Abstract
An operational and informational semantics for the ternary relation R is explored as a framework for modeling informational relevance. We extend this framework into robustly epistemic terrain. We take a new perspective on the problem of logical omniscience, using informationalised operational semantics to model the properties of the epistemic actions that underpin the epistemic relevance of certain explicit epistemic states of an epistemic agent as that agent executes said actions.
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- 1.
Announcements and observations may be run together as a single type of epistemic action if you assume that announcements are always truthful, always believed, and always non-noisy, van Ditmarsh et al. (2008).
- 2.
This point is similar to the one made by Frege in his letter to Jourdain. Frege entertains an agent who is able, in principle, to comprehend every atomic sentence, but does not have the ability to execute any semantic composition. Given language’s essential productivity, such an ability is, according to Frege, of little general interest. I am indebted to an anonymous referee for bringing this to my attention.
- 3.
Assuming logical omniscience for the epistemic agents in one’s model makes perfect sense insofar as one wants to idealize away from variables.
- 4.
The motivation here is similar to that of (Duc 1997). The difference is that Duc has models for what the agent knows after she has executed some rule of inference or other, whereas here we will be modelling the properties of the epistemic actions which underpin the execution of such rules.
- 5.
- 6.
The role of a partial or pre-order in the Routley–Meyer semantics for relevance logic is well-known and explored in some detail in (Bimbó and Dunn 2008, Chap. 2).
- 7.
This is not guaranteed, since there is no sensible requirement on an epistemic state that the state in question be itself epistemically relevant to another epistemic state that subsumes the information carried by the original state. For example, my knowing that grass is green at some point in time does not have to be an epistemically relevant episode to every future epistemic state or action involving the information that grass is green. That is, our epistemic states are not totally ordered.
- 8.
Of course, we could write “\(\alpha \) knows/believes explicitly that A” as \(x\Vdash _\alpha A\) or some such, but typographical rigour has a tendency to get in the way of readability.
- 9.
As well might be the case, given that both x and y carry A.
- 10.
Although cognitive epistemic actions may, and often do, involve the combination more that two premises, the treatment of the two-premise case is privileged on several fronts. Firstly, it is the simplest possible case. Given this, any model of cognitive epistemic actions needs to be shown to handle such cases before being applied to more complex cases. Secondly, it seems to be at least plausible that the majority of deductive episodes do proceed via two-premise combinations. Witness the standard natural deduction rules and classical syllogisms as examples. A third reason is simply that the two-premise case is hard enough.
- 11.
Note that Association is given here in its readable, abbreviated form. The full form of Association is \(\exists u((x \bullet y\sqsubseteq u)\wedge (w\bullet u\sqsubseteq z))\Longleftrightarrow \exists t((w\bullet x\sqsubseteq t)\wedge (t\bullet y\sqsubseteq z))\). This makes sense if you think about it. In the abbreviated form above, we are merely cutting out explicit reference to the states u and t, which are the results of composing w and x on the one hand, and x and y on the other, respectively.
- 12.
Dunn uses “data” to refer to static information p, q, etc., and “programs” to refer to dynamic information, or conditionals, \(p\rightarrow q\), etc. As we will see in Sect. 5, agents may treat programs as data.
- 13.
There is a lot to say here about dynamic negation and negative information. One way to go is to say that there is a null object \(\mathbf {0}\) such that \(x\Vdash \mathbf {0}\text { for no }x\). The way is then clear to define a dynamic negation \(A^\mathbf {0}\) in terms of \(A\rightarrow \mathbf {0}\), which will type information of the type that can never be combined with information of type A. Classical and other static negations rule out truth, whilst dynamic negations rule out certain operations or combinatorial procedures. See Dunn (1993, 1996) and Sequoiah-Grayson (2009).
- 14.
Since, as the category theory folks are fond of saying, “Arrows associate!”.
- 15.
Although both formulas are classically (and non-classically in certain logics) equivalent, recall that our epistemic agent \(\alpha \) is not logically omniscient.
- 16.
For an investigation into the epistemic role of explicit conjunctions, especially with respect to the closure axiom and related modal-epistemic phenomena, see Sequoiah-Grayson (2013).
- 17.
Although Dunn’s remarks are not framed in explicitly epistemic terms, all of his examples concerning disjunction are epistemic/doxastic in nature. This is presumably no mere coincidence!
- 18.
I would like to thank two anonymous referees for their invaluable feedback. I would like to thank Katalin Bimbó also, for her tireless efforts and superhuman patience.
References
Bilkova, M., Majer, O., Pelis, M., & Restall, G. (2010). Relevant Agents. Advances in Modal Logic (Vol. 8). London: College Publications.
Bimbó, K., & Dunn, J. M. (2008). Generalized Galois logics. Relational semantics of nonclassical logical calculi, vol. 188 of CSLI Lecture Notes. Stanford, CA: CSLI Publications.
Duc, H. N. (1997). Reasoning about rational, but not logically omniscient, agents. Journal of Logic and Computation, 7(5), 633–648.
Dunn, J. M. (1993). Star and perp: Two treatments of negation. Philosophical Perspectives, 7, 331–357. (Language and Logic, J. E. Tomberlin (Ed.)).
Dunn, J. M. (1996). Generalised ortho negation. In H. Wansing (Ed.), Negation: A notion in focus (pp. 3–26). New York, NY: Walter de Gruyter.
Dunn, J. M. (2015). The relevance of relevance to relevance logic. In M. Banerjee & S. N. Krishna (Eds.), Logic and its applications: 6th Indian conference, ICLA 2015, Mumbai, India, January 8–10, 2015, number 8923 in lecture notes in computer science (pp. 11–29). Heidelberg: Springer.
Dunn, J. M., & Hardegree, G. M. (2001). Algebraic methods in philosophical logic, vol. 41 of Oxford logic guides. Oxford, UK: Oxford University Press.
Majer, O., & Pelis, M. (2009). Epistemic logic with relevant agents. In M. Pelis & V. Puncochar (Eds.) The Logica Yearbook 2008 (pp. 123–135).
Mares, E. D. (1996). Relevant logic and the theory of information. Synthese, 109, 345–360.
Mares, E. D. (2004). Relevant logic: A philosophical interpretation. Cambridge: Cambridge University Press.
Restall, G. (1996). In J. Seligman & D. Westerstahl (Eds.), Logic, language, and computation (Vol. 1, pp. 463–477)., Information flow and relevant logics Stanford: CSLI Publications.
Roy, O., & Hjortland, O. T. (2014). Dynamic consequences for soft information. Journal of Logic and Computation, 1–22.
Sedlar, I. (2012). Boxes are relevant. In M. Pelis & V. Puncochar (Eds.), The Logica Yearbook 2011 (pp. 265–278). London: College Publications.
Sedlar, I. (2014). Epistemic extensions of modal distributive substructural logics. Journal of Logic and Computation, (online first).
Sequoiah-Grayson, S. (2009). Dynamic negation and negative information. Review of Symbolic Logic, 2(1), 233–248.
Sequoiah-Grayson, S. (2013). Epistemic closure and commuting, nonassociating residuated structures. Synthese, 190(1), 113–128.
van Ditmarsh, H., van der Hoek, W., & Kooi, B. (2008). Dynamic epistemic logic. Springer.
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Sequoiah-Grayson, S. (2016). Epistemic Relevance and Epistemic Actions. In: Bimbó, K. (eds) J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-29300-4_8
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