An Inferentially Many-Valued Two-Dimensional Notion of Entailment

Carolina Blasio; João Marcos; Heinrich Wansing

Bulletin of the Section of Logic (2017)

  • Volume: 46, Issue: 3/4
  • ISSN: 0138-0680

Abstract

top
Starting from the notions of q-entailment and p-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized q-matrices referred to as B-matrices. After showing that every purely monotonic singleconclusion consequence relation is characterized by a class of B-matrices with respect to q-entailment as well as with respect to p-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of B-entailment, a two-dimensional multiple-conclusion notion of entailment based on B-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a B-consequence relation is defined, and an abstract characterization of such relations by classes of B-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko’s Thesis and the corresponding reduction to show that any B-consequence relation is, in general, inferentially four-valued.

How to cite

top

Carolina Blasio, João Marcos, and Heinrich Wansing. "An Inferentially Many-Valued Two-Dimensional Notion of Entailment." Bulletin of the Section of Logic 46.3/4 (2017): null. <http://eudml.org/doc/295524>.

@article{CarolinaBlasio2017,
abstract = {Starting from the notions of q-entailment and p-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized q-matrices referred to as B-matrices. After showing that every purely monotonic singleconclusion consequence relation is characterized by a class of B-matrices with respect to q-entailment as well as with respect to p-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of B-entailment, a two-dimensional multiple-conclusion notion of entailment based on B-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a B-consequence relation is defined, and an abstract characterization of such relations by classes of B-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko’s Thesis and the corresponding reduction to show that any B-consequence relation is, in general, inferentially four-valued.},
author = {Carolina Blasio, João Marcos, Heinrich Wansing},
journal = {Bulletin of the Section of Logic},
keywords = {Inferential many-valuedness; two-dimensional entailment; B-matrices; B-consequence relations; monotonic consequence relations; q-entailment; p-entailment; Suszko Reduction},
language = {eng},
number = {3/4},
pages = {null},
title = {An Inferentially Many-Valued Two-Dimensional Notion of Entailment},
url = {http://eudml.org/doc/295524},
volume = {46},
year = {2017},
}

TY - JOUR
AU - Carolina Blasio
AU - João Marcos
AU - Heinrich Wansing
TI - An Inferentially Many-Valued Two-Dimensional Notion of Entailment
JO - Bulletin of the Section of Logic
PY - 2017
VL - 46
IS - 3/4
SP - null
AB - Starting from the notions of q-entailment and p-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized q-matrices referred to as B-matrices. After showing that every purely monotonic singleconclusion consequence relation is characterized by a class of B-matrices with respect to q-entailment as well as with respect to p-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of B-entailment, a two-dimensional multiple-conclusion notion of entailment based on B-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a B-consequence relation is defined, and an abstract characterization of such relations by classes of B-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko’s Thesis and the corresponding reduction to show that any B-consequence relation is, in general, inferentially four-valued.
LA - eng
KW - Inferential many-valuedness; two-dimensional entailment; B-matrices; B-consequence relations; monotonic consequence relations; q-entailment; p-entailment; Suszko Reduction
UR - http://eudml.org/doc/295524
ER -

References

top
  1. [1] C. Blasio, Revisitando a lógica de Dunn-Belnap, Manuscrito 40 (2017), pp. 99–126. 
  2. [2] A. Bochman, Biconsequence relations: A general formalism of reasoning with inconsistency and incompleteness, Notre Dame Journal of Formal Logic 39 (1998), pp. 47–73. 
  3. [3] C. Caleiro, W. Carnielli, M. Coniglio and J. Marcos, Suszko’s Thesis and dyadic semantics, Research Report. 1049-001 Lisbon, PT: CLC, Department of Mathematics, Instituto Superior Técnico, 2003. http://sqig.math.ist.utl.pt/pub/CaleiroC/03-CCCM-dyadic1.pdf 
  4. [4] C. Caleiro, W. Carnielli, M. Coniglio and J. Marcos, Two’s company: “The humbug of many logical values”, [in:] J.-Y. Béziau (ed.), Logica Universalis, Birkhäuser, Basel, 2005, pp. 169–189. 
  5. [5] C. Caleiro, J. Marcos and M. Volpe, Bivalent semantics, generalized compositionality and analytic classic-like tableaux for finite-valued logics, Theoretical Computer Science 603 (2015), pp. 84–110. 
  6. [6] J. M. Dunn and G. M. Hardegree, Algebraic Methods in Philosophical Logic, Oxford Logic Guides, Vol. 41, Oxford Science Publications, Oxford, 2001. 
  7. [7] S. Frankowski, Formalization of a plausible inference, Bulletin of the Section of Logic 33 (2004), pp. 41–52. 
  8. [8] S. Frankowski, p-consequence versus q-consequence operations, Bulletin of the Section of Logic 33 (2004), pp. 41–52. 
  9. [9] S. Frankowski, Plausible reasoning expressed by p-consequence, Bulletin of the Section of Logic 37 (2008), pp. 161–170. 
  10. [10] R. French and D. Ripley, Valuations: bi, tri and tetra, Under submission (2017). 
  11. [11] L. Humberstone, Heterogeneous logic, Erkenntnis 29 (1988), pp. 395–435. 
  12. [12] T. Langholm, How different is partial logic?, [in:] P. Doherty (ed.), Partiality, Modality, and Nonmonotonicity, CSLI, Stanford, 1996, pp. 3–43. 
  13. [13] G. Malinowski, q-consequence operation, Reports on Mathematical Logic 24 (1990), pp. 49–59. 
  14. [14] G. Malinowski, Towards the concept of logical many-valuedness, Folia Philosophica 7 (1990), pp. 97–103. 
  15. [15] G. Malinowski, Many-Valued Logics, Oxford Logic Guides, Vol. 25, Clarendon Press, Oxford, 1993. 
  16. [16] G. Malinowski, Inferential many-valuedness, [in:] Jan Woleński (ed.), Philosophical Logic in Poland, Kluwer Academic Publishers, Dordrecht, 1994, pp. 75–84. 
  17. [17] G. Malinowski, Inferential paraconsistency, Logic and Logical Philosophy 8 (2001), pp. 83–89. 
  18. [18] G. Malinowski, Inferential intensionality, Studia Logica 76 (2004), pp. 3–16. 
  19. [19] G. Malinowski, That p + q = c(onsequence), Bulletin of the Section of Logic 36 (2007), pp. 7–19. 
  20. [20] G. Malinowski, Beyond three inferential values, Studia Logica 92 (2009), pp. 203–213. 
  21. [21] G. Malinowski, Multiplying logical values, Logical Investigations 18 (2012), Moscow–St. Petersburg, pp. 292–308. 
  22. [22] J. Marcos, What is a non-truth-functional logic, Studia Logica 92 (2009), pp. 215–240. 
  23. [23] D. J. Shoesmith and T. J. Smiley,Multiple-Conclusion Logic, Cambridge University Press, 1978. 
  24. [24] Y. Shramko and H. Wansing, Truth and Falsehood. An Inquiry into Generalized Logical Values, Trends in Logic, Vol. 36, Springer, Berlin, 2011. 
  25. [25] R. Suszko, The Fregean axiom and Polish mathematical logic in the 1920’s, Studia Logica 36 (1977), pp. 373–380. 
  26. [26] A. Urquhart, Basic many-valued logic, [in:] D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. 2 (2nd edition), Kluwer, Dordrecht, 2001, pp. 249–295. 
  27. [27] H. Wansing and Y. Shramko, Suszko’s Thesis, inferential many-valuedness, and the notion of a logical system, Studia Logica 88 (2008), pp. 405–429, 89 (2008), p. 147. 
  28. [28] R. Wójcicki, Some remarks on the consequence operation in sentential logics, Fundamenta Mathematicae 68 (1970), pp. 269–279. 
  29. [29] R. Wójcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations, Kluwer, Dordrecht, 1988. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.