1st SNS Pisa-MCMP-Meeting (27 May 2019)
|10:00 - 10:05||Introduction|
|10:05 - 10:55||Benedict Eastaugh: Hyperarithmetical Analysis and Kripke–Feferman Truth Theories|
|11:00 - 11:55||Mario Piazza: Partial Truths in Classical Logic|
|11:55 - 13:00||Lunch Break|
|13:00 - 13:55||Martin Fischer: A Look at Modalities in Arithmetic|
|14:05 - 15:00||Hannes Leitgeb: HYPE and Possible States Semantics|
|15:05 - 16:00||Gabriele Pulcini: How Many (Classically) Unprovable Sequents are There?|
A theory of hyperarithmetical analysis is one whose omega-models are closed under hyperarithmetic reducibility, and which holds in the class of hyperarithmetical sets HYP. Such theories have been studied in mathematical logic since the late 1950s; Kreisel’s 1962 paper that isolated the axioms of hyperarithmetical comprehension and Σ-1-1 choice was an important early landmark. These theories are interesting from the point of view of axiomatic theories of truth since they are proof-theoretically equivalent to the classical Kripke–Feferman theory KF. Perhaps surprisingly, few mathematical statements from areas outside logic have been found to be theorems of hyperarithmetical analysis. One exception is the statement INDEC that every countable scattered indecomposable linear order is either indecomposable to the left, or indecomposable to the right. In this talk I will give a tour of the landscape of hyperarithmetical analysis, with a focus on separation results, the role of induction, and the relationship to axiomatic theories of truth.top
Modalities have been employed within mathematical theories, especially set theories but also arithmetical theories. In the latter case one of the motivations is to provide an interpretation of 'potential infinity'. In this talk I want to reconsider the role of modalities within arithmetical theories. Whereas a suitable interpretation of 'potential infinity' might not be easy to achieve, modalities could play a different role, namely provide a motivation for arithmetical principles. One of the applications I intend to focus on are modal notions within reflection principles, especially modal predicates.top
This talk will extend the system HYPE of hyperintensional logic and semantics to a possible-states-semantics for
various kinds of modalities, including (one type of) 'because'.top
In this talk one provides a precise proof-theoretic articulation to the elusive notion of partial truth within the confines of classical logic: a non-true classical proposition is not necessarily completely false. Truth-values become discrete elements of the set of rational numbers Q in the interval [0; 1]. The main feature of this approach is its purely syntactical dimension: truth-values decorate the axioms and the rules of classical sequent calculus in order to keep record, along the proof, of the number of occurrences of identity axioms. Some applications of this approach will be addressed.top
Two proof-systems P and P* are said to be complementary when one proves exactly the non-theorems of the other. Complementary systems come as a particular kind of refutation calculi whose patterns of inference always work by inferring unprovable conclusions from unprovable premises.
In the first part of my talk, I will focus on LK*, the sequent system complementing Gentzen system LK for classical logic. I will show, then, how to enrich LK* with two admissible (unary) cut rules, which allow for a simple and efficient cut-elimination algorithm. In particular, two facts will be highlighted: 1) for any given provable sequent, complementary cut-elimination always returns one of its simplest proofs, and 2) provable LK* sequents turn out to be "deductively polarized" by the empty sequent.
This latter fact seems to be of a certain philosophical relevance. In fact, from a proof-theoretic viewpoint, whereas (classically) valid sequents form a wide and multifarious galaxy of logical objects, the complementary set of (classically) invalid sequents is, so to speak, organized like a gravitationally bound system in which provable sequents all 'orbit' the empty sequent.
Room C 113
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