Mathematical Fuzzy Logic in Reasoning and Decision Making under Uncertainty Hykel Hosni http://www.filosofia.unimi.it/~hosni/ Prague, 17 June 2016 Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 1 / 20
The uncertain reasoner’s point of view UNCERTAINTY Decision-making Modelling (epistemic) (ontic) Natural Sciences Social Sciences (statistical mechanics) (vagueness) Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 2 / 20
However... In a variety of reasoning and decision-making – especially policy-making epistemic and ontic uncertainty tend to overlap Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 3 / 20
However... In a variety of reasoning and decision-making – especially policy-making epistemic and ontic uncertainty tend to overlap Intergovernamental Panel on Climate Change “By 2080, an increase of 5 to 8% of arid and semi-arid land in Africa is projected under a range of climate scenarios (high confidence).” Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 3 / 20
However... In a variety of reasoning and decision-making – especially policy-making epistemic and ontic uncertainty tend to overlap Intergovernamental Panel on Climate Change “By 2080, an increase of 5 to 8% of arid and semi-arid land in Africa is projected under a range of climate scenarios (high confidence).” Some references IPCC 2007 Synthesis Report Summary for Policymakers, WGII Box TS.6, 9.4.4. [ http://www.ipcc.ch/pdf/assessment-report/ar4/syr/ar4_syr.pdf ] Smith, L., & Stern, N. (2011). “Uncertainty in science and its role in climate policy”. Philosophical Transactions of the Royal Society. Series A, Mathematical, Physical, and Engineering Sciences , 369(1956), 4818–41. T. Aven, and O. Renn (2015) “An Evaluation of the Treatment of Risk and Uncertainties in the IPCC Reports on Climate Change”, Risk Analysis , 35: 701–712). Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 3 / 20
The Bayesian-statistical view [ . . . ] it must be borne in mind that whenever we say that an event is something that surely turns out to be either true or false, we are making a limiting assertion as it is not always possible to tell the two cases apart so sharply. Suppose, for instance, that a certain baby was born exactly at midnight on 31 December 1978 and we had to tell which was the year of his birth. How can it be decided in which year the baby was born, if his birth started in 1978 and ended in 1979? It will have to be decided by convention . [. . . ] There is always a margin of approximation, which we can either take into account or not if we say ‘1978’ or ‘1979’. 1 1 de Finetti, B. (2008). Philosophical lectures on probability. Springer Verlag. Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 4 / 20
We can do better than this! Put the framework of MFL to work in Reasoning and Decision-making with the view to 1 Disentangle ontic/epistemic uncertainty 2 Make the most of the formal overlap between MFL and the mathematics of reasoning and decision-making under uncertainty Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 5 / 20
We can do better than this! Put the framework of MFL to work in Reasoning and Decision-making with the view to 1 Disentangle ontic/epistemic uncertainty 2 Make the most of the formal overlap between MFL and the mathematics of reasoning and decision-making under uncertainty The idea Import problems to MFL and export theoretical solutions (rather than “fuzzifying everything”) Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 5 / 20
Three work-packages Highly relevant and highly interdisciplinary topics in which “fuzzy methods” are being used but the theoretical power of MFL is not 1 Rational preferences ◮ mathematical economics (decision, games, social choice) ◮ psychology of reasoning ◮ psychophysics 2 Ambiguity (aka Knightian or Model Uncertainty) ◮ decision under uncertainty ◮ policy-making ◮ psychology and cognition 3 Judgment aggregation ◮ social choice theory ◮ policy-making Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 6 / 20
1 Rational preferences 2 Ambiguity 3 Judgment Aggregation Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 7 / 20
Rationality and preferences Rationality is defined choice-theoretically preference relations, e.g. x � y ⇔ x ∈ C ( { x, y } ) Preferences are consistent if they satisfy standard ordering conditions (Reflexivity, Completeness, Transitivity) This carries through the whole of mathematical economics, including ◮ Debreu representation ◮ von-Neuman Morgenstern representation ◮ Savage’s representation ◮ Arrow’s impossibility theorem ◮ Fundamental theorem of welfare economics ◮ . . . A problem If � is transitive, so is ∼ defined as ( x � y and y � x ) Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 8 / 20
Luce’s coffee (1956) Suppose an agent has a preference for no sugar in coffee and is not able to distinguish between no sugar and 1 grain of sugar enough grains and there will be a preference reversal! Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 9 / 20
Luce’s coffee (1956) Suppose an agent has a preference for no sugar in coffee and is not able to distinguish between no sugar and 1 grain of sugar enough grains and there will be a preference reversal! An old problem in psychophysiology Ernest Weber in 1834 points out that there is a limit to a persons’ ability to discern among perceptual stimuli and called just noticeable difference the minimal increase required to discern the difference Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 9 / 20
The role of MFL Early literature in economics focussed on inventing orderings with intransitive indifference More recent work on decision theory insists on on “fuzzy methods” inspired by this problem, but little or no work uses the full power of MFL. This work-package must be highly interdisciplinary given the clear importance of the cognitive Given how central this is to mathematical economics, the potential for significant impact is huge it brings together the mathematical and philosophical sides of MFL Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 10 / 20
Key references Luce, R. (1956), “Semiorders and a theory of utility discrimination”, Econometrica 24: 178–191 Fishburn, P.(1970), “Intransitive indifference with unequal indifference intervals”, Journal of Mathematical Psychology 7: 144–149 Applications to welfare economics Sen, A. (1992) Inequality Reexamined . Oxford: Clarendon Press;. Boorme, J. (1997) “Is Incommensurability Vagueness?”, reprinted as Chapter 8 of Ethics out of Economics , Cambridge University Press 1999 W. Rabinowicz, (2009) “Incommensurability and Vagueness” Proceedings of the Aristotelian Society , Supplementary Volumes Vol. 83, pp. 71-94 Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 11 / 20
1 Rational preferences 2 Ambiguity 3 Judgment Aggregation Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 12 / 20
The Ellsberg problem 30 60 R Y B 1 R 1 0 0 1 Y 0 1 0 1 R ∨ B 1 0 1 1 Y ∨ B 0 1 1 Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 13 / 20
The Ellsberg problem 30 60 R Y B 1 R 1 0 0 1 Y 0 1 0 1 R ∨ B 1 0 1 1 Y ∨ B 0 1 1 Ellsberg’s preference 1 R ≻ 1 Y and 1 Y ∨ B ≻ 1 R ∨ B which is inconsistent with: 1 the probabilistic representation of uncertainty 2 the maximisation of (subjective) expected utility as the norm of rational decision Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 13 / 20
A well-known solution Suppose we had a measure of uncertainty µ such that µ ( R ) = 1 / 3 µ ( Y ∨ B ) = 2 / 3 µ ( Y ) = µ ( B ) = 0 Then the Ellsberg preferences would be satisfied by letting µ ( Y ∨ B ) = 2 / 3 > µ ( R ∨ B ) = 1 / 3 Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 14 / 20
A well-known solution Suppose we had a measure of uncertainty µ such that µ ( R ) = 1 / 3 µ ( Y ∨ B ) = 2 / 3 µ ( Y ) = µ ( B ) = 0 Then the Ellsberg preferences would be satisfied by letting µ ( Y ∨ B ) = 2 / 3 > µ ( R ∨ B ) = 1 / 3 Choquet expected utility Schmeidler, D. (1989). “Subjective Probability and Expected Utility without Additivity”. Econometrica , 57(3), 571–587. Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 14 / 20
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