Yde Venema. Meeting strength in substructural logics. Logic Group Preprint Series No. 86, Department of Philosophy, Utrecht University, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and the subformula property Before turning to the semantics of we will prove the Cut elimination theorem and subformula prop erty for it, since the latter is essential for the completeness proof, and a corollary to the former. 380 As we remarked earlier, our proof rules are adapted from [Venema, 1993b] Therefore, we can refer the reader to that paper for most of the Cutelimination proof. The only notable difference between both systems lies in the structural rules they allow. Note that resource preservation implies that for any [ j) inference we have the following two sim ple but important ....

Yde Venema. Meeting strength in substructural logics. Logic Group Preprint Series No. 86, Department of Philosophy, Utrecht University, 1993.

....and commutativity hold when one of the participating elements belongs to the relevant subalgebra. We shall discuss such interpretation here, and yet present truly modal proof rules which are incomplete for the subalgebra interpretation. 14 The remaining two sides of this square are covered by Venema (1993), which provides a complete proof theory for the subalgebra interpretation of structural operators, and by Kurtonina (1993) which provides a modal interpretation of structural operators. Example: Associativity in NL For controlled associativity in NL we interpret in an algebra hL; L 0 i ....

....modality. 5.3 Embedding Girard (1987) shows how a particular translation faithfully embeds intuitionistic logic into linear logic with a (contraction and weakening) modality. Morrill (1992) conjectured that the same translation works generally for the subalgebra interpretations; this is proved in Venema (1993). 5.4 Limitation: Island Constraints We turn now to the respect in which associative Lambek calculus is linguistically too strong. As it stands it cannot respect constraints such as the fact that coordinate stuctures are islands to extraction (Coordinate Structure Constraint) and that so also are ....

Venema, Yde: 1993, `Meeting strength in substructural logics', preprint 86, Department of Philosophy, Utrecht.

....In x4.2 we developed a multiplicative theory of the unary operators 3; 2 # , presenting them as truncated forms of ffl and a residual implication. In a language with u, one can develop an alternative additive account of the unary control operators. We present the proposals of [Venema 93] for the introduction of a unary operator r, decomposable as AuQ, where Q is a type constant picking out a subset of the interpretation domain W a subset of elements which count as special in a sense indicated by the constant Q. The r operator resolves model theoretic problems for the ....

....hWQ ; Deltai. Notice that the rule [Qffi] is in fact a compiled Cut. But being a Cut on a constant, it does not threaten decidability. Finally, there is a set of r controlled structural rules. Controlled Permutation is given as an example. Definition 5.6. The additive structural operator r ( Venema 93] Logical rules, structural rules (i 2 f1; 2g) The [rLi] and [rR] rules are derivable if one defines rA as A u Q. rL1] Gamma[A] B Gamma[rA] B Gamma[Q] B Gamma[rA] B [rL2] Delta ) A Delta ) Q Delta ) rA [rR] Delta 1 ) Q Delta 2 ) Q Gamma[Q] A Gamma[ Delta 1 ; Delta ....

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Venema, Y. (1993), `Meeting strength in substructural logics'. UU Logic Preprint. To appear in Studia Logica.

....logic 1 [Gir87, Ben91] which has attracted considerable attention in computer science [Gir87] natural language [Ben91] and planning [Hol92] communities recently. The choice of logic is not completely arbitrary. It is chosen for specific reasons: 1) it has applications in linguistics (see e.g.[Ven93]) 2 . The mathematical aspects have also been studied recently [Kur92, Ven93, Roo92] 2) the tableaux method works well with intensional logics, while Sequent calculus is often used for resource logics. We deliberately chose a logic which has features from two ends of the scale to test the ....

....computer science [Gir87] natural language [Ben91] and planning [Hol92] communities recently. The choice of logic is not completely arbitrary. It is chosen for specific reasons: 1) it has applications in linguistics (see e.g. Ven93] 2 . The mathematical aspects have also been studied recently [Kur92, Ven93, Roo92]; 2) the tableaux method works well with intensional logics, while Sequent calculus is often used for resource logics. We deliberately chose a logic which has features from two ends of the scale to test the generality of LDS; 3) modal operators have already been used in Linear Logic [Gir87] to ....

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Y. Venema. Meeting strength in substructural logics. Logic Group Preprint Series 38, Dept. of Phil., U. Utrecht,, Jan 1993.

....Lambek style In Figures 2 and 3 we juxtapose two equivalent axiomatic presentations of the pure logic of residuation for the extended language. The Lambek style presentation of Fig. 2 is based on the residuation 3 For an additive alternative to our multiplicative view on unary operators, see [46]. id A : A A f : A B g : B C g ffi f : A C unit2 : 2A A co unit2 : A 2A unit = A=B ffl B A co unit = A (A ffl B) B unit n : B ffl BnA A co unit n : A Bn(B ffl A) f : A B (f) Pi : A B f : A B (f) 2 : 2A 2B f : A B g : C D f Delta g : A ffl C B ....

Venema, Y. (1993) `Meeting strength in substructural logics'. UU Logic Preprint. To appear in Studia Logica.

.... and modal substructural logics for natural language (Parsing as Deduction paradigm) and categorial theorem proving [Moo91] The mathematical aspects of modal resource logics (using modality for controlling restricted permutation for categorial grammar in linguistics) have been studied recently [Kur92, Ven93, Roo92]; ffl the tableaux method works well with intensional logics, but has not been previously applied to weaker, resource sensitive, substructural logics. On the other hand, Sequent calculus is often used for the latter cases. We deliberately choose a logic with features from two ends of the scale ....

....semantics and proof theory are presented, completeness (if true) should be obtainable fairly directly. 3. Adding modality to a weak Concatenation Logic is by no means simple. On the contrary, because we have no structural rules, standard classical techniques need not be applicable. Others, e.g. [Roo92, Ven93, Kur92] have used sophisticated techniques for completeness. 4. The definition of Canonical Proof Sequences is very important, as explained by Gabbay [Gab90, pages 135] While it may look complicated, the resulting structure does simplify the proof of lemma 7.3 (Fundamental Lemma) It plays a similar ....

Y. Venema. Meeting strength in substructural logics. Logic Group Preprint Series 38, Dept. of Phil., U. Utrecht,, Jan 1993.

No context found.

Venema, Y. (1993), `Meeting strength in substructural logics'. UU Logic Preprint. To appear in Studia Logica.

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Venema, Y. (1995) `Meeting strength in substructural logics'. Studia Logica 54, 3--32.

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Venema, Y. 1993a. `Meeting Strength In Substructural Logics.' OTS Working Paper, Onderzoekinstituut voor Taal en Spraak, Universiteit Utrecht, Netherlands.

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