The document discusses comparing degrees of truth in a quantitative way using concepts from utility theory. It proposes bringing key concepts like ordinal vs. cardinal, certainty vs. risk vs. uncertainty, and preferences vs. choice to analyze truth. A representation theorem is presented showing that real-valued truth valuations can arise from certain qualitative comparisons between sentences' degrees of truth under some conditions. This approach provides philosophical insights and suggests new ways to address old questions about fuzzy and many-valued logics.

1. Comparing degrees of truth
Lessons from utility theory
Rossella Marrano
Scuola Normale Superiore
Joint work with Hykel Hosni
Rome, 19 June 2014
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2. Motivation
Degrees of truth as real numbers
We shall assume that the truth degrees are linearly ordered, with 1
as maximum and 0 as minimum. Thus truth degrees will be coded
by (some) reals. And even if logics of finitely many truth degrees
can be developed we choose not to exclude any real number from the
set of truth degrees. We shall always take the set [0; 1] with its
natural (standard) linear order. (Petr Hájek, Metamathematics of
Fuzzy 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= true?
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3. 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 fanno
provare all’uomo. Se una cosa soddisfa bisogni o desideri dell’uomo
si disse che aveva un valore d’uso, un’utilità. (Pareto)
Our proposal
Bringing 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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4. Ordinal foundations
Jeremy Bentham (1748-1832)
I The amount of pleasure or pain
caused by a certain good is
measurable
I Agents have utils in their heads
Vilfredo Pareto (1848-1923)
I Agents can only tell between two
goods which one they prefer
I Utility has an ordinal meaning
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5. Representation theorems
I comparative judgments:
preferences or indifference
I pairwise evaluation
I X2
I numerical analysis: utility
function
I point-wise evaluation
I u: X ! R
Representation theorems
If satisfies certain conditions then there exists u such that for all x; y 2 X
x y () u(x) u(y):
[von Neumann Morgenstern (1947), Savage (1954), Debreu (1954)]
I ‘behavioural’ foundation of measurement
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6. 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
Problem
Lay down sufficient conditions for the relation ‘more or less true’ to be
represented by a real-valued valuation
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7. Formally: the case of Łukasiewicz infinite-valued logic
Language
I L = fp1; p2; : : : g
I :, _
I SL
I ?;
I `
Łukasiewicz valuation functions
v : SL ! [0; 1]
1. v(?) = 0.
2. v(:) = 1 v()
3. v( _ ) = minf1; v() + v()g
Ordinal valuations (‘no less true than’)
SL SL
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8. Representation theorem for truth1
(T.1) SL2 is complete and transitive
(T.2) , ?
(T.3) `Ł =)
(T.4) 1 2; 1 2 =) 1 _ 1 2 _ 2
(T.5) =) : :
Theorem
If satisfies axioms (T.1)–(T.5) then there exists a unique Łukasiewicz
valuation v : SL ! [0; 1] such that for all ; 2 SL:
=) v() v():
1Ongoing work with H. Hosni and V.Marra
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9. Philosophical implications
Generalizations
I other fuzzy logics
I many-valued logics
I classical logic
Feedback
I real-valued valuation functions arise from certain comparisons between
degrees 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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10. Further work
1. Introducing uncertainty in the outcomes
Certainty perfect information regarding the outcome
Risk no perfect information, the probabilities are known
Uncertainty no perfect information, unknown probabilities
Expected degrees of truth
EDT() = p(v) v()
() EDT() EDT()
2. Choice-based approach
Revealed preferences Let X6= ;, x; y 2 X and let C() be a choice
function over X.
x y () x = C(fx; yg):
Truth by choice (conventionalism)
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11. Conclusion
I Relating truth with utility theory
1. ordinal – cardinal
2. certainty – risk – uncertainty
3. preferences – choice
I Ordinal foundations for many-valued semantics
I the case of Łukasiewicz real-valued logic
I many-valued valuations can be proved to arise from
truth-comparisons under certain conditions
I Philosophical relevance
I Methodological lessons
I New answers to old questions
I New questions!
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12. 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. Savage
The Foundations of Statistics.
Wiley, 1954.
George J. Stigler.
The Development of Utility Theory. I
The Journal of Political Economy, Vol. 58, No. 4. (Aug., 1950), pp. 307-327.
George J. Stigler.
The Development of Utility Theory. II
The Journal of Political Economy, Vol. 58, No. 5. (Oct., 1950), pp. 373-396.
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