Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-31T20:25:50.371Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LOGIC OF RESOURCES AND CAPABILITIES

Published online by Cambridge University Press:  22 May 2018

MARTA BÍLKOVÁ ,
GIUSEPPE GRECO ,
ALESSANDRA PALMIGIANO ,
APOSTOLOS TZIMOULIS  and
NACHOEM WIJNBERG
MARTA BÍLKOVÁ*
Affiliation:
Department of Logic, Faculty of Arts, Charles University
GIUSEPPE GRECO*
Affiliation:
Department of Languages, Literature and Communication, University of Utrecht
ALESSANDRA PALMIGIANO*
Affiliation:
Faculty of Technology, Policy and Management, Delft University of Technology; Department of Pure and Applied Mathematics, University of Johannesburg
APOSTOLOS TZIMOULIS*
Affiliation:
Faculty of Technology, Policy and Management, Delft University of Technology
NACHOEM WIJNBERG*
Affiliation:
Faculty of Economics and Business, University of Amsterdam; College of Business and Economics, University of Johannesburg
*
*DEPARTMENT OF LOGIC, FACULTY OF ARTS CHARLES UNIVERSITY PRAGUE, CZECH REPUBLIC E-mail: [email protected]
†DEPARTMENT OF LANGUAGES, LITERATURE AND COMMUNICATION UNIVERSITY OF UTRECHT UTRECHT, THE NETHERLANDS E-mail: [email protected]
‡FACULTY OF TECHNOLOGY, POLICY AND MANAGEMENT DELFT UNIVERSITY OF TECHNOLOGY DELFT, THE NETHERLANDS and DEPARTMENT OF PURE AND APPLIED MATHEMATICS UNIVERSITY OF JOHANNESBURG JOHANNESBURG, SOUTH AFRICA E-mail: [email protected]
§FACULTY OF TECHNOLOGY, POLICY AND MANAGEMENT DELFT UNIVERSITY OF TECHNOLOGY DELFT, THE NETHERLANDS E-mail: [email protected]
**FACULTY OF ECONOMICS AND BUSINESS UNIVERSITY OF AMSTERDAM AMSTERDAM, THE NETHERLANDS and COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF JOHANNESBURG JOHANNESBURG, SOUTH AFRICA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the logic LRC, designed to describe and reason about agents’ abilities and capabilities in using resources. The proposed framework bridges two—up to now—mutually independent strands of literature: the one on logics of abilities and capabilities, developed within the theory of agency, and the one on logics of resources, motivated by program semantics. The logic LRC is suitable to describe and reason about key aspects of social behaviour in organizations. We prove a number of properties enjoyed by LRC (soundness, completeness, canonicity, and disjunction property) and its associated analytic calculus (conservativity, cut elimination, and subformula property). These results lay at the intersection of the algebraic theory of unified correspondence and the theory of multitype calculi in structural proof theory. Case studies are discussed which showcase several ways in which this framework can be extended and enriched while retaining its basic properties, so as to model an array of issues, both practically and theoretically relevant, spanning from planning problems to the logical foundations of the theory of organizations.

Keywords

03B4203B2003B6003B4503F0303G1003A99display calculuslogics for organizationsmultitype calculusalgebraic proof theory
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 2 , June 2018 , pp. 371 - 410
Copyright
Copyright © Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Baltag, A., Moss, L. S., & Solecki, S. The logic of public announcements, common knowledge and private suspicious. Technical Report SEN-R9922, CWI, Amsterdam, 1999.Google Scholar
Barney, J. (1991). Firm resources and sustained competitive advantage. Journal of Management, 17(1), 99–120.CrossRefGoogle Scholar
Belnap, N. (1982). Display logic. Journal of Philosophical Logic, 11(4), 375–417.CrossRefGoogle Scholar
Belnap, N. (1991) Backwards and forwards in the modal logic of agency. Philosophy and Phenomenological Research, 51(4), 777–807.CrossRefGoogle Scholar
Belnap, N. & Perloff, M. (1988). Seeing to it that: A canonical form for agentives. Theoria, 54(3), 175–199.CrossRefGoogle Scholar
Birkhoff, G. & Lipson, J. D. (1970). Heterogeneous algebras. Journal of Combinatorial Theory, 8(1), 115–133.CrossRefGoogle Scholar
Blok, W. & Van Alten, C. (2005). On the finite embeddability property for residuated ordered groupoids. Transactions of the American Mathematical Society, 357(10), 4141–4157.CrossRefGoogle Scholar
Brown, M. A. (1988). On the logic of ability. Journal of Philosophical Logic, 17(1), 1–26.CrossRefGoogle Scholar
Chellas, B. F. (1969). The Logical form of Imperatives. Ph.D. Thesis, Stanford University.Google Scholar
Chellas, B. F. (1995). On bringing it about. Journal of Philosophical Logic, 24(6), 563–571.CrossRefGoogle Scholar
Ciabattoni, A., Galatos, N., & Terui, K. (2012). Algebraic proof theory for substructural logics: Cut-elimination and completions. Annals of Pure and Applied Logic, 163(3), 266–290.CrossRefGoogle Scholar
Ciabattoni, A. & Ramanayake, R. (2016). Power and limits of structural display rules. ACM Transactions on Computational Logic (TOCL), 17(3), 17.CrossRefGoogle Scholar
Ciabattoni, A., Straßburger, L., & Terui, K. (2009). Expanding the realm of systematic proof theory. In Grädel, E., and Kahle, R., editors. International Workshop on Computer Science Logic. Berlin: Springer, pp. 163–178.CrossRefGoogle Scholar
Conradie, W. & Craig, A. (2017). Canonicity results for mu-calculi: An algorithmic approach. Journal of Logic and Computation, 27(3), 705–748.CrossRefGoogle Scholar
Conradie, W., Craig, A., Palmigiano, A., & Zhao, Z. (2017). Constructive canonicity for lattice-based fixed point logics. In Kennedy, J., and de Queiroz, R. J. G. B., editors. Logic, Language, Information, and Computation. LNCS 10388. Berlin: Springer-Verlag, pp. 92–109.CrossRefGoogle Scholar
Conradie, W., Fomatati, Y., Palmigiano, A., & Sourabh, S. (2015). Algorithmic correspondence for intuitionistic modal mu-calculus. Theoretical Computer Science, 564, 30–62.CrossRefGoogle Scholar
Conradie, W., Frittella, S., Palmigiano, A., Piazzai, M., Tzimoulis, A., & Wijnberg, N. (2016). Categories: How I learned to stop worrying and love two sorts. In Väänänen, J., Hirvonen, Å., and de Queiroz, R., editors. Logic, Language, Information, and Computation. LNCS 9803. Berlin: Springer-Verlag, pp. 145–164.CrossRefGoogle Scholar
Conradie, W., Frittella, S., Palmigiano, A., Piazzai, M., Tzimoulis, A., & Wijnberg, N. (2017). Towards an epistemic-logical theory of categorization. Electronic Proceedings in Theoretical Computer Science, 251, 167–186. ArXiv:1707.08743.CrossRefGoogle Scholar
Conradie, W., Ghilardi, S., & Palmigiano, A. (2014). Unified correspondence. In Baltag, A. and Smets, S., editors. Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, Vol. 5. Dordrecht: Springer International Publishing, pp. 933–975.Google Scholar
Conradie, W. & Palmigiano, A. (2012). Algorithmic correspondence and canonicity for distributive modal logic. Annals of Pure and Applied Logic, 163(3), 338–376.CrossRefGoogle Scholar
Conradie, W. & Palmigiano, A. Algorithmic correspondence and canonicity for non-distributive logics, preprint, arXiv: 1603.08515.Google Scholar
Conradie, W. & Palmigiano, A. Constructive canonicity of inductive inequalities, preprint, arXiv: 1603.08341.Google Scholar
Conradie, W., Palmigiano, A., & Sourabh, S. (2017). Algebraic modal correspondence: Sahlqvist and beyond. Journal of Logical and Algebraic Methods in Programming, 91, 60–84.CrossRefGoogle Scholar
Conradie, W., Palmigiano, A., Sourabh, S., & Zhao, Z. Canonicity and relativized canonicity via pseudo-correspondence: An application of ALBA, preprint, arXiv: 1511.04271.Google Scholar
Conradie, W., Palmigiano, A., & Zhao, Z. Sahlqvist via translation, preprint, arXiv: 1603.08220.Google Scholar
Conradie, W. & Robinson, C. (2017). On Sahlqvist theory for hybrid logics. Journal of Logic and Computation, 27(3), 867–900.Google Scholar
Cross, C. B. (1986). ‘Can’ and the logic of ability. Philosophical Studies, 50(1), 53–64.CrossRefGoogle Scholar
Dignum, V. (2003). A Model for Organizational Interaction: Based on Agents, Founded in Logic. Ph.D. Thesis, The Netherlands: University of Utrecht.Google Scholar
Dignum, V. & Dignum, F. (2009). A logic of agent organizations. Logic Journal of IGPL, 20(1), 283–316.CrossRefGoogle Scholar
Elgesem, D. (1997). The modal logic of agency. Nordic Journal of Philosophical Logic, 2, 1–46.Google Scholar
Engberg, U. H. & Winskel, G. (1993). Linear logic on Petri nets. In de Bakker, J. W., de Roever, W. P., and Rozenberg, G., editors. REX School/Symposium: A Decade of Concurrency, Reflections and Perspective. LNCS 3580, Vol. 803. Berlin: Springer, pp. 176–229.CrossRefGoogle Scholar
Frittella, S., Greco, G., Kurz, A., & Palmigiano, A. (2016). Multi-type display calculus for propositional dynamic logic. Journal of Logic and Computation, 26(6), 2067–2104.CrossRefGoogle Scholar
Frittella, S., Greco, G., Kurz, A., Palmigiano, A., & Sikimić, V. (2014). Multi-type sequent calculi. In Indrzejczak, A., Kaczmarek, J., and Zawidski, M., editors. Proceedings of Trends in Logic XIII. Lodz: Lodz University Press, pp. 81–93.Google Scholar
Frittella, S., Greco, G., Kurz, A., Palmigiano, A., & Sikimić, V. (2016a). Multi-type display calculus for dynamic epistemic logic. Journal of Logic and Computation, 26(6), 2017–2065.CrossRefGoogle Scholar
Frittella, S., Greco, G., Kurz, A., Palmigiano, A., & Sikimić, V. (2016b). A proof-theoretic semantic analysis of dynamic epistemic logic. Journal of Logic and Computation, 26(6), 1961–2015.CrossRefGoogle Scholar
Frittella, S., Greco, G., Palmigiano, A., & Yang, F. (2016). A multi-type calculus for inquisitive logic. In Väänänen, J., Hirvonen, Å., and de Queiroz, R., editors. Logic, Language, Information, and Computation. LNCS 9803. Berlin: Springer, pp. 215–233.CrossRefGoogle Scholar
Frittella, S., Palmigiano, A., & Santocanale, L. (2017). Dual characterizations for finite lattices via correspondence theory for monotone modal logic. Journal of Logic and Computation, 27(3), 639–678.Google Scholar
Gehrke, M. & Jónsson, B. (1994). Bounded distributive lattices with operators. Mathematica Japonica, 40(2), 207–215.Google Scholar
Gehrke, M. & Jónsson, B. (2004). Bounded distributive lattice expansions. Mathematica Scandinavica, 94, 13–45.CrossRefGoogle Scholar
Gibbons, R. & Roberts, J. (2013). The Handbook of Organizational Economics. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Girard, J.-Y. (1995). Linear logic: Its syntax and semantics. In Girard, J.-Y., Lafont, Y., and Regnier, L., editors. Advances in Linear Logic. London Mathematical Society Lecture Note Series 222. Cambridge: Cambridge University Press, pp. 1–42.CrossRefGoogle Scholar
Greco, G., Kurz, A., & Palmigiano, A. (2013). Dynamic epistemic logic displayed. In Huang, H., Grossi, D., and Roy, O., editors. Proceedings of the 4th International Workshop on Logic, Rationality and Interaction (LORI-4). LNCS, Vol. 8196. Berlin: Springer, pp. 135–148.CrossRefGoogle Scholar
Greco, G., Liang, F., Moshier, A., & Palmigiano, A. (2017). Multi-type display calculus for semi De Morgan logic. In Kennedy, J. and de Queiroz, R. J. G. B., editors. Logic, Language, Information, and Computation. LNCS 10388. Berlin: Springer-Verlag, pp. 199–215.CrossRefGoogle Scholar
Greco, G., Ma, M., Palmigiano, A., Tzimoulis, A., & Zhao, Z. (2016). Unified correspondence as a proof-theoretic tool. Journal of Logic and Computation, https://doi.org/10.1093/logcom/exw022.CrossRefGoogle Scholar
Greco, G. & Palmigiano, A. (2017). Lattice logic properly displayed. In Kennedy, J. and de Queiroz, R. J. G. B., editors. Logic, Language, Information, and Computation. LNCS 10388. Berlin: Springer-Verlag, pp. 153–169.CrossRefGoogle Scholar
Greco, G. & Palmigiano, A. Linear logic properly displayed, preprint, arXiv: 1611.04181.Google Scholar
Haniková, Z. & Horćik, R. (2014). The finite embeddability property for residuated groupoids. Algebra Universalis, 72(1), 1–13.CrossRefGoogle Scholar
Harel, D., Kozen, D., & Tiuryn, J. (2000). Dynamic Logic. Cambridge, MA: MIT Press.Google Scholar
Hiroakira, O. & Komori, Y. (1985). Logics without the contraction rule. The Journal of Symbolic Logic, 50(1), 169–201.Google Scholar
Kopylov, A. P. (2001). Decidability of linear affine logic. Information and Computation, 164(1), 173–198.CrossRefGoogle Scholar
Kurz, A. & Palmigiano, A. (2013). Epistemic updates on algebras. Logical Methods in Computer Science, 9(4:17), 1–28.CrossRefGoogle Scholar
le Roux, C. (2016). Correspondence Theory in Many-Valued Modal Logics. Master’s Thesis. South Africa: University of Johannesburg.Google Scholar
Ma, M., Palmigiano, A., & Sadrzadeh, M. (2014). Algebraic semantics and model completeness for intuitionistic public announcement logic. Annals of Pure and Applied Logic, 165(4), 963–995.CrossRefGoogle Scholar
Ma, M. & Zhao, Z. (2017). Unified correspondence and proof theory for strict implication. Journal of Logic and Computation, 27(3), 921–960.Google Scholar
Mahoney, J. T. (1995). The management of resources and the resource of management. Journal of Business Research, 33(2), 91–101.CrossRefGoogle Scholar
Marté-Oliet, N. & Meseguer, J. (1991). From Petri nets to linear logic. Mathematical Structures in Computer Science, 1(1), 69–101.CrossRefGoogle Scholar
Mol, J. M. & Wijnberg, N. M. (2011). From resources to value and back: Competition between and within organizations. British Journal of Management, 22(1), 77–95.CrossRefGoogle Scholar
Okada, M. & Terui, K. (1999). The finite model property for various fragments of intuitionistic linear logic. The Journal of Symbolic Logic, 64(2), 790–802.CrossRefGoogle Scholar
Ono, H. (1998). Decidability and finite model property of substructural logics. In Ginzburg, J., Khasidashvili, Z., Vogel, C.-W., Lévy, J.-J., and Vallduví, E., editors. The Tbilisi Symposium on Logic, Language and Computation. Stanford, CA: CSLI Publications, pp. 263–274.Google Scholar
Palmigiano, A., Sourabh, S., & Zhao, Z. (2017). Jónsson-style canonicity for ALBA-inequalities. Journal of Logic and Computation, 27(3), 817–865.CrossRefGoogle Scholar
Palmigiano, A., Sourabh, S., & Zhao, Z. (2017). Sahlqvist theory for impossible worlds. Journal of Logic and Computation, 27(3), 775–816.Google Scholar
Plaza, J. (2007). Logics of public communications. Synthese, 158(2), 165–179.CrossRefGoogle Scholar
Pym, D. & Tofts, C. (2006). A calculus and logic of resources and processes. Formal Aspects of Computing, 18(4), 495–517.CrossRefGoogle Scholar
Pym, D. J., O’Hearn, P. W., & Yang, H. (2004). Possible worlds and resources: The semantics of BI. Theoretical Computer Science, 315(1), 257–305.CrossRefGoogle Scholar
Ricardo, D. (1891). Principles of Political Economy and Taxation. London: G. Bell and Sons.Google Scholar
Scott, W. G. (1961). Organization theory: An overview and an appraisal. The Journal of the Academy of Management, 4(1), 7–26.Google Scholar
Segerberg, K. (1982). The logic of deliberate action. Journal of Philosophical Logic, 11(2), 233–254.CrossRefGoogle Scholar
Shamsie, J. & Mannor, M. J. (2013). Looking inside the dream team: Probing into the contributions of tacit knowledge as an organizational resource. Organization Science, 24(2), 513–529.CrossRefGoogle Scholar
Troelstra, A. S. (1992). Lectures on Linear Logic. Lecture Notes, No. 29. Stanford, CA: CSLI Publications.Google Scholar
Tsoukas, H. & Knudsen, C. (2005). The Oxford Handbook of Organization Theory. New York: Oxford University Press.Google Scholar
van der Hoek, W., van Linder, B., & Meyer, J. J. C. (1994). A logic of capabilities. In Nerode, A., and Matiyasevich, Y. V., editors. International Symposium on Logical Foundations of Computer Science. Lecture Notes in Computer Science, Vol. 813. Berlin: Springer, pp. 366–378.CrossRefGoogle Scholar
Wansing, H. (1998). Displaying Modal Logic. Norwell, MA: Kluwer.CrossRefGoogle Scholar
Wernerfelt, B. (1984). A resource-based view of the firm. Strategic Management Journal, 5(2), 171–180.CrossRefGoogle Scholar