**Workshop on
Logical Constants**

August 7-12, 2011, Ljubljana, Slovenia

part of ESSLLI 2011

Organizers

Denis Bonnay (Paris Ouest University)

Dag Westerståhl (University of Gothenburg)

Co-organizer

Uli Sauerland (Zentrum für Allgemeine Sprachwissenschaft, Berlin)

Workshop Description

All logical systems make a distinction between logical and non-logical symbols, and the meaning of the former needs to be specified in detail and in effect defines the logic in question. This distinction is usually stipulated (though it can be argued that natural languages make a similar distinction), but the issue of the grounds for it, i.e. of what characterizes a logical constant, is a central question in logic, cutting across the huge variety of logical systems existing today. This question has been tackled from various sides, ranging from historical investigation to formal criteria spelled out within different logical frameworks. The main aim of the workshop is to gain a better understanding of the problem by bringing together complementary approaches coming from various fields:

*** Logic. Given a particular logical framework (e.g. semantic, proof-theoretic, game-theoretic), is there a systematic way to characterize the class of logical constants within that framework? Is it possible to provide an integrated account covering the various frameworks?

*** Philosophy of Logic and Mathematics. What are the relevant conceptual analyses of logicality? What is the philosophical significance of the distinction for the philosophy of logic and philosophy of mathematics, e.g. regarding the success or failure of logicism?

*** Linguistics. Is there a linguistic or psycholinguistic « natural kind » corresponding to logical words?

*** History of Logic. How did the question emerge? What are the relationships between historical forerunners of the distinction (such as the distinction between categoremata and syncategoremata) and the contemporary idea of logical constants?

The workshop is organized as part of ESSLLI, August 1-12, 2011 (http://esslli2011.ijs.si). Participants are required to register at ESSLLI 2011, and can attend any other ESSLLI courses and workshops of their choice.

Supported by

*** European Science Foundation (ESF)

*** The Swedish Research Council

*** ESSLLI 2011

Abstracts

J. van Benthem (Amsterdam and Stanford University)

Logical Constants in a Dynamical Perspective

Handout + relevant paper "Logical Constants, the various fortunes of an elusive notion" (in W. Sieg, R. Sommer, and C. Talcott eds,

Abstract: In logical systems for information flow and many-agent interaction, logical constants play two different roles. They structure propositions, but they also structure actions of various sorts. We will discuss this broader theatre in general terms, trying to extend existing approaches to what makes logical constants so special.

D. Bonnay (Paris Ouest University) and Dag Westerståhl (University of Stockholm)

Constants and Consequences

Slides

Abstract: Given an interpreted language and a set of logical constants, Tarski's semantic definition of logical consequence yields a consequence relation. But given a consequence relation, is there a natural way to extract from it a set of logical constants? In this talk, we will compare two ways of doing so, one purely syntactical, which is based on the idea that an expression is logical if it is essential to the validity of at least one inference, and one semantical, which is based on the idea that an expression is logical if its interpretation is fully determined by the rules for its use.

K. Došen (Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, and Faculty of Philosophy, University of Belgrade)

Logical Constants and Adjunction

Slides + bibliography

Abstract: The idea that logical constants are those that can be characterized by proof-theoretical means is related to Lawvere's thesis that logical constants are tied to functors in adjoint situations. From the perspective of categorial proof theory, Lawvere's thesis accords rather well with intuitionistic logic ("rather well" and not "perfectly" because of distribution of conjunction over disjunction). Its status with other logics is questionable. In particular, it does not fare very well in classical logic, although that logic can be understood sensibly in categorial proof theory.

C. Dutilh Novaes (University of Groningen)

Syncategoremata are not logical constants

Slides + relevant paper "Reassessing logical hylomorphism..."

Abstract: The medieval concept of syncategorematic terms is often viewed as a forerunner of the modern concept of logical constants. However, there are significant differences between how medieval authors conceived the role of syncategoremata within logical theorizing and modern conceptions of logical constants. Reflecting on the differences between the two frameworks allows us to evaluate critically some of the assumptions and theoretical commitments underlying current discussions on logical constants, and this will be the goal of the present contribution.

F. Engström (University of Gothenburg)

Slides

Abstract: Dependence logic, proposed by Väänänen, is the logic of first-order logic plus atomic formulas for functional dependencies between variables. It is similar to Hintikka’s Independence Friendly logic even though there is no compositional translation between them. Still, dependence logic and IF-logic both have the strength of existential second-order logic on the level of sentences.

To define a compositional semantics for dependence logic we need to consider sets of assignments called teams. There are several types of dependencies that could be investigated in this setting: functional, multivalued, and embedded multivalued dependence.

Dependence logic uses functional dependence as its notion of dependence. The truth condition for the dependence atom is more or less the same as above. However care is needed when stating the truth conditions for complex formulas as we are dealing with sets of assignments.Recently arguments for logics with (embedded) multivalued dependence as its notion of dependence has been proposed. One of the arguments is that multivalued dependence handles generalized quantifiers better than functional dependence. We will give some background on dependence in logic and arguments for and against the different dependence atoms, hopefully shedding some light on the differences between these variants of IF-logic.

S. Feferman (Stanford University)

Which quantifiers are logical? A combined semantical and inferential criterion

Slides + relevant paper "Which quantifiers are logical? A combined semantical and inferential criterion"

Abstract: The aim of logic is to characterize the forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter; thus its concerns combine semantical and inferential notions in an essential way. Up to now most proposed characterizations of logicality of sentence generating operations have been given either in semantical or inferential terms. This paper offers a combined semantical and inferential criterion for logicality (improving one originally proposed by Jeffery Zucker) and shows that the quantifiers that are logical according to that criterion are exactly those definable in first order logic.

P. Galliani (University of Amsterdam)

Some denotationally indistinguishable operators

Slides

Abstract: When reasoning about the definitions of the operators for a formal system, it can be necessary to distinguish between the denotations of their semantic definitions - that is, the truth conditions that they induce when applied to suitable expressions of the language - and their connotations, that is, the formal descriptions of the procedures to be used in order to compute these conditions.

In this talk, I will argue that logicality cannot be defined in terms of denotation only, and, in particular, that operators which induce the same satisfiability condition may behave differently with respect to the property of logicality.

In order to do so, I will discuss two equivalent semantics for disjunction and for existential quantification in Dependence Logic, and I will show that only one of these two semantics respects locality with respect to extensions of Dependence Logic.

A. Giordani (Catholic University of the Sacred Heart)

Three Ways to Logicality

Slides

Abstract: Tarski characterizes logical constants in terms of invariance under the group of all permutations on a fixed universal domain. It is known that this account of logicality is subjected to the objection that operators that intuitively are not considered logical, such as numerical quantifiers, turn out to be logical on the base of it. In the present talk three independent ways for identifying logical operators are sketched and are shown to be consistent with Tarski’s account in as much as second order quantifiers are acknowledged as logical operators.

R. Kahle (New University of Lisbon), joint work with J. Alama

Logical Constants: Classical, Intuitionistic, Dialogical ?

Slides

Abstract: Logical constants are a controversial issue in the philosophy of logic. The discussion focuses largely on generalized quantifiers and modal operators (to name two examples) and investigates possible criteria of demarcation. We will confine ourselves in this work to propositional logical constants, a seemingly innocent area: the standard sentential connectives, as "and", "or", "if . . . then" count as logical constants, and \nothing else". But one can challenge the distinction of these constants, when one considers the differences in classical, intuitionistic, and dialogical logic.

M. Petrolo (University Paris VII) & A. Naibo (University Paris I) & T. Seiller (University Aix-Marseille 2)

A computational analysis of logical constants

Slides

Abstract: We investigate the conditions used to establish what counts as a logical constant from a proof-theoretical point of view. First, we present some limitations of Prawitz's inversion principle. Our criticism naturally leads to an additional requirement (eta-expansion) based on the Curry-Howard correspondence, showing that the properties enjoyed by logical constants have a computational nature. Nevertheless, lambda-calculus does not allow a satisfactory reconstruction of logical inferentialism on omputational basis rather than linguistic ones. What is needed is a combinatorial and untyped setting in which logical types emerge through a notion of interaction, allowing the definition of logical constants by means of constructions on untyped objects.

T. Piecha (University of Tübingen) & W. de Campos Sanz (Federal University of Goias)

On the constructive meaning of implication in classical and intuitionistic logic

Slides

Abstract: We scrutinize two approaches to constructive semantics put forward by Prawitz and, more recently, by Sandqvist. Both semantics depend on extensions by

systems of production rules only. We argue that the restriction to production rules leads to a conflation of admissibility with derivability, generating a justification of Peirce's law and thus of classical logic. Such a justification is avoided if basic rules discharging basic assumptions are allowed. However, also in this wider setting the Mints rule would be justified, implying incompleteness of natural deduction for minimal and intuitionistic logic with respect to the proposed semantics. Allowing for non-basic extensions prevents this justification, but jeopardizes central aims of constructive semantics.

G. Restall (University of Melbourne)

Logical Constants, Sequent Structures and Speech Acts

Slides

Abstract: In this talk I will consider the connections between the logicality of elements of the vocabulary of an object language, the structural features available in a proof system for a logic, and the different ways those structures can be connected to kinds of speech acts (such as assertion and denial), structural features of speech acts (including substitution and other transformations) and transitions between speech acts (such as the making of different kinds of suppositions).

The focus of my attention will be the hypersequent calculus for the two-dimensional modal logic of Humberstone and Davies [1], and I will consider the different ways that kinds of modal operators could be taken to be either logical or non-logical parts of our vocabulary, depending on what features of our structures of our discourse we hold constant.

[1] Greg Restall “A Cut-Free Sequent System for Two Dimensional Modal Logic, and why it matters,” Annals of Pure and Applied Logic, to appear.

G. Sher (UC San Diego)

Logical Constants, Mathematical Constants, and Truth

Friday 12. 11.00-12.00

Abstract: In this talk I will consider the following questions:

1. What is the relation between logical and mathematical constants?

2. Are logical and mathematical constants denoting constants?

3. If they are, are their denotations “real” or fictional?

4. If their denotations are fictional, are logical and mathematical laws false?

M. Weiss (University of British Columbia)

The early Wittgenstein's critique of Frege and Russell

Handout

Abstract: After outlining Frege's and Russell's conceptions of the subject of logic, I reconstruct the basis of the early Wittgenstein's response that logical constants are not representational.