Wien Workshop

(preliminary list)

• Rachel Boddy (IUSS Pavia)

• Michele Contente (Czech Academy of Sciences)

• Fiona Doerthy (University of Notre Dame)

• Fernando Ferreira (University of Lisboa)

• Salvatore Florio (University of Oslo)

• Leon Horsten (University of Konstanz)

• Nils Kürbis (Ruhr University of Bochum)

• Richard Lawrence (University of Wien)

• Matteo Pascucci (Central European University)

• Alexander Paseau (University of Oxford)

• Mario Piazza (Scuola Normale Superiore of Pisa)

• Antonio Piccolomini d'Aragona (University of Tübingen)

• Michael Potter (University of Cambridge)

• Andrea Sereni (IUSS Pavia)

• Georg Schiemer (University of Wien)

• Emelia Stanley (University of Vienna)

• Jan von Plato (University of Helsinki)

• Daiana Zavate (University of Konstanz)

Under construction

Michele Contente

Generalized predicativity and the stability of classical predicativism

By classical predicativism I mean the influential conception of predicativity that takes the structure of the natural numbers, together with the principle of induction, as given. This philosophical stance is supported by the proof-theoretic analysis of predicative provability developed by Kreisel, Feferman, and Sch¨utte, which has the merit, among other things, of providing a rigorous demarcation of predicative mathematics thus conceived. While acknowledging the merits of this approach, the aim of this talk is to assess whether the philosophical proposal based on it is stable. More specifically, stability may be threatened because the arguments put forward in support of the view that the structure of the natural numbers can be taken as given overgenerate, in the sense that they also inadvertently justify extensions of the predicative standpoint that go beyond the limits of predicativity established by the proof-theoretic analysis. Indeed, such extended or generalized conceptions of predicativity have been formulated in the literature and are typically characterized by the acceptance of certain forms of generalized inductive definitions. I will discuss some relevant examples and assess whether such extensions can be considered genuinely predicative. To strengthen this point, I will consider the case of the constructive conception of predicativity as articulated within Martin-Löf type theory. This conception provides a powerful example of generalized predicativity whose justification is based on an entirely new approach.

Fiona Doerthy

Neo-Fregean logicism: structuralism in Scottish clothing

In this talk I will argue Hale and Wright's Neo-Fregean Logicism depends on structuralist commitments to undergird its account of abstraction. I have argued elsewhere that the account of definition by abstraction central to Hale and Wright's neo-Fregean logicist project precludes the possibility of providing a referential route to the numbers as Fregean objects. In this talk I suggest that the implicit ontology of Hale and Wright's logicism is a form of non-eliminative structuralism and that Hale and Wright's structural commitments are essential to their account lest they fail to avoid decisive objections.

Fernando Ferreira

What went wrong? The limitations of Frege’s logicism

The contradiction in Frege's system of the Grundgesetze der Arithmetik stems from the occurrence of impredicative parameters in value-range terms. We inquire about the impact of this verdict on Frege's logicist program.  We conclude that the impact is devastating for the more ambitious claim of logicism as a foundational program for mathematics. However, the impact still allows for some equivocal success in founding arithmetic.

Salvatore Florio

Sets and pluralities

There is a close but puzzling connection between sets and pluralities. A set is fully characterized by the plurality of its elements but, it seems, not all pluralities correspond to sets. I examine different ways in which theories of pluralities have been used to explain sets. Then I discuss how some Cantorian ideas can provide a successful answer to the question under which conditions a plurality corresponds to a set. This answer leads to a simple, consistent, and historically significant theory of set abstraction.

Leon Horsten

Structures in arbitrary object theory

I discuss two approaches that seek to explain ante rem mathematical structures in terms of arbitrary objects, and seek to go beyond them. The first approach is a theory that was developed by Kit Fine (“Cantorian Abstraction”, 1998), and the second is a view that was more recently developed by me (“Generic Structures”, 2019). I have criticised Fine’s approach, and he has criticised mine. In the talk, I make a plea for a synthesis between the two approaches, taking on board valid critical points that have been made on both sides.

Nils Kürbis

Proof-theoretic logicism. Some thoughts and questions

Logicism is the view, originally put forward and developed by Frege, that arithmetic is nothing but logic: the laws of arithmetic are analytic truths and derivable solely from the laws of logic and definitions (together with proofs that establish the legitimacy of these definitions, see Grundlagen §3.)

Deductive Semantics (aka proof-theoretic semantics or inferentialism) is the view that the meanings of the logical constants are defined by the rules of inference governing them, provided they satisfy certain criteria. It originates with Gentzen and was developed by Dummett, Prawitz and Schroeder-Heister. 

Proof-theoretic Logicism combines deductive semantics and logicism. That is, according to it, some fundamental concepts of arithmetic are logical constants the meanings of which are defined by the rules of inference governing them. (Others may be defined explicitly.) This is an attractive position: deductive semantics provides a neat account of logic, which, paired with logicism, would extend to a neat account of arithmetic. 

My aim in this presentation is to assess the prospects of defending proof-theoretic logicism and to raise three questions that would need to be addressed. More precisely, my question is whether we can combine the logicism put forward by Frege in Grundlagen and modified, developed and defended by Bob Hale and Crispin Wright (and a couple of others) with (my version of) deductive semantics. 

Proof-theoretic logicism is not new. Neil Tennant has proposed precisely such a view. I do, however, have views on deductive semantics that, I think, diverge from Tennant's. My views on the matter are admittedly austere. The aim in my talk is less to assess Tennant's approach, but to see whether relative to my own views on deductive semantics, some (plausible) form of proof theoretic logicism is defensible. 

Richard Lawrence

Frege, Grassmann, and Schleiermacher on induction

In his 1872 /Formenlehre oder Mathematik/, Robert Grassmann offered a "Proposition of progressive (inductive) proof for magnitudes" stating a principle of induction, for which he also offered a proof. Intriguingly, Frege was aware of this text, and signed it out of the library just after the publication of /Begriffsschrift/.  Lothar Kreiser has noted the parallels in the two texts but argued that Frege's own proof of an induction principle goes well beyond Grassmann's, proving a stronger principle that fills a gap in Grassman's logic.  Yet I think Kreiser misses something important: rather than being a failed attempt to implement 'complete' (Fregean) induction, Grassmann's principle of induction is instead an attempt to capture a description of /empirical/ induction found in Schleiermacher.  Recognizing this not only clarifies how exactly Frege's logic went beyond Grassmann's; it also suggests that Frege's thinking about abstraction principles was indirectly inspired by discussions in German idealism.

Matteo Pascucci

A multimodal logic for the empirical analysis of necessary and sufficient conditions

Necessary conditions and sufficient conditions play a fundamental role in many domains of argumentation, such as causal reasoning, legal reasoning, moral reasoning, etc. Their logical analysis is typically reduced to a relationship between two formulas that is associated with the intensional connective of strict implication in normal modal logic. According to this approach, for instance, A is sufficient for B if and only if in all possible states in which A holds, B holds too. However, in everyday reasoning, necessary conditions and sufficient conditions are often defined in terms of empirical tests that involve both deductive and non-deductive arguments. For instance, starting with a dataset of observations D, the claim that A is a sufficient condition for B can be taken to constitute the conclusion of a defeasible argument based on D, whereas the claim that A is not a sufficient condition for B is the conclusion of an indefeasible argument based on D. In this talk, I will present a multimodal logic that allows one to capture such an empirical and more fine-grained analysis of necessary and sufficient conditions.

Alexander Paseau

Metaphysical problems for plenitudinous Platonism

How many types of mathematical object are there? In recent and contemporary philosophy of mathematics one finds three kinds of answer: zero, one and all possible ones. The third answer is the doctrine of Plenitudinous Platonism (also known as Full-Blooded Platonism).  It holds that mathematical objects are abstract and that there as many types of them as are consistently possible. Plenitudinous Platonists for example maintain that there are as many self-standing set universes as there are consistent set theories. Some of these satisfy the axioms of ZFC as well as the Continuum Hypothesis (CH), others the axioms of ZFC plus not-CH. In fact, according to Plenitudinous Platonists, the mathematical realm is as large as it could possibly be: every consistent mathematical theory corresponds to its own type(s) of mathematical object. My talk will raise some problems for the view. As the title indicates, I will focus on the metaphysical side of things.

Mario Piazza

Fractions from proofs: rethinking classical contradictions

Fractional semantics for classical logic assigns formulas a value that tracks the “quantity of identity” (intuitively, quantity of truth) revealed by proof search in a disciplined sequent calculus. The value is computed as a ratio between axiomatic initial sequents and all initial sequents in any derivation, yielding a multivalued refinement that breaks the usual symmetry between tautologies and contradictions. We exploit this asymmetry to define a paraconsistent consequence relation whose explosiveness depends on the fractional “falsity” of contradictions: the lower the value, the stronger the explosive power. Finally, we show how the same graded view of contradictions induces an informational stratification of tautologies via an equivalence on their negations, yielding “degrees of truth” anchored in classical derivability.Fractions from proofs: rethinking classical contradictions.

Antonio Piccolomini d’Aragona

Some variants of proof-theoretic semantics and their relations with intuitionistic logic

The content of the talk is drawn from a joint paper with Dag Prawitz. I first of all discuss the relationship between different variants of proof-theoretic semantics, particularly those stemming from Dag Prawitz's initial ideas and some subsequent developments due to Peter Schroeder-Heister. I clarify that the known proofs of incompleteness of intuitionistic logic with respect to (monotonic or non-monotonic) proof-theoretic semantics of the latter kind do not apply to Prawitz's original semantics. Nevertheless, I settle negatively the question of the completeness of intuitionistic logic with respect to Prawitz's approach, thus refuting a conjecture he made. Finally, I point to some features of the discussed variants of proof-theoretic semantics that we consider to be philosophically unsatisfactory.

Michel Potter

How to be a structuralist

A central motivation for structuralism is that mathematicians often talk of structures as if they are unique, when they are only unique up to isomorphism (e.g. the cyclic group of order 5 or the 2-element directed graph) . Yet they do not seem to talk analogously about sets (e.g. the 4-element set). Why not? I shall suggest an answer.

Emelia Stanley

One articulation of logical coventionalism is that facts about logical validity in a language are wholly explained by linguistic conventions of that language (Warren 2020, p. 33), and are therefore independent of “extralinguistic” facts. While somewhat overlooked in the recent conventionalist literature,1 in Convention, A Philosophical Study (1969), David Lewis provides a powerful account of conventionality in terms of arbitrary equilibria in coöperative games. Specifically, Lewis articulates the particular sense in which languages are conventional, using the formal framework of signalling games. With a goal to develop an account of conventionalism in Lewis’ game‐theoretic terms, I present original research showing that signalling equilibria define a consequence relation, giving rise to facts about logical validity unique to certain linguistic conventions.  I connect these results to conjectures about logical pluralism and inferentialism, including the (game‐theoretic) purpose of logical reasoning.

Andrea Sereni

Structures, Abstractions and Explications

Abstraction principles are powerful tools for the reconstruction of mathematical theories in the Fregean and neo-Fregean traditions. On the other hand, structuralist views adopt axiomatic presentations, in closer connection with mathematical practice. Both abstractions and axioms can be seen as varieties of implicit definitions of primitive mathematical notions, and their respective merits or limits take center-stage in the debate between structuralists and neologicists. Neologicism is primarily a programs of conceptual analysis, or rational reconstruction, aimed at recovering a uniquely correct characterization of the intended mathematical domain. However, the availability of alternative reconstructions of what could count as the same mathematical theory casts doubt on the tenability of such projects - due to alternative abstractive definitions of the same numerical notions, or alternative background logics, or to the inability to uniquely characterize e.g. pre-theoretical arithmetic. We discuss how such indeterminacy impacts on the Neologicism-Structuralism debate. Faced with the limits of reconstructive epistemology, we suggest adopting a Carnapian conception of explication to reconsider the intended epistemological import of abstractive definitions and, more generally, the epistemology of the neo-logicist enterprise

Jan Von Plato

Can root-first proof search direct the  construction of a type-theoretical context for a formal proof?

Formal proofs found by a root-first search in sequent calculus  are translatable into ones  in natural deduction. When these latter tree-form proofs are in turn translated into proofs in a linear form, an order of assumptions is constrained that determines the correct contexts of proofs, in the sense of lists of variable declarations in type-theoretic semantics.

Daiana Zavate

A metaphysical reading of abstraction principles

What does it mean for two abstraction principles to coexist? Is it simply that their conditions do not overlap or restrict what either claims to be possible, or that the abstract entities they “generate” do not conflict, so that one principle neither overrides nor contradicts what can be derived from the other? Reducing inconsistencies between abstraction principles to purely logical or epistemic considerations risks overlooking unexplored metaphysical implications.

What kinds of relationships can abstraction principles bear to one another beyond mere non-interference or conflict avoidance? It is also often unclear what positive theoretical work such “good companions” are meant to achieve together.

This presentation explores deeper issues concerning “companions” (good and bad) that remain hidden when abstraction principles are read solely as implicit definitions. In particular, Hume’s Principle (HP) need not be understood as providing only a flattened conception of cardinality; rather, attention can be given to its ontological status. On a metaphysical reading, HP may shed light on the broader role and scope of abstraction principles as metaphysical principles, and on the different kinds of relationships between such principles and abstracta.