The following Workshop is organized as the 5th one of the workshop series co-organized with the philosophy of logic group of University of Paris-1 (of Department of Philosophy and of IHPST).
Logic and mathematics are commonly thought as domains in which disagreement is hardly possible. Indeed, on the one hand, if logic provides the norms of thought, it seems that one has to follows the rules of logic if one is to think at all. And on the other hand, the existence of a proof seems to be a universal criterion for truth in mathematics. In this talk, it will be argued that disagreement is nonetheless possible, both in logic and in mathematics, but for different reasons. Examples of disagreement will be provided and discussed in both fields. It will also be shown how such analysis help us deepen our understanding of the very notion of disagreement.
"Disagreement" and "language games"Playing a language game" is often defined by Wittgenstein simply as "following a rule" : "disagreement" between speakers should consist of a misunderstanding of or a refusal to follow a rule. But whereas Wittgenstein often understands "disagreement" as the point at which two speakers stop playing a same language game, he nevertheless reckons that to disagree implies not only the existence of a highly codified protocol of interaction (arguments, proofs), but also that speakers share -at least, and more fundamentally- the grammar of the question discussed. Challenging thus an axiomatic approach and all throughout exemples from his Lectures on the Foundations of Mathematics (where such an approach could have been expected), we would try to inquiry if the raising of a problem, as well as its resolution -or dissolution- signs the abandon of a particular game or if it is, on the contrary, a necessary step in any language game "match".
In this presentation we formalize the Gettier problem by an extended modal logic. As a result, we can give modal formalization to one of the post-Gettier proposals on definition of knowledge: o No Defeat Proposal (Lehrer and Paxson, 1969). On the result, we present a new interpretation on the Gettier’s scenario: Smith, the protagonist of the story, believes that he knows the proposition at issue. Then we suggest that ‘belief’ is not one of constituents that constitute definiens of ‘knowledge’. Rather, in at least certain cases like the Gettier ones, notion of ‘belief’ depends on that of ‘knowledge’. Accordingly, the role of material from which ‘knowledge’ is to be refined is undertaken not by ‘belief’, but by “might possibility”, or, “hypothetical possibility”, which is, in turn, generated by abduction.
We normally think individuals are authoritative when it comes to their own preferences, beliefs and values. In recent years, however, this principle of ‘first-person authority’ seems to have come under threat. Data-driven algorithms are increasingly relied on to make decisions about individuals in contexts ranging from various recommendation systems, employment, education to criminal justice. The assumption that justifies such deference to algorithms is that that the latter are equally or more authoritative about what drives individuals than these individuals themselves. But is this assumption true? In this paper, I address the question of what type of authority algorithms might have, and how it differs from the type of authority individuals have about themselves. I suggest that the main difference concerns the grounds of authority: predictions based on behaviour on the one hand, in the case of algorithms, versus agential commitments, which are often not based on evidence at all, on the other. I end by suggesting how this new conceptualization can help to detect and diagnose potential clashes between 'algorithmic authority' and first-person authority, and clarify what is at stake, normatively speaking. I end by suggesting that algorithmic authority can cause a conservative bias, as algorithmic predictions are based on individuals' earlier behaviour and scores. It therefore potentially jeopardises individuals’ spontaneity, which in itself is importantly connected to freedom.
We first discuss how the traditional “universal” logical grammars could be understood to emerge through a (strong sense of) agreement-disagreement distinctio, in the referentialist and other standpoints. We then review Husserl’s and Wittgenstein’s three-layered distinction on the nonsense-sense distinction to reconsider “disagreement-agreement” for situated (logical) grammars, and discuss how proof-formations/argumentation takes an important role for formation of (networking of) norms.
We discuss shortly some examples of formation of some primitive concepts and norms in elementary arithmetic and elementary in algorithmic ethics.
To provide mathematics with a firm foundation, Hilbert proposed Hilbert's Program, whose aim was to show from the finitary standpoint that a formal system of analysis is consistent, namely, a finite number of logical steps in this system never leads to contradictory results. A lesson from Gödel's incompleteness theorems was as follows: If second-order arithmetic, which is a formal system of analysis, is consistent, its consistency cannot be proved from the finitary standpoint in the original form. This lesson also applied to first-order arithmetic, which is a subsystem of second-order arithmetic. In this background, Gentzen proved the consistency of first-order arithmetic by extending the finitary standpoint. Gentzen eventually gave three proofs for the consistency of first-order arithmetic, and he disagreed with Bernays about the finiteness of Gentzen's 1935 consistency proof. In this presentation, we first explain that there are two forms of the disagreement between Gentzen's norm about finiteness and Bernays' one. Next, we argue that type assignment rules in type theory enable us to formulate these two forms of disagreement in a uniform way.
We have proposed a novel interpretation of Brouwer’s argument for bar induction via a tool in traditional proof-theory called the Omega-rule. In this talk, we discuss a disagreement between this interpretation and other ones which been proposed so far.