Download Relational semantics of the Lambek calculus

Technion - Computer Science Department - Tehnical Report CS-2012-09 - 2012
Relational semantics of the Lambek calculus
extended with classical propositional logic
Michael Kaminski
Nissim Francez
Department of Computer Science
Technion– Israel Institute of Technology
Haifa 32000
Israel
Abstract
We show that the relational semantics of the Lambek calculus,
both nonassociative and associative, is also sound and complete for
its extension with classical propositional logic. Then, using filtrations,
we obtain the finite model property for the nonassociative Lambek
calculus extended with classical propositional logic.
Keywords: Lambek calculus, relational semantics, classical propositional
logic
1
Introduction
Adding propositional connectives to the language of the Lambek calculus is
not new. A sequent calculus for the associative Lambek calculus extended
with conjunction and disjunction was introduced in [8]. In contrast with
intuitionistic and classical logic, in the above extension, conjunction and disjunction are not mutually distributive.1 Therefore, the numerical semantics
1
In fact, only the sequent (A ∧ C) ∨ (B ∧ C) → (A ∨ B) ∧ C is derivable in that calculus.
Technion - Computer Science Department - Tehnical Report CS-2012-09 - 2012
of the extended calculus in [8, Section 4],2 is not complete. A natural deduction system for the extension of the associative Lambek calculus with
conjunction and disjunction, where these connectives are mutually distributive can be found in [16].
The nonassociative Lambek calculus extended with conjunction, disjunction, negation, ⊤ and ⊥ was studied in [5, 6], where it was shown that this
extension is sound and complete with respect to distributive lattice-ordered
residuated groupoids augmented with boolean negation It was also shown there
that this algebraic semantics possesses the finite model property.3 The proof
is based on the proof of the finite embeddability property that, as was shown
in [5, 6], holds for several related systems as well. Thus, the extension of
the nonassociative Lambek calculus with classical propositional logic is decidable.
In this paper we also study the extension of both nonassociative and
associative Lambek calculi with classical propositional logic, presented as a
Hilbert-style system,4 but from a different perspective. We show that the
relational semantics (see [7]) is sound and complete for these extensions and
that the latter are conservative extensions of both the corresponding Lambek
calculi and propositional logic. Then, using filtrations, we obtain the finite
model property of the relational semantics for the extended nonassociative
Lambek calculus that yields an alternative proof of decidability of the latter.
One of the features of the extended nonassociative Lambek calculus and
its relational semantics is that the latter is the “standard” semantics of classical negation, cf. [4] and [15]. It appears that the presence of conjunction and
disjunction allows one to describe negation in a sound and complete form.
This paper is organized as follows. In the next section we introduce the
extension of the nonassociative Lambek calculus with classical propositional
logic. Then, in Section 3 we define the relational semantics of the extended
nonassociative Lambek calculus and, in Section4 we prove the corresponding
completeness theorem. In Section 5, we show that the extended nonassociative Lambek calculus is decidable. Finally, in Section 6 we define the
(associative) relational semantics of the extended associative Lambek calculus and prove the corresponding completeness theorem.
2
This semantics is, actually, the string semantics of the associative Lambek calculus
over a one letter alphabet, cf. [2, 13].
3
In [5, 6] this property is called the strong finite model property (SFMP), because it
also holds for the consequence relation.
4
Cf. [5, 6], where the extension is presented as a sequent system with additional axioms.
2
Technion - Computer Science Department - Tehnical Report CS-2012-09 - 2012
2
Nonassociative Lambek calculus extended
with classical propositional logic
The extension PNL of the nonassociative Lambek calculus (NL) with classical propositional logic (P ) is defined as follows. Formulas are constructed
from propositional variables (atomic formulas) by means of the Lambek connectives \, /, ·, and the propositional connectives ∧ (conjunction), ∨ (disjunction), ⊃ (implication), and ¬ (negation).
Formulas constructed from propositional variables by means of the Lambek connectives only are called formulas of the pure Lambek calculus and
formulas of the form A ⊃ B, where A and B are formulas of the pure Lambek calculus are called Lambek implications.5
The rules of inference and the axioms PNL are the rules of inference of
the nonassociative Lambek calculus NL
A·B ⊃C
B ⊃ A\C
(1)
B ⊃ A\C
A·B ⊃C
(2)
A·B ⊃C
A ⊃ C/B
(3)
A ⊃ C/B
A·B ⊃C
(4)
modus ponens (MP)
A, A ⊃ B
B
and the axioms of classical propositional calculus P
A ⊃ (B ⊃ A)
(A ⊃ (B ⊃ C)) ⊃ ((A ⊃ B) ⊃ (A ⊃ C))
5
We replace the Lambek notation → with ⊃.
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(¬A ⊃ ¬B) ⊃ ((¬A ⊃ B) ⊃ A)
A ⊃ (B ⊃ (A ∧ B))
(A ∧ B) ⊃ A
(A ∧ B) ⊃ B
A ⊃ (A ∨ B)
B ⊃ (A ∨ B)
(A ⊃ C) ⊃ ((B ⊃ C) ⊃ (A ∨ B) ⊃ C))
Here A, B, and C range over all PNL formulas.6 , whereas propositional
variables will be denoted by P, Q, R, etc.
A formula A is derivable from a set of formulas (assumptions) Θ, denoted
Θ ⊢ A, if there exists a sequence of formulas A1 , A2 , . . . , An = A, such that
for all i = 1, 2, . . . , n one the following holds.
• Ai is an axiom of PNL; or
• Ai ∈ Θ; or
• for some i′ < i, Ai is obtained from Ai′ by one of the rules of inference (1)–(4); or
• for some i′ , i′′ < i, Ai is obtained from Ai′ and Ai′′ by MP.
6
Since we deal with classical propositional logic only, we (equivalently) replaced schemes
1b, 7, and 8 from the axiomatization in [9, p. 82] with the schemes (A2) and (A3) from [12,
p. 27].
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Let Θ(P1 , . . . , Pn ) and ϕ(P1 , . . . , Pn ) be a set of PNL formulas and a
PNL formula, respectively. If
Θ(P1 , . . . , Pn ) ⊢PNL ϕ(P1 , . . . , Pn ) ,
then for all PNL formulas A1 , . . . , An ,
Θ(A1 , . . . , An ) ⊢PNL ϕ(A1 , . . . , An )
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as well, because PNL is structural.
If Θ(P1 , . . . , Pn ) and ϕ(P1 , . . . , Pn ) are a set of propositional formulas and
a propositional formula, respectively, and
Θ(P1 , . . . , Pn ) ⊢P ϕ(P1 , . . . , Pn ),
we shall say that ϕ(A1 , . . . , An ) is derivable from Θ(A1 , . . . , An ) by means of
P , or just derivable by means of P , if Θ(P1 , . . . , Pn ) = ∅.
Example 1 Axioms of the Lambek calculus A ⊃ A are derivable (in PNL)
by means of P .
Example 2 The implication A ⊃ C is derivable from {A ⊃ B, B ⊃ C} by
means of P . That is, the transitivity rule
A ⊃ B, B ⊃ C
A⊃C
of the Lambek calculus is admissible in PNL.
Similarly, if Θ(P1 , . . . , Pn ) and ϕ(P1 , . . . , Pn ) are a set of Lambek implications and a Lambek implication, respectively, and
Θ(P1 , . . . , Pn ) ⊢NL ϕ(P1 , . . . , Pn ) ,
we shall say that ϕ(A1 , . . . , An ) is derivable from Θ(A1 , . . . , An ) by means of
NL, or just derivable by means of NL, if Θ(P1 , . . . , Pn ) = ∅.
We proceed with a series of examples which will be used for the proof of
the completeness theorem in Section 4.
7
Of course, Θ(A1 , . . . , An ) = {ψ(A1 , . . . , An ) : ψ(P1 , . . . , Pn ) ∈ Θ(P1 , . . . , Pn )}.
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Example 3 (Cf. [5, Equation (4)].) The formulas
A · (B ∨ C) ⊃ A · B ∨ A · C
(5)
(B ∨ C) · A ⊃ B · A ∨ C · A .
(6)
and
are derivable in PNL. We present the derivation of (5) only. The derivation
of (6) is symmetric.
1.
2.
3.
4.
5.
6.
A·B ⊃A·B∨A·C
B ⊃ A\(A · B ∨ A · C)
A·C ⊃A·B∨A·C
C ⊃ A\(A · B ∨ A · C)
B ∨ C ⊃ A\(A · B ∨ A · C)
A · (B ∨ C) ⊃ A · B ∨ A · C
axiom
follows
axiom
follows
follows
follows
from 1 by (1)
from 3 by (1)
by means of P from 2 and 4
from 5 by (2)
Example 4 The formulas
A · B ⊃ ((C ∧ A) · B ∨ (¬C ∧ A) · B)
(7)
B · A ⊃ (B · (C ∧ A) ∨ B · (¬C ∧ A))
(8)
and
are derivable in PNL. The derivation of (7) follows and the derivation of (8)
is symmetric.
1.
2.
3.
4.
A ⊃ ((C ∧ A) ∨ (¬C ∧ A))
A · B ⊃ ((C ∧ A) ∨ (¬C ∧ A)) · B
((C ∧ A) ∨ (¬C ∧ A)) · B ⊃ ((C ∧ A) · B ∨ (¬C ∧ ¬A) · B)
A · B ⊃ ((C ∧ A) · B ∨ (¬C ∧ A) · B)
derivable by means of P
derivable from 1 by means of NL
(6)
follows from 2 and 3 by means
of P (transitivity)
Example 5 The formulas
A · (B ∧ C) ⊃ A · B
(9)
(B ∧ C) · A ⊃ B · A
(10)
and
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are derivable in PNL. Like in the previous examples, we present only one
derivation (of (9)). The other derivation is symmetric and is omitted.
1.
2.
3.
4.
5.
B∧C ⊃B
A·B ⊃A·B
B ⊃ A\A · B
B ∧ C ⊃ A\A · B
A · (B ∧ C) ⊃ A · B
axiom
Example 1
follows from 2 by (1)
derivable from 1 and 3 by means of P
follows from 4 by (2)
Example 6 If ⊢ ¬A, then ⊢ ¬(A · B) and ⊢ ¬(B · A). The derivation of
¬(A · B) is as follows.
1. A ⊃ A/B derivable from ¬A by means of P
2. A · B ⊃ A follows from 1 by (4)
3. ¬(A · B) derivable from 2 and ¬A by means of P
The derivation of ¬(B · A) is symmetric and is omitted.
3
The relational semantics of PNL
The semantics of PNL we consider here is the relational semantics from [7],
cf. [1, Definition 1.30], extended with the standard interpretation of the
propositional connectives. Namely, an interpretation is a triple I = hW, R, V i,
where W is a nonempty set of (possible) worlds, R is a ternary (accessibility) relation on W , and V is a (valuation) function from W into sets of
propositional variables (propositional interpretations).
The satisfiability relation |= between worlds in W and PNL formulas is
defined as follows. Let u ∈ W .
• If A is a propositional variable, then I, u |= A, if A ∈ V (u);
• I, u |= A · B, if there are v, w ∈ W such that R(u, v, w), I, v |= A and
I, w |= B;
• I, u |= A/B, if for all v, w ∈ W such that R(w, u, v), I, v |= B implies
I, w |= A;
• I, u |= A\B, if for all v, w ∈ W such that R(v, w, u), I, w |= A implies
I, v |= B;
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• I, u |= A ∨ B, if I, u |= A or I, u |= B;
• I, u |= A ∧ B, if I, u |= A and I, u |= B;
• I, u |= A ⊃ B, if I, u 6|= A or I, u |= B; and
• I, u |= ¬A, if I, u 6|= A.
An interpretation I = hW, R, V i is finite if W is finite and for each world
u ∈ W , V (u) is a subset of the same finite set of propositional variables.
A formula A is satisfiable if I, u |= A for some interpretation I = hW, R, V i
and some u ∈ W . Also, we shall say that I satisfies a formula A, denoted
I |= A, if I, u |= A, for all u ∈ W and we say that I satisfies a set of formulas
Θ, denoted I |= Θ, if I |= A, for all A ∈ Θ. Finally, a set of formulas Θ
semantically entails a formula A, denoted Θ |= A, if for each interpretation I,
I |= Θ implies I |= A.
It can be readily verified by the usual induction on the derivation length
that the relational semantics is sound for PNL,8 i.e., Θ ⊢ A implies Θ |= A,
cf. [7, proposition 1]. In the next section we show that this semantics is also
complete.
4
Completeness
The proof of the completeness theorem, i.e., that Θ |= A implies Θ ⊢ A,
follows the standard proof in, e.g., [10, Section 3] (that is based on the Henkin
construction), but is more involved because of propositional connectives.
We shall need the following extension of PNL.
Let Θ be a set of PNL formulas. The calculus PNLΘ results from PNL
in augmenting it with the set of axioms Θ. Derivability in PNLΘ is denoted
⊢Θ . Thus, ⊢Θ A, if and only if Θ ⊢ A.
Remark 7 Obviously, Examples 3–6 extend to PNLΘ .
Definition 8VA set of formulas Γ is called Θ-consistent, if for no finite subset
Ψ of Γ, ⊢Θ ¬ Ψ.9
8
9
Therefore,VPNL is consistent.
As usual, Ψ is the conjunction of all elements of Ψ.
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In
V what follows we write ⊢Θ Γ ⊃ A if for some finite subset Ψ of Γ,
Ψ ⊃ A. For example, in this notation, a set of formulas Γ is Θ⊢Θ
consistent if and only if there is no formula A such that ⊢Θ Γ ⊃ A and
⊢Θ Γ ⊃ ¬A.
Below we shall freely use well-known properties of Θ-consistent sets of
formulas. These properties are derivable from Θ by means of P . For example,
• if Γ is a set of formulas, and A is a formula such that 6⊢Θ Γ ⊃ ¬A, then
Γ ∪ {A} is Θ-consistent;
• if Γ is a Θ-consistent set of formulas, then, for each formula A, either
Γ ∪ {A} or Γ ∪ {¬A} is Θ-consistent;
• if Γ is a maximal (with respect to inclusion) Θ-consistent set of formulas, then, for each formula A, either A ∈ Γ or ¬A ∈ Γ;
• if Γ is a maximal (with respect to inclusion) Θ-consistent set of formulas
and ⊢Θ Γ ⊃ A, then A ∈ Γ;
etc.
Example 9 Let Θ be a set of formulas, I = hW, R, V i be an interpretation
satisfying Θ, and let u ∈ W . Then the set of formulas
[[u]]I = {A : I, u |= A}
(11)
is maximal and Θ-consistent.
For the proof of the completeness theorem (and Theorem 24 in Section 6)
we shall need Definition 10 and Proposition 11 below.
Definition 10 A set of formulas Φ is called Θ-conjunctively completeVif for
each finite subset Ψ of Φ there is a formula A ∈ Φ such that ⊢Θ A ⊃ Ψ.
Proposition 11 involves the following notation. For two sets of formulas
Γ′ and Γ′′ , we denote the set of formulas
{A′ · A′′ : A′ ∈ Γ′ and A′′ ∈ Γ′′ }
by Γ′ · Γ′′ .
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Proposition 11 Let Γ be a maximal Θ-consistent set of formulas, Φ′ be a
Θ-conjunctively complete set of formulas, and
let Φ′′ be a set of formulas
V
such that for all finite subsets Ψ of Φ′′ , Φ′ · Ψ ⊆ Γ. Then, there exists a
maximal Θ-consistent set of formulas Γ′′ including Φ′′ such that Φ′ · Γ′′ ⊆ Γ.
The proof of Proposition 11 is based on the following lemma.
Lemma 12 Let Γ be a maximal Θ-consistent set of formulas, Φ′ be a Θconjunctively complete set of formulas, V
and let Φ′′ be a set of formulas such
that for all finite subsets Ψ of Φ′′ , Φ′ · Ψ ⊆ Γ. Then, for each formula C
one of the following holds.
V
• For all finite subsets Ψ of Φ′′ , Φ′ · (C ∧ Ψ) ⊆ Γ
or
V
• for all finite subsets Ψ of Φ′′ , Φ′ · (¬C ∧ Ψ) ⊆ Γ.
Proof Assume to the contrary that there are finite subsets Ψ1 and Ψ2 of Φ
such that
^
Φ′ · (C ∧
Ψ1 ) 6⊆ Γ
and
Φ′ · (¬C ∧
implying, by (9), that
Φ′ · (C ∧
and
^
Φ′ · (¬C ∧
^
Ψ2 ) 6⊆ Γ ,
Ψ1 ∧
^
^
Ψ1 ∧
^
Ψ1 ∧
Ψ2 ) 6⊆ Γ
^
Ψ2 ) 6⊆ Γ
^
Ψ2 ) 6∈ Γ .
either, because Γ is maximal.
That is, there are formulas B1 , B2 ∈ Φ′ such that
^
^
B1 · (C ∧ Ψ1 ∧ Ψ2 ) 6∈ Γ
and
B2 · (¬C ∧
Then,
¬(B1 · (C ∧
^
Ψ1 ∧
10
^
Ψ2 )) ∈ Γ
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and
¬(B2 · (¬C ∧
^
Ψ′ ∧
^
Ψ2 )) ∈ Γ ,
because Γ is maximal, and, by the contraposition of (10), we also have
^
^
¬((B1 ∧ B2 ) · (C ∧ Ψ1 ∧ Ψ2 )) ∈ Γ
(12)
and
¬((B1 ∧ B2 ) · (¬C ∧
We contend next that
(B1 ∧ B2 ) ·
^
^
Ψ′ ∧
Ψ1 ∧
^
^
Ψ2 )) ∈ Γ .
Ψ2 ∈ Γ .
(13)
(14)
Indeed, since Φ′ is Θ-conjunctively complete, there is a formula A ∈ Φ′
such that
⊢Θ A ⊃ B1 ∧ B2 ,
implying, by means of NL,
^
^
^ ^ ⊢Θ A ·
Ψ1 ∧ Ψ2 ⊃ (B1 ∧ B2 ) ·
Ψ1 ∧ Ψ2 .
(15)
Since A ∈ Φ′ and Ψ′ ∪ Ψ2 is a finite subset of Φ′′ , by the lemma prerequisite,
^
^ A·
Ψ1 ∧ Ψ2 ∈ Γ .
This, together with (15) implies our contention (14), because Γ is maximal.
Now, it follows from (14) that
^
^
^
^
((B1 ∧B2 )·(C ∧ Ψ1 ∧ Ψ2 ))∨((B1 ∧B2 )·(¬C ∧ Ψ1 ∧ Ψ2 )) ∈ Γ . (16)
V
V
This is because because (8), with A being Ψ1 ∧ Ψ2 and B being B1 ∧ B2 ,
is derivable in PNL and Γ is maximal.
However, (12), (13), and (16) contradict Θ-consistency of Γ.
Proof of Proposition 11 Let A0 , A1 , . . . be a complete listing of the set
fi ∈
of all formulas. We shall recursively construct a sequence of formulas A
{Ai , ¬Ai }, i = 0, 1, . . ., such that for all n = −1, 0, 1, . . . and all finite subsets
Ψ of Φ′′ ,
!
n
^
^
fi ⊆ Γ .
Ψ∧
Φ′ ·
A
(17)
i=0
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By the proposition prerequisite, the boundary case n = −1 is immediate,
fi ∈ {Ai , ¬Ai }, i = 0, 1, . . . , n, such that (17)
and assume that formulas A
is satisfied have been constructed. Then, by Lemma 12, for some formula
′′
]
A
n+1 ∈ {An+1 , ¬An+1 } and all finite subsets Ψ of Φ ,
!
n
^
^
fi ⊆ Γ ,
]
Ψ∧
Φ′ · A
A
n+1 ∧
i=0
which completes the construction description.
The desired set of formulas Γ′′ is
fi : i = 0, 1, ...} .
Γ′′ = {A
Since n
Γ is o
Θ-consistent, by (17), Example 6, and Example 5 (9), all sets of
f
formulas Ai
, n = 0, 1, . . ., are Θ-consistent. Thus, by compactness
i=0,1,...,n
of consistency, Γ′′ is Θ-consistent. Therefore, by (17) (and Example 5 (9), of
course), Φ′′ ⊆ Γ′′ . In addition, since, by our construction, for each formula
C, either C ∈ Γ′′ or ¬C ∈ Γ′′ , Γ′′ is maximal.
Finally, the inclusion Φ′ · Γ′′ ⊆ Γ follows from (17), Example 5 (9), and
maximality of Γ.
Corollary 13 Let Γ be a maximal Θ-consistent set of formulas and let A and
B be formulas such that A · B ∈ Γ. Then, there exist maximal Θ-consistent
sets of formulas Γ′ containing A and Γ′′ containing B such that Γ′ · Γ′′ ⊆ Γ.
Proof Since the set of formulas consisting of B only is Θ-conjunctively complete, similarly to the proof of Proposition 11, one can show that there exists
a maximal consistent sets of formulas Γ′ containing A such that Γ′ · B ⊆ Γ.
Then, since maximal sets of formulas are Θ-conjunctively complete, by the
same proposition, there exists a maximal consistent sets of formulas Γ′′ containing B such that Γ′ · Γ′′ ⊆ Γ.
Definition 14 Let Θ be a set of PNL formulas. The Θ-canonical interpretation IΘ = hWΘ , RΘ , VΘ i is defined as follows.
• WΘ consists of all maximal Θ-consistent sets of formulas,
• RΘ = {(Γ, Γ′ , Γ′′ ) ∈ WΘ3 : Γ′ · Γ′′ ⊆ Γ}, and
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• VΘ (Γ) = Γ ∩ P, where P is the set of all propositional variables (atomic
formulas of PNL).
Theorem 15 Let Γ ∈ WΘ . Then, for each formula C, IΘ , Γ |= C if and
only if C ∈ Γ.
Proof The proof is by induction on the complexity of C. The basis (i.e.,
the case of a propositional variable) and the cases of classical propositional
connectives are immediate. The cases of the Lambek connectives are also
treated in the standard manner and, for the sake of completeness, we consider
them below.
• Let C be of the form A · B and let IΘ , Γ |= A · B. That is, there are
Γ′ , Γ′′ ∈ WΘ such that IΘ , Γ′ |= A, IΘ , Γ′′ |= B, and Γ′ · Γ′′ ⊆ Γ. By the
induction hypothesis, A ∈ Γ′ and B ∈ Γ′′ , which, together with Γ′ · Γ′′ ⊆ Γ,
implies A · B ∈ Γ.
Conversely, let A · B ∈ Γ. Then, by Corollary 13, there are Γ′ , Γ′′ ∈ WΘ
such that A ∈ Γ′ , B ∈ Γ′′ , and Γ′ · Γ′′ ⊆ Γ. By the induction hypothesis,
IΘ , Γ′ |= A and IΘ , Γ′′ |= B, which, together with Γ′ · Γ′′ ⊆ Γ, implies
IΘ , Γ |= A · B.
• Let C be of the form A\B and let IΘ , Γ |= A\B. We contend that
⊢Θ A · Γ ⊃ B .
(18)
To prove (18), assume to the contrary that 6⊢Θ A · Γ ⊃ B. Then, A · Γ ∪ {¬B}
is Θ-consistent. Let Γ′ be a maximal Θ-consistent set of formulas including
A · Γ ∪ {¬B}. Since Γ is maximal, it is Θ-conjunctively complete. Thus, by
the “left counterpart” of Proposition 11, there exists a maximal Θ-consistent
set of formulas Γ′′ containing A such that Γ′′ · Γ ⊆ Γ′ . In addition, by the
induction hypothesis, IΘ , Γ′′ |= A.
Therefore, since IΘ , Γ |= A\B, by the definition of the satisfiability relation |=, IΘ , Γ′ |= B, and by the induction hypothesis, B ∈ Γ′ . This, however,
contradicts ¬B ∈ Γ′ , which proves our contention (18).
Now, by (1), ⊢Θ A · Γ ⊃ B implies ⊢Θ Γ ⊃ A\B which, in turn, implies
A\B ∈ Γ, because Γ is maximal.
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Conversely, let A\B ∈ Γ and let Γ′ and Γ′′ be such that Γ′′ · Γ ⊆ Γ′ and
IΘ , Γ′′ |= A. We have to show that IΘ , Γ′ |= B.
By the induction hypothesis, A ∈ Γ′′ which, together with A\B ∈ Γ and
′′
Γ · Γ ⊆ Γ′ , implies A · (A\B) ∈ Γ′ . Since A · (A\B) ⊃ B is derivable by
means of NL and Γ′ is maximal, B ∈ Γ′ . Thus, by the induction hypothesis,
IΘ , Γ′ |= B.
• The case of / is symmetric to that of \ and is omitted.
Theorem 16 (Completeness) If Θ |= A, then Θ ⊢ A.
Proof Assume to the contrary that Θ 6⊢ A. Then, {¬A} is Θ-consistent.
Therefore, there is a maximal Θ-consistent set of formulas Γ containing ¬A.
By Theorem 15, IΘ , Γ 6|= A, which contradicts this theorem prerequisite.
An immediate corollary to the completeness theorem is that the classical
and the Lambek calculi are orthogonal, i.e., PNL is a conservative extension
of both NL and P .
Corollary 17 PNL is a conservative extension of NL.
Proof Assume to the contrary that PNL is not a conservative extension of
NL and let Θ and A be a finite set of Lambek implications and a Lambek
implications, respectively, such that Θ ⊢PNL A, but Θ 6⊢NL A. Then, by the
“Θ-extension” of the completeness theorem in [7] (see also, e.g., [10]), there
is an NL interpretation I such that I |= Θ, but I 6|= A. This, however,
contradicts the soundness of the relational semantics with respect to PNL,
because I is a PNL interpretation as well.
Corollary 18 PNL is a conservative extension of P .
Proof Assume to the contrary that PNL is not a conservative extension of P
and let Θ and A be a finite set of propositional formulas and a propositional
formula, respectively, such that Θ ⊢PNL A, but Θ 6⊢P A. Then, by the
classical completeness theorem there is a propositional interpretation I such
that I |= Θ, but I 6|= A.
Extend I to an interpretation I = hW, R, V i, where W = {u} is a one
element world, R = ∅, say, and V (u) = I. Then I |= Θ, but I 6|= A either.
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This, however, contradicts the soundness of the relational semantics with
respect to PNL.
We conclude this section with the canonical embedding of interpretation
satisfying a set of formulas Θ into IΘ .
Definition 19 Let Θ be a set of formulas and let I = hW, R, V i be an
interpretation satisfying Θ. The canonical embedding ιI : W → WΘ is defined
by ιI(u) = [[u]]I.10,11
Corollary 20 Let Θ be a set of formulas, I = hW, R, V i be an interpretation satisfying Θ, and let u, v, w ∈ W be such that R(u, v, w). Then
RΘ (ιI(u), ιI(v), ιI(w)).
Proof Assume R(u, v, w) and let A ∈ [[v]]I and B ∈ [[w]]I. We have to show
that A · B ∈ [[u]]I.
By definition, I, v |= A and I, w |= B, which, together with R(u, v, w),
implies I, u |= A · B. Thus, by definition, we have the desired containment
A · B ∈ [[u]]I.
5
Decidability of PNL
In this section we show that PNL is decidable. That is, for a finite set of
PNL formulas Θ and a PNL formula A, it is decidable whether Θ ⊢ A.
Decidability of PNL follows from the finite model property of relational
semantic, see Corollary 22 below. The proof of the latter is based on the
filtration technique, see [14, Chapter I, Section 7].
For an interpretation I = hW, R, V i and a subformula closed set of formulas Φ,12 we define an equivalence relation ∼Φ on W by u ∼Φ v, if for each
formula C ∈ Φ the following holds.
• I, u |= C if and only if I, v |= C.
Next, we define the interpretation IΦ = hWΦ , RΦ , VΦ i by
10
See (11) for the definition of [[u]]I .
Since I |= Θ, ιI is well-defined.
12
That is, if A ∈ Φ and B is a subformula of A, then also B ∈ Φ.
11
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• WΦ = W/ ∼Φ , i.e., is the set of equivalence classes of ∼Φ ;
• RΦ = {([u], [v], [w]) ∈ WΦ3 : there are u′ ∈ [u], v ′ ∈ [v], and w′ ∈ [w]
such that (u′ , v ′ , w′ ) ∈ R};13
and
• VΦ ([u]) = V (u) ∩ P|Φ , where P|Φ is the set of all propositional variables
which occur in formulas from Φ.14
Theorem 21 Let I = hW, R, V i be an interpretation and let Φ be a subformula closed set of formulas. Then, for each formula C ∈ Φ and each u ∈ W ,
I, u |= C if and only if IΦ , [u] |= C.
Proof The proof is by induction on the complexity of C. The basis (i.e., the
case where C is a propositional variable (an atomic formula) and the cases
of classical connectives are immediate.
• Let C be of the form A · B and let I, u |= A · B. That is, there are v, w ∈ W
such that I, v |= A, I, w |= B, and R(u, v, w). By the induction hypothesis,
IΦ , [v] |= A and IΦ , [w] |= B, and, by the definition of RΦ , RΦ ([u], [v], [w]).
Thus, by the definition of |=, IΦ , [u] |= A · B.
Conversely, let IΦ , [u] |= C. That is, there are v, w ∈ W such that
RΦ ([u], [v], [w]), IΦ , [v] |= A, and IΦ , [w] |= B. By the definition of RΦ ,
there are u′ ∈ [u], v ′ ∈ [v], and w′ ∈ [w] such that R(u′ , v ′ , w′ ) and, by the
induction hypothesis, I, v ′ |= A and I, w′ |= B. Therefore, by the definition
of |=, I, u′ |= A · B, and by the definition of ∼Φ , I, u |= A · B, as well.
• Let C be of the form A\B, I, u |= A\B, and let v, w ∈ W be such that
RΦ ([v], [w], [u]) and IΦ , [w] |= A. We have to show that IΦ , [v] |= B.
By the definition of RΦ , there are u′ ∈ [u], v ′ ∈ [v], and w′ ∈ [w] such that
R(v ′ , w′ , u′ ). By the definition of ∼Φ , I, u′ |= A\B, and, by the induction
hypothesis, I, w |= A. Applying the definition of ∼Φ one more time, we
obtain I, w′ |= A. Therefore, by the definition of |=, I, v ′ |= B, and, by the
induction hypothesis, IΦ , [v ′ ] |= B. Since v ′ ∈ [v], [v ′ ] = [v] and IΦ , [v] |= B
follows.
13
14
We write [u] for the ∼Φ equivalence class of u.
As Φ is subformula closed, VΦ is well defined.
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Conversely, let IΦ , [u] |= A\B, and let v, w ∈ W be such that R(v, w, u)
and I, w |= A. We have to show that I, v |= B.
By the definition of RΦ , RΦ ([v], [w], [u]), and, by the induction hypothesis,
IΦ , [w] |= A. Therefore, by the definition of |=, IΦ , [v] |= B, and, by the
induction hypothesis, I, v |= B, as desired.
• The case of / is symmetric to that of \ and is omitted.
Corollary 22 Let Θ be a finite set of formulas. If Θ 6⊢ A, then there is a
finite interpretation satisfying Θ but not satisfying A.
Proof By Theorem 16 there is an interpretation I satisfying Θ such that
I 6|= A. Let Φ be the subformula closure of Θ ∪ {A}.15 Then, by Theorem 21,
IΦ |= Θ, but IΦ 6|= A. Since Θ ∪ {A} is finite, Φ is also finite and, therefore,
IΦ is finite as well.
6
Associative Lambek calculus extended with
propositional logic
Associative Lambek calculus L and propositional associative Lambek calculus
PL result in adding to NL and PNL, respectively the axioms
(A · B) · C ⊃ A · (B · C)
(19)
A · (B · C) ⊃ (A · B) · C .
(20)
and
In this section we prove that the associative interpretation semantics defined below is sound and complete for PL.
Definition 23 A ternary relation R on a set W is associative, if for all
u, v, w, x ∈ W the following holds.
• There exists y such that R(y, v, w) and R(u, y, x) if and only if there
exists z such that R(z, w, x) and R(u, v, z).
15
That is, Φ is the minimal (with respect to inclusion) subformula closed set of formulas
including Θ ∪ {A}.
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An interpretation I = hW, R, V i is associative, if R is associative.
By [7, Proposition 2], the axioms (19) and (20) are satisfied by associative
interpretations. Thus, the latter is sound for PL.
The notion of Θ-canonical interpretation IΘ (Definition 14) extends to
PL in a natural manner.16 Thus, for the proof of completeness of the associative relational semantics with respect to PL, it suffices to show that the
canonical PL interpretation IΘ is associative.
Theorem 24 The interpretation IΘ is associative.
Proof We prove only the “only if” condition of Definition 23. The proof of
the “if” condition is similar.
Let Γu , Γv , Γw , Γx , and Γy be maximal Θ-consistent sets of formulas such
that Γv · Γw ⊆ Γy and Γy · Γx ⊆ Γu . We have to show that there is a maximal
Θ-consistent sets of formulas Γz such that Γw · Γx ⊆ Γz and Γv · Γz ⊆ Γu .
It follows from Γv · Γw ⊆ Γy and Γy · Γx ⊆ Γu that (Γv · Γw ) · Γx ⊆ Γu .
Therefore, by (19),
Γv · (Γw · Γx ) ⊆ Γu
(21)
as well.
We contend that for all finite subsets Ψ of Γw · Γx ,
^ Ψ ⊆ Γu .
Γv ·
(22)
Let
Ψ = {B1 · C1 , B2 · C2 , . . . , Bn · Cn } .
n
n
V
V
Ci ∈ Γx , because both Γw and Γx are maximal
Bi ∈ Γw and
Then,
i=1
i=1
and, by Example 5
⊢
n
^
i=1
Bi
!
·
n
^
i=1
Ci
!
⊃
^
Ψ.
Now, for A ∈ Γv , by means of NL, we have
!!
!
n
n
^
^
^
Ci
⊃ A · Ψ,
Bi ·
⊢A·
i=1
16
i=1
Alternatively, we may assume that Θ includes both (19) and (20).
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Technion - Computer Science Department - Tehnical Report CS-2012-09 - 2012
implying, by Example 5,
⊢ A · (Γw · Γx ) ⊃ A ·
^
Ψ.
(23)
Thus, our contention (22) follows from (23), (21), and maximality of Γu .
Now, since Γv is maximal, it is Θ-conjunctively complete, and, by Proposition 11, there exists a maximal Θ-consistent sets of formulas Γz including
Γw · Γx such that Γv · Γz ⊆ Γu .
Now, exactly like in the proof of Corollaries 17 and 18, it can be shown
that PL is a conservative extension of L and P , respectively.
It is known from [3] that, in contrast with NL, the general problem of
derivability from a set of assumptions in L is undecidable. Therefore, the
same problem for PL is undecidable either.17
However, the problem of derivability of a formula in L is decidable, because L admits cut elimination, see [11], and we conclude this paper with
the problem whether derivability in PL, i.e., without assumptions, is also
decidable.
Acknowledgment
The authors are grateful to Wojciech Buszkowski for his comments on earlier
versions of this paper.
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