Comparing degrees of truthLessons from utility theory

Rossella Marrano

Scuola Normale Superiore

Joint work with Hykel Hosni

Rome, 19 June 2014

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Motivation

Degrees of truth as real numbers

We shall assume that the truth degrees are linearly ordered, with 1as maximum and 0 as minimum. Thus truth degrees will be codedby (some) reals. And even if logics of finitely many truth degreescan be developed we choose not to exclude any real number from theset of truth degrees. We shall always take the set [0, 1] with itsnatural (standard) linear order. (Petr Hájek, Metamathematics ofFuzzy Logic, 1998)

Artificial precision

I arbitrariness of the choice

how can we justify the choice of the truth value 0.24 over 0.23?

I implausibility of the interpretation

what does it mean for a sentence to be 1/π true?

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Lessons from utility theory

Bisogna trovare il modo di sottoporre i gusti degli uomini al calcolo.Perciò si ebbe l’idea di dedurli dal piacere che certe cose fannoprovare all’uomo. Se una cosa soddisfa bisogni o desideri dell’uomosi disse che aveva un valore d’uso, un’utilità. (Pareto)

Our proposalBringing key concepts of utility theory to bear on the analysis of truth

1. ordinal – cardinal

2. certainty – risk – uncertainty

3. preferences – choice

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Ordinal foundations

Jeremy Bentham (1748-1832)

I The amount of pleasure or paincaused by a certain good ismeasurable

I Agents have utils in their heads

Vilfredo Pareto (1848-1923)

I Agents can only tell between twogoods which one they prefer

I Utility has an ordinal meaning

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Representation theorems

I comparative judgments:preferences or indifference

I pairwise evaluation

I � ⊆ X2

I numerical analysis: utilityfunction

I point-wise evaluation

I u : X → R

Representation theoremsIf � satisfies certain conditions then there exists u such that for all x, y ∈ X

x � y ⇐⇒ u(x) ≥ u(y).

[von Neumann & Morgenstern (1947), Savage (1954), Debreu (1954)]

I ‘behavioural’ foundation of measurement

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Back to truth

Graded notions:

I interest in a numerical analysis (quantitative)

I comparative judgements (qualitative) are more plausible

I representation theorems

Qualitative or ordinal

I ‘more or less true’

I ranking alternatives

Quantitative or cardinal

I ‘degrees of truth’

I numerical evaluation

ProblemLay down sufficient conditions for the relation ‘more or less true’ to berepresented by a real-valued valuation

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Formally: the case of Łukasiewicz infinite-valued logic

Language

I L = {p1, p2, . . . }I ¬, ∨I SLI ⊥,>I `

Łukasiewicz valuation functionsv : SL → [0, 1]

1. v(⊥) = 0.

2. v(¬θ) = 1− v(θ)

3. v(θ ∨ φ) = min{1, v(θ) + v(φ)}

Ordinal valuations (‘no less true than’)� ⊆ SL × SL

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Representation theorem for truth1

(T.1) � ⊆ SL2 is complete and transitive

(T.2) > � θ, > � ⊥

(T.3) `Ł θ =⇒ θ ∼ >

(T.4) θ1 � θ2, φ1 � φ2 =⇒ θ1 ∨ φ1 � θ2 ∨ φ2(T.5) θ � φ =⇒ ¬φ � ¬θ

TheoremIf � satisfies axioms (T.1)–(T.5) then there exists a unique Łukasiewiczvaluation v : SL → [0, 1] such that for all θ, φ ∈ SL:

θ � φ =⇒ v(θ) ≥ v(φ).

1Ongoing work with H. Hosni and V.MarraRossella Marrano (SNS) Comparing degrees of truth 6/06/2014 8 / 12

Philosophical implications

Generalizations

I other fuzzy logics

I many-valued logics

I classical logic

Feedback

I real-valued valuation functions arise from certain comparisons betweendegrees of truth of sentences

I natural appeal of the notion ‘no less true than’

I axioms as properties

I independence from the mathematical apparatus

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Further work

1. Introducing uncertainty in the outcomes

Certainty perfect information regarding the outcomeRisk no perfect information, the probabilities are known

Uncertainty no perfect information, unknown probabilities

Expected degrees of truth

EDT (θ) = p(v) · v(φ)

θ � φ⇐⇒ EDT (θ) ≥ EDT (φ)

2. Choice-based approachRevealed preferences Let X 6= ∅, x, y ∈ X and let C(·) be a choicefunction over X.

x � y ⇐⇒ x = C({x, y}).

Truth by choice (conventionalism)

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Conclusion

I Relating truth with utility theory

1. ordinal – cardinal2. certainty – risk – uncertainty3. preferences – choice

I Ordinal foundations for many-valued semanticsI the case of Łukasiewicz real-valued logicI many-valued valuations can be proved to arise from

truth-comparisons under certain conditions

I Philosophical relevance

I Methodological lessonsI New answers to old questionsI New questions!

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References

Roberto L. O. Cignoli, Italia M. L. D’Ottaviano and Daniele Mundici.Algebraic foundations of many-valued reasoning,Trends in Logic – Studia Logica Library, Kluwer Academic Publishers, 2000.

Petr Hájek.Metamathematics of Fuzzy Logic,Kluwer Academic Publishers, 1998.

J. von Neumann, & O. Morgenstern.The Theory of Games and Economic Behavior (2nd ed).Princeton: Princeton University Press, 1947

L. J. SavageThe Foundations of Statistics.Wiley, 1954.

George J. Stigler.The Development of Utility Theory. IThe Journal of Political Economy, Vol. 58, No. 4. (Aug., 1950), pp. 307-327.

George J. Stigler.The Development of Utility Theory. IIThe Journal of Political Economy, Vol. 58, No. 5. (Oct., 1950), pp. 373-396.

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