Mathematical Fuzzy Logic in Reasoning and Decision Making under Uncertainty
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1 Mathematical Fuzzy Logic in Reasoning and Decision Making under Uncertainty Hykel Hosni Prague, 17 June 2016 Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and17/07/16 Decision Making 1 / 20 und
2 The uncertain reasoner s point of view UNCERTAINTY Decision-making (epistemic) Modelling (ontic) Natural Sciences (statistical mechanics) Social Sciences (vagueness) Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and17/07/16 Decision Making 2 / 20 und
3 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 and17/07/16 Decision Making 3 / 20 und
4 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 and17/07/16 Decision Making 3 / 20 und
5 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, [ 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), 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: ). Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and17/07/16 Decision Making 3 / 20 und
6 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 de Finetti, B. (2008). Philosophical lectures on probability. Springer Verlag. Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and17/07/16 Decision Making 4 / 20 und
7 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 and17/07/16 Decision Making 5 / 20 und
8 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 and17/07/16 Decision Making 5 / 20 und
9 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 and17/07/16 Decision Making 6 / 20 und
10 1 Rational preferences 2 Ambiguity 3 Judgment Aggregation Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and17/07/16 Decision Making 7 / 20 und
11 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 and17/07/16 Decision Making 8 / 20 und
12 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 and17/07/16 Decision Making 9 / 20 und
13 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 and17/07/16 Decision Making 9 / 20 und
14 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 17/07/16 Decision Making 10 / 20 und
15 Key references Luce, R. (1956), Semiorders and a theory of utility discrimination, Econometrica 24: Fishburn, P.(1970), Intransitive indifference with unequal indifference intervals, Journal of Mathematical Psychology 7: 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 Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 11 / 20 und
16 1 Rational preferences 2 Ambiguity 3 Judgment Aggregation Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 12 / 20 und
17 The Ellsberg problem R Y B 1 R Y R B Y B Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 13 / 20 und
18 The Ellsberg problem R Y B 1 R Y R B Y B 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 17/07/16 Decision Making 13 / 20 und
19 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 17/07/16 Decision Making 14 / 20 und
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), Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 14 / 20 und
21 Key references Whereas capacities and related measures are widely used in economic theory, very little stock is taken there of the MFL contribution to this Chateauneuf, A. (2000). Choquet Expected Utility Model: A New Approach to Individual Behavior Under Uncertainty And To Social Welfare. In M. Grabisch, T. Murofushi, & M. Sugeno (Eds.), Fuzzy Measures and Integrals (pp ). Physica-Verlag. Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 15 / 20 und
22 Key references Whereas capacities and related measures are widely used in economic theory, very little stock is taken there of the MFL contribution to this Chateauneuf, A. (2000). Choquet Expected Utility Model: A New Approach to Individual Behavior Under Uncertainty And To Social Welfare. In M. Grabisch, T. Murofushi, & M. Sugeno (Eds.), Fuzzy Measures and Integrals (pp ). Physica-Verlag. Ghirardato, P. (2010). Ambiguity. In R. Cont (Ed.), Encyclopaedia of Quantitative Finance. Wiley and Sons. Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 15 / 20 und
23 Key references Whereas capacities and related measures are widely used in economic theory, very little stock is taken there of the MFL contribution to this Chateauneuf, A. (2000). Choquet Expected Utility Model: A New Approach to Individual Behavior Under Uncertainty And To Social Welfare. In M. Grabisch, T. Murofushi, & M. Sugeno (Eds.), Fuzzy Measures and Integrals (pp ). Physica-Verlag. Ghirardato, P. (2010). Ambiguity. In R. Cont (Ed.), Encyclopaedia of Quantitative Finance. Wiley and Sons. Gilboa, I., and Marinacci, M. (2013). Ambiguity and the Bayesian Paradigm. In D. Acemoglu, M. Arellano, E. Dekel (Eds.), Advances in Economics and Econometrics: Theory and Applications, Tenth World Congress of the Econometric Society. Cambridge University Press. Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 15 / 20 und
24 Key references Whereas capacities and related measures are widely used in economic theory, very little stock is taken there of the MFL contribution to this Chateauneuf, A. (2000). Choquet Expected Utility Model: A New Approach to Individual Behavior Under Uncertainty And To Social Welfare. In M. Grabisch, T. Murofushi, & M. Sugeno (Eds.), Fuzzy Measures and Integrals (pp ). Physica-Verlag. Ghirardato, P. (2010). Ambiguity. In R. Cont (Ed.), Encyclopaedia of Quantitative Finance. Wiley and Sons. Gilboa, I., and Marinacci, M. (2013). Ambiguity and the Bayesian Paradigm. In D. Acemoglu, M. Arellano, E. Dekel (Eds.), Advances in Economics and Econometrics: Theory and Applications, Tenth World Congress of the Econometric Society. Cambridge University Press. Smith, L., & Stern, N. (2011). Uncertainty in science and its role in climate policy. Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences, 369(1956), Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 15 / 20 und
25 Key references Whereas capacities and related measures are widely used in economic theory, very little stock is taken there of the MFL contribution to this Chateauneuf, A. (2000). Choquet Expected Utility Model: A New Approach to Individual Behavior Under Uncertainty And To Social Welfare. In M. Grabisch, T. Murofushi, & M. Sugeno (Eds.), Fuzzy Measures and Integrals (pp ). Physica-Verlag. Ghirardato, P. (2010). Ambiguity. In R. Cont (Ed.), Encyclopaedia of Quantitative Finance. Wiley and Sons. Gilboa, I., and Marinacci, M. (2013). Ambiguity and the Bayesian Paradigm. In D. Acemoglu, M. Arellano, E. Dekel (Eds.), Advances in Economics and Econometrics: Theory and Applications, Tenth World Congress of the Econometric Society. Cambridge University Press. Smith, L., & Stern, N. (2011). Uncertainty in science and its role in climate policy. Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences, 369(1956), Binmore, K., Stewart, L., & Voorhoeve, A. (2012). How much ambiguity aversion? Finding indifferences between Ellsberg s risky and ambiguous bets, Journal of Risk and Uncertainty 45 Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 15 / 20 und
26 1 Rational preferences 2 Ambiguity 3 Judgment Aggregation Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 16 / 20 und
27 The Doctrinal Paradox p q r (p q) r Judge Judge Judge where p stands for the defendant is in breach of the contract q for the contract is valid and r for the defendant is liable. As usual r (p q) is short for (r (p q)) ((p q) r). Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 17 / 20 und
28 The Doctrinal Paradox where p q r (p q) r Judge Judge Judge Majority p stands for the defendant is in breach of the contract q for the contract is valid and r for the defendant is liable. As usual r (p q) is short for (r (p q)) ((p q) r). List, C., & Pettit, P. (2002). Aggregating Sets of Judgments: An Impossibility Result. Economics and Philosophy, 18(1), Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 17 / 20 und
29 A general phenomenon p q p q Majority p q p q Majority p q p q Majority A number of Arrow-like theorems show a logical tension between Natural aggregation rules The semantics of classical logic Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 18 / 20 und
30 Ways out of the impossibility C. Duddy, and A. Piggins, (2013) Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary, Journal of Economic Theory 148, Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 19 / 20 und
31 Ways out of the impossibility C. Duddy, and A. Piggins, (2013) Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary, Journal of Economic Theory 148, Idea The problem (formulated for conjunction) is solved in L which has a t-norm, the one with zero divisors, i.e. x, y (0, 1) such that f (x, y) = 0 Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 19 / 20 und
32 Ways out of the impossibility C. Duddy, and A. Piggins, (2013) Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary, Journal of Economic Theory 148, Idea The problem (formulated for conjunction) is solved in L which has a t-norm, the one with zero divisors, i.e. x, y (0, 1) such that f (x, y) = 0 Great start but no general MFL perspective 1 there is a lot more to MFT than t-norms! (alternative connectives with zero divisors?) 2 no proper algebraic logic setting is used 3 natural to investigate graded closures Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 19 / 20 und
33 Summing up Reasoning and Decision-making under Uncertainty abound with problems which can be tackled with the full-fledged theory of MFL This is worth doing also for the expected impact of MFL in a variety of disciplines, from theoretical economics, to experimental psychology Ultimately this will contribute to making logic more relevant in reasoning and decision-making under uncertainty! Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and 17/07/16 Decision Making 20 / 20 und
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