Didier Dubois - "Some remarks on truth-values and degrees of belief"
Abstract:
There has been a long-lasting misunderstanding in the literature of artificial intelligence and uncertainty modeling, regarding the role of many-valued
logics (and fuzzy logic). The recurring question is that of the mathematical and pragmatic meaningfulness of a compositional calculus and the validity of
the excluded middle law. This confusion even pervades the early developments of probabilistic logic, despite early warnings of some philosophers of
probability. This talk discusses some aspects of this misunderstanding.
It suggests that the root of the controversies lies in the unfortunate confusion between degrees of belief and what logicians call "degrees of truth". The
latter are usually compositional, while the former cannot be so. It is recalled that any belief representation where compositionality is taken for granted
is bound to at worst collapse to a Boolean truth assignment and at best lead to a poorly expressive tool. We display the non-compositional belief
representation embedded in the standard propositional calculus. It turns out to be an all-or-nothing version of possibility theory. This framework is then
extended to discuss the case of fuzzy logic versus possibility logic. Moreover other kinds of logics seem to suffer from the same difficulties, like Belnap
4-valued logics, and partial logic, as well as some attempts to cast rough sets into a three-valued logic.
Richard Zach - "Semantics for vagueness vs. logics for vagueness: The case of fuzzy
logics"
Abstract:
It is a commonplace in the recent philosophical literature on vagueness
that degree-theoretic approaches to vagueness, such as those based on
fuzzy logic, suffer from major difficulties. I will survey these
objections and evaluate the prospects of taking fuzzy logic as the right
approach to analyzing inference in the presence of vagueness. I will
argue that the difficulties discussed affect the semantics of vague
propositions, but that a case can nevertheless be made for thinking that
fuzzy logic is a contender as a logic of vagueness. I will focus
specifically on Gödel logics, since they, I think, are the best
candidates for fuzzy logics of vagueness that escape the objections.
Daniele Mundici - "Regularity conditions for nonboolean partitions
and projective lattice-ordered groups"
Abstract:
Up to logical equivalence, the elements of
free MV-algebras are just formulas in the
infinite-valued Lukasiewicz calculus.
For these algebras we present a characterization
that does not mention piecewise linear functions
and free variables.
Peter Vojtas - "Relations between deductive and inductive models of
imperfect information and knowledge"
Abstract:
We consider the problem of querying and combining imperfect
information in a distributed environment, typicaly in semantic web. We
present some motivation examples and tools for solving respective
querying problems, mainly based on different quantitative models of
logic programming. Main points are monotonicity of preference modeled by
comparative notion of truth, aggregation of different witnesses and/or
features, relations between fuzzy and annotated logic based models and
algorithms for combining information in a restricted mode of access. We
consider learning of user dependent aggregation function by an ILP
method and comment on some experiments.
Viorica Sofronie - "Representation theorems for lattice-ordered structures
and automated theorem proving in non-classical logics"
Abstract:
We present several classes of non-classical logics (ranging from
many-valued and fuzzy logics to modal and description logics)
which can be translated into tractable and relatively simple fragments
of classical logic. We show that, in many cases, such translations
can be obtained using representation theorems for the algebraic
models of the logic under consideration.
In this context, we show that refinements of resolution can often be
used successfully for automated theorem proving, and in many cases
yield optimal decision procedures.
Francesc Esteva - "Rational extensions of some Weak Nilpotent Minimum logics"
Abstract:
In this paper we investigate some extensions of propositional Weak
Nilpotent Minimum logic, a weaker logic than both G\"odel and
Nilpotent Minimum logics, by adding rational truth-values as truth
constants in the language and by adding corresponding book-keeping
axioms for the truth-constants. Weak and strong standard
completeness of these logics are studied in the general case and when we
restrict ourselves to formulas of the kind $\overline{r} \to
\varphi$, where $r$ is a rational in [0, 1] and $\varphi$ is a
formula without rational truth-constants. The rational extensions of
G\"odel and Nilpotent Minimum logics appear as interesting particular
cases of the above extensions.
Albert Visser - "The Janus Faced Nature of Polish Notation"
Abstract:
We show how to provide Polish Notation with an unstructured language and
semantics: any string of symbols of the alfabet will have an
interpretation.
Concatenation of strings will correspond to a meaningful semantical
operation. Our analysis is a theoretical articulation of a way of
looking at
Polish Notation that is familiar from e.g. the use of RPN in
Hewlett-Packard
calculators.
Our analysis leads to the construction of a monoidal operation
on the arrows of a category extending composition-of-arrows.
We can look at this construction as exploiting systematic typical ambiguity.
We may use our analysis to formulate a semantics for Dynamic Predicate
Logic,
DPL. The version of DPL, so obtained, is simultaneously a version of
Discourse Representation Theory, DRT. Thus we provide one possible
analysis of the relationship between DPL and DRT.
In my talk I will concentrate on the basic construction for the
case of Polish notation. Tempore volente, I will sketch the connections
with DPL and DRT.
Sándor Jenei - "On monoidal operations for non- classical logics"
Abstract:
The rotation construction and the
rotation-annihilation construction are
defined. Some theorems concerning
the continuity points of left-
continuous t-norms are presented.
Then the standard completeness for non-commutative
monoidal t-norm logic is mentioned. Finally, we
introduce the notion of involutive
elements of residuated l-
semigroups. We prove some
interesting theorems related to
involutive elements, which seem to
support the importance of this new
notion.
Reiner Haehnle - "Many-Valued logic in formal specification languages"
Abstract:
Many-valued, to be more precise, 3- or 4-valued logics have long been
used to model partiality in formal specification languages. In fact,
this application is often mentioned as a justification to use multiple
truth values. I give an overview and critical evaluation of different
approaches taken to model partially defined functions in such
formalisms as Z, B, VDM, OCL, JML, CASL, plus ad-hoc suggestions for
many-valued first-order logics coming from the automated reasoning
community. As will be seen, the available usage scenarios of
many-valued logic in specification are not very satisfying. I explain
why, and I try to suggest what could be done about it.
Arnon Avron - "Non-deterministic multiple-valued structures - a general semantic framework"
Abstract:
The standard method of providing semantics for logics is by using
what is known as multi-valued matrices. The idea behind this method
is that every formula of a given logic is assigned a value from a
certain set of possible "truth-values", so that the value of a complex
formula is completely determined by the values assigned to its immediate
subformulas (according to the operations of the matrix which is used
for that particular logic). The main idea we are going to present
is to use a generalization of ordinary matrices in which
non-deterministic computations of truth-values are allowed, so that given
the valued assigned to its subformulas, the
value that a valuation assigns to a complex formula
can be chosen non-deterministically from a certain nonempty set of options.
We call structures which are based on this idea Non-deterministic
matrices (Nmatrices). We will present basic results concerning logics
based on Nmatrices (like a general compactness theorem), and many examples
of applications of Nmatrices for logics related to uncertainty.
Norbert Preining - "Semantics of Gödel logics"
Abstract:
We discuss the semantics of Gödel logics, characterize which are
axiomatizable and which are not r.e. and present preliminary results on
the number of Gödel logics.
Christian Fermueller - "Revisiting Giles:
On Bets, Dialogue Games, Fuzzy logics, and Hypersequents"
Abstract:
Already in the 1970s Robin Giles has shown that a combination of
Lorenzen's dialogue game with a specific way to calculate expected losses
associated with bets on the results of `atomic experiments' leads
to a characterization of {\L}ukasiewizc logic.
We extend Giles remarkable result in several ways that
lead to a characterization of other important fuzzy logics,
including G\"odel and Product logic. In particular we show
how one can obtain uniform hypersequent calculi from a systematic
search for winning strategies in appropriate versions of the game.
Dov Gabbay - "Temporal Dynamics of Argumentation Networks"
Abstract:
We consider networks where nodes carry numerical values and can attack
other nodes by propagating their values.
Some combinations of values give known fuzzy combining functions,
otherwise the set-up is much more general and should be viewed as
arguments attacking or supporting otner arguments.
George Metcalfe - "Uninorm based Fuzzy Logics"
Abstract:
Uninorms are a generalization of t-norm and t-conorm operators
commonly used to interpret "ands" and "ors" in fuzzy logics i.e.
commutative, associative, increasing binary functions on the real unit
interval [0,1], with an identity element e that can appear anywhere in
the interval (e=1 giving a t-norm, e=0 giving a t-conorm). The idea of
this talk is to generalize the well-known construction of fuzzy logics
based on t-norms to fuzzy logics based on uninorms. To this end
we give an axiomatization for a new basic fuzzy logic U which can be
viewed both as Monoidal t-norm based fuzzy logic MTL without weakening,
and as a kind of ``linear linear logic''. We then investigate axiomatic
extensions of U which correspond to logics based on particular classes of
uninorms, including a logic of idempotent uninorms which turns out to be the
relevant logic RM, and a logic of continuous uninorms related to
the combining function of the expert system MYCIN.
Lluis Godo - "Reasoning with partial degrees of truth in t-norm based fuzzy
logics"
Abstract:
We survey known extensions 'a la Pavelka of different t-norm based propositional logics
by adding into the language rational truth-values as truth constants and by adding corresponding book-keeping
axioms for the truth-constants. These include recent results on the extensions of some parametric families of
Weak Nilpotent Minimum logics, weaker than both Goedel and Nilpotent Minimum logics. Weak and strong standard
completeness of these logics are studied in general and in particular when we restrict ourselves to formulas of the
kind r* -> P, where r is a rational in [0, 1] and P is a formula without rational truth-constants. We also comment
about applications of these rational extensions to define logics for reasoning under uncertainty.
Arnold Beckmann - "On a connection between Gödel logics and linearly ordered Kripke frames"
Abstract:
We will explain that first order Gödel logics exactly correspond
to the first order logics characterized by countable linearly ordered Kripke frames with constant domain.
We will draw some corollaries coming from this correspondence.
