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Dear colleagues,

in my PhD thesis I'm using the concept of "validity degree" which

seems to be strange or even outright revolting to logicians ;-)

Let me try to explain:

Classically, in many-valued logics, validity is two-valued. The model

relation is usually defined using a set of designated truth values,

calling an interpretation a model of a formula iff the truth value of

the formula under the interpretation falls into the set of designated

truth values.

Obviously, for a non-graded model relation, validity is two-valued: a

formula is said to be valid iff all interpretations are models for it.

It can be argued that for knowledge modelling purposes, a yes-or-no

definition of validity is less than satisfying. Truth values are not

really looked at in mathematical logic, but quantified over when

defining validity, semantic consequence or semantic

equivalence. Without a graded model relation, all other logical

concepts stay a little too `crisp' for the resulting logic to be

called `fuzzy'.

As soon as the model relation becomes graded, however, the validity of

a formula naturally becomes graded too: It is the infimum(*) of the

`modelness degrees' of all interpretations for this formula.

How to define a graded model relation?

Some approaches known from the literature:

----------------------------------------------------------------------

a) [does this approach have a name to it?]

In a many-valued logic, just define the `degree of modelness' to be

the truth value of a formula under the given interpretation.

This approach is problematic in two ways.

First, it doesn't really allow to distinguish between truth values and

validity degrees, so it can't be used for analysing the relationship

between these concepts.

Secondly, it doesn't lead to a logic of very high expressive

power. I'm not aware of a lot of literature where this approach is

used (though I'd be interested to hear of others), one of the earlier

references seems to be [1].

----------------------------------------------------------------------

----------------------------------------------------------------------

b) Possibilistic logic.

In the simplest variant, formulae of two-valued logic are labelled

with elements from the real unit interval called "neccessity

degrees". The label of a formula represents the "trust" in the

information represented by the formula. The higher the value of the

label, the more trust is placed in the source of the information.

The `degree of modelness' of a two-valued interpretation I for a

labelled formula <F,d> is defined as follows:

. If I is a (classical) model for F, the degree is 1.

. If I is no (classical) model for F, the degree is 1-d.

The interpretation of this definition is as follows: If I is no model

of F, but the information represented by F is not fully trusted (d<1),

then <F,d> is given the `benefit of the doubt', i.e. it is accepted

that <F,d> could still be valid up to the degree 1-d.

This definition is especially interesting when drawing semantic

consequences from a set of labelled formulae. In classical logic, when

trying to establish that a formula F follows from a set X of formulae,

the set X acts as a `constraint' on the set of interpretations to be

considered. All interpretations in the constrained set have to be

models of F for F to be a consequence of X. The larger X becomes, the

smaller the set of interpretations, and the easier it becomes for F to

be a consequence of X.

In possibilistic logic, this constraint becomes a fuzzy one and the

set of interpretations to be considered is a fuzzy set. Using the

canonical definition of semantic consequence in this case, it turns

out that the label d of a labelled formula <F,d> acts as a threshold

value for the trust in the information to be considered in deriving F:

<F,d> follows from a set X of labelled formulae iff (essentially) F

follows classically from all formulae labelled in X with a degree at

least as high as d.

Possibilistic logic has been studied intensively in the literature;

compare [2].

----------------------------------------------------------------------

----------------------------------------------------------------------

c) Similarity-based logic.

This approach is also based on two-valued logic. The set of all

interpretations of some given two-valued logic is equipped with a

"similarity relation", i.e. a binary fuzzy relation (mapping pairs of

interpretations to elements of the real unit interval) satisfying the

`canonically fuzzified' properties of a classical equivalence relation

(relexivity, symmetry, transitivity).

The `degree of modelness' of a two-valued interpretation I for a

Formula F is then the infimum of the degrees of similarity of I with

all classical models of F, i.e. the `degree of existence' of a

classical model of F similar with I.

I won't discuss this approach any further here, see for instance [3].

----------------------------------------------------------------------

I've put a relevant section of my thesis at

http://lrb.cs.uni-dortmund.de/~lehmke/pub/sec34.pdf

Now for my first question: Are there any fundamentally different ways

of defining a graded model relation? As far as I know, probabilistic

logic and `uncertainty logics' (based on Dempster-Shafer theory) are

similar to possibilistic logic, only the way of calculating with

labels is different.

Are there any references which absolutely have to be part of a survery

on this subject?

----------------------------------------------------------------------

Next, some words on the approach taken in my thesis:

How can `uncertainty' expressed by truth values (i.e. vagueness) be

combined with `uncertainty' expressed by validity degrees (i.e. graded

illknowledge)?

The approach a described above doesn't offer a lot of possibilities,

because, as already remarked, it mixes truth values and validity

degrees in an unfortunate way.

For similarity-based logic (approach c), several open questions

remain. First of all, what exactly is the semantical meaning of the

degrees of modelness in the current version of this approach?

Secondly, how should the graded model relation be defined when the

underlying logic is many-valued? The definition of the graded model

relation is based on the classical non-graded model relation, so

either the classical definition of a model in many-valued logic is

used (using a set of designated truth values), or the similarity

relation used for defining the graded model relation is modified to

incorporate the truth value of a formula under a given interpretation

(i.e. the similarity relation maps a 4-tuple of two interpretations

and two truth values to a validity degree).

While it would be interesting to develop this further and compare it

to my approach sketched below, it won't be pursued any further here.

Possibilistic logic (approach b) offers a very straightforward

extension to many-valued logic, to be described in the following.

First, a little excurse: What happens if we use a _fuzzy_set_ of

designated truth values in many-valued logic? The degree of modelness

of an interpretation for a formula could be defined to be the degree

of membership of its truth value in the fuzzy set of designated truth

values.

This approach has the great advantage that it's immediately obvious

that the domain and range of the fuzzy set of designated truth values

do not need to be the same.

In fact, by choosing two arbitrary lattices (**) T, D of truth values

and validity degrees, respectively, and defining the fuzzy set of

designated truth values to be a D-fuzzy set on T(***), it is made

absolutely sure from the outset that no confusion between truth values

and validity degrees can arise.

Interestingly, choosing D to be two-valued yields classical

many-valued logic as a special case, and choosing T=D and the fuzzy

set of designated truth values to be identity yields approach a) as a

special case.

Another interesting special case is to choose T to be two valued,

yielding a logic where only validity degrees are many-valued.

Unfortunately, this doesn't make much sense when only one fuzzy set of

designated truth values is considered.

The obvious solution to this dilemma is the way taken in Pavelka's

logic. In Pavelka's logic [4], the set of designated truth values is

_localized_ to formulae by effectively labelling formulae with sets of

truth values (which are restricted to principal filters of the truth

value lattice).

Now, finally, coming to the approach taken in my PhD thesis: Label

formulae with D-fuzzy sets(***) on T, and define the degree of

modelness of a many-valued interpretation I for a labelled formula

<F,L> as L(I(F)) (where I(F) denotes the truth value of F under the

interpretation I).

Obviously, all examples so far (including Pavelka's logic) apart from

approach c) are special cases of this approach. Furthermore, choosing

both T and D to be many-valued yields a straightforward generalization

of possibilistic logic to the case of an underlying many-valued logic.

The main advantage I see in this approach is that it makes absolutely

clear that truth values and validity degrees are distinct and

independent concepts, yet defines a simple, understandable and

straightforward way for getting from the truth value of a formula to a

validity degree.

In my PhD thesis, I'm studying the semantics of logics of this type in

depth. Special cases (taking T to be two-valued leads to possibilistic

logic, taking D to be two-valued leads to [a generalization of]

Pavelka-type logic) are discussed and compared. Part of this is

already published, see for instance [5].

----------------------------------------------------------------------

Now for my second questions: Are there any other studies of logics of

this kind from a mathematical logic point of view?

I know that fuzzy sets of truth values are mentioned at several places

under different names (`truth qualifications' in [6]; `truth value

restrictions' in [7]), but I haven't seen a study in the context of

mathematical logic yet.

Furthermore, the term "fuzzy possibilistic logic" is well-known, but

it is only mentioned, not yet formally studied by Dubois and Prade

(compare [2]). Other mentions for instance in [8] are in a completely

different setting (fuzzy modal logic).

Lastly: Does this lengthy explanation establish a notion of "validity

degree" acceptable to logicians?

Is this kind of logic interesting as a formal tool for reasoning?

Can anyone (but me) see any merit/potential in it?

With many thanks in advance for any comments, references or pointers

Stephan

----------------------------------------------------------------------

(*) When truth values are used to model vagueness, the most sensible

choice of truth value structure is a complete lattice, the unit

element of which means "total truth" and the zero element of which

means "total falsity". This choice will be implicitly assumed in the

following. The most popular particular lattice used is the real unit

interval [0,1] with the usual order of real numbers.

(**) Concerning the choice of a lattice structure for truth values,

see footnote (*). In fact, a lattice is probably the most general

sensible structure for truth values as well as degrees of validity,

when the intention is to model uncertainty.

(***) In my thesis, I'm making the additional assumption that the

fuzzy set is a _D-fuzzy_filter_ of T, in particular, it's monotone and

maps 1 to 1. The same assumption is made in this posting.

There's a lot of justification for this choice, but it'd lead too far

to expand on this here.

----------------------------------------------------------------------

[1]

@ARTICLE{Lee/Chang71,

language = "USenglish",

author = "Richard C. T. Lee and Chin-Liang Chang",

title = "Some Properties of Fuzzy Logic",

journal = {Information and Control},

year = 1971,

volume = 19,

number = 1,

pages = "417-431"}

[2]

@INCOLLECTION{Dubois/Lang/Prade94,

language = "USenglish",

editor = "Dov M. Gabbay and C. J. Hogger and J. A. Robinson",

series = {Handbook of Logic in Artificial Intelligence and Logic Programming},

booktitle = {Nonmonotonic Reasoning and Uncertain Reasoning},

publisher = "Claredon Press",

address = "Oxford",

volume = 3,

year = 1994,

author = "Didier Dubois and J{\'e}r{\^o}me Lang and Henri Prade",

title = {Possibilistic Logic},

pages = "439-513"}

[3]

@INCOLLECTION{Esteva/Garcia/Godo99,

language = "USenglish",

editor = "Didier Dubois and Erich Peter Klement and Henri Prade",

booktitle = "Fuzzy Sets, Logics and Reasoning about Knowledge",

publisher = "Kluwer Academic Publishers",

series = {Applied Logic},

volume = 15,

year = "1999",

author = {Francesco Esteva and Pere Garcia and Llu{\'\i}s Godo},

title = "About Similarity-Based Logical Systems",

pages = "269-287"

}

[4]

@ARTICLE{Pavelka79I,

language = "USenglish",

author = "Jan Pavelka",

title = "On Fuzzy Logic {I} --- {M}any-valued rules of inference",

journal = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der Mathematik},

year = 1979,

volume = 25,

pages = "45-52"}

[5]

@INCOLLECTION{Lehmke99a,

language = "USenglish",

editor = "Vil\'em Nov\'ak and Irina Perfilieva",

booktitle = "Discovering the World with Fuzzy Logic",

publisher = "Physica-Verlag",

address = "Heidelberg",

series = "Studies in Fuzziness and Soft Computing",

volume = 57,

year = "2000",

author = {Stephan Lehmke},

title = "Degrees of Truth and Degrees of Validity --- Two Orthogonal Dimensions of Representing Fuzziness in Logical Systems",

pages = "192-236"

}

[6]

@ARTICLE{Zadeh78,

language = "USenglish",

author = "Lotfi A. Zadeh",

title = "{PRUF} --- A Meaning Representation Language for Natural Languages",

journal = {International Journal of Man Machine Studies},

year = 1978,

volume = 10,

pages = "395-460",

note = "Reprinted in \cite{ZadehSelectedPapers}."}

[7]

@INCOLLECTION{Baldwin81,

language = "USenglish",

editor = "E. H. Mamdani and B. R. Gaines",

booktitle = "Fuzzy Reasoning and its Applications",

publisher = "Academic Press",

address = "London",

year = 1981,

author = "J. F. Baldwin",

title = {Fuzzy logic and fuzzy reasoning},

pages = "133-148"}

[8]

@ARTICLE{Hajek/Harmancova/Verbrugge94,

language = "USenglish",

author = {Petr H{\'a}jek and Dagmar Harmancov\'a and Rineke Verbrugge},

title = "A Qualitative Fuzzy Possibilistic Logic",

journal = {International Journal of Approximate Reasoning},

year = 1995,

volume = 12,

pages = "1-9"}

-- Stephan Lehmke Stephan.Lehmke@cs.uni-dortmund.de Fachbereich Informatik, LS I Tel. +49 231 755 6434 Universitaet Dortmund FAX 6555 D-44221 Dortmund, Germany############################################################################ This message was posted through the fuzzy mailing list. (1) To subscribe to this mailing list, send a message body of "SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at (2) To unsubscribe from this mailing list, send a message body of "UNSUB FUZZY-MAIL" or "UNSUB FUZZY-MAIL yoursubscription@email.address.com" to listproc@dbai.tuwien.ac.at (3) To reach the human who maintains the list, send mail to fuzzy-owner@dbai.tuwien.ac.at (4) WWW access and other information on Fuzzy Sets and Logic see http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info (5) WWW archive: http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html

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