Summer School: A map through the land of dragons. Syntax, Truth, and Paradox
Volker Halbach and Lorenzo Rossi will provide an introduction to the theory of the semantic paradoxes and discuss their proposed solutions, analyses, and their philosophical relevance. The liar and related paradoxes do not only affect the notion of truth, but others that are central to philosophy such as necessity, apriority, and future truth. We will study the properties of such sentences as The sentence in italic on this page is not true in a precise formal setting. Participants are not expected to be familiar with techniques such as arithmetization and diagonalization. Bypassing the unnecessary mathematical tools by using an axiomatic syntax theory, we will still provide a formally precise account and proceed swiftly to the philosophical core of the discussions around the paradoxes. In particular, we will discuss the effect of paradoxes on the expressive power of truth, necessity, and related notions, as well as their impact on the foundations of semantics. We will also cover a wide variety of paradoxes such as the Visser-Yablo, the Knower, McGee's, and the No Future paradox. We will critically discuss a gamut of conceptions of paradoxicality, applying them to the study of the various kinds of paradoxes we have introduced, and we will investigate their connection with some of the main formal theories of truth.
Most of the content will be based on Graham Leigh's and Volker Halbach's new book The Road to Paradox: A Guide to Syntax, Truth, and Modality, which is scheduled to appear with Cambridge University Press in January 2024. The summer school, however, will be self-contained and all relevant teaching materials will be supplied to participants. Moreover, the course givers are happy to adjust their plans to the wishes and preferences of the audience to some extent. Hence there may be some deviations, especially in the later parts.
Participants who would like to prepare themselves in the best way for the summer school can have a look at this annotated bibliography and at the slides uploaded on this page.
Volker Halbach
Volker Halbach is Professor of Philosophy at the University of Oxford and Fellow of New College, Oxford. His main interests are in logic and especially the theory of truth. He is the author of the Axiomatic Theories of Truth (2011), The Logic Manual (2010), and other books. His new book The Road to Paradox: A Guide to Syntax, Truth, and Modality, which is close to the topics of the summer school, is scheduled to be published in early 2024.
Lorenzo Rossi
Lorenzo Rossi is Assistant Professor at the University of Turin (Department of Philosophy and Education, and Center for Logic, Language, and Cognition (LLC)). His main interests are in logic, and he has worked on theories of truth, semantic paradoxes, the semantics of quantifiers, indicative conditionals, and vagueness. His first monograph (Truth and Paradox in Context, with J. Murzi) is under contract with OUP.
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Schedule of the Summer School
Here is a provisional schedule of the summer school.
Monday 17
3.30pm-4pm
Welcome
4pm-5.30pm
Introduction. A Map through the land of dragons (Halbach & Rossi)
6pm
Drinks at TBD
Tuesday 18
9.30am-10.30am
A theory of expressions (Halbach)
I introduce a theory of syntax that is sufficient for proving Gödel's diagonal lemma. No previous knowledge of the techniques commonly used in the lemma is assumed (such as coding of syntax, the theory of computability etc.).
10.30am-11am
Coffee break
11am-1pm
A theory of expressions (Halbach)
I will look at the connection of syntax-free paradoxes such as Russell's paradox and the paradoxes with diagonalization. I discuss whether truth, necessity, and other notions should be attributed to propositions, sentences, or other objects.
1pm-3pm
Lunch break
3pm-4pm
What is a semantic paradox? Some preliminary coordinates (Rossi)
I will present the difference between paradoxical sentences and paradoxical arguments, and that between proof-theoretic and model-theoretic characterizations of paradoxicality. I will also stress the philosophical significance of semantic paradoxes, for both the expressive power of truth (blind ascriptions, blind arguments), and truth-conditional semantics. Finally, I will present the Inclosure approach to paradoxes.
4.30pm-6.00pm
A. Frenzel - The Relevancy of Carnap's Syntax to the Study of Formal Languages and Semantic Paradoxes
P. Valencia Equy - Axiomatic Truthmaker Theory
A. Lupo - Another Paradox of Property Grounding
Wednesday 19
9.30am-10.30am
The paradoxes (Halbach)
The diagonal lemma is used to prove some basic results such as the liar paradox,Tarski's theorem on the undefinability of truth, and Montague's paradox.
10.30am-11am
Coffee break
11am-1pm
The paradoxes (Halbach)
Some refinements of the diagonal lemma are employed to prove Yablo's paradox, Visser's paradox of ill-founded truth hierarchies, and McGee's omega-inconsistency theorem.
1pm-3pm
Lunch break
3pm-4pm
Conceptions of paradoxicality (I) (Rossi)
The naive conception of paradoxicality, the discussion of paradoxicality and derivability, and the relations with Löb's Theorem are discussed before turning to the proof-theoretic conception, i.e. paradoxicality as non-normalizability.
4.30pm
Exercise and discussion session
Thursday 20
9.30am-10.30am
Possible-worlds analysis of the paradoxes (Halbach)
I discuss a possible-worlds analysis of the paradoxes.
10.30am-11am
Coffee Break
11am-12pm
Possible-worlds analysis of the paradoxes (Halbach)
The previously introduced analysis will be used to argue that ill-foundedness is at the root of many paradoxes.
12pm-2pm
Lunch break
2pm-4pm
Conceptions of paradoxicality (II) (Rossi)
After having introduced the graph-theoretic conception (structural features of paradoxical sentences), I turn to a discussion of ill-foundedness, focusing on circularity and infinitely descending chains.
4.30pm-5.30pm
T. Schuff - Weak Kleene Grounding and Revenge
A. Giglia - Was Cantor Right? How Potentialism Risks a Collapse Into Actualism
Friday 21
9.30am-10.30am
Conceptions of paradoxicality (III) (Rossi)
The Kripkean conception of paradoxicality is introduced, followed by the revision-theoretic one.
10.30am-11am
Coffee Break
11am-1pm
Truth! (Halbach)
A more powerful syntax theory is introduced that permits the study of axiomatic theories of truth such as basic disquotational theories, the Kripke-Feferman and Friedman-Sheard theory.
1pm-3pm
Lunch break 3pm-4pm
Wrap-up (Halbach & Rossi)
Besides the one on the first day, a second drink will be organized and offered on another evening, after the sessions. The precise day will be decided later on depending on the weather, for there is a nice walk to reach and come back from that place.
