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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

(epistemic)

Modelling

(ontic)

Natural Sciences

(statistical mechanics)

Social Sciences

(vagueness)

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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).

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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

1de 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”)

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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

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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)

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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!

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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

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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

• f 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

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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

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1 Rational preferences 2 Ambiguity 3 Judgment Aggregation

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The Ellsberg problem

30 60 R Y B 1R 1 1Y 1 1R∨B 1 1 1Y ∨B 1 1

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The Ellsberg problem

30 60 R Y B 1R 1 1Y 1 1R∨B 1 1 1Y ∨B 1 1 Ellsberg’s preference 1R ≻ 1Y and 1Y ∨B ≻ 1R∨B which is inconsistent with:

1 the probabilistic representation of uncertainty 2 the maximisation of (subjective) expected utility as the norm of

rational decision

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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.

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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. 289–313). Physica-Verlag.

Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 15 / 20

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. 289–313). 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 Decision Making under 17/07/16 15 / 20

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. 289–313). 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 Decision Making under 17/07/16 15 / 20

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. 289–313). 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), 4818–41.

Hykel Hosni (UniMi) Mathematical Fuzzy Logic in Reasoning and Decision Making under 17/07/16 15 / 20

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. 289–313). 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), 4818–41. 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

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1 Rational preferences 2 Ambiguity 3 Judgment Aggregation

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The Doctrinal Paradox

p q r ↔ (p ∧ q) r Judge 1 1 1 1 1 Judge 2 1 1 Judge 3 1 1 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).

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The Doctrinal Paradox

p q r ↔ (p ∧ q) r Judge 1 1 1 1 1 Judge 2 1 1 Judge 3 1 1 Majority 1 1 1 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).

List, C., & Pettit, P. (2002). Aggregating Sets of Judgments: An Impossibility

• Result. Economics and Philosophy, 18(1), 89–110.

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A general phenomenon

p q p ∧ q 1 1 1 1 2 1 3 1 Majority 1 1 p q p ∨ q 1 2 1 1 3 1 1 Majority 1 p q p → q 1 1 1 1 2 1 3 1 Majority 1 1 A number of Arrow-like theorems show a logical tension between Natural aggregation rules The semantics of classical logic

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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, 2 793–805

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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, 2 793–805 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 Decision Making under 17/07/16 19 / 20

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, 2 793–805 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 Decision Making under 17/07/16 19 / 20

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!

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