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From Degrees of Truth to Degrees of Consequence
Rossella Marrano
Scuola Normale Superiore, Pisa
Joint work with Hykel Hosni
17 June 2013

Aristotle, Metaphysics, Γ 1008b 31–37
Again, however much all things may be ‘so
and not so’, still there is a more and a less
in the nature of things; for we should not
say that two and three are equally even,
nor is he who thinks four things are ﬁve
equally wrong with him who thinks they
are a thousand. If then they are not
equally wrong, obviously one is less wrong
and therefore more right.
(Translated by W. D. Ross)
Rossella Marrano (SNS) Degrees of consequence 17 June 2013 2 / 24

Isaac Asimov, The Relativity of Wrong, 1989
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.

The problem
intuitive plausibility of the relation “less wrong than”;
under the assumption that it is not simply matter of rhetoric, this
relation seems to concern the notion of truth;
more or less wrong less or more true truth comes in degrees.
Main question
How can we interpret and logically model the resulting graded notion of
truth?
inﬁnite-valued logic.

Łukasiewicz logic Ł∞
L = {p1, . . . , pn}.
C = {¬, →}.
SL.
∞
v : L → [0, 1].
f¬(x) = 1 − x, f→(x, y) = min{1, 1 − x + y}.
Γ |=∞ φ ⇐⇒ ∀v ∈ V if v(Γ) = 1 then v(φ) = 1.
a conclusion follows logically from some premises if and only if,
whenever the premises are true, the conclusion is also true.
(Tarski, 1936)

Why Ł∞ is not a good model?
1. The ﬁne-grain problem:
what does it mean for a sentence to be 0.704638366 true?
to what extent a sentence mapped to 0.704638366 is truer than one
mapped to 0.704638346?
2. General epistemological problems:
what does it mean to be ‘partially true’ or ‘true to a certain degree’?
hidden philosophical assumption about the nature of truth.

Suszko Reduction I
Łukasiewicz is the chief perpetrator of a magniﬁcent conceptual
deceit lasting out in mathematical logic to the present day. [. . . ]
Obviously, any multiplication of logical values is a mad idea.
(Suszko, 1977)
Reminder (Tarskian consequence relation and Tarskian logic)
A Tarskian consequence relation (TCR) for L is a relation |=⊆ 2SL × SL
that satisﬁes the following conditions:
(REF) θ ∈ Γ ⇒ Γ |= θ,
(MON) Γ ⊆ ∆, Γ |= θ ⇒ ∆ |= θ,
(TR) Γ |= θ, Γ, θ |= φ ⇒ Γ |= φ.
A logic L, |= is Tarskian if |= is so.

Suszko Reduction II
Reduction Theorem (Suszko, 1977)
Every Tarskian logic has a bivalent semantics.
TCRs preserve a value from the premises to the conclusion.
Let A be such that v : SL → A and D ⊆ A,
Γ |= φ ⇐⇒ ∀v ∈ V if v(Γ) ∈ D then v(φ) ∈ D.
SR: TCRs are logically bivalent.
Suszko’s Thesis
There are only two logical truth-values: true and false.
Intermediate values are just algebraic truth-values.
Incompatibility between Tarskian consequence relation and
many-valuedness.

Reconsider the notion of many-valuedness
meta-logical bivalence:
There is some metalinguistic bivalence that one will not easily get rid
of: either an inference obtains or it does not, but not both.
Caleiro et al. (2007)
logical many-valuedness:
a logical value may be seen as a value that is used to deﬁne in a
canonical way an entailment relation on a set of formulas. By a
canonical deﬁnition of entailment we mean [. . . ] a relation that
preserves membership in a certain set of algebraic values.
Wansing and Shramko (2008)
|=∞ is 1-preserving.
SR: 1 and 0 are the only logical truth-values.
Each truth-value should be logical!
Consequence should be a−preserving for each a ∈ A.

Taking degrees seriously
Truth-values: a ∈ [0, 1]
Truth-degrees: [a) ⊆ [0, 1], i.e. [a) = { x ∈ [0, 1] | a ≤ x }.
Deﬁnition (Font, 2009)
Γ |=∞ φ ⇐⇒ ∀v ∈ V, for each a ∈ [0, 1]
if ∀γ ∈ Γ v(γ) ≥ a ∈ [a) then v(φ) ≥ a ∈ [a).
[. . . ] a conclusion follows logically from some premises if and only
if, whenever the premises attain a certain degree of truth, the
conclusion also attains the same degree.
(Font, 2009)
|=∞ is canonically deﬁned, whence Tarskian.
|=∞ preserves all subsets [a).

Feedback on the main problem
By shifting the focus from degrees of truth to degrees of consequence we
can trigger a positive feedback which sharpens our intuition on the initial
problem.
1. relational perspective,
2. cardinal vs. ordinal account,
3. intermediate values.

1. Relational perspective
a−preserving consequence leads to a relational perspective.
Compare the following:
Question A
Find x, y ∈ [0, 1] such that
v(θ) = x and v(φ) = y.
Question B
v(φ) ≥
?
v(θ)
B doesn’t run into the ﬁne-grain problem.
cognitive plausibility (analogy with qualitative probability).

2. Cardinal vs. ordinal
“How much is it truer?”
Shift from cardinal to ordinal
The exact position of values on the unit interval does not matter, what
really matters is whether one exceeds the other at some point.
the cardinal account reduces to rhetoric;
it does not make sense to consider a multiplicative factor: expressions
like “doubly true” or “ten times truer” are just ﬁgures of speech.

3. Intermediate values
Reminder: main question
How can we interpret and logically model a graded notion of truth?
inﬁnite-valued logic
graded consequence relation
Feedback
Intermediate values are degrees of consequence, therefore they are logical!
The main problem reﬁned
How should the resulting degrees of consequence be interpreted?

A probabilistic semantics (Knight, Paris, Picado-Muiño)
A probability function on L is a map P : SL → [0, 1] such that for all
θ, φ ∈ SL,
(P1) if |= θ then P(θ) = 1,
(P2) if |= ¬(θ ∧ φ) then P(θ ∨ φ) = P(θ) + P(φ).
Let η, ζ ∈ [0, 1],
Γ η
ζ φ ⇐⇒ for all P on SL,
if P(γ) ≥ η for all γ ∈ Γ then P(φ) ≥ ζ.

KPP Interpretation
η, ζ are degrees of belief (subjective probabilities),
Γ η
ζ φ single rational agent who, having accepted Γ with a threshold
η, is forced to accept φ with a threshold ζ.
Logico-semantical level
v ∈ V
v : SL → {0, 1}
|=
Logico-epistemic level
P ∈ P
P : SL → [0, 1]
η
ζ

De Finetti, B. (1980). Probabilità. Enciclopedia Einaudi, 1146-1187.

Is KPP enough?
That handkerchief which I so loved and gave thee. Thou gavest
to Cassio.
Othello
T(heft) Desdemona’s handkerchief was stolen.
L(oss) Desdemona lost her handkerchief.
G(ift) Desdemona gave away her handkerchief.
Othello doesn’t know (from an epistemic point of view he is
uncertain),
suppose Othello ranks scenarios according to the relation “no less
probable than” >p as follows:
G >p L >p T.

Shakespeare does know how things are:
p(G) = p(T) = 0 e p(L) = 1.
Shakespeare knows that:
1. Othello is uncertain,
2. Othello is wrong in believing
G >p L >p T,
3. he’d be less wrong had his ranking been:
T >p L >p G,
why is that? Shakespeare evaluates according to an objective ordering
<w representing the relation “less wrong than”
L <w T <w G.

Lessons from Shakespeare
Degrees of truth = degrees of belief
Degrees of truth as captured by the relation <w are
agent-independent.
With respect to de Finetti scheme, degrees of truth should be
modelled at the logical level.
And how should they be interpreted?
Proposal:
Degrees of objective probability.

Logical interpretation of probability
Keynes A treatise on probability, 1921
Jeﬀreys Scientiﬁc inference, 1931
Theory of probability, 1939
Carnap Logical foundations of probability, 1950
Probability is interpreted as a logical relation between sentences;
since it is logical is objective.
Classical entailment
θ |= φ ⇒ P(φ|θ) = 1,
θ |= ¬φ ⇒ P(φ|θ) = 0.
Partial entailment
θ |=p φ ⇒ P(φ|θ) = p ∈ [0, 1].

Rehabilitation
To those who speak of objective probability we should say: for
any event E the only objective probability is P(E|E) = 1 if E
obtains and P(E|˜E) = 0 if E doesn’t obtain.
de Finetti (1980)
Objective probability is nothing more than sentences’ truth-value,
according to de Finetti there exist just two logical values, 1 and 0.
If we have logical intermediate values we can recover the idea of degrees of
objective probability.

Conclusion
1. Degrees of truth are not obviously meaningless (plausibility of the
relation “less wrong than”),
2. standard many-valued logics are not adequate: degrees of truth call
for degrees of consequence,
3. an interpretation in terms of subjective probability doesn’t capture the
agent-independency of degrees,
4. proposal: a−preserving consequence whose semantics is objective
probability.

Key references
Caleiro C., Carnielli W., and Coniglio, M. E. (2005). Two’s company:
The humbug of many logical values.
de Finetti, B. (1980). Probabilità. Enciclopedia Einaudi, 1146–1187.
Font, J.M. (2009). Taking Degrees of Truth Seriously, Studia Logica:
An International Journal for Symbolic Logic, 91(3):383–406.
Knight, K. (2002). Measuring inconsistency, Journal of Philosophical
Logic, 31: 77–98.
Łukasiewicz, J. (1970). Selected works (ed. Borkowski, L.),
North-Holland Publishing Company, Amsterdam, London.
Williamson, J. (2010). In Defence of Objective Bayesianism, Oxford
University Press, Oxford, 2010.
Paris, J.B. and Picado Muiño, D. and Roseﬁeld, M. (2009).
Inconsistency as qualiﬁed truth: A probability logic approach.
International Journal of Approximate Reasoning, 50 (8), 1151–1163.

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