In this talk I will isolate a class of logics which I shall call Variable Designated Values (VDV) Logics, and consider some of their properties. VDV logics are many-valued logics in which different sets of designated values are used for the premises and conclusions. The idea goes back, as far as I know, to Malinowski (1990) and (1994), though much use of the idea has been made by logicians recently in the form of the logics ST and TS (S= Strict; T = Tolerant).
This talk investigates the odd fact that one may add connexive theses to relevant logics, giving rise to contraclassical systems, and obtain logics which are not trivial, still obey many of the desired relevance properties, and yet allow one to prove every negated implication. I’ll show why this is the case, and investigate alternative connexive relevant logics in the area that don’t have this undesirable property.
In this talk I will argue that we should care more for and be more careful with the history of computability making a plea for a more diverse and informed understanding. The starting point will be the much celebrated Turing machine model. Why is it that within the computability community, this model is often considered as the model? In the first part of this talk I review some of those reasons, showing how and why they are in need of a revision based, mostly, on historical arguments. On that basis I argue that, while surely, the Turing machine model is a basic one, part of its supposed superiority over other models is based on socio-historical forces. In part II then, I consider a number of historical, philosophical and technical arguments to support and elaborate the idea of a more diversified understanding of the history of computability. Central to those arguments will be the differentiation between, on the one hand, the logical equivalence between the different models with respect to the computable functions, and, on the other hand, some basic intensional differences between those very same models. To keep the argument clear, the main focus will be on the different models provided by Emil Leon Post but I will also include references to the work by Alonzo Church, Stephen C. Kleene and Haskell B. Curry.
Supported by the PROGRAMme project, ANR-17-CE38-0003-01.
People often reason contrary to the prescriptions of classical logic. In the talk I will discuss some cases of divergence between everyday and logical-mathematical reasoning and propose that they are a consequence of a tendency in human cognition to neglect models which verify sentences by virtue of an empty configuration [neglect-zero tendency, Aloni 2022]. I will then introduce a bilateral state-based modal logic (BSML) which formally represents the neglect-zero tendency and can be used to rigorously study its impact on reasoning and interpretation. After discussing some of the applications, I will compare BSML with related systems (truthmaker semantics, possibility semantics, and inquisitive semantics) via translations into Modal Information Logic [van Benthem 2019].
Maria Aloni. Logic and conversation: The case of free choice, Semantics and Pragmatics, vol 15 (2022)
Johan van Benthem. Implicit and Explicit Stances in Logic, Journal of Philosophical Logic, vol 48, pages 571–601 (2019)
This talk introduces a systematic way of analyzing the semantics of causative linguistic expressions, and of how natural languages express causal relationships. For this purpose, I will employ the Structural Equation Modeling (SEM) framework and demonstrate how this method offers a rigorous model-theoretic approach to examining the distinct semantics of causal expressions. This paper introduces formal logical definitions of different types of conditions using SEM networks, and illustrates how this proposal, along with its formal tools, can help to clarify the asymmetric entailment relationship among different causative constructions.
We present a general approach to quantified modal logics (QML) that can simulate most other approaches. The language is based on operators indexed by terms which allow to express de re modalities and to control the interaction of modalities with the first-order machinery and with non-rigid designators. The semantics is based on a primitive counterpart relation holding between n-tuples of objects inhabiting possible worlds. This allows an object to be represented by one, many or no object in an accessible world. Moreover by taking as primitive a relation between n-tuples we avoid the shortcomings of standard individual counterparts. Finally, we use cut-free labelled sequent calculi to give a proof-theoretic characterisation of the quantified extensions of each first-order definable propositional modal logic. In this way we show how to complete many axiomatically incomplete QML.
Many versions of the Axiom of Choice (AC), though equivalent in ZF set theory, are inequivalent from the computational point of view. When we consider polynomial-time analogues of AC, many of these different versions can be shown to be equivalent to other more standard questions about the relationship between complexity classes. We will use some of these formulations of AC to motivate several complexity questions that might otherwise seem a bit bespoke and unrelated from one another.
Next, as many versions of AC are about cardinals, in the second half of the talk we introduce a polynomial-time version of cardinality, in the spirit of polynomial-time model theory. As this is a new theory, we will discuss some of the foundational properties of polynomial-time cardinality, some of which may be surprising when contrasted with their set-theoretic counterparts. The talk will contain many open questions, and the paper contains even more! Based on arXiv:2301.07123 [cs.CC].
The goal of this talk is to use the sorites paradox to illustrate the methodology of “reverse philosophy”—i.e. the application of methods from reverse mathematics to study the mathematical involvement of recognized arguments in analytic philosophy. After briefly motivating such a program, I will focus on the following: 1) the role of measurement and representation theorems in the linguistic formulation of various forms of the sorites; 2) the role of a weak form of Hölder’s Theorem in the formulation of the conditional sorites for predicates such as “tall”; 3) the role of a stronger form of Hölder’s Theorem in the formulation of the so-called continuous sorites for predicates such as “red” of Weber & Colyvan 2010/Weber 2021. Contrasts will be drawn between the constructivity of the weaker form (as observed by Krantz 1968 and formalized in RCA_0 by Solomon 1998) and the non-constructivity of the latter form (due to its apparent dependence on Arithmetical Comprehension).
I’ll propose an application to logic of the “neopragmatist” program. Neopragmatists argue that inquiry into the nature of what we think and talk about can be fruitfully replaced by inquiry into the functions of concepts and expressions. Logical vocabulary can serve as a particular target for neopragmatist theorizing, but it has also been taken to pose obstacles to neopragmatist accounts more generally. I’ll argue that a neopragmatist approach to logical relations (such as logical consequence), as well as to ascriptions of content, undermines two constraints on neopragmatist accounts of logical connectives (such as “and”, “or”, and “not”). Freed from these constraints, I’ll sketch a simple version of such an account, on which logical connectives express dialectical attitudes. The resulting approach is deflationary in two ways: it’s based on deflationism about logical relations and it aims to deflate some of neopragmatists’ usual theoretical ambitions.
Among the numerous logics that are studied and applied nowadays, many can be described via a proof system, which captures the valid reasoning that can be carried out in the logic, and there are many examples of well-known concrete proof systems that occur in various areas in logic and beyond.
As is well-known, a single logic has many different proof systems, and not every system is as useful as another. Given the goal of the proof-theoretic formalization, be it philosophical or mathematical or of a different nature, one can often distinguish among the proof systems of the logic a class of good proof systems, proof systems with properties that makes them useful in the investigation of the logic. A well-known example are analytic sequent calculi and their use in proving properties of the logic such as decidability or interpolation. But there are many other examples, some of which will be discussed during the talk.
Given a class of such useful proof systems the questions naturally arise whether a certain logic has such a good proof system or not, and what it means for a logic to have such proof systems. These are no easy questions to answer and stated in such generality they may not always have a reasonable answer. But for concrete classes of logics and proof systems something insightful can be said. In this talk the aim is to do so for intermediate and (intuitionistic) modal logics, where the proof systems that are considered are sequent calculi. And although no full answer is obtained, some small steps in that direction are taken.