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Inhabitants of Intuitionistic Implicational Theorems Read chapter Read first chapter

2018 | OriginalPaper | Chapter

The Effort of Reasoning: Modelling the Inference Steps of Boundedly Rational Agents

Authors : Sonja Smets, Anthia Solaki

Published in: Logic, Language, Information, and Computation

Publisher: Springer Berlin Heidelberg

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Abstract

In this paper we design a new logical system to explicitly model the different deductive reasoning steps of a boundedly rational agent. We present an adequate system in line with experimental findings about an agent’s reasoning limitations and the cognitive effort that is involved. Inspired by Dynamic Epistemic Logic, we work with dynamic operators denoting explicit applications of inference rules in our logical language. Our models are supplemented by (a) impossible worlds (not closed under logical consequence), suitably structured according to the effect of inference rules, and (b) quantitative components capturing the agent’s cognitive capacity and the cognitive costs of rules with respect to certain resources (e.g. memory, time). These ingredients allow us to avoid problematic logical closure principles, while at the same time deductive reasoning is reflected in our dynamic truth clauses. We finally show that our models can be reduced to awareness-like plausibility structures that validate the same formulas and a sound and complete axiomatization is given with respect to them.

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previous chapter Beliefs Based on Evidence and Argumentation
Appendix
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Footnotes
1
Notice that \(\mathrm {c}\) is an r-tuple. The choice of r is discussed in the next subsection.
 
2
We will focus on sound inference rules, i.e. rules whose translation is a tautology, because (a) the agent’s state is naturally built on truth-preserving inferences, and (b) it would be infeasible to (empirically) assign a cost to arbitrary arrays of premises and conclusions. This task is meaningful due to the experimental results on how humans handle rule-schemas and on how the logical complexity of the formulas involved in their instantiations relates to their difficulty (although determining the exact relation between the complexity of formulas and the cognitive difficulty of a rule-application depends on empirical input and is left for future work). The cost assigned to non-sound rules is thus irrelevant and will not affect our constructions.
 
3
Numerical assignments might also pertain to the use of pupil assessment and eye-tracking as measures of attention and indicators of cognitive effort [18, 23, 37].
 
4
We will assume that worlds are valuation-wise unique, i.e. we view the valuation as \(V:= V_p \cup V_i\), where the functions \(V_p\) and \(V_i\) taking care of possible and impossible worlds are injective. This assumption is not vital but it serves the simplicity of the setting because we avoid a multiplicity of worlds unnecessary for our purposes.
 
5
These properties, which follow from the definition of ord, will not force unnecessarily strong (introspective) validities for non-ideal agents because of the presence of impossible worlds.
 
6
Note that \(=\) between formulas stands for syntactic identity. It is used due to Minimal Consistency and the fact that \(V^*\) is given directly by V in impossible worlds.
 
7
See, for example, [9, 17, 27‐29]. In fact, different schools in psychology of reasoning attribute inferential asymmetries to different causes. However, the very observation that not all inferences require equal cognitive effort is common ground.
 
8
\({ Ineq }\) is of course slightly adapted as terms are interpreted as r-tuples. This makes no difference for the axioms in [15], with the exception of dichotomy which is not needed given our reading of inequality.
 
9
More on this terminology can be found in the Appendix.
 
10
Notice that the fact that the interpretations of \(\langle RAD \rangle _\rho \) and \([RAD]_\rho \) are not arbitrary in impossible worlds is important in this proof.
 
11
In fact, we can claim that this logic is weakly complete with respect to ALPS where \(\ge \) is conversely well-founded. This is because our structures have the finite model property (via filtration theorem, [8]) so there are no infinite > chains of more and more plausible worlds.
 
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Metadata
Title
The Effort of Reasoning: Modelling the Inference Steps of Boundedly Rational Agents
Authors
Sonja Smets
Anthia Solaki
Copyright Year
2018
Publisher
Springer Berlin Heidelberg
Book
Logic, Language, Information, and Computation

Print ISBN: 978-3-662-57668-7

Electronic ISBN: 978-3-662-57669-4

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-662-57669-4_18

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