ABSTRACT. I discuss a collection of problems in relevance logic.

The main problems discussed are: the decidability of the positive semilattice system,

decidability of the fragments of R  in a restricted number of variables,

and the complexity of the decision problem for the implicational fragment of R.

Some related problems are discussed along the way.

ABSTRACT. In the first talk of this tutorial I offer an introduction to Lukasiewicz propositional logic that differs from
the standard ones in that it does not start from real-valued valuations as a basis for the semantical definition of the
system. Rather, I show how a necessarily informal but rigorous analysis of the semantics of certain vague predicates naturally
leads to axiomatisations of Lukasiewicz logic. It is then the deductive system itself, now motivated by the intended
semantics in terms of vagueness, that inescapably leads to magnitudes — the real numbers or their non-Archimedean
generalisations

ABSTRACT. This is a talk in the branch of logic known as model theory, more precisely, in o-minimality. The first example of an o-minimal structure is the ordered algebraic structure on the set of real numbers and I shall focus on expansions of this structure. Being o-minimal means that the first order definable sets in the structure do not exhibit wild phenomena (this will be made precise). After discussing some basic theory of such structures I shall present some recent applications to diophantine geometry.

ABSTRACT. The study of modal extensions of main systems of mathematical fuzzy logic is a currently ongoing research topic. Nonetheless, several recent works have been already published on these modal extensions, covering different aspects. Among others, Hansoul and Teheux have investigated modal extensions of Lukasiewicz logic, Caicedo and Rodriguez have focused on the study of modal extensions of Godel fuzzy logic, while Bou et al. have studied modal logics over finite residuated lattices. However, the study of modal extensions over the product fuzzy logic has remained unexplored. In this paper we present some results that partially fill this gap by studying modal extensions of product fuzzy logic with Kripke style semantics where accessibility relations are crisp, and where the underlying product fuzzy logic is expanded with truth-constants, the Monteiro-Baaz Delta operator, and with two infinitary inference rules. We also explore algebraic semantics for these fuzzy modal logics.

ABSTRACT. Decidability of the validity problem is established for a family of many-valued modal logics (including modal logics based on infinite-valued G̈odel logics), where propositional connectives are evaluated locally at worlds according to the order of values in a complete chain and box and diamond modalities are evaluated as infima and suprema of values in (many-valued) Kripke frames. When the chain is infinite and the language is sufficiently expressive, the standard semantics for such a logics lacks the finite model property. However, the finite model property does hold for a new equivalent semantics for the same logics and thus decidability is obtain.

ABSTRACT. Recent work in many-valued modal logic has introduced a very general method for defining the (least) many-valued modal logic over a given finite residuated lattice. The logic is defined semantically by means of Kripke models that are many-valued in two different ways: the valuations as well as the accessibility relation among possible worlds are both many-valued. Providing complete axiomatizations for such logics, even if we enrich the propositional language with a truth-constant for every element of the given lattice, is a non-trivial problem, which has been only partially solved to date. In this presentation we report on ongoing research in this direction, focusing on the contribution that the theory of natural dualites can give to this enterprise. We show in particular that duality allows us to extend the above method to prove completeness with respect to local modal consequence, obtaining completeness for global consequence, too. Besides this, our study is also a contribution towards a better general understanding of quasivarieties of (modal) residuated lattices from a topological perspective.

ABSTRACT. We propose a framework in terms of domain theory for semantic information models. We show how an artificial agent (the computer) can operate within such a model in a multiple attitude environment (fuzziness) where information is conveyed. The approximation relation in some of these models can be expressed in terms of deducibility in some relevant logics.

ABSTRACT. The notion of {\it fuzzy quantifier} as a generalization of the classical `for all'' and `there exists' was introduced by L.A. Zadeh in 1975. This provided a semantics for fuzzy modifiers such as {\it most, many, few, almost all}, etc. and introduced the idea of reasoning with syllogistic arguments along the lines of `$\!${\it Most} men are vain; Socrates is a man; therefore, it is {\it likely} that Socrates is vain', where vanity is given as a fuzzy predicate. This and numerous succeeding publications developed well-defined semantics also for {\it fuzzy probabilities} (e.g., {\it likely, very likely, uncertain, unlikely}, etc.) and fuzzy {\it usuality modifiers} (e.g., {\it usually, often, seldom}, etc.). In addition, Zadeh has argued at numerous conferences over the years that these modifiers offer an appropriate and intuitively correct approach to nonmonotonic reasoning.

The matter of exactly how these various modifiers are interrelated, however, and therefore of a concise semantics for such syllogisms, was not fully explored. Thus while a new reasoning methodology was suggested, it was never developed. The present work has grown out of an effort to realize this goal. A previous paper in Artificial Intelligence defined a formal system {\bf Q} of `qualified syllogisms', together with a detailed discussion of how the system may be used to address some well-known issues in the study of default-style nonmonotonic reasoning. That system falls short of the overall goal, however, in that it deals only with crisp predicates. A research note recently submitted to Artificial Intelligence takes the next step by creating a system that accommodates fuzzy predicates. The present abstract overviews these works.

ABSTRACT. During the last decades Fuzzy Set Theory has become an important method in dealing with vagueness in engineering, economics and many other applied sciences. Alongside this development, there has been significant segregation: fuzzy logic in broad sense include everything that is related to fuzziness and is mostly oriented to real-world applications, while fuzzy logic in narrow sense, also called mathematical fuzzy logic, develops mathematical methods to model vagueness and fuzziness by well-defined logical tools. These two approaches do not always meet each other. This is unfortunate, since theory should always reflect practice, and practice should draw upon the best theories. In this work, we try to bridge the gap between practical applications of Fuzzy Set Theory and mathematical fuzzy logic. We demonstrate how continuous [0,1]-valued fuzzy sets can be naturally interpreted as open formulas in a simple first order fuzzy logic of Pavelka style; a logic whose details we discuss here. The main idea is to understand truth values as continuous functions; for single elements x the truth values are constant function defined by the membership degree, for open formulas they are the membership functions, where the base set X is scaled to the unit interval [0,1], for universally closed formulas truth values are definite integrals understood as constant functions. We also introduce constructive existential quantifiers. We show that this logic is complete in Pavelka's sense and generalize all classical tautologies that are definable in the language of this logic. However, all proves and many details are omitted.

ABSTRACT. We characterize the class of partially ordered sets that are embeddable into distributive lattices.

ABSTRACT. Duality theory provides well-understood and parametrizable machinery for relating logics (more generally algebraic structures) of various kinds and their topological semantics. Stone and Priestley duality are of course the best known examples, but many others, including some versions of multi-valued logic have been profitably sent through this machinery. The results, however, are inherently two-valued in that the resulting topological structures are always Stone spaces. Any ``multi-valuedness'' is carried by the structure of a discrete dualizing (truth value) object. So the topological structure of the semantics is still essentially Boolean.

In this work, we develop techniques to deal directly in three-valued semantics whereby a proposition may be affirmed, denied or neither. The semantics is bitopological. One topology provides the possibilities for affirming propositions and another topology provides the possibilities for denying them. The two topologies need not by identical (when negation is not present in the langauge). Even more importantly, it need not be the case that any two distinct models can be separated by a single classical proposition. So the underlying toplogical semantics is not based on Stone spaces.

ABSTRACT. In this contribution, we investigate logics which expand the four-valued Belnap-Dunn logic by a notion of inconsistency. In order of increasing expressive power, this can take the form of an expansion by an inconsistency predicate, an inconsistency constant, or a reductio ad contradictionem negation. After briefly motivating the study of inconsistency and reviewing some related work, we introduce these logics as the quasiequational logics of a certain class of algebras and then discuss their relational semantics. This leads us to a logic which is a natural conservative extension of both classical and intuitionistic logic.

ABSTRACT. During the meta-mathematical period of logic flourishing in the 1930s, the paradox of Liar gave way to proofs of classical limitative theorems—e.g., Gödel’s [1930] Incompleteness Theorems, Church’s [1936] proof of the Unsolvability of the Entscheidungsproblem, and Tarski’s [1936] proof of the Undefinability of Truth. Ways of overcoming these limitations were initially explored by Kleene [1938] using partial recursive functions. The semantic equivalent of Kleene’s approach uses truth-value gaps to overcome Tarski’s Undefinability Theorem. Formal languages with truth-representing truth predicates were constructed by van Fraassen [1968], Woodruff and Martin [1975], and Kripke [1975]. By weakening the assumption of bivalence, these formal solutions exploited meta-language reasoning to prove paradoxical sentences were quarantined by forced assignment to truth-value gaps. Skepticism as to whether these truth-value gap theories actually “solve” the paradoxes is supported by strengthened versions of the Liar formalizing the semantic concepts used to block the paradoxes and showing that fundamental semantic principles cannot be expressed without reintroducing paradox [Mar 1985]. An alternative approach to solving the paradoxes with truth-value gaps is seeking richer semantic patterns of paradox using many-valued and infinite-valued fuzzy logics with degrees of truth (Mar and Grim [1991], Grim, Mar and St. Denis [1992], and Mar [2001]). This approach can be seen as diverging from Tarski’s classic analysis of the Liar by

(1) generalizing bivalent logical connectives to an infinite-valued logic with degrees of truth /~p/ = 1 − /p/ /(p ∧ q)/ = MIN {/p/, /q/} /(p ∨ q)/ = MAX {/p/, /q/} /(p → q)/ = MIN {1, 1 − /p/ + /q/} /(p ↔ q)/ = 1 − ABS { /p/ − /q/}

(2) replacing Tarski’s bivalent (T) schema with Rescher’s Vvp schema for many-valued logics /Tp/ = 1 − ABS {t − /p/} /Vvp/ = 1 − ABS {v − /p/}

(3) modeling self-reference as semantic feedback thus allowing us to embed the semantics in the mathematics and geometry of dynamical systems theory. This is done by replacing the constant truth-value v in the Vvp schema with expressions S(xn) representing the value the sentence attributes to itself in terms of a previously estimated values xn.

• Continuous-Valued Liars (“I am as true a the truth-value v”) with S(xn) = v, yielding the Classical Liar for v = 0, Rescher’s fixed-point “solution” to the Liar for v = ½, and Kripke’s Truth-Teller for v = 1.

• The Cautious Truth-Teller (“I am half as true as I am estimated to be true”) with S(xn) = xn/2.

• The Contradictory Liar (“I am as true as the conjunction of my estimated value and the estimated value of my negation”) with S(xn) = MIN{ xn, 1 − xn}.

The semantic differences among these sentences can be made visually perspicuous using a web diagram. The web diagrams for the Continuous-Valued Liars appears as nested series of simple squares ranging from the Classical Liar to the Truth-Teller with a singular fixed-point at ½. The web diagram for the Cautious Truth-Teller is a fixed-point attractor: no matter what initial value with which we begin other than precisely the fixed-point 2/3: the successively revised estimated values are inevitably drawn toward that fixed-point. The web diagram for the Contradictory Liar, in contrast, is a fixed-point repellor: for any values other than the fixed-point 2/3, the successively revised values are repelled away from 2/3 until the values settle on the oscillation between 1 and 0, characteristic of the Classical Liar. In short, the Cautious Truth Teller and the Contradictory Liar, while identical to the Classical Liar on the values 0 and 1, exhibit diametrically opposed semantic behavior in the interval (0,1). This example provides a justification for degrees of truth in an infinite-valued logic approach: bivalence masks intriguing semantic differences. Instead of taming patterns of semantic paradox by excluding semantic cycles (Barwise and Etchemendy [1987]) or seeking semantic stability (Gupta [1982] and Herzberger [1982]), this approach seeks semantic complexity and chaotic instability. The simplest generalizations of the classical bivalent Liar in the context of fuzzy logic with degrees of truth generate semantic chaos. The Chaotic Liar (“I am as true as I am estimated to be false”) is geometrically represented by the chaotic tent function. Using the squaring function for the modifier ‘very’ (Zadeh [1975]), we obtain the Logistic Liar (“I am as true as I am estimated to be very false”) represented by another paradigmatically chaotic function. These semantic generalizations of the paradox of the Liar are chaotic in a precise mathematical sense. Consider a pair of paradoxical statements known as the Dualist Liar:

Aristotle, “What Epimenides says is true.” Epimenides, “What Aristotle says is false.”

We can model the Dualist Liar as a pair of dynamical systems:

xn+1 = 1 − ABS {yn} yn+1 = 1 − ABS {(1 − xn) − yn }

Counting the number of iterations required for the ordered pairs (xn, yn) to exceed a threshold of the unit circle centered at (0,0), we obtain an escape-time diagram. Self-symmetry on descending scales characteristic of Zeno’s paradoxes (Mar and St. Denis [1999], Stewart [1992]) yields fractal images of semantic chaos. Following the lead of the limitative theorems of Gödel [1930] and Tarski [1936], what is initially a paradox of semantic chaos can be turned into a proof. Using a Strengthened Chaotic Liar, we can use well-known methods to prove that the set of functions chaotic on the [0,1] interval is not definable (Mar and Grim [1991]), Grim, Mar and St. Denis [1992], Mar [2001]). Paradox is not illogicality, but it has been a trap for logicians: the semantic paradoxes look just a little simpler and more predictable than they actually are. Our goal here is to offer glimpses into the infinitely complex, chaotic, and fractal patterns of semantic instability that have gone virtually unexplored.

References

Anderson, C. Anthony and Mikhail Zeleny (eds.) [2001]. Logic, Meaning and Computation: Essays In Memory of Alonzo Church (Netherlands: Kluwer Academic Publishers). Barwise, Jon and John Etchemendy [1987]. The Liar (New York, New York: Oxford University Press). Church, Alonzo [1936]. A note on the Entscheidungsproblem, J. Symb. Log. 1, 40-1, correct., ibid., 101-2. Devaney, Robert [1989]. An Introduction to Chaotic Dynamical Systems, 2nd edition (Menlo Park, California: Addison-Wesley). van Fraassen, Bas [1968]. Presupposition, implication, and self-reference, J. of Phil. 65, 136-152. Gödel, Kurt [1931] Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38, 173-198. Grim, Patrick, Gary Mar and Paul St. Denis [1998]. The Philosophical Computer: Exploratory Essays in Philosophical Computer Modeling (Cambridge, Mass.: MIT Press). Herzberger, Hans [1982]. Notes on naive semantics, J. of Phil. Logic 11, 61-102. Kleene, Stephen [1938]. On notations for ordinal numbers, J. of Sym. Logic 3, 150-55. Kripke, Saul [1975]. Outline of a theory of truth, J. of Phil. 72, 690-716. Mar, Gary [1985]. Liars, Truth-Gaps, and Truth, Dissertation (UCLA), director Alonzo Church (Ann Arbor, Michigan: University Microfilms International). Mar, Gary [1992]. Chaos, fractals, and the semantics of paradox. Newsletter on Computers Use in Philosophy, American Philosophical Association Newsletters 91 (Fall), 30-34. Mar, Gary [2001]. Church’s Theorem and Randomness in Anderson and Zeleny [2001], 479-90. Mar, Gary and Patrick Grim [1991]. Pattern and chaos: new images in the semantics of paradox, Noûs 25, 659-93. Mar, Gary and Paul St. Denis [1999]. What the liar taught Achilles, J. of Phil. Logic 28, 29-46. Martin, Robert [1970]. A category solution to the Liar, in Robert L. Martin (editor) The Liar Paradox (New Haven, Conn.: Yale University Press). Martin, Robert and Peter Woodruff [1975]. On representing ‘true-in-L’ in L, Philosophia 5, 217-21. Rescher, Nicholas [1969]. Multi-Valued Logic (New York: McGraw-Hill). Stewart, Ian [1993]. A partly true story, Scientific American 268 (Feb.), 110-112 discusses Mar and Grim [1991]. Tarski, Alfred [1936]. Der wahrheitsbegriff in den formalisierten sprachen, Studien Philosophica I, 261-405 (translated by J. H. Woodger [1983], The concept of truth in formalized languages, and anthologized in J. Corcoran (ed.) [1983] Logic, Semantics and Metamathematics (Indianapolis, Indiana: Hackett). Zadeh, Lofti [1975]. Fuzzy logic and approximate reasoning, Synthese 30, 407-38.

ABSTRACT. We describe some central facets of the foundational system \pounds and give an account which supports an interpretation which assigns truth degrees to its sentences in the rational interval [-1,1],

ABSTRACT. Every first-order set of sentences T, viewed as a set of axioms, gives rise to a classical theory T_c when T is closed under consequences of classical logic, or an intuitionistic one, T_i, when T is closed under consequences of intuitionistic logic. While it is clear that every sentence which is intuitionistically provable is also classically provable, the question when the converse holds leads us to the so-called conservativity problem. More precisely, given a class G of formulas, we say that the theory T_c is G-conservative over its intuitionistic counterpart T_i iff for all formulas A from the class G, A is provable in T_i whenever A is provable in T_c.

A typical example is that of classical Peano Arithmetic, PA and its intuitionistic counterpart, Heyting Arithmetic HA. The well-known result concerning these two theories states that PA is Pi_2-conservative over HA.

In our talk we consider possible generalizations of this result. However, instead of using syntactic methods, we exploit semantic methods and present some new conservativity results proven by means of Kripke models for first-order theories.