Marta Esteves and Rodrigo Nicolau Almeida
The problem of vagueness – how gradual predicates can be understood in a semantic and logical sense – is a longstanding question in philosophy, with a number of recent discussions in the literature and implications in fields such as the philosophy of science, epistemology, natural language processing, amongst others (Boolos, 1991; Fine, 1975; Lewis, 1982, 1988). Authors such as Stewart Shapiro (2006) defend a semantic account, arguing that vagueness stems from different contexts in which meaning is attributed, borrowing from David Lewis’ (Lewis, 1969) notion of convention. He argues that a a metalanguage for a logic of vagueness which is not fully based on classical logic, can be given using conversational scores as defining a notion of logical consequence. These scores constitute contexts of utterances in which agents actualise each others use of a given predicate. However, in order to give substance to the consequence relation, the agent’s perceptions in those scores should converge to a given truth-value.
In parallel, the field of opinion dynamics has ever since its inception been focused on questions of consensus formation, posing agents influencing each other in a given context: a lattice, a network, etc . Models based on sociophysics such as the Ising and DeGroot models, as well as more culturally inclined evolutionary models (Castellano, Fortunato, & Loreto, 2007) have contributed to understanding how opinions change, and have been subject to many relevant extensions. However, as noted by (Dong, Zhan, Kou, Ding, & Liang, 2018), the processing of opinions based on natural language has been lacking (with some recent exceptions, cf. (Crosscombe & Lawry, 2017)), in part due to the vague nature of opinions formulated in linguistic terms, and requires a proper framework in order to be accounted for in such models.
In this work, we propose to both analyse how convergence develops in conversational scores, and to provide a preliminary basis for the development of fuzzy opinion dynamics. We consider a pragmatic theory of mental states according to which they can be inferred from the agent’s actions (taking from Pierce’s Semiotics (1958), Rasmey’s theory of action (1931) and more recent success semantical accounts (such as Milikan, (1984) and Blackburn, (2005)). Towards this end, we develop a novel framework for the analysis of convergence by considering a structured network where agents interact with their neighbours to adjust their classificatory scheme. In the model, following suggestions from Lawry (Lawry, 1998, 2006), agents are given inputs which they classify against a “parameter of scepticism”, which details what level of the specific input must exist in order for them to attribute a classical (crisp) truth-value to the proposition referring to the input. They then exchange this information with their links in the network, updating their parameter of scepticism. Further parameters such as topology and resistance to change are also considered as extensions.
In the process, we also consider the existence of multiple types of vague predicates which carry distinct implications for the convergence of conversational scores (Grinsell, 2012; Michels, 2018). We propose that while these predicates may be complex they are passible to be reduced to a fuzzy-logical first order expression, with distinct importance scores governing the general importance granted by any given agent to any of the constituents – in that sense attempting to contribute to generalise semantic accounts of vagueness to the case of multidimensional predicates.
This work provides an exploration of these problems, and seeks to explore some hypothesis via simulation: that “simple” vague predicates have a tendency to converge (understood here as a group standard deviation in agent scepticism values) towards the average of the distribution of agent scepticism score; that predicates involving logical connectives and multiple dimensions can also converge under some assumptions on topology and number of connections.
In this work we present some preliminary results of simulation which show the way in which this can be considered, as well as point the philosophical implications which this carries into a semantic account of vagueness. More so, we suggest that the mechanism of fuzzy communication between agents can in turn be imported to models of opinion dynamics, allowing for complex phrases to be decomposed into fuzzy logic predicates and analysed according as such in a truth-functional setting.
- Blackburn, S. (2005). Success Semantics. In H. Lillehammer & D. H. Mellor (Eds.), Ramsey’s Legacy (pp. 22–36). Oxford & New York.
- Boolos, G. (1991). Zooming Down the slippery slope. https://doi.org/10.2307/2215638, Noûs, 25(5), 695.
- Castellano, C., Fortunato, S., & Loreto, V. (2007). Statistical physics of social dynamics. https://doi.org/10.1103/RevModPhys.81.591
- Crosscombe, M., & Lawry, J. (2017). Exploiting Vagueness for Multi-agent Consensus (pp. 67–78). https://doi.org/10.1007/978-981-10-2564-8_5
- Deffuant, G. (2006). Comparing Extremism Propagation Patterns in Continuous Opinion Models. Journal of Artificial Societies and Social Simulation, 9(3), 8. Retrieved from http://jasss.soc.surrey.ac.uk/9/3/8.html
- Dong, Y., Zhan, M., Kou, G., Ding, Z., & Liang, H. (2018). A survey on the fusion process in opinion dynamics. Information Fusion, 43, 57–65. https://doi.org/10.1016/j.inffus.2017.11.009
- Fine, K. (1975). Vagueness, Truth and Logic. Synthese, 30(3/4), 265–300.
- Grinsell, T. W. (2012). Avoiding predicate whiplash: social choice theory and linguistic vagueness. In Proceedings of SALT 22 (pp. 424–440). Chicaco, Illinois: University of Chicago Press. Retrieved from https://journals.linguisticsociety.org/proceedings/index.php/SALT/article/view/2628/237
- Herrera, F., Alonso, S., Chiclana, F., & Herrera-Viedma, E. (2009). Computing with words in decision making: foundations, trends and prospects. Fuzzy Optimization and Decision Making, 8(4), 337–364. https://doi.org/10.1007/s10700-009-9065-2
- Lawry, J. (1998). A voting mechanism for fuzzy logic. International Journal of Approximate Reasoning, 19(3–4), 315–333. https://doi.org/10.1016/S0888-613X(98)10013-0
- Lawry, J. (2006). Modelling and Reasoning with Vague Concepts (Vol. 12). Boston: Springer US. https://doi.org/10.1007/0-387-30262-X
- Lewis, D. (1969). Convention: A Philosophical Study. Cambridge, M.A.: Harvard University Press.
- Lewis, D. (1982). Logic for Equivocators. Noûs, 16(3), 431. https://doi.org/10.2307/2216219
- Lewis, D. (1988). Vague Identity: Evans Misunderstood. https://doi.org/10.2307/3328214, Analysis, 48(3), 128.
- Michels, R. (2018). Multidimensional Vagueness – A Supervaluational Approach. Retrieved from https://sopha2018.sciencesconf.org/190171/document
- Millikan, R. G. (1984). Language, Thought, and Other Biological Categories: New Foundations for Realism. Cambridge, M.A.: MIT Press.
- Peirce, C. S. (1958). The Collected Papers (Vol. 1–6). Cambridge, M.A.: Harvard University Press.
- Ramsey, F. (1931). Truth and Probability. In R. . Braithwaite (Ed.), The Foundations of Mathematics and other Logical Essays (1999 Elect, pp. 156–198). London: Kegan, Paul, Trench, Trubner & Co.
- Sahlgren, M. (2008). The Distributional Hypothesis. Rivista Di Linguistica, 20(1), 33–53.
- Shapiro, S. (2006). Vagueness in Context. Oxford: Clarendon Press.
- Williamson, T. (1996). Vagueness. New York: Routledge.