Variable Sharing Properties and Their Discontents

About

Event: Variable Sharing Properties and Their Discontents

Date: November 14-16, 2024

Venue: Rensselaer Polytechnic Institute (RPI) (Link to Website)

Location: Low Center for Industrial Innovation (CII), RPI, Troy, NY (Link to Map)

Room Number: 3206 (The room is on the main entrance floor.)

Format: Hybrid - all the talks can be accessed remotely in the WebEx room.

Link: https://rensselaer.webex.com/meet/fergut5

Organizers and Contacts

Thomas Ferguson (Rensselaer Polytechnic Institute) fergut5@rpi.edu

Jitka Kadlecikova (Rensselaer Polytechnic Institute) kadlej@rpi.edu

Shay Logan (Kansas State University) shay.a.logan@gmail.com

Shawn Standefer (National Taiwan University) standefer@ntu.edu.tw

Program

Thursday, November 14

Time	Speaker	Title
11:00 - 12:00	Registration
		WebEx: LINK
12:00 - 12:40	Gemma Robles	On Acceptable Connexive Logics with the Variable Sharing Property — remote
12:40 - 13:20	Katalin Bimbó	Tweaking Variable Sharing — remote
13:20 - 14:00	Nicholas Ferenz	Vector Spaces as Topic Structures in Topic-Sensitive Logics — remote
14:00 - 14:20	Coffee break
14:20 - 15:00	Brandon Rozek	VSPursuer: A VSP Verification Tool for MaGIC
15:00 - 15:55	Roundtable discussion

Friday, November 15

Time	Speaker	Title
9:00 - 9:45	Shawn Standefer	The Significance of Variable-Sharing
9:50 - 10:35	Sandra López	The Variable-Sharing Property and the Quasi-Relevance Property in 4-Valued Logics
10:40 - 11:00	Coffee break
11:00 - 11:45	Thomas Ferguson	Qua, Per Se, and Other Topic-Theoretic Operators
11:50 - 14:00	Lunch break
14:00 - 14:45	Martina Zirattu	On Weak Kleene Generalizations of Classical Logic
14:50 - 15:35	Damian Szmuc	More Semantics for Angell’s Logic of Analytic Containment
15:40 - 15:50	Coffee break
15:50 - 16:35	Blane Worley	Decorating Classical Logic
16:40 - 17:25	Tore Øgaard	Entailment or Implication
17:30 - 18:15	Shay Logan	Nonuniform Substitutions, Variable Sharing, and the Enigma of Contraction
19:00 - 22:00	Conference dinner

Saturday, November 16

Time	Speaker	Title
9:00 - 9:45	Mina Young Pedersen	Hybrid Relevant Logic
9:50 - 10:35	Franci Mangraviti	Variable Sharing Beyond Logic: Two Case Studies
10:40 - 11:00	Coffee break
11:00 - 11:45	Yale Weiss	Shallow Syntactic Properties, Deep Semantic Correlates, and Variable Sharing
11:50 - 12:35	Andrew Tedder	Representing Topics

Abstracts

On Acceptable Connexive Logics with the Variable Sharing Property
Gemma Robles (Universidad de León)

It is known that addition of Aristotle’s thesis to a relevant logic does not necessarily result in an acceptable connexive logic. In fact, there are not acceptable connexive extensions of basic relevant logic B. On the other hand, it is also well known that a necessary property of any relevant logic is the variable sharing property (VSP). In this paper, a number of acceptable connexive logics with the VSP (i.e., acceptable relevant connexive logics) is presented together with a series of results about them. (This work is supported by the Spanish Ministry of Science and Innovation MCIN/AEI/ 10.13039/501100011033 [Grant PID2020-116502GB-I00].)

Tweaking Variable Sharing
Katalin Bimbó (University of Alberta)

I will consider versions of the variable sharing property and discuss whether variable sharing is a desirable feature of relevance logics.

Vector Spaces as Topic Structures in Topic-Sensitive Logics
Nicholas Ferenz (Universidade de Lisboa)

In this talk I motivate using vector spaces in topic-sensitive logics, given several logics as semantics, and sketch routes to completeness proofs (i.e., building vector spaces from sets of formulas). The motivation for vector spaces is related to their use in representing both syntactic and semantic aspects of words for use in natural language processing. In particular, formal relations between vectors corresponds to semantic relations between topics. These vector space models further open the possibility of logics that represent disagreement between agents in terms of disagreements of topics: e.g., “do A and B really mean the same thing by this formula?” The Variable Sharing Property is then reconceptualized as a special case of topic preservation. This is spelled out using vector spaces and projections.

VSPursuer: A VSP Verification Tool for MaGIC
Brandon Rozek (Rensselaer Polytechnic Institute)

For a given logic, there may be thousands of satisfiable matrix models. Finding models which satisfy variable sharing is a labor-intensive process. In this talk, we introduce VSPursuer, a software tool that automatically finds witnesses for VSP. The tool is designed on top of MaGIC, leveraging existing algorithms to generate the initial candidate set of satisfiable models. We provide an overview on how to use the new tool, the techniques used to automate the search process, and discuss avenues for future developments. This includes joint work with Andrew Tedder.

The Significance of Variable-Sharing
Shawn Standefer (National Taiwan University)

In previous work, I suggested that we can use variable-sharing to define the class of relevant logics. In this talk, I will expand on this idea and explore the significance of variable-sharing. The goal will be to identify some of its prominent philosophical consequences. I will also use it as a foil to help identify features of the usual suspects in the relevant logic literature, e.g. B, R, T, etc., that distinguish them as interesting and important logics.

The Variable-Sharing Property and the Quasi-Relevance Property in 4-Valued Logics
Sandra López (University of Valladolid)

The variable-sharing property (VSP) is considered the main condition for a formal system to be a relevance logic, that is, to be free of paradoxes of relevance i.e., conditional formulas where the antecedent and the consequent do not share any common semantic content. The VSP can be defined as follows: a logic L has the VSP iff, for all theorems in L of the form A → B, A and B share at least a propositional variable. In Robles (2023), a general method to build binary expansions of FOUR (i.e., the system also known as Belnap-Dunn logic B4, which is equivalent to FDE) with the VSP is shown; in addition, an interesting logic determined by one of these expansions is fully developed and its fundamental properties are explored. On the other hand, the quasi-relevant property (QRP) is characteristic of logics in the family of relevance such as R-Mingle and may be defined as follows: a logic L has the QRP iff, for all theorems in L of the form A → B, either (i) A and B share a propositional variable or (ii) both ¬A and B are theorems. The aim of this talk is twofold: (1) to extend the method in Robles (2023) for the purpose of building implicative expansions of FOUR with the QRP; (2) to establish different criteria to choose among these matrices in order to develop logics that may be of interest from the point of view of relevance as something else than mere artificial constructs.

This work is supported by the Spanish Ministry of Science and Innovation MCIN/AEI/ 10.13039/501100011033 [Grant PID2020-116502GB-I00]

Qua, Per Se, and Other Topic-Transformative Operators
Thomas Ferguson (Rensselaer Polytechnic Institute)

Recent work challenging principles of topic transparency in topic-sensitive logics has relied on providing accounts of connectives that are topic-transformative, that is, which non-trivially influence the overall topic assigned to a complex. This leads naturally to the question of what operators in natural language might also act as topic-transformative functions. This talk reviews work in progress studying “qua”, “per se”, and other topic-transformative operators. After discussing ways to analyze these operators, we will emphasize how such analyses are likely to assist in a parallel project of updating Richard Sylvan’s work on relevant containment logic. (This is joint work with Pietro Vigiani and Jitka Kadlečíková.)

On Weak Kleene Generalizations of Classical Logic
Martina Zirattu (Università di Torino)

In an article that appeared some years ago, Stefan Wintein analyzed the effect that adopting different definitions of logical consequence had on three- and four-valued strong Kleene generalizations of Classical Logic. In a nutshell, these generalizations of Classical Logic consist of keeping the same truth and falsity conditions for the complex logical expressions, while allowing for truth-value gaps or gluts. Our work aims to study the effect of adopting different definitions of logical consequence on weak Kleene generalizations of Classical Logic. The characteristic feature of weak Kleene logics is their ability to handle grammatical but otherwise meaningless, nonsensical, or off-topic sentences in a formally systematic way. Thus, our generalizations keep the truth and falsity conditions of Classical Logic the same for the complex logical expressions, while allowing for truth-value gaps or gluts—and, most importantly, for meaningless, nonsensical, or off-topic sentences to be around.

More Semantics for Angell’s Logic of Analytic Containment
Damian Szmuc (CONICET)

This presentation aims to explore new semantics for Angell’s logic of Analytic Containment through the discussion of the topic-transformativeness of negation. For this purpose, we review some new developments by Song, Omori, Arenhart, and Tojo on two-address valuations for topic-transparent logics related to content inclusion, and extend their techniques for Angell’s logic of Analytic Containment. In particular, we present a 4-valued non-deterministic and a 16-valued deterministic semantics, both obtained through direct products of De Morgan lattices and involutive semilattices. (Joint work with Martina Zirattu)

Decorating Classical Logic
Blane Worley (UC Davis)

I investigate the closure under non-uniform substitutions of consecutions in bunched systems. In particular, I characterize the sublogic of classical logic that is maximally closed under depth substitutions. I then provide a tableau semantics for each of these systems and demonstrate their formal equivalence. Finally, I speculate whether these maximal sublogics are known logics and whether they enjoy the variable sharing property.

Entailment or Implication
Tore Øgaard (University of Bergen)

Relevant logics can be motivated in roughly two ways; (A) as a theory about a conditional—“relevant implication”—or (B) as a theory of logical consequence. The first first will typically appeal to natural language conditionals in order to substantiate the claim that certain conditionals fail to be true if the antecedent and consequent are about distinct topics. The other theory will typically claim that there is a notion of logical consequence—“entailment”—which require for validity that the premises and conclusion topically overlap. Note that neither of these two motivational ways need exclude the other.

In addition to the rather stable view that relevance require some form of topical overlap, relevant logics have been motivated using a notion of proof-theoretic utility: that the correctness of a deduction of a conclusion for a collection of premises require that the premises have been used in some substantial way. I will try to motivate the claim that this intuition can be accounted for using a topic-overlap-inspired property, namely that the consequence relation satisfy what Avron has called the basic relevance criterion (BRC). If the entailment consequence relation for a logic satisfies (BRC) then it follows that the logic satisfies the variable sharing property (VSP). There are, however, logics which satisfy (VSP) without satisfying (BRC) for entailment. Using Fine’s semantics for relevant logics I’ll sketch a proof that (BRC) holds not only for the entailment for standard relevant logics between B and R, but also for the standard/”Official” consequence relation and that the property holds for Official consequence for non-relevant logics such as RM₃. (BRC), therefore, fails to yield relevance in the standard sense.

There are two topics that I wish to highlight during my talk:

First of all, whereas the variable sharing property has lately been strengthened in various ways—ways motivated by considerations of whether negation and relevant implication are themselves topically transparent contexts—the investigation into the BRC-property remains in its infancy. The other is to emphasize the importance of how logics with such strengthened variable sharing properties are motivated. Even though the relevant conditional is often called an “entailment conditional”, I point to the fact that

modus ponens fails for entailment for logics with stronger variable sharing properties—“depth relevance” and beneath
motivational cases from natural language are typically of contingently true conditionals, and
that theories formulated in relevant logics are typically investigated as being closed under “Official” consequence

in order to suggest that

the motivational case for these logics is best view as of type A
that at least on of the intended consequence relations for these logics is the “Official” one
point to the fact that (2) is in tension with how much of current research practice

I’ll end by suggesting that to combination of some form of hyperformality and the BRC-property might be a viable way to argue that there is an additional notion of logical consequence which require for validity not only truth-preservance but also topical overlap in virtue of logical form.

Nonuniform Substitutions, Variable Sharing, and the Enigma of Contraction
Shay Logan (Kansas State University)

Relevant logics share variables. Weak relevant logics are also closed under interesting classes of nonuniform substitutions. In this talk I will show that in a range of interesting cases, the latter entails the former. This allows us give easy proofs of (new and exciting!) strong variable sharing results without relying on Belnap’s variable sharing results for the logic R. But there’s an enigma to address. R admits contraction. So variable sharing and contraction are clearly compatible. Yet (as I will show) contraction seems entirely incompatible with the whole business of nonuniform substitutions. If time permits after surveying this much, I will say a few words about how we might overcome this obstacle.

Hybrid Relevant Logic
Mina Young Pedersen (University of Bergen)

This talk concerns a work-in-progress project on adding nominals and the operator @_i from hybrid logic to the language of relevant logic. Hybrid logic is a modal logic that includes special propositional atoms, called nominals, where each nominal is true at only one state in the model. In effect, nominals let us name individual states and refer to them by formulas on the form @_i Φ which we read as “Φ holds in the state called i”. The project is motivated both by exploring modal logics in a relevant setting and by investigating the interesting mathematical structures that arise. The addition of nominals to relevant logic is not straightforward as a direct transfer of the valuation function from hybrid logic conflicts with the Heredity Lemma. We therefore propose that nominals name hereditarily closed sets instead of states in the model. In the talk, I show that the hybrid semantics of @_i further conflicts with relevant motivations and discuss some non-standard alternatives.

Variable Sharing Beyond Logic: Two Case Studies
Franci Mangraviti (University of Padua)

Is the notion of variable sharing central to relevant logic, or is it a mere byproduct? And how should it be expressed and understood? In this talk I want to explore a particular approach to addressing such questions, which is to look at the possible roles of the property under consideration (in this case, variable sharing) within practices outside of (the discipline of) logic itself. The idea is that, if a property P is core to a logic, then this should manifest in the way said logic partakes in other practices; conversely, looking at out-of-logic practices should provide insights (or at least support existing insights) into the best way to formulate P. Here I will focus on two (time permitting) case studies, inspired by the literature on relevant mathematics and feminist relevant logic:

if relevant logic is (in some sense) a good logic for mathematics, what is the mathematical value (as in: the value from the perspective of working mathematicians) of variable sharing?
if relevant logic is (in some sense) a good liberatory logic, what is the liberatory value of variable sharing?

Shallow Syntactic Properties, Deep Semantic Correlates, and Variable Sharing
Yale Weiss (Philosophy Program and Saul Kripke Center, The Graduate Center, CUNY)

In this talk I examine the Variable Sharing Property and variants and restrictions thereof from a semantic (i.e., model-theoretic) perspective. I am interested in identifying intuitively significant semantic properties in select model-theoretic frameworks which correspond to species of variable sharing. Some forms of variable sharing in some logics are robustly associated with natural semantic properties, while other forms appear (by “pessimistic induction”) not to correspond to any semantically significant notion. Various technical results and their possible philosophical upshots are discussed.

Representing Topics
Andrew Tedder (Ruhr University Bochum)

I (Tedder, forthcoming) have proposed a representation of topics as subalgebras, inspired by the recent work by Berto and his collaborators (2022). Berto’s framework, however, is apparently much more general than mine – he allows topics to come structured in terms of any old join semilattice, whereas the subalgebra lattice of any given algebra has significantly more structure. In this talk we investigate the question of when a Berto frame can be represented within a pair of an algebra and its subalgebra lattice. The answer turns on an algebraic property (JEP – the joint embedding property) which corresponds, in extensions of FL, to a version of VSP which holds at the level of logical consequence (the pseudo-relevance property, as investigated by Maksimova). We consider the significance of this fact for the topical reading of VSP, and glimpse around at related territory. (Joint work with Wesley Fussner (CAS-ICS)).

Conference Proceedings

The participants will have the opportunity to submit full papers for inclusion in the conference proceedings in a venue and publisher to be determined.

Transportation

We recommend moving around the campus and Troy by the RPI Shuttle which is free of charge. The distance from the hotel to the venue is about 1 mile. Please note that the RPI shuttle does not have a fixed schedule but it circulates the campus continuously.

You can also make use of the local bus lines from the Hilton Garden Inn Hotel to the venue. The local bus charges $1.50 per ride and you can find the schedules in the links below or integrated into Google Maps.

For line 87, use the hotel stop Hoosick St & Burdett Ave and the venue stop 15th St & RPI Walk Over Bridge.
For line 289, use the hotel stop 15th St & Hoosick St and the venue stop 15th St & Sage Ave.

Conference Dinner

The conference dinner will take place on Friday November 15th at the Dinosaur BBQ.
The venue will open at 18:30 and dinner will be served at 19:00.