Vagueness, graded truth and pairwise valuationscintula/slides/fmfl_marrano.pdf ·...

Vagueness, graded truth and pairwise valuations

Rossella Marrano

Scuola Normale Superiore, Pisa

The Future of Mathematical Fuzzy LogicPrague, 17 June 2016

MFL and vagueness

A possible diagnosisThe philosophical discussion on the role and significance of MFLremains focused on real-valued semantics and numerical degrees oftruth.

Rossella Marrano (SNS) The future of MFL 17/6/2016 1 / 15

MFL and vagueness

A possible diagnosisThe philosophical discussion on the role and significance of MFLremains focused on real-valued semantics and numerical degrees oftruth.

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. (Hájek, 1998)

The original cathedralEmploying a well-understood and familiar mathematical structure inorder to

1. model the idea of truth as a graded notion,

2. provide a formal theory of vagueness.

The reality1. DoT have been considered philosophically implausible,

2. MFL has been considered inadequate as a theory of vagueness.

Intrinsic philosophical implausibility of DoT

First, it is necessary to say something about what a degree of truth is.Second, some account must be given of the source and justificationof the numbers that are to be assigned as degrees. (Sainsbury, 1995)

Absence of some substantial philosophical account of what degreesof truth are. (Graff, 2000)

Questions about the nature of degrees of truth:

1. how can truth be measured to an exact extent?

2. how do we interpret the fact that a sentence is true to a certaindegree?

3. what are degrees of truth?

Inadequacy of DoT

[T]he degree theorist’s assignments impose precision in a form thatis just as unacceptable as a classical true/false assignment. [. . . ]All predications of “is red” will receive a unique, exact value, but itseems inappropriate to associate our vague predicate “red” withany particular exact function from objects to degrees of truth. Fora start, what could determine which is the correct function, settlingthat my coat is red to degree 0.322 rather than 0.321? (Keefe,1998)

Artificial precision objectionA semantics based on functions from sentences to degrees of truthcoded by real numbers misrepresents the phenomenon of vagueness.

...and many other objectionsAmbiguity of degrees objection, higher-order vagueness objection,linearity objection and truth-functionality objection.

Inadequacy of DoT

[T]he degree theorist’s assignments impose precision in a form thatis just as unacceptable as a classical true/false assignment. [. . . ]All predications of “is red” will receive a unique, exact value, but itseems inappropriate to associate our vague predicate “red” withany particular exact function from objects to degrees of truth. Fora start, what could determine which is the correct function, settlingthat my coat is red to degree 0.322 rather than 0.321? (Keefe,1998)

Artificial precision objectionA semantics based on functions from sentences to degrees of truthcoded by real numbers misrepresents the phenomenon of vagueness.

...and many other objectionsAmbiguity of degrees objection, higher-order vagueness objection,linearity objection and truth-functionality objection.

No easy way out

I Logic is concerned with valid inferences, rather than with concretetruth-values assignments.

I Structuralist position: all that matters are the structuralproperties of the set.

I Logic as modelling view: the unwanted precision of themathematical structure is treated as an artefact of the model.

Until valuation functions remain the single most important element ofthe formal semantics, issues related to the philosophical status of truthvalues cannot be left out of consideration.

No easy way out

Renovating the cathedral

Instead of concluding that

I graded truth is a meaningless notion,

I graded truth does not make a good theory of vagueness,

we could focus on the comparative aspect of graded truth.

Graded truth (preformal)

I quantitative: φ is true x, with x ∈ [0, 1],

I qualitative or comparative: φ is more true than ψ.

Renovating the cathedral

Instead of concluding that

I graded truth is a meaningless notion,

I graded truth does not make a good theory of vagueness,

we could focus on the comparative aspect of graded truth.

Graded truth (preformal)

I quantitative: φ is true x, with x ∈ [0, 1],

I qualitative or comparative: φ is more true than ψ.

Truth from comparison I

Shift in focus from pointwise valuations, assigning degrees of truthto sentences, to pairwise valuations based on comparativejudgements of the form

“the sentence φ is less (or more) true than the sentence ψ”.

Formalisations of graded truth

I pointwise valuations: v : SL → [0, 1],

I pairwise valuations: � ⊆ SL2.

Truth from comparison I

Shift in focus from pointwise valuations, assigning degrees of truthto sentences, to pairwise valuations based on comparativejudgements of the form

“the sentence φ is less (or more) true than the sentence ψ”.

Formalisations of graded truth

I pointwise valuations: v : SL → [0, 1],

I pairwise valuations: � ⊆ SL2.

Truth from comparison II

Main problem: representation resultsDetermine which conditions a pairwise valuation � should satisfy inorder to guarantee the existence of a pointwise valuation v : SL → [0, 1]representing it, namely such that for all φ, ψ ∈ SL

φ � ψ ⇒ v(φ) ≤ v(ψ).

Pairwise valuations act as

(i) method for evaluating sentences of a given formal language,

(ii) foundation for the standard truth-value approach.

If sentences can be compared ‘well enough’ with respect to their truththen it is as if we attach them a specific truth value.

Truth from comparison II

Main problem: representation resultsDetermine which conditions a pairwise valuation � should satisfy inorder to guarantee the existence of a pointwise valuation v : SL → [0, 1]representing it, namely such that for all φ, ψ ∈ SL

φ � ψ ⇒ v(φ) ≤ v(ψ).

Pairwise valuations act as

(i) method for evaluating sentences of a given formal language,

(ii) foundation for the standard truth-value approach.

If sentences can be compared ‘well enough’ with respect to their truththen it is as if we attach them a specific truth value.

Benefits

I Catching up with the mathematical work in MFL that emphasisesthe comparative aspect of fuzzy logics: real-valued algebras arenot the only intended semantics for these logics!

I Philosophical value:I no recourse to problematic objects,I structuralism taken seriously,I more fundamental level.

I Instrumental value: representation results reintroduce DoT andmake the whole machinery of MFL available.

Degrees of truth explained away

Philosophical underpinningThe numerical aspect of mapping sentences into degrees of truth can beconsidered secondary with respect to the comparative aspect, whichconsists in ranking sentences by performing pairwise comparisons.

Feedback on the nature of degrees of truth:

I they are formal constructions encoding the relative positions ofsentences with respect to the primitive relation more or less truethan;

I they are interpreted as possible measures (or cardinalisations) of acomparative notion of truth.

Degrees of truth explained away

Philosophical underpinningThe numerical aspect of mapping sentences into degrees of truth can beconsidered secondary with respect to the comparative aspect, whichconsists in ranking sentences by performing pairwise comparisons.

Feedback on the nature of degrees of truth:

I they are formal constructions encoding the relative positions ofsentences with respect to the primitive relation more or less truethan;

I they are interpreted as possible measures (or cardinalisations) of acomparative notion of truth.

There is nothing essentially numerical about graded truth, nothing thatforces us to formalise it by using real numbers.

The semantics based on pairwise valuations

I meets our intuitions on the connection between vagueness andgraded truth,

I is an alternative semantics for the logic (strongly sound andcomplete),

I mitigates some of the weak points of the degree-theoreticsemantics,

I provides a possible formal theory of vagueness.

There is nothing essentially numerical about graded truth, nothing thatforces us to formalise it by using real numbers.

The semantics based on pairwise valuations

I meets our intuitions on the connection between vagueness andgraded truth,

I is an alternative semantics for the logic (strongly sound andcomplete),

I mitigates some of the weak points of the degree-theoreticsemantics,

I provides a possible formal theory of vagueness.

Sorites and pairwise judgements

Shift of focus from the predicates P and non-P to the binary relation‘more or less P than’ defined over the objects in the domain of P .

[P]roponents of the standard analysis conceive of borderline cases interms of a certain kind of ordering. They suppose that for any pre-dicate ‘Φ’ having borderline cases, there is some linear ordering ofitems (values) on a dimension decisive of the application of ‘Φ’, pro-gressing from an item that is definitely Φ to an item that is definitelynot-Φ. Call such an ordering a Φ-ordering. (Raffman, 2014)

If P is vague, then the P -ordering is non-trivial, linear, with someintermediate elements (borderline cases) and there may be indefinitelymany steps between the top and the bottom element.

Sorites as a chain in which each object is no P -er than the previousone, with at least some significant differences.

Adjacency

Adjacency: For any objects x and y, if they are adjacent in theP -ordering, then ‘P (x)’ and ‘P (y)’ are adjacent in thetruth ordering or they are in the same equivalence class,‘P (x)′ ∼ ‘P (y)’.

Closeness: For any objects x and y, if they are very close inP -relevant respects, then ‘P (x)’ and ‘P (y)’ are very closein respect of truth.

We are not justified to conclude in general for any given predicate Pthat if two elements xi and xj are adjacent with respect to theP -ordering then they are close in P -relevant respects, unless we knowthat P is vague.

Adjacency 6= Closeness (Smith, 2005)

Not a definition of vagueness

I No notion of almost true, closeness, small step can be expressed incomparative terms,

I the topology over real numbers is necessary in order to be able toformulate distance considerations.

The theory does not have the resources to distinguish fine-graded seriesfrom coarse-graded series and to distinguish vague predicates fromgraded predicates that are not vague (not even from precise predicates!)

Pairwise valuations provide a theory of graded predicates, and if weconsider those that are vague, then it provides a formal theory ofvagueness.

What’s next?

I Better understanding of graded truth as philosophical notion:I is it subjective, objective, intersubjective?I how is it related to graded belief?

I Fruitful generalisations:I predicative case,I graded consequence relations.

I Build bridges with psychology and improve on the cognitiveplausibility of the model.

Děkuji!