Completeness for non-distributive propositional case
5.1. INTERPRETATION DUALIZATION 81 Inductive step for the co-satisfaction of ← :
yψ←ϕ i y≥V(ψ←ϕ) Then we have: yψ←ϕ i y≥V(ψ←ϕ) i y≥W {x|x∈J∞(F+)andx≤V(ψ←ϕ)} i ∀x[x≤V(ψ←ϕ)⇒x≤y] i ∀x[xψ←ϕ⇒x≤y]
Where xψ ← ϕin the last line can be replaced by the associated satisfaction conditions as established
above.
Given the previous unraveling of satisfaction conditions, for any given interpretation V : AtProp −→ G((X, Y,≤))into the complex algebraF+, and with
M= (F, V)andF= ((X, Y,≤), R, R♦, R., R/, R◦, R→, R←),
we can dene the associated two-sorted relational semantics by induction as follows: for x∈X, y ∈Y and p∈AtPropwe let M, xp i x≤V(p) M, yp i y≥V (p)
. Forϕ, ψ∈F orm(AtProp)
(5.1.11) M, yV ♦ψ i ∀x0[x0V ψ ⇒ yR♦x0] (5.1.12) M, xV ♦ψ i ∀y[yV ♦ψ ⇒ x≤y]. (5.1.13) M, xV ψ i ∀y[yV ψ ⇒ xRy]. (5.1.14) M, yV ψ i ∀x[xV ψ⇒ x≤y]. (5.1.15) M, yV Cψ i ∀y0[y0 ψ ⇒ yRCy0]. (5.1.16) M, xV Cψ i ∀y[yV Cψ ⇒ x≤y]. (5.1.17) M, xV Bψ i ∀x0[x0V ψ ⇒ xRBx0]. (5.1.18) M, yV Bψ i ∀x[xV Bψ⇒ x≤y]. (5.1.19) M, yV ϕ◦ψ i ∀x1, x2[(x1ϕandx2ψ)⇒R◦(y, x1, x2)]
5.2. PROPOSITIONAL SUBSTRUCTURAL LOGIC COMPLETENESS ON NON-DISTRIBUTIVE SETTING 82 (5.1.20) M, xV ϕ◦ψ i ∀y[yV ϕ◦ψ⇒x≤y] (5.1.21) M, xV ϕ→ψ i ∀x0, y[[x0V ϕandyV ψ]⇒R→(x0, x, y)] (5.1.22) M, yV ϕ→ψ i ∀x[xV ϕ→ψ⇒x≤y] (5.1.23) M, xV ψ←ϕ i ∀x0, y[[x0V ϕandyV ψ]⇒R←(x, x0, y)] (5.1.24) M, yV ψ←ϕ i ∀x[xV ψ←ϕ⇒x≤y]
We now proceed to build the canonical model for non-distributive logic with modal/substructural signature type.
5.2. Propositional substructural logic completeness on non-distributive setting A model is a pairM= (F, V)whereFis a frame,V :AtProp−→ G((X, Y,≤))is an interpretation.
Our canonical model will be based on an RS-polarity (X, Y,≤) where the elements of X will be certain
theories Σ and the elements of Y will be certain co-theories ∆ (see claim 115 below). For every theory Σ∈X and every co-theory∆∈Y we setΣ≤∆ i Σ`∆. The truth lemma now will have to treat both
satisfaction and co-satisfaction relations, and reducing them to the set-theoretic belong-to relation, in short
(c,c) =truth lemma(3,3).
Definition 112. Let Mc = (Fc, Vc)be the canonical model for non-distributive substructural modal logic
(SML henceforth), based on the canonical frameFc = (X, Y,≤), Rc, Rc♦, Rc., Rc/, Rc◦, Rc→, Rc←
, where: • X is the set of all optimal theories, i.e. X ={Σ| hΣ,∆iis a maximal lter-ideal pair for some∆}; • Y is the set of all optimal co-theories, i.e. Y ={∆| hΣ,∆iis a maximal lter-ideal for some Σ}; • ≤⊆X×Y, is s.t. Σ≤∆ i Σ∩∆6=∅;
• The canonical relationsRc
,R
c
♦ ,R
c
. ,Rc/ andRc◦ are dened as follows:
Rc ♦⊆Y ×X, s.t. ∆Rc♦Σ i ♦[Σ]∩∆6=∅; Rc ⊆X×Y, s.t. ΣR c ∆ i [∆]∩Σ6=∅; RcC⊆Y ×Y, s.t. ∆RcC∆0 i /[∆0]∩∆6=∅; Rc B⊆X×X, s.t. ΣRcBΣ0 i .[Σ0]∩Σ6=∅; Rc◦⊆Y ×X×X, s.t. ∆R◦cΣ,Σ0 i Σ◦Σ0∩∆6=∅, withΣ◦Σ0={ϕ◦ψ|ϕ∈Σ &ψ∈Σ0}
• The canonical valuationVc:AtProp→
F+ is s.t.Vc(p) :=W{Σ|Σ∈X andp∈Σ}=V{∆|∆∈
Y andp∈∆}. Claim 113. The equalityW
{Σ|Σ∈X andp∈Σ}=V
5.2. PROPOSITIONAL SUBSTRUCTURAL LOGIC COMPLETENESS ON NON-DISTRIBUTIVE SETTING 83 Proof. LetA={Σ|Σ∈X andp∈Σ}andB ={∆|∆∈Y andp∈∆}. Clearly, for anyΣ∈A and
any∆∈Bwe haveΣ∩∆6=∅withpas witness. SoΣ≤∆by denition. Thus,Au=B andBl=A. Hence WAis the least element in B andVB is the greatest element inA. ThereforeWA=VB, as desired.
Remark 114. Our denitions closely follow [Gehrke 2006] work. We have swapped Y coordinate onRc◦
to look more alike the usual approach with diamond like interpretation. Likewise we omit Rc
→ and Rc←
denitions as they are simply swappings ofRc
◦. We refer the reader to [Gehrke 2006]:267-68 for the proofs
that the canonical frame is of the right kind (based on a polarity which is anRS-frame) and whose relations
are compatible (this is proven forRc
◦but it is straightforward to see it carries over to unary modal relations).
Claim 115. GivenhΣ,∆ia maximal pair, thenΣis a theory and∆ a co-theory.
Proof. To show that Σis a theory amounts to prove it is closed under derivability. So suppose that Σ`ϕand assume towards a contradiction thatϕ /∈Σ. ThenΣ0 := Σ∪ {ϕ} is a proper extension ofΣ. Now
observe that Σ`ϕ impliesC(Σ) = C(Σ0)-they have the same set of consequences-. Then Σ0∆ implies
Σ0 0∆. But this contradicts the assumption thathΣ,∆iis maximal. Thereforeϕ∈Σ. The proof that∆ is
a co-theory is order-dual.
The following lemma will need later on to create witness points. An optimal theory is a generalization of the notion of prime theory (cf. denition 39), and simply states that the theory fullls the denition of lter for posets (denition 77). Co-theory is the dual generalization.
Lemma 116. IfΣis an optimal theory and ∆ is an optimal cotheory then
• −1[Σ] is an optimal theory and
• .−1[Σ] an optimal cotheory.
• ♦−1[∆]is an optimal cotheory and
• /−1[∆]is an optimal theory.
• ∆1={χ0|χ0◦ψ∈∆} is an ideal (an optimal cotheory).
• ∆2={ψ0|∃χ0(χ0∈Σ &χ0◦ψ0∈∆)} is an ideal (an optimal cotheory).
Remark 117. To prove such lemma we will need to consider the residual operations for all the unary modal operations -fusion already has it own residuals-. So let ,,J,I be the residual operations of ,♦, ., /
respectively. As [Gehrke 2006] points out, although,♦, ., /are not generally stipulated to be residuated,
they become so in the canonical extension.
Proof. Suppose Σis a theory. Then it is closed under derivability, i.e. ifΣ`ϕthenϕ∈Σ, and it is
downdirected.
We show−1[Σ] is closed under derivability. So suppose−1[Σ]`ψ. Then there areϕ1, . . . , ϕn∈−1[Σ]
such that Vn
i=1
ϕi ` ψ. But then n
V
i=1
ϕi ` ψ and thus n
V
i=1
ϕi ` ψ since preserves meets. From ϕ1, . . . , ϕn ∈ −1[Σ] it immediately follows that ϕ1, . . . ,ϕn ∈ Σ and hence Σ ` ψ , with Σ being a
theory. Thenψ∈Σand thusψ∈−1[Σ] . Sinceψwas an arbitrary formula, we proved that
−1[Σ] is
closed under derivability, i.e. it is a theory. It is downdirected: Suppose ϕ, ψ∈−1[Σ], then
ϕ,ψ∈Σ
and since Σ is downdirected, then there is some β ∈ Σ with
β`ϕ β`ψ . Therefore β`ϕ β`ψ with β ∈−1[Σ], as
=−1. Sinceϕ, ψ were arbitrary, this shows that−1[Σ]is downdirected.
We show.−1[Σ]is closed under inverse of derivability. Now supposeψ`.−1[Σ]. Then there areϕ
1, . . . , ϕn ∈ .−1[Σ] such that ψ ` Wn
i=1
ϕi. But then . n
W
i=1
ϕi ` .ψ and thus n
V
i=1
5.2. PROPOSITIONAL SUBSTRUCTURAL LOGIC COMPLETENESS ON NON-DISTRIBUTIVE SETTING 84 From ϕ1, . . . , ϕn ∈ .−1[Σ] it immediately follows that .ϕ1, . . . , .ϕn ∈ Σ and henceΣ ` .ψ, with Σbeing
a theory. Then .ψ ∈ Σand thus ψ ∈ .−1[Σ] . Since ψ was an arbitrary formula, we proved that .−1[Σ]
is closed under inverse of derivability, i.e. it is a cotheory. It is updirected: Suppose ϕ, ψ ∈ .−1[Σ], then
.ϕ, .ψ∈Σand sinceΣis downdirected, then there is someβ ∈Σwith
β`.ϕ β`.ψ . Therefore ϕ`Jβ ψ`Jβ withJβ∈.−1[Σ], as
J=.−1. Sinceϕ, ψ were arbitrary, this shows that.−1[Σ]is updirected.
Suppose∆ is a co-theory. Then it is closed under inverse of derivability, i.e. if ϕ`∆ thenϕ∈∆ and it is
updirected.
So suppose ψ` ♦−1[∆]. Then there are ϕ
1, . . . , ϕn ∈♦−1[∆]such thatψ`
n
W
i=1
ϕi. But then ♦ψ `♦
n W i=1 ϕi and thus ♦ψ ` n W i=1
♦ϕi since ♦ preserves joins. From ϕ1, . . . , ϕn ∈ ♦−1[∆] it immediately follows that
♦ϕ1, . . . ,♦ϕn ∈∆ and hence♦ψ ` ∆ , with ∆ being a co-theory. Then ♦ψ ∈ ∆ and thus ψ ∈♦−1[∆] .
Since ψwas an arbitrary formula, we proved that ♦−1[∆]is closed under inverse of derivability, i.e. it is a
co-theory. It is updirected: Supposeϕ, ψ∈♦−1[∆], then
♦ϕ,♦ψ∈∆and since∆ is updirected, then there
is some β ∈ ∆ with ♦ϕ`β ♦ψ`β . Therefore ϕ`β ψ`β with β ∈♦−1[∆], as =♦−1. Sinceϕ, ψ were
arbitrary, this shows that♦−1[∆]is updirected.
Now suppose/−1[∆]`ψ. Then there areϕ
1, . . . , ϕn ∈/−1[∆]such that
n
V
i=1
ϕi `ψ. But then /ψ`/ n V i=1 ϕi and thus/ψ` n W i=1
/ ϕisince/ turns meets into joins. Fromϕ1, . . . , ϕn∈/−1[∆]it immediately follows that /ϕ1, . . . , /ϕn∈∆ and hence/ψ`∆, with ∆ being a cotheory, i.e. closed under the inverse of derivability.
Then /ψ ∈∆ and thus ψ ∈/−1[∆] . Since ψ was an arbitrary formula, we proved that /−1[∆] is closed
under derivability, i.e. it is a theory. It is downdirected: Supposeϕ, ψ∈/−1[∆], then/ϕ, /ψ∈∆ and since ∆ is updirected, then there is someβ ∈∆ with
/ϕ`β /ψ`β . Therefore Iβ `ϕ Iβ `ψ withI β ∈/−1[∆], as
I=/−1. Sinceϕ, ψ were arbitrary, this shows that/−1[∆]is downdirected.
Now we show that ∆1 ={χ0|χ0◦ψ∈∆} is an ideal (a cotheory). It is down-closed. Leta∈∆1 and b`a,
then a◦ψ∈∆ andb◦ψ `a◦ψ since fusion is order preserving on both coordinates. Thereforeb◦ψ∈∆
since ∆ is a cotheory. Then b ∈ ∆1. It is updirected: ifa, b ∈ ∆1 then there is somez ∈ ∆1 with a` z
andb`z. To see this, leta, b∈∆1 then
a◦ψ∈∆ b◦ψ∈∆ and thus a◦ψ`c b◦ψ`c for some c∈∆because∆ is
updirected. Then by residuation, a`c←ψ b`c←ψ and (c←ψ)◦ψ=c∈∆. Thus c←ψ∈∆1 andc←ψ is our witnessz.
Finally, we show∆2={ψ0|∃χ0(χ0∈Σ &χ0◦ψ0 ∈∆)} is an ideal (a cotheory). Supposeα∈∆2, then there
is some z∈Σs.t. z◦α∈∆, which is a downset. Suppose further that β ` α, thenz◦β `z◦αas fusion
is order preserving on both coordinates. Thereforez◦β ∈∆and β ∈∆2. Thus∆2 is a downset. Now we
show it is updirected. Assume ψ0, ψ” ∈ ∆2, then there exist ϕ0, ϕ” ∈ Σ with (∗)
ϕ0◦ψ0∈∆
ϕ”◦ψ”∈∆ . Since Σ is down-directed then there existsθ ∈Σ such that
θ`ϕ0 θ`ϕ” . But then (∗∗) θ◦ψ0`ϕ0◦ψ0 θ◦ψ”`ϕ”◦ψ” since