Kosta Dosen, Modal translations in substructural logics, Journal of Philosophical Logic 21 (1992), 283--336.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....To allow such embeddings, we introduce a new judgment, A is valid , which we write A valid. We view validity not as a primitive judgment, but as a categorical judgment derived from truth in the absence of linear hypotheses. Similar notions of categorical judgment apply in a wide variety of logics [11, 23]. If # A true, then A valid. In the resource interpretation, A valid means that we can achieve A true without consuming any resources. Dually, an assumption A valid allows generation of as many copies of A true as needed, including none at all. Stated structurally, hypotheses of the form A ....

K. Dosen. Modal translations in substructural logics. Journal of Philosophical Logic, 21:283--336, 1992.

....rule available wherever needed for that derivation. Such a global modalisation strategy is stated as an embedding translation. For example, the translation (4. 2) embeds a fragment of LP within L, so that Gamma ) A is a theorem of the former iff 4j Gammaj ) jAj is a theorem of the latter (see [5]) Another example is that intuitionistic logic can be embedded within (intuitionistic) linear logic, via an embedding translation using the exponentials and . jAj : A (where A is atomic) jA Omega Bj : 4jAj) ffl (4jBj) jB GammaffiA)j : 4jBj) n jAj jAffi GammaB)j : jAj (4jBj) ....

Dosen, K. 1990. `Modal translations in substructural logics'. Report 10--90, Universit'at Konstanz.

....type formulae F freely generated from a set of atomic formulae A: F : A j F=F j F ffl F j FnF . The most general interpretation for such a language can be given in terms of Kripke style relational structures ternary relational structures hW; R 3 i in the case of the binary connectives (cf. Dosen 1992). W here is to be understood as the set of linguistic resources (signs) and the accessibility relation R as representing linguistic composition. From a ternary frame we obtain a model by adding a valuation V sending prime formulae to subsets of W and satisfying the clauses below for compound ....

....rely heavily on semantic argumentation. The cornerstone of the approach is the completeness of the logics compared, which guarantees that syntactic derivability A B and semantic inclusion V (A) V (B) coincide for the classes of models we are interested in. For the F( ffl; n) fragment, Dosen 1992 shows that NL is complete with respect to the class of all ternary models, and L, NLP, LP with respects to the classes of models satisfying the frame constraints for the relevant packages of structural postulates. The completeness results are obtained on the basis of a simple canonical model ....

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Dosen, K. (1992) `Modal translations in substructural logics'. Journal of Philosophical Logic 21, 283--336.

....linguistic uses. Structural modalities allow that stronger logics may be embedded within weaker ones, via embedding translations. For example, the translation (2) embeds a fragment of LP within L, so that Gamma ) A is a theorem of the former iff 4j Gammaj ) jAj is a theorem of the latter (see [2] for discussion) Another example is that intuitionistic logic can be embedded within (intuitionistic) linear logic, via an embedding translation using the exponentials and . Such an embedding shows, of course, that when the weaker logic is augmented with the appropriate structural modalities, ....

Dosen, K. (1990) `Modal translations in substructural logics'. Report 10-90, Universit'at Konstanz.

....marked with a : X Gamma B X; A Gamma B [W ] and X; A; A Gamma B X; A Gamma B [C ] It was a very natural move for researchers interested in developing other hybrid substructural logics to look at linear logic for inspiration. The issue was addressed in wide scope in Do sen (cf. [5, 6]) who discusses the general picture, concentrating on proof theoretical properties like embeddability. Independently, the idea was taken up by Morrill et al..ii in [17] who were interested in an extension of the Lambek Calculus with restricted Permutation, and in Yetter [25] where an extension of ....

....= X and (A 0 ) Gamma = A, so by ( we find that S 0 X Gamma A, which is what we wanted to prove indeed. 2 Remark Note that the non modal kind of embedding defined here, deviates from the tradition in the literature. The connections between our theorem and the results in e.g. Do sen [5, 6] remains to be investigated. 7 Conclusions Accepting the idea to use operators for the task of strengthening a substructural logic, we have asked ourselves the question what the meaning of a formula rA (r the operator) in a resourcebounded derivation system might be. Our answer was, that a ....

Dosen, K., "Modal Translations in Substructural Logics", Journal of Philosophical Logic, 21 (1992) pp. 283--336.

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Kosta Dosen, Modal translations in substructural logics, Journal of Philosophical Logic 21 (1992), 283--336.

No context found.

Kosta Dosen, Modal translations in substructural logics, Journal of Philosophical Logic 21 (1992), 283--336.

No context found.

Kosta Dosen, Modal translations in substructural logics, Journal of Philosophical Logic 21 (1992), 283--336.

No context found.

Dosen, K. 1990. `Modal translations in substructural logics'. Report 10-90, Universit'at Konstanz.

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