Workshop on Truth and Paradox

April 9

Buenos Aires, Argentina

Luca Castaldo (LLM-Munchen)

Jonathan Erenfryd (IIF-SADAF-CONICET)

Camillo Fiore (IIF-SADAF-CONICET)

Camila Gallovich (IIF-SADAF-CONICET)

Thursday, April 9

14:00 to 19:00 (GMT-03, Buenos Aires Local Time)

14:00 - 15:00: Camila Gallovich

15:00 - 15:15: Coffee break

15:15 - 16:15: Luca Castaldo

16:15 - 16:45: Coffee break

16:45 - 17:45: Jonathan Erenfryd

17:45 - 18:00: Coffee break

18:00 - 19:00: Camillo Fiore

Luca Castaldo: "Least VS Greatest intrinsic fixed-points: philosophically and proof-theoretically"

How can we explain the word 'true' to someone who does not yet understand it? An influential answer is given by Kripke in the Outline. There, in a dialogue with an imaginary interlocutor, Kripke proposes that "we are entitled to assert (or deny) of any sentence that it is true precisely under the circumstances when we can assert (or deny) the sentence itself". By repeated application of these simple rules, the interlocutor is eventually able to decide, for most statements, whether truth applies to them or not. Kripke's minimal fixed-point model mirrors this learning process, and it is often regarded---in his own words---as "probably the most natural model for the intuitive concept of truth". Contrary to this widely accepted view, in the talk I develop a different perspective. Starting from a revision of Kripke's dialogue, I argue that the most natural model for the intuitive concept of truth is not the minimal, but rather the greatest intrinsic fixed-point (GIFP). Moreover, I introduce an axiomatic system whose intended model is GIFP and show that it is proof-theoretically at least as strong as Burgess' axiomatization of minimal fixed-point, known as KF_\mu.

This is joint work with Mateusz Łełyk and Konstantinos Papafilippou.

Jonathn Erenfryd: "Radical Recapture"

Non-classical theories of truth address the semantic paradoxes by rejecting the validity of certain classical principles. A paracomplete approach, for instance, rejects the Law of Excluded Middle. But rejecting a principle's validity need not involve rejecting all of its instances. This gives rise to the project of classical recapture. An important question that those engaged in the project must face concerns which instances can be recovered. It is natural to think that only paradox-inducing instances should be given up, and so a natural desideratum is that every non-paradoxical instance should remain available. However, this has been called into question: there are instances of LEM that are unproblematic on their own but jointly unsatisfiable, giving rise to many maximal consistent collections with no principled way to choose among them. This has been taken to show that the non-classical logician must settle for weaker forms of recapture. I will argue that these limitations do not undermine the strong form of recapture, and that there is a unique, non-arbitrary way of capturing the commitment to retaining every non-paradoxical instance.

Camillo Fiore: "No No-No"

The No-No pair comprises two statements, each of which claims that the other is false. The perfect symmetry of these statements  suggests that they should be assigned the same truth value; however, such an assignment cannot be done on pain of triviality. In a thought-provoking paper, Roy Cook has maintained two theses concerning the No-No pair. First, that its statements are genuine paradoxes, on an equal footing with the liar. Second, that symmetry considerations are not needed to establish their paradoxicality. The aim of this talk is to discuss and ultimately reject each of these theses. We claim that symmetry considerations are essential for regarding the statements in the No-No pair paradoxical. Moreover, these considerations are unjustified upon closer inspection. We conclude that the statements in the No-No pair are closer to prototypical hypodoxes, like the truth-teller. In this way, we re-arrive at the diagnosis that Roy Sorensen had initially made about the No-No pair.

This is joint work with Camila Gallovich and Lucas Rosenblatt.

Camila Gallovich: "Truth, give me strength"

One of the most challenging arguments against non-classical logics is that they give rise to weaker theories than classical logic. This concern is especially pressing in mathematics, where classical reasoning is often seen as indispensable. The thought is that, other things being equal, if a non-classical theory is mathematically weaker than the corresponding classical theory, then the latter should be preferred. In this work we offer two responses to the argument from strength, taking the classical theory Kripke-Feferman (KF) and the non-classical theory Partial Kripke-Feferman (PKF) as our test case. Although both theories aim to axiomatize Kripke’s semantic construction for truth over Peano arithmetic, KF is arithmetically stronger than PKF. Our first response to the argument is that comparing KF to PKF requires adopting not only a conception of truth but also a conception of arithmetic. After reviewing various conceptions, we conclude that PKF fares at least as well as KF under most of them. Our second response is based on an argumentative strategy known as classical recapture. PKF can be strengthened by selectively adding safe instances of excluded middle. Together, these responses suggest that, contrary to what is often assumed, the argument from strength does not undermine non-classical logics. 

This is joint work with Camillo Fiore and Lucas Rosenblatt.

Lucas Rosenblatt  (IIF-SADAF-CONICET)
Bruno Da Re (IIF-SADAF-CONICET)
Edson Bezerra (IIF-SADAF-CONICET)
Aylén Bavosa Castro (IIF-SADAF-CONICET)
Juan Manuel Gagino Di Leo (UBA)

TBA