Titles and short abstracts

Marianna Antonutti Marfori: Arithmetical Completeness and Absolute Undecidability

Abstract: Feferman [1962] proved a completeness theorem for transfinite progressions of formal theories of arithmetic. It is generally accepted that this result does not answer the question of absolute undecidability (whether there are true mathematical statements that are undecidable relative to any justified set of axioms) due to the role of intuition in “selecting" the complete path through the tree of ordinal notations. For this reason, the significance of Feferman's result has not been substantively discussed in the philosophical literature. In this talk, I will argue that this view is too simplistic. It is not well known that Feferman’s proof constructivises [Sündholm 1983], and the use of constructively or finitistically acceptable methods seems to obviate the need for intuition. In concluding, I will assess the prospects for a more compelling argument that Feferman's completeness theorem does not answer the question of absolute undecidability.


Jean-Yves Béziau: Beyond Truth and Proof

Abstract: Proof and truth are generally presented as strongly opposed. This opposition is often expressed as the duality syntax-semantics and the completeness theorem is presented as a non-trivial bridge between the two. However completeness can in a way be trivialized and then the gap between truth and proof vanishes. This is done by working out some abstract versions of this theorem in the spirit of universal logic: extracting the general from the particular. This is what I will explain in this talk, also making some historico-philosophical comments, e.g. about the fact that truth-tables are considered as syntax by Chang and Keisler in their seminal book on model theory.


Pablo Cobreros: Identity and the Sorites Paradox

Abstract: It is generally agreed that vague predicates are sorites-susceptible. We can therefore construct a sorites argument involving the identity predicate. But what is exactly the reason why we can construct a sorites argument involving identity? Is it that the terms we use fail to determinately refer or is it that the objects referred to are themselves vague? The "received view" takes it that identity can be vague just in the first sense, and resort to Evans argument to make a case for this claim. If we concede the underlying rationale for Evans argument (that metaphysical indeterminacy cannot appeal to failure or either Leibniz's Law or Abstraction) the scope of solutions to the identity sorites for the defender of metaphysical indeterminacy of identity is extremely narrow.


Marcos Cramer & Jérémie Dauphin: Modelling Arguments about the Liar Paradox in Structured Argumentation Theory

Abstract: Structured argumentation theory is a methodologigal approach originating in the field of Artificial Intelligence, which studies how to systematically construct and evaluate defeasible arguments and counterarguments from a logical language. We apply this methodology to formally model both the reasoning involved in semantic paradoxes like the Liar paradox, and the reasoning involved in the expo­sition of various proposed solutions to these paradoxes and in the construction of various arguments for and against certain proposed solutions. For this purpose, we propose the framework ASPIC-END, which modifies and extends an existing framework of structured argumentation, called ASPIC+, in a way that is suitable to the modelling of argumentative and explanatory reasoning about semantic paradoxes.


Bogdan Dicher: LP, ST and tolerant metainferences

Abstract: The strict-tolerant (ST) approach to paradox promises to erect theories of naïve truth and tolerant vagueness on the bedrock of classical logic. We assess the extent to which this claim is founded. We show that the usual proof-theoretic formulation of propositional ST in terms of the classical sequent calculus without primitive Cut is incomplete with respect to ST-valid metainferences, and exhibit a complete calculus for the same class of metainferences. We also argue that the latter calculus, far from coinciding with classical logic, is a close kin of Priest' s LP.


Jonathan Dittrich: Motivating noncontractive approaches by Cut-inadmissibility

Abstract TBA


Giulia Felappi: Truth, Content, and how the Content is Reached

Abstract: The notion of validity is usually spelled out in terms of necessary preservation of truth. But not everybody agrees that truth is the central and unique notion when it comes to defining validity. In this talk, I will discuss the prima facie threats to the thesis that truth is all that matters put forward by Kit Fine and David Kaplan, and their conclusion that we should (or should also) go back to the Mediaeval notion of validity in terms of preservation of content or information. I will show why defenders of validity in terms of truth might not be particularly impressed by these cases. I will moreover urge that if we want to have a notion of validity that goes beyond truth, content and information do not seem enough.


Volker Halbach: The substitutional analysis of logical consequence

Abstract: A substitutional account of logical truth and consequence is developed and defended against competing accounts such as the model-theoretic definition of validity. Roughly, a substitution instance of a sentence is defined as the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category and possibly relativizing quantifiers. In particular, predicate symbols can be replaced with formulae possibly containing additional free variables. A sentence is defined to be logically true iff all its substitution instances are satisfied by all variable assignments. Logical consequence is defined analogously. Satisfaction is introduced axiomatically. The notion of logical validity defined in this way is universal, that is, validity is not defined in some metalanguage, but rather in the language in which validity is defined. It is shown that for every model-theoretic interpretation there is a corresponding substitutional interpretation in a sense ot be specified. Conversely, however, there are substitutional interpretations -- in particular the trivial intended interpretation -- that lack a model-theoretic counterpart. Thus the definition of substitutional validity overcomes the weaknesses of more restrictive accounts of substitutional validity. In Kreisel's squeezing argument the formal notion of substitutional validity naturally slots into the place of informal intuitive validity.


João Marcos: Consequence Beyond Truth and Proof

Abstract: In this contribution I shall experiment with an analysis of logical consequence done neither in terms of preservation of truth nor in terms of preservation of warrant to assert. The analysis will be done instead from an abstract viewpoint, taking the perspective of judgmental agents that entertain certain cognitive attitudes towards the informational content of given sentences. In this approach, neither truth-values nor inference rules need to be taken as primitive, for they can be fruitfully explicated in terms of a conceptually prior notion of compatibility between possibly overlapping cognitive attitudes of a given agent. I will discuss some effects of such an approach on the provision of a satisfactory theory of meaning that goes beyond truth and proof, but will also show how the announced analysis maps naturally into a four-valued non-deterministic semantics, and into an analytic bi-dimensional proof system. If time permits, I will also discuss how logical consequence so understood may pave the way towards a novel approach concerning the understanding of gappy and glutty reasoning.


Eugenio Orlandelli: A relevant solution for stability

Abstract: Dummett and Tennant agree that meaning-conferring rules must be harmonious and stable, and that one key element of stability is the requisite of maximality. In the present paper we argue that the requisite of maximality is not completely convincing and we propose a modification of Dummett's notion of stability. Maximality is needed to ensure the unique determination of the harmonious and stable rules of inference---it disallows operators like quantum disjunction, but it doesn't seem compatible with a pluralistic perspective on logic. The modified notion of stability that we propose differs from Dummett's one in that (i) it rules out patently problematic operators such as knot without having to assume maximality, and (ii) it justifies a logic (MR) that is a relevant fragment of minimal logic. Finally, we will propose a further modification of Dummett's notion of stability that justifies rules which allow us to get rid not only of the paradoxes of material implication, but also of semantic paradoxes like Curry's one.


Mattia Petrolo: Paradoxal activity

Abstract: We challenge the idea that the lack of a normalization procedure can be taken as a distinctive feature of paradoxes. In particular, we present a counterexample to Tennant’s proof-theoretic characterization of paradoxes in intuitionistic natural deduction with standard and generalized rules. Our counterexample follows a simple technique for eliminating cuts introduced by Kreisel and Takeuti.


Paolo Pistone: The geometry of vicious circles: what typability in the lambda-calculus tells us about paradoxes

Abstract: From a lambda-calculus perspective, the usual identification of paradoxes with not-normalizing derivations can be put in this form: a system of rules generates a paradox when its associated derivation corresponds to the typing of a looping combinator. Many questions about the proof-theory of paradoxes can be then translated into questions about the typability of pure lambda-terms. In particular, the usual analysis of paradoxes in terms of so-called "id est inferences" (Tennant 1982) can be translated into the theory of recursive or fixpoint types. A combinatorial and decidable characterization of the typability conditions for cycling combinators, based on first-order unification, accounts for most paradoxes discussed in the philosophical logic literature. However, this analysis does not characterize all paradoxes. For instance, Girard's paradox, which corresponds to a looping though not cycling combinator, requires a more sophisticated background, like the one provided by the investigations of typability in the polymorphic type discipline. In this field, based on second-order unification or semi-unification, many problems have been shown to be undecidable. Hence, once we accept the identification of paradoxes with looping combinators, it seems that many theoretical questions about paradoxes might turn out to be undecidable and that the most interesting paradoxes are yet to be discovered.


Lucas Rosenblatt: Noncontractive Classical Logic

Abstract: One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes. Both Contraction and Cut play a crucial role in typical paradoxical arguments. In this talk I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach offers a lot of benefits that are not available in the noncontractive setting. I sketch a new noncontractive account that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language and it retains every classically valid argument.


Chris Scambler: Arithmetic and the proof theory of cardinality quantifiers

Abstract: The study of generalized quantifiers is standardly pursued in model-theoretic terms. One reason this might be the case is that the standard validity relations for such quantifiers as they are usually defined in model theory aren't recursively enumerable, and so don't admit of nice complete axiomatizations. However I'll argue that these facts shouldn't dissuade proof-theorists from giving partial deductive systems for generalized quantifiers, and that indeed there are often good reasons to do so. As a specific example, I'll give a partial deductive system for some generalized cardinality quantifiers, and show that it can prove translations of many intuitively true claims about primitive recursive properties and relations on the natural numbers. The fact that the system is explicitly axiomatized is argued to yield various philosophical benefits, incompleteness notwithstanding, especially in support of nominalism in the philosophy of mathematics.


Kordula Świętorzecka: Is Bolzano's Inbegriff of All Adherences Paradoxical?

Abstract The point of the analysis will be a fragment of ontology of B. Bolzano, employing two ontological categories: substance, adherence. They are related to certain Leibnizian and Platonic ideas but still keeping original features. The notion of adherence is used in the argument for the existence of substances given by Bolzano in Athanasia (1838). We propose its formalization in frame of the theory of abstract objects of E. Zalta (1983). The key concept of the argument is Inbegriff of all adherences. Our ”maximal” notion breaks the restriction on comprehension scheme, taken by Zalta to protect his theory against Clark’s paradox. Our formalism is weaken by the requirement of the existence of Inbegriff . We also sketch semantics in which our weakening can be deleted but we keep general Zalta’s restriction.