Lucas Champollion

24 Oct, 2pm-3:30pm, LH 201

Why can I tell you that I ‘ran for five minutes’ but not that I ‘*ran all the way to the store for five minutes’? Why can you say that there are ‘five pounds of books’ in this package if it contains several books, but not ‘*five pounds of book’ if it contains only one? What keeps you from using ‘*sixty degrees of water’ to tell me the temperature of the water in your pool when you can use ‘sixty inches of water’ to tell me its height? And what goes wrong when I complain that ‘*all theants in my kitchen are numerous’?

The constraints on these constructions involve concepts that semanticists usually study separately: aspect, plural and mass reference, measurement, distributivity. I will provide a unified perspective on these concepts, formalize it within algebraic semantics, and use it connection to transfer insights across unrelated bodies of literature. In particular, I will show how to generalize previous insights by Dowty, Krifka, and Schwarzschild by formulating a single constraint that explains the judgments above and improves on existing characterizations of distributivity, aspect, and measurement.

This talk is self-contained. You don’t need to be an expert in semantics to be able to follow it.

Fenner Tanswell

17 Oct, 2pm-3:30pm, LH 201

In this talk I will discuss the kinds of answers we can give to questions of the correctness, rigour and logic of informal proofs if we do not want to reduce them to formal proofs. I will consider the impacts of accepting a wider class of content-dependent inferences, re-examine the importance of practical knowledge of mathematics and see how informal rigour can be developed in accordance with these.

Fenner Tanswell

19 Sep, 2pm-3:30pm, LH 201

In this talk I will consider the problems posed by the “informal” proofs of actual mathematical practice. I shall focus on the family of views which aim to account for informal proofs in terms of underlying formal proofs (such as Jody Azzouni’s derivation-indicator view), discussing desiderata that such theories are aiming to satisfy. I will argue that the kind of formalisation invoked by these accounts gives rise to a problem of associating informal proofs with too many different formal proofs, undermining the ways in which the accounts were hoping to satisfy the desiderata. Next, I argue that the success of formalisation projects, surprisingly, also do not provide support for such accounts. I will conclude that the formalisability of informal proofs is a red herring and that an alternative account is needed.

Chris Barker

2013/2014 Annual Logic Lecture

28 Mar, 1:30pm-3:30pm, Class of 1947 room

Scope-taking is one of the most dramatic, as well as one of the most characteristic, phenomena in natural language. In scope-taking, a deeply embedded constituent controls (take scope over) the interpretation of surrounding material. For instance, when we gloss the sentence “Mary called everyone yesterday” as `for every person x, Mary called x yesterday’, we are claiming that the embedded direct object “everyone” controls the interpretation of the entire surrounding sentence.

50 years ago, Lambek provided a substructural logic called NL for reasoning about ordinary function-argument combination (`merge’) in natural language. He analyzed argument\function combination and function/argument combination as the left and right adjoints of string concatenation. In Linear Logic terms, his merge is a (noncommutative) multiplicative conjunction (tensor).

In order to extend Lambek’s logic to scope-taking, we need to residuate not on concatenation, but on the part-whole relation. The adjoints then are subpart\whole and whole/subpart. This characterizes a syntactic relationship not of left or right adjacency, but of being-surrounded-by, and of surrounding—exactly what is needed for characterizing scope-taking.

I will present a substructural logic called NL_lambda in which the relationship between the merge mode and the scope-taking mode is characterized by a single structural inference rule. Reporting on joint work with Chung-chieh Shan, I will show that the logic is sound and complete with respect to the usual class of relational models. I will also show that the logic is conservative with respect to Lambek’s original logic. That is, a sequent in the language of NL is a theorem in NL_lambda iff it is a theorem in NL. In addition, I will show that NL_lambda is decidable.

Illustrative applications of the logic to natural language phenomena will include not only ordinary scope-taking and scope ambiguity, but more exotic phenomena such the parasitic scope analysis for words such as “same” and “different”.

Johanna Franklin

11 Apr 2014, 2pm-3:30pm, Oak Hall 408

When shown a binary sequence, most people can intuitively describe it as “random” or “not random.” In this talk, I will characterize randomness formally using three different intuitive approaches as starting points: unpredictability, incompressibility, and a lack of distinguishing properties. If time permits, I will discuss different formalizations within each approach that result in different kinds of randomness and how well these formalizations fit our intuitions about other properties a random sequence should have.

Tamar Lando

6 Dec 2013, 2pm-4pm, Oak Hall 408

Long before Kripke semantics became standard in modal logic, Tarski showed us that the basic propositional modal language can be interpreted in topological spaces. In Tarski’s semantics for the modal logic S4, each propositional variable is evaluated to an arbitrary subset of a fixed topological space. I develop a closely related, measure theoretic semantics for modal logics, in which modal formulas are interpreted in the Lebesgue measure algebra, or algebra of Borel subsets of the real interval [0,1], modulo sets of measure zero. This semantics was introduced by Dana Scott in the last several years. I discuss some of my own completeness results, and a philosophical application to the way we understand physical space.

Zach Weber

15 Nov 2013, 2pm-4pm, Oak Hall 408

Many solutions to the sorties paradox use non-classical logic. But what becomes of the sorties when we turn the tables, and pose the problem using non-classical logics? Is the candidate non-classical logic too weak to be able to formulate the sorites paradox to begin with? There would seem to be something awry if a logic were not strong enough to express the very problems that logic was invoked to address. We will look at some basic tools to demonstrate what a sorties paradox looks like when fully recast in paraconsistent mathematics.

Carl Mummert

1 Nov 2013, 2pm-4pm, Oak Hall 408

The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the “logical” content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice. Beyond clarifying the nature of implication in Reverse Mathematics, this work has potential applications in automated theorem proving, such as the Reverse Mathematics Zoo.

Shawn Standefer

25 Oct 2013, 2pm-4pm, Oak Hall 408

The revision theory is a general theory of circular definitions. We give a brief history and overview of the revision theory. We motivate extending the revision theory with a new unary operator, and show how to modify the basic framework of the revision theory to accommodate it. We provide some examples to show how the extended theory works. A modification of the calculus C_0 from The Revision Theory of Truth is sound and complete with respect to one notion of validity for the revision theory. This permits us to demonstrate how the addition of the operator strengthens the calculus.

There is a stronger notion of validity for which the calculus is merely sound. In the basic revision theory, completeness can be regained by restricting to a certain class of definitions, the finite definitions. These are definitions for which the revision process is, in a sense, over in finitely many steps. In the modified theory, the notion of finiteness needs to be refined, but the new notion retains many of the nice properties of the old one. We sketch how to obtain completeness for definitions that are finite in the new sense.

Time permitting, we will also make connections to a modal logic and sketch a proof of a Solovay-style completeness theorem.

Relevant background material: Gupta and Belnap 1993 “The Revision Theory of Truth”, chapter 4 sections 1-4, chapter 5B

Paul Égré

18 Oct 2013, 2pm-4pm, Oak Hall 408

Theories of indicative conditionals differ on whether sentences of the form “if A then C” have as their negation the conjunction “A and not C” or the conditional negation “if A then not C”. We argue that this opposition is unduly restrictive. We adopt a version of Kratzer’s modal analysis of conditionals on which the default negation of an indicative conditional is “if A then possibly not C”, a conditional negation weaker than “if A then not C”. We present empirical evidence in favor of this assumption and for the hypothesis that both conjunctive negation and strong conditional negation can be retrieved from this weak negation based on pragmatic assumptions regarding the information available to the contradictor of a conditional.