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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 ‘soand not so’, still there is a more and a lessin the nature of things; for we should notsay that two and three are equally even,nor is he who thinks four things are fiveequally wrong with him who thinks theyare a thousand. If then they are notequally wrong, obviously one is less wrongand 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 wasflat, they were wrong. When peoplethought the Earth was spherical, they werewrong. But if you think that thinking theEarth is spherical is just as wrong asthinking the Earth is flat, then your view iswronger than both of them put together.


The problem

I intuitive plausibility of the relation “less wrong than”;

I under the assumption that it is not simply matter of rhetoric, thisrelation seems to concern the notion of truth;

I more or less wrong less or more true truth comes in degrees.

Main questionHow can we interpret and logically model the resulting graded notion oftruth?

I infinite-valued logic.


Łukasiewicz logic Ł∞

I L = {p1, . . . , pn}.I C = {¬,→}.I SL.I `∞

I v : L → [0, 1].I f¬(x) = 1− x , f→(x , y) = min{1, 1− x + y}.

I 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:

I what does it mean for a sentence to be 0.704638366 true?

I to what extent a sentence mapped to 0.704638366 is truer than onemapped to 0.704638346?

2. General epistemological problems:

I what does it mean to be ‘partially true’ or ‘true to a certain degree’?

I hidden philosophical assumption about the nature of truth.


Suszko Reduction I

Łukasiewicz is the chief perpetrator of a magnificent conceptualdeceit 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 × SLthat 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.I 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 ThesisThere are only two logical truth-values: true and false.Intermediate values are just algebraic truth-values.

I Incompatibility between Tarskian consequence relation andmany-valuedness.


Reconsider the notion of many-valuednessI meta-logical bivalence:

There is some metalinguistic bivalence that one will not easily get ridof: either an inference obtains or it does not, but not both.

Caleiro et al. (2007)

I logical many-valuedness:a logical value may be seen as a value that is used to define in acanonical way an entailment relation on a set of formulas. By acanonical definition of entailment we mean [. . . ] a relation thatpreserves membership in a certain set of algebraic values.

Wansing and Shramko (2008)

I |=∞ is 1-preserving.I SR: 1 and 0 are the only logical truth-values.I Each truth-value should be logical!I Consequence should be a−preserving for each a ∈ A.


Taking degrees seriouslyTruth-values: a ∈ [0, 1]

Truth-degrees: [a) ⊆ [0, 1], i.e. [a) = { x ∈ [0, 1] | a ≤ x }.

Definition (Font, 2009)

Γ |=6∞ 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 onlyif, whenever the premises attain a certain degree of truth, theconclusion also attains the same degree.

(Font, 2009)

I |=6∞ is canonically defined, whence Tarskian.

I |=6∞ preserves all subsets [a).


Feedback on the main problem

By shifting the focus from degrees of truth to degrees of consequence wecan trigger a positive feedback which sharpens our intuition on the initialproblem.

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 AFind x , y ∈ [0, 1] such thatv(θ) = x and v(φ) = y .

Question Bv(φ) ≥︸︷︷︸

?

v(θ)

I B doesn’t run into the fine-grain problem.I cognitive plausibility (analogy with qualitative probability).


2. Cardinal vs. ordinal

“How much is it truer?”

Shift from cardinal to ordinalThe exact position of values on the unit interval does not matter, whatreally matters is whether one exceeds the other at some point.

I the cardinal account reduces to rhetoric;I 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 questionHow can we interpret and logically model a graded notion of truth?

I infinite-valued logicI graded consequence relation

FeedbackIntermediate values are degrees of consequence, therefore they are logical!

The main problem refinedHow should the resulting degrees of consequence be interpreted?


A probabilistic semantics (Knight, Paris, Picado-Muiño)

I A probability function on L is a map P : SL → [0, 1] such that for allθ, φ ∈ SL,(P1) if |= θ then P(θ) = 1,

(P2) if |= ¬(θ ∧ φ) then P(θ ∨ φ) = P(θ) + P(φ).

I Let η, ζ ∈ [0, 1],

Γ η Bζ 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 Bζ φ single rational agent who, having accepted Γ with a thresholdη, is forced to accept φ with a threshold ζ.

Logico-semantical levelI v ∈ V

I v : SL → {0, 1}

I |=

Logico-epistemic levelI P ∈ P

I P : SL → [0, 1]

I ηBζ


De Finetti, B. (1980). Probabilità. Enciclopedia Einaudi, 1146-1187.Rossella Marrano (SNS) Degrees of consequence 17 June 2013 17 / 24

Is KPP enough?

That handkerchief which I so loved and gave thee. Thou gavestto Cassio.

Othello

T(heft) Desdemona’s handkerchief was stolen.L(oss) Desdemona lost her handkerchief.G(ift) Desdemona gave away her handkerchief.

I Othello doesn’t know (from an epistemic point of view he isuncertain),

I suppose Othello ranks scenarios according to the relation “no lessprobable than” >p as follows:

G >p L >p T.


I Shakespeare does know how things are:

p(G) = p(T) = 0 e p(L) = 1.

I 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,

I 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

I Degrees of truth 6= degrees of belief

I Degrees of truth as captured by the relation <w areagent-independent.

I With respect to de Finetti scheme, degrees of truth should bemodelled at the logical level.

I And how should they be interpreted?

Proposal:Degrees of objective probability.


Logical interpretation of probability

Keynes A treatise on probability, 1921Jeffreys Scientific inference, 1931

Theory of probability, 1939Carnap Logical foundations of probability, 1950

I Probability is interpreted as a logical relation between sentences;I since it is logical is objective.

Classical entailmentθ |= φ⇒ P(φ|θ) = 1,θ |= ¬φ⇒ P(φ|θ) = 0.

Partial entailmentθ |=p φ⇒ P(φ|θ) = p ∈ [0, 1].


Rehabilitation

To those who speak of objective probability we should say: forany event E the only objective probability is P(E |E ) = 1 if Eobtains and P(E |E ) = 0 if E doesn’t obtain.

de Finetti (1980)

I Objective probability is nothing more than sentences’ truth-value,I 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 ofobjective probability.


Conclusion

1. Degrees of truth are not obviously meaningless (plausibility of therelation “less wrong than”),

2. standard many-valued logics are not adequate: degrees of truth callfor degrees of consequence,

3. an interpretation in terms of subjective probability doesn’t capture theagent-independency of degrees,

4. proposal: a−preserving consequence whose semantics is objectiveprobability.


Key references

I Caleiro C., Carnielli W., and Coniglio, M. E. (2005). Two’s company:The humbug of many logical values.

I de Finetti, B. (1980). Probabilità. Enciclopedia Einaudi, 1146–1187.I Font, J.M. (2009). Taking Degrees of Truth Seriously, Studia Logica:

An International Journal for Symbolic Logic, 91(3):383–406.I Knight, K. (2002). Measuring inconsistency, Journal of Philosophical

Logic, 31: 77–98.I Łukasiewicz, J. (1970). Selected works (ed. Borkowski, L.),

North-Holland Publishing Company, Amsterdam, London.I Williamson, J. (2010). In Defence of Objective Bayesianism, Oxford

University Press, Oxford, 2010.I 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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