Workshop on Truthmakers, Possibilities, and Information States

Invitation pdf

Programme

Tuesday, November 18

9:40-10:00 Introduction (Vít Punčochář)

10:00-10:50 Ivano Ciardelli: Dependence and Arbitrariness

10:50-11:10 coffee break

11:10-12:00 Simone Conti: Completeness for Weak Inquisitive First-Order Logic

12:00-14:00 lunch

14:00-14:50 Guillaume Massas: Two Applications of Possibility Structures in the Philosophy of Mathematics

15:00-15:50 Tadeusz Litak: TBA

15:50-16:20 coffee break

16:20-17:10 Søren Knudstorp: Truthmakers and Information States: Inclusion, Containment, Duality 

Wednesday, November 19

9:40-10:30 Vita Saitta: A Norm-Based Truthmaker Semantics for Modal Logic

10:30-10:50 coffee break

10:50-11:40 Mark Jago: Stating the Obvious

11:40-13:40 lunch

13:40-14:30 Francisca Silva: Truthlikeness, Exact Similarity and Counterfactuals

14:40-15:30 Vít Punčochář: Truthmaker Semantics, Inquisitive Logic, and Curry-Howard Correspondence

15:30-16:00 coffee break

16:00-16:50 Kit Fine (online): The Trivialization of Exact Equivalence

17:00-17:50 Wesley Holliday (online): Fundamental Logic with Necessity

Abstracts

Ivano Ciardelli

Title: Dependence and Arbitrariness
Abstract: In previous work, I have proposed an analysis of dependence claims as strict conditionals whose antecedent and consequent are questions: in a nutshell, “Q1 determines Q2” is true if, within a certain range of possibilities, any answer to Q1 implies some answer to Q2. In this talk, I consider dependence statements in mathematics, such as (1).

(1) The number of sides of a polygon determines the sum of its internal angles.
I argue that (1) fits the general analysis provided we take an indefinite like “a polygon x” to introduce a range of possibilities, one for each permissible value of x. Within this range of possibilities, an answer to the question "how many sides does x have" yields an answer to the question "what is the sum of the internal angles of x". Thus, the semantics of dependence statements like (1) involves a kind of intensionality that has its source in arbitrary reference.
I implement this idea in a team-based extension of first-order logic that comprises (i) an operator [x] which introduces an arbitrary referent and (ii) questions concerning the value of this referent. In this logic, the above dependence claim can be regimented as follows:

(2) [x]( polygon(x) -> ( value-of(num-sides(x)) -> value-of(sum-angles(x)) )
I then study how, and to what extent, such intensional claims can be reduced to extensional claims in first-order logic, such as (3):

(3) ∀x∀y( polygon(x) & polygon(y) -> ( num-sides(x)=num-sides(y) -> sum-angles(x)=sum-angles(y))
As I will show, the relation between the two ways of expressing dependencies is far from straightforward.
If time permits, I will also consider a modal extension of the logic and discuss how it offers a unified analysis of different varieties of supervenience.

Simone Conti

Title: Completeness for Weak Inquisitive First-Order Logic
Abstract: While complete proof systems are known for propositional inquisitive logic, it remains open whether the standard system of first-order inquisitive logic (InqBQ) admits one. In this talk, we will describe a sound and complete natural deduction proof system for InqWQ, a fragment of InqBQ that excludes the inquisitive existential quantifier and occupies an intermediate position between propositional inquisitive logic and InqBQ. 

Kit Fine

Title: The Trivialization of Exact Equivalence
Abstract:  I show that there are some relatively trivial ways of reducing classical logical equivalence to exact truthmaker equivalence and consider some of the philosophical implications of these reductions.

Wesley Holliday

Title: Fundamental Logic with Necessity
Abstract: Fundamental logic is a non-classical logic based only on the introduction and elimination rules for the connectives in a Fitch-style proof system (see https://arxiv.org/abs/2207.06993). Building on previous work that added modalities to fundamental logic (https://arxiv.org/abs/2403.14043), in this talk I will discuss the addition of a necessity operator to fundamental logic that makes possible a full and faithful translation of intuitionistic logic into fundamental logic. The main results concern relational semantics for this fundamental logic with necessity.

Mark Jago

Title: Stating the Obvious
Abstract: The standard approach in semantics is to treat content in terms of sets of situations (worlds, proofs, states). Standard truthmaker semantics inherits this approach, adding mereological structure to states. Reasoning about content is thus a hybrid enterprise, partly about algebraic structure and partly about set membership. In this talk, I tentatively propose a purely algebraic alternative, on which we reason about contents and the relationships between them directly. This has a number of advantages, some conceptual, some logical, and some practical. 

Søren Knudstorp

Title: Truthmakers and Information States: Inclusion, Containment, Duality
Abstract: I highlight points of contact between truthmaker semantics and information semantics as studied in the tradition of inquisitive and team semantics, such as Aloni's (2022) BSML. First, I show how to obtain sound and complete semantics for Angell's (1977) Analytic Containment by tweaking the semantics of BSML. This also achieves an information semantics complete for replete truthmaker entailment. Next, I present a full and faithful translation of BSML, inquisitive logic, dependence logic, and other propositional team logics into a modal logic. This modal logic can be taken to be that of van Benthem's (2019) translation of truthmaker semantics [cf. SBK (2023)], thereby enabling comparison of the two traditions within a common framework. Lastly, if time permits, I note further points of comparison, including (i) regular truthmaker propositions vs. meanings in BSML, (ii) conjunctive vs. disjunctive states, and (iii) potential benefits of combining the traditions through sets of truthmakers (this is joint work in progress with Robert van Rooij).

Tadeusz Litak

Title: Possibilities, Complete Additivity and GQMs
Abstract: The talk presents the story of two related papers with Wesley Holliday. In the first of them (RSL 2019), we revisited
the question of modal incompleteness with respect to completely additive algebras, posed by Yde Venema in the first
edition of the "Handbook of Modal Logic" (and in my 2005 PhD Thesis). For Wesley, quite obviously, the real motivation
was completeness with respect to possibility semantics. Joining forces, we found out not only that such incompleteness
is "just as common" as Kripke incompleteness and covers some pretty famous examples, but also that complete additivity
of modal algebras is an elementary condition (an observation later generalized by the Hungarian school). Attempts to
capture it by means of a single axiom in a suitable extension of the modal syntax lead us to invent the language of
"global quantificational modalities" or GQMs (AiML 2018). Unlike all other propositional quantifiers in the literature,
they can be interpreted in an arbitrary general (Henkin-style) frame, avoid so-called Kaplan paradox, and allow a
recursive axiomatization and reduction of arbitrary modal logics (including incomplete ones) to universal theories over
this language. As a side bonus, almost the entire GQM paper has been formalized in the Coq proof assistant. I will
conclude with a list of open problems posed by both papers; to the best of my knowledge, rather little progress has been
made on most.

Guillaume Massas

Title: Two Applications of Possibility Structures in the Philosophy of Mathematics
Abstract: Extending the framework of possibility semantics to quantified languages, possibility structures provide a semantics for classical logic that generalizes Tarskian model theory. The distinctive feature of possibility structures is that formulas are evaluated in a partially ordered set of viewpoints, rather than at a single point. In this talk, I will review some recent results on the model theory of possibility structures, and discuss two applications to the philosophy of mathematics. The first one is related to recent debates on non-Cantorian notions of size for infinite sets of natural numbers, and the second one connects to well-known issues regarding the semantics of second-order logic. 

Vít Punčochář

Title: Truthmaker Semantics, Inquisitive Logic, and Curry-Howard Correspondence
Abstract: Kit Fine characterized his truthmaker semantics for intuitionistic logic as a “cross between the construction-oriented semantics of Brouwer–Heyting–Kolmogorov and the condition-oriented semantics of Kripke.” While the relation to Kripke semantics is well understood, the connection to construction-oriented, type-theoretic frameworks that more directly implement the BHK interpretation has remained less clear. In this talk, I clarify this relationship by developing a truthmaker semantics for the simply typed lambda calculus with products and sums. The resulting framework provides a fully compositional interpretation of proofs and derivations within the algebra of truthmakers. I further show that, using the close connections between truthmaker semantics and inquisitive semantics, one can extend this approach to a modified lambda calculus adapted to inquisitive logic, thereby shedding new light on its type-theoretic features.

Vita Saitta

Title: A Norm-Based Truthmaker Semantics for Modal Logic
Abstract: This paper contributes to that development of a truthmaker semantics for modal logic. The basic idea we introduce is that truthmakers of modal sentences are norms. This allows us to derive two interesting results: first, we introduce a new modal semantics for minimal monotonic logics; second, we provide a truthmaker semantics for every modal logic characterized by a class of Kripke frames.

The core of the idea rests on the interplay between norms and the modal profile of a world, i.e., the modal truths valid within that world. A norm is introduced as primitive entity having a content that determines the set of worlds it considers possible. In this setting, given the verification clauses for the modal operators, truthmaker semantics is connected to standard possible worlds semantics. In more detail, necessity is defined so that all the worlds that comply with the norms grounding a necessary proposition extend some truthmaker for that proposition. Accordingly, the present semantics, while aligning with the basic tenets of a truthmaker approach, also preserves the intuitive connection between what is necessary and what is accessible across worlds, which motivated the original development of Kripke semantics.

Francisca Silva

Title: Truthlikeness, Exact Similarity and Counterfactuals
Abstract: In many contexts where someone has uttered a false claim, we’re nonetheless still inclined to say that “there is something true” in what the speaker has said. In the literature (Humberstone, 2003; Yablo, 2014; Fine, 2025) this is captured by the notion of partial truth, and that in the case someone has said something false, only a proper part of what is said is true. A partially true claim may still have very little truthlikeness or, put in other words, resemble the truth to a very small extent. Fine (2019, 2021) has also done some work on capturing the notion of comparative verisimilitude within the framework of truthmaker semantics. According to some accounts of verisimilitude, completely false claims are completely unlike the truth. One might think that some examples run against this, however. In my talk, I want to introduce and characterize a notion of what we may call semblance of truth according to which completely false claims nonetheless may resemble the truth. The notion of semblance of truth will be based on a relation of exact similarity between states, much in the same way as there is a primitive relation of similarity between worlds in the Lewis-Stalnaker analysis of counterfactuals. Instead of taking the notion to be primitive, however, I define it, and then derive some of the usual constraints on similarity from the definition of exact similarity.