Categorial Grammar and Discourse



[t/u ]P τ

= P [t/u ]τ

[t/u ]τ 1 Rτ 2

= [t/u ]τ 1 R[t∕u ]τ 2

[t/u](τ 1 is τ2 )

= [t/u]τ 1 is [t/u]τ2

[t/u ]not Φ

= not [t/u]Φ

[t/u ](Φ or Ψ)

= [t/u ]Φ or [t/u ]Ψ

[t/u](φψ)

= [t/u[t/u]Ψ

if u ∈∕ adr (Φ)

[t/u](φψ)

= [t/uΨ

if u adr (Φ)

[t∕u][~u | γ1, . . . ,γm]

= [~u | [t∕u]γ1 , . . . , [t∕u]γm]

if u ∈∕ {~u}

[t∕u][~u | γ1, . . . ,γm]

= [~u | γ1 , . . . , γm]

if u ∈ {~u}

[t∕u](Φ; Ψ)

= [t∕u]Φ; [t∕u

if u ∈∕ adr(Φ)

[t∕u](Φ; Ψ)

= [t∕u]Φ; Ψ

if u ∈ adr(Φ)

The next definition gives our translation function from boxes and condi-
tions to first-order formulae. The variable
x that is appearing in the sixth
and eighth clauses is supposed to be fresh in both cases, i.e. it is defined to
be the first variable in some fixed ordering that does not occur (at all) in
Φ or in Ψ. Note that the sequencing operation ; is associative: Φ; (Ψ; Ξ) is
equivalent with (Φ; Ψ); Ξ for all Φ, Ψ and Ξ. This means that we may assume
that all boxes are either of the form [
~u | ~γ]; Φ or of the form [~u | ~γ]. We shall
use the form [
~u | ~γ]; Φ to cover both cases, thus allowing the possibility that
Φ is empty. If Φ is empty, Φ
Ψ denotes Ψ.

(P τ )t                         =  P (τ )t

(τ 1 Rτ 2 )t                        =   R(τ 1 )t(τ 2 )t

(τ 1 is τ2)t                      =   (τ 1 )t = (τ2)t

(not Φ)t                  =  (Φ)t

or Ψ)t                =  Φt Ψt

(([u~ | ~]; Φ) Ψ)t        =  ∀x([x∕u](([~ | ~]5Φ) Ψ))t

(([ | γι,...,Ym]5Φ) Ψ)t =  (7ιt ∧ ... ∧ .) → Ψ)t

([u~u | ~γ]; Φ)t                   =   ∃x([x∕u]([~u | ~γ]; Φ))t

([ | γ1, . . . , γm]; Φ)t            =   γ1t ∧ . . . ∧γmt Φt

By way of example, the reader may verify that the function f sends (14) to
(15).

(14) [ | [u1 u2 | man u1 , woman u2, u1 bores u2] [ | u2 ignores u1]]

(15) x1 x2 ((man(x 1 ) woman(x2) bores (x 1 )(x2)) ignores (x 2)(x 1))

It is clear that the function f is algorithmic: at each stage in the reduction of
a box or condition it is determined what step should be taken. The following

12



More intriguing information

1. The name is absent
2. Spatial agglomeration and business groups: new evidence from Italian industrial districts
3. The name is absent
4. The name is absent
5. Manufacturing Earnings and Cycles: New Evidence
6. The name is absent
7. A Multimodal Framework for Computer Mediated Learning: The Reshaping of Curriculum Knowledge and Learning
8. Innovation and business performance - a provisional multi-regional analysis
9. Nurses' retention and hospital characteristics in New South Wales, CHERE Discussion Paper No 52
10. Disentangling the Sources of Pro-social Behavior in the Workplace: A Field Experiment
11. Tourism in Rural Areas and Regional Development Planning
12. Family, social security and social insurance: General remarks and the present discussion in Germany as a case study
13. The name is absent
14. The name is absent
15. Washington Irving and the Knickerbocker Group
16. The name is absent
17. Stillbirth in a Tertiary Care Referral Hospital in North Bengal - A Review of Causes, Risk Factors and Prevention Strategies
18. Temporary Work in Turbulent Times: The Swedish Experience
19. Optimal Taxation of Capital Income in Models with Endogenous Fertility
20. Inflation and Inflation Uncertainty in the Euro Area