RossellaMarrano_G&D2

Analogies between truth and utility
Rossella Marrano
Scuola Normale Superiore
Joint work with Hykel Hosni
Pisa, 7 July 2014
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 1 / 15

Motivation
Degrees of truth Utility
similar criticisms
Ordinal revolution
Rigorous notion of utility

Motivation
similar criticisms
Ordinal revolution
Our proposal
Bringing key concepts and methods of utility theory to bear on the analysis of
truth
1. cardinal – ordinal (representation theorems)
2. certainty – risk – uncertainty – ambiguity
3. preferences – choices

“Truth is rarely pure and never simple”
To say of what is that it is not, or of what is not that it is, is false,
while to say of what is that it is, and of what is not that it is not, is
true. (Aristotle)
Veritas est adaequatio rei et intellectus (Truth is the equation of
thing and intellect). (Thomas Aquinas)
The opinion which is fated to be ultimately agreed to by all who
investigate, is what we mean by the truth. (Peirce)

Classical semantics
I L = fp1; p2; : : : g is a propositional language
I C = f:;^;_;!g is the set of propositional connectives
I SL = f; ; : : : g is the set of sentences on L
Propositional valuations on L are functions v : L ! f0; 1g.

Classical semantics
I L = fp1; p2; : : : g is a propositional language
I C = f:;^;_;!g is the set of propositional connectives
I SL = f; ; : : : g is the set of sentences on L
Propositional valuations on L are functions v : L ! f0; 1g.
Remark
Valuations extend uniquely to SL.
Suppose ; 2 SL, then there exists a function f_ : f0; 1g f0; 1g ! f0; 1g
such that v( _ ) = f_(v(); v()):
p q : p p ^ q p _ q p ! q
1 1 0 1 1 1
1 0 0 0 1 0
0 1 1 0 1 1
0 0 1 0 0 1

Modelling truth (classically)
I truth as property of sentences
I truth as specific value assigned to sentences (truth-value)

Modelling truth (classically)
I truth as property of sentences
I truth as specific value assigned to sentences (truth-value)
I there are exactly two truth-values: true and false [Bivalence]
I each sentence is given exactly one truth-value [Non-contradiction]
I the truth-value of a compound sentence is determined by the
truth-values of its components [Truth-functionality]

Beyond true and false (1920)
Jan Łukasiewicz (1878-1956)
To me, personally, the principle of bivalence
does not appear to be self-evident. Therefore
I am entitled not to recognize it, and to
accept the view that besides truth and
falsehood there exist other truth-values,
including at least one more, the third
truth-value.
One class of such systems seems to have the
same relation to ordinary logic that geometry in
a space of an arbitrary number of dimensions
has to the geometry of Euclid. [. . . ] In these
systems instead of the two truth-values + and
, we have m distinct ‘truth-values’ tl, t2, . . . ,
tm where m is any positive integer.
Emil Post (1897-1954)

Relaxing bivalence
I v : SL ! f0; 1; ?g

Relaxing bivalence
I v : SL ! f0; 1; ?g
I possible, undetermined [Łukasiewicz (1920)]
I shall be in Warsaw at noon on 21 December of the next year
I undetermined by means of algorithms [Kleene (1938)]
Non terminating processes
I nonsense or meaningless [Bochvar (1938)]
This sentence is false

Relaxing bivalence
I v : SL ! f0; 1; ?g
I possible, undetermined [Łukasiewicz (1920)]
I shall be in Warsaw at noon on 21 December of the next year
I undetermined by means of algorithms [Kleene (1938)]
Non terminating processes
I nonsense or meaningless [Bochvar (1938)]
This sentence is false
I v : SL ! [0; 1]
I degrees of truth
What do “degrees of truth” mean? Interpretation and measurability

Cardinal – Ordinal
Jeremy Bentham (1748-1832)
Agents have “utils” in their heads
The amount of pleasure or pain
felt for a good is measurable
Vilfredo Pareto (1848-1923)
Utility has an ordinal meaning
Agents can only tell between two
goods which one they prefer

Representation theorems
[I]n order to examine general economic equilibrium, this measurement [of
the degrees of utility] is unnecessary. It is sufficient to ascertain if one
pleasure is larger or smaller than another. This is the only fact we need
to build a theory. (Pareto, 1898)
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
[I]n order to examine general economic equilibrium, this measurement [of
the degrees of utility] is unnecessary. It is sufficient to ascertain if one
pleasure is larger or smaller than another. This is the only fact we need
to build a theory. (Pareto, 1898)
I comparative judgments:
preferences or indifference
I pairwise evaluation
I X2
I numerical analysis: utility
function
I point-wise evaluation
I u: X ! R
General form of the representation
Necessary and sufficient conditions on a preference relation for the
existence of a(n equivalence class of a) real-valued utility function u such that
x y () u(x) u(y):
Plausibility (behavioural foundations) and mathematical convenience

Back to truth
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. [. . . ] We shall always take the set [0; 1] with its
natural (standard) linear order. (Petr Hájek, Metamathematics of
Fuzzy Logic, 1998)

Back to truth
Fuzzy Logic, 1998)
Artificial precision (Pareto strikes back)
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?

Back to truth
Fuzzy Logic, 1998)
Artificial precision (Pareto strikes back)
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?
The ordinal bell rings!

Ordinal foundations for many-valued semantics
1. natural appeal of the notion being more or less true than
I closeness to the truth
Compare “a square is round” with “a triangle is round”.
I scientific fallibilism
John, when people thought the Earth was flat, they were wrong.
When people thought the Earth was spherical, they were wrong.
But if you think that thinking the Earth is spherical is just as
wrong as thinking the Earth is flat, then your view is wronger
than both of them put together. (Isaac Asimov, The Relativity
of Wrong, 1989)
I mathematical modelling
All models are wrong, but some models are more wrong than others.

Ordinal foundations for many-valued semantics
1. natural appeal of the notion being more or less true than
I closeness to the truth
Compare “a square is round” with “a triangle is round”.
I scientific fallibilism
John, when people thought the Earth was flat, they were wrong.
When people thought the Earth was spherical, they were wrong.
But if you think that thinking the Earth is spherical is just as
wrong as thinking the Earth is flat, then your view is wronger
than both of them put together. (Isaac Asimov, The Relativity
of Wrong, 1989)
I mathematical modelling
All models are wrong, but some models are more wrong than others.
2. axioms as properties

Representation theorems for many-valued semantics
Many-valued valuations can be proved to arise from truth-comparisons under
specific conditions.
The case of Łukasiewicz infinite-valued logic1
(T.1) SL2 is reflexive and transitive
(T.2) , ?
(T.3) `Ł ! =)
(T.4) 1 2; 1 2 =) 1 1 2 2
(T.5) =) : :
If satisfies axioms (T.1)–(T.5) then there exists a Łukasiewicz valuation
v : SL ! [0; 1] such that for all ; 2 SL:
=) v() v():
1Ongoing work with Hykel Hosni and Vincenzo Marra

Pushing the analogy

Pushing the analogy
1. Certainty – Risk – Uncertainty – Ambiguity

Pushing the analogy
Expected truth-value of a sentence
[. . . ] the average of its truth in all the worlds the agent has not ruled out,
weighted according to how likely the agent thinks it is that each of those
worlds is the actual one. (Smith, Vagueness and Degrees of Truth, 2008)

Pushing the analogy
2. Preferences – Choice

Pushing the analogy
2. Preferences – Choice
Truth by choice or ‘revealed truth’
[. . . ] the truth of which we regard as absolute [because we have] freely
conferred this certainty on it by looking upon it as a convention.
Conventions, yes; arbitrary, no.
(Poincaré, Science and Hypothesis, 1905)

Conclusion
similar criticisms
Ordinal revolution

Conclusion
similar criticisms
Ordinal revolution
1. cardinal – ordinal

Conclusion
similar criticisms
Ordinal revolution
I along the way: many disanalogies!

Conclusion
similar criticisms
Ordinal revolution
I along the way: many disanalogies!
True and good as primitive notions

References
Pierpaolo Battigalli, Simone Cerreia-Vioglio, Fabio Maccheroni, Massimo Marinacci
Mixed Extensions of Decision Problems under Uncertainty
2013
Rosanna Keefe.
Theories of vagueness,
Cambridge University Press, 2000.
Petr Hájek.
Metamathematics of Fuzzy Logic,
Kluwer Academic Publishers, 1998.
Roberto Marchionatti and Enrico Gambino
Pareto and Political Economy as a Science: Methodological Revolution and Analytical
Advances in Economic Theory in the 1890s
Journal of Political Economy, Vol. 105, No. 6 (December 1997), pp. 1322-1348.
J. von Neumann, O. Morgenstern.
The Theory of Games and Economic Behavior (2nd ed).
Princeton: Princeton University Press, 1947
George J. Stigler.
The Development of Utility Theory. I
The Journal of Political Economy, Vol. 58, No. 4. (Aug., 1950), pp. 307-327.

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