3 and as x3 6≤x2 we have thatU3 6⊆V2. Finally if

U2∩U3⊆P1\↑f−1(W1\W0)∪V1

then sincex2∈U2∩U3 we must have thatx2belongs toP1\↑f−1(↓f(x1))∪f−1(↓f(x1)). Now as x1 < x2 and x1 ∈ f−1(↓f(x1)) it follows that x2 ∈ ↑f−1(↓f(x1)) whence we must have that x2 ∈ f−1(↓f(x1)). Hence f(x2) ≤ f(x1). It then follows from the fact that f is order preserving that f(x2) = f(x1). However as x2 6≤ x1 and (P1, P0, f) is conservative we must have that f(↓x2) 6⊆ f(↓x1). Consequently we must havex≤x2 such thatf(x)6∈f(↓x1). In particular, we must have thatx < x2 and that f(x2)(=f(x1))6=f(x). But then we have thatx2∈ ↑f−1(↓f(x)) andx2 6∈f−1(↓f(x)). Whencex26∈P1\↑f−1(↓f(x))∪f−1(↓f(x)).

Therefore lettingV₁0 :=f−1_{(↓f(x)) andW}0

1 :=↓f(x), we see that

U1∩U3⊆V3 and U2∩U3∩f−1(W10)⊆f−1(W0)∪V10 and U2∩U3⊆V2.

But

U16⊆V3 and U2∩U3 6⊆P1\↑f−1(W10\W0)∪V10 and U3 6⊆V2.

Whence (P1, P0, f) fails to validate the hypersequent rule (HBd3).

From the above proposition it is clear that Theorem 4.10 applies to finite conservative step frames validatingHBd3 whence we obtain the following:

Corollary 5.11. The hypersequent calculus HJ L0 + (HBd3) has the bounded proof property and the finite model property.

Now sinceHJ L0+(HBd3) has the finite model property, the universal classU(HBd3) of Heyting algebras validatingHBd3 will be generated by its finite subdirectly irreducible

elements. A finite Heyting algebra is subdirectly irreducible iff its dual is rooted. Con- sequently it follows from Proposition 5.10 that the universal class of Heyting algebras validating (HBd3) will be generated by the class of finite Heyting algebras the duals of which are finite rooted frames not containing a 4-chain. Since the logic BD3 has the finite model property we know that the variety of Heyting algebras validating bd3 is generated by the the class of finite subdirectly Heyting algebras validating bd3. From this we may conclude that the universal classes U(HBd3) and U(BD3) generates the same variety and thereby thatHJ L0+ (HBd3) is a hypersequent calculus for the logic

BD3.

We have thus found a calculus with the bounded proof property for a logic above P3. It of course still remains to be shown whether or not the calculus HJ L0+ (HBd3) has cut-elimination.