Skip to main content
Log in

Graded consequence: an institution theoretic study

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

We develop a general study of graded consequence (of many-valued logic) in an institution theoretic (in the sense of Goguen and Burstall) style. This means both syntax and semantics are considered fully abstract, as well as the satisfaction between them. Our approach contrasts to other approaches on many-valued logic in that it is a multi-signature one, in the spirit of institution theory. We consider graded consequence at three different conceptual levels: entailment, semantic, and closure operators, and explore several interpretations between them. We also study logical connectors and quantifiers both at the entailment and semantic level, compactness and soundness properties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. What is called ‘external’ here and in Mossakowski et al. (2005) corresponds to what is called ‘internal’ in Diaconescu (2003, 2008). ‘Internal’ suggests that the properties characterising the connectives/quantifications are defined as properties of the respective entailment system, while ‘external’ suggests the involvement of the models as an entity which is outside to the entailment system. One may be tempted to use ‘semantic’ instead of ‘external’ and ‘syntactic’ of ‘proof theoretic’ instead of ‘internal’, but this would not be a good choice of terminology because the ‘syntactic’ stuff for the semantic entailment would have a semantic essence.

References

  • Béziau J-Y (2006) 13 questions about universal logic. Bul Sect Log 35(2/3):133–150

    Google Scholar 

  • Béziau J-Y (ed) (2012) Universal logic: an anthology. Studies in universal logic. Springer, Basel

  • Burmeister P (1982) Partial algebra–an introductory survey. Algebra Univers 15:306–358

    Article  MATH  MathSciNet  Google Scholar 

  • Chakraborty MK (1988) Use of fuzzy set theory in introducing graded consequence in multiple valued logic. In: Gupta MM, Yamakawa T (eds) Fuzzy logic in knowledge-based systems. Decision and control. Elsevier Science Publishers B.V., North Holland, pp 247–257

    Google Scholar 

  • Chakraborty MK (1995) Graded consequence: further studies. J Appl Non-Class Log 5(2):127–137

    Article  Google Scholar 

  • Cintula P, Hájek P (2006) On theories and models in fuzzy predicate logic. J Symb Log 71(3):832–863

    Google Scholar 

  • Diaconescu R (2000) Grothendieck institutions. Appl Categor Struct 10(4):383–402, 2002. Preliminary version appeared as IMAR Preprint 2-2000, ISSN:250–3638, February 2000

  • Diaconescu R (2003) Institution-independent ultraproducts. Fundamenta Informaticæ 55(3–4):321–348

    MATH  MathSciNet  Google Scholar 

  • Diaconescu R (2006) Proof systems for institutional logic. J Logic Comput 16(3):339–357

    Google Scholar 

  • Diaconescu R (2008) Institution-independent Model Theory. Birkhäuser, Boston

  • Diaconescu R (2011) On quasi-varieties of multiple valued logic models. Math Logic Q 57(2):194–203

    Article  MATH  MathSciNet  Google Scholar 

  • Diaconescu R (2011) Structural induction in institutions. Info Comput 209(9):1197–1222

    Article  MATH  MathSciNet  Google Scholar 

  • Diaconescu R (2012) Three decades of institution theory. In: Béziau J-Y (ed) Universal logic: an anthology. Springer, Basel, pp 309–322

  • Diaconescu R (2013) Institutional semantics for many-valued logics. Fuzzy Sets Syst 218:32–52

    Article  MATH  MathSciNet  Google Scholar 

  • Dutta S, Basu S, Mihir K (2013) Chakraborty. Many-valued logics, fuzzy logics and graded consequence: a comparative appraisal. In: Lodaya K (ed) ICLA 2013, vol 7750 of lecture notes in artificial intelligence. Springer, Berlin, pp 197–209

  • Eklund P, Helgesson R (2010) Monadic extensions of institutions. Fuzzy Sets Syst 161:2354–2368

    Article  MATH  MathSciNet  Google Scholar 

  • Fiadeiro JL, Costa JF (1996) Mirror, mirror in my hand: a duality between specifications and models of process behaviour. Math Struct Comput Sci 6(4):353–373

    MATH  MathSciNet  Google Scholar 

  • Fiadeiro JL, Sernadas A (1988) Structuring theories on consequence. In: Sannella D, Tarlecki A (eds) Recent trends in data type specification, vol 332 of lecture notes in computer science. Springer, Berlin, pp 44–72

  • Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics. Elsevier, Amsterdam

    Google Scholar 

  • Gerla G (2001) Fuzzy logic: mathematical tools for approximate reasoning. Kluwer, Dordrecht

  • Goguen J (1968) The logic of inexact concepts. Synthese 19:325–373

    Article  Google Scholar 

  • Goguen J, Burstall R (1984) Introducing institutions. In: Clarke E, Kozen D (eds) Proceedings of the logics of programming workshop, vol. 164 of lecture notes in computer science. Springer, Berlin, pp 221–256

  • Goguen J, Burstall R (1992) Institutions: abstract model theory for specification and programming. J Assoc Comput Mach 39(1):95–146

    Article  MATH  MathSciNet  Google Scholar 

  • Grätzer G (2011) Lattice theory: foundation. Springer, Basel

  • Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht

  • Hussmann H (1993) Nondeterminism in algebraic specifications and algebraic program. Birkhaüser, Boston

  • Lamo Y (2003) The institution of multialgebras—a general framework for algebraic software development. Ph.D. thesis, University of Bergen

  • Mac Lane S (1998) Categories for the working mathematician, 2nd edn. Springer, Berlin

  • Mayoh B (1985) Galleries and institutions. Technical Report DAIMI PB-191, Aarhus University

  • Meseguer J (1989) General logics. In: Ebbinghaus H-D et al (eds) Proceedings, logic colloquium, 1987. North-Holland, pp 275–329

  • Meseguer J (1998) Membership algebra as a logical framework for equational specification. In: Parisi-Pressice F (ed) Proceedings of the WADT’97. Lecture notes in computer science, vol 1376. Springer, Berlin, pp 18–61

  • Mossakowski T, Goguen J, Diaconescu R, Tarlecki A (2005) What is a logic? In: Béziau J-Y (ed) Logica universalis. Birkhäuser, Boston, pp 113–133

  • Ono H (2003) Substructural logics and residuated lattices—an introduction. In: Hendricks VF, Malinowski J (eds) 50 Years of studia logica. Trends in logic, vol 20. Kluwer, Springer, Dordrecht, Berlin, pp 177–212

  • Pavelka J (1979) On fuzzy logici—many-valued rules of inference. Zeitscher for Math. Logik und Grundlagen d. Math 25:45–52

    Google Scholar 

  • Rabe F (2013) A framework for combining model and proof theory. Math Struct Comput Sci 23(5):945–1001

    Google Scholar 

  • Sannella D, Tarlecki A (2012) Foundations of algebraic specifications and formal software development. Springer, Berlin

  • Scott D (1974) Completeness and axiomatizability in many-valued logic. In: Henkin L et al (ed) Proceedings, tarski Symposium. American Mathematical Society, pp 411–435

  • Tarlecki A (1986) Bits and pieces of the theory of institutions. In: Pitt D, Abramsky S, Poigné A, Rydeheard D (eds) proceedings, summer workshop on category theory and computer programming. Lecture notes in computer science, vol 240. Springer, Berlin, pp 334–360

  • Tarski A (1956) On some fundamental concepts of metamathematics. In: Woodger JH (ed) Logic, semantics, metamathematics. Oxford University Press, NY, USA, pp 30–37

  • Voutsadakis G (2002) Categorical abstract algebraic logic: algebrizable institutions. Appl Categorical Struct 10:531–568

    Article  MATH  MathSciNet  Google Scholar 

  • Walicki M (1993) Algebraic specification of nondeterminism. Ph.D. thesis, Department of Informatics, University of Bergen

  • Walicki M, Meldal S (1997) Algebraic approaches to nondeterminism —an overview. ACM Comput Surv 29:30–81

    Google Scholar 

Download references

Acknowledgments

This work has been supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0439. Thanks to Ionuţ Ţuţu for enlightening discussions around the temporal logic example.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Răzvan Diaconescu.

Additional information

Communicated by L. Spada.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diaconescu, R. Graded consequence: an institution theoretic study. Soft Comput 18, 1247–1267 (2014). https://doi.org/10.1007/s00500-014-1231-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-014-1231-y

Keywords

Navigation