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.

Magdalena Kaufmann

26 Apr 2013, 2pm-3:30pm, Manchester 227

The truth and appropriateness of statements and instructions about what is the best thing to do (as expressed by declarative sentences containing deontic modals and imperative sentences) can depend on the information that is or will be available to the relevant agent at the time of choice. Moreover, antecedents of conditionals can sometimes behave as if supplying the relevant information for modals or imperatives in the consequent. These interactions between what counts as the best or optimal action for an agent and what information is available poses problems for standard treatments of deontic modals and conditionals in formal semantics. I will start by introducing Cariani, Kaufmann & Kaufmann’s (forthcoming) treatment of the miners’ paradox (Kolodny & MacFarlane 2010), and I will then focus on the influence of subjunctive marking on German modals and point out possible connections to recent work on the distinction between weak and strong necessity modals (like ‘ought’ vs. ‘have to’).

Igor Yanovich

29 Mar 2013, 2pm-3:30pm, Manchester 227

The proper analysis of “weak” vs. “strong” necessity deontics (should and ought vs. must and have to) and the scopal interaction of different deontics with negation are two topics much discussed in the recent literature. In this talk, I will present data from Russian that shed new light on both, underscoring the importance of using cross-linguistic data when considering universal semantic issues. The thread uniting the two topics will be the importance of advice modality.

Russian features a rich deontic system, with more than half a dozen of necessity deontics. I will show that this rich system does not contain a single categorical contrast between weak and strong necessity, as in English. Instead, different tests commonly used for English as well as other languages (cf. von Fintel and Iatridou 2008) as all targeting the same distinction turn out to divide the Russian deontic space into different groupings. Thus the category of weak necessity is no more than an illusion created by the interplay of several independent properties which do not have much to do with the strength of modal force.

One of such properties is the ability of a given modal to be used as (for lack of a better term) an advice modal. “Advice modality” seems to be a special type of modality, comparable to such categories as teleological or metaphysical modality. It seems to be characterized by three properties: 1) it chooses one of the possible alternative courses of actions, thus solving a certain decision problem; 2) it needs to be endorsed by the speaker in matrix contexts, and by the attitude bearer under attitudes; 3) it always scopes over clausemate negation. All English modals that can be used as advice modals have other readings as well, but Russian features modal stoit that conveniently can only be used in advice. Furthermore, for most Russian necessity deontics, relative scope with negation is not fixed, unlike in English, Greek, Dutch, German, Spanish (cf. Iatridou and Zeijlstra 2013). This creates a convenient testing ground. I will describe that modal’s behavior compared to other modals of Russian and English, and discuss how the three properties of advice modality may be related to each other.

Stewart Shapiro

14 Mar 2013, 4pm-5:30pm, Manchester 227

We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary actual infinity. Also, in contrast to intuitionistic analysis, Bishop’s constructive analysis, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our “gunky line” into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence of “indecomposability” from a non-punctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and the interval analogue of Dedekind completeness (least-upper-bound principle), and that they determine an isomorphism with the Dedekind-Cantor structure of R as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

Keith Simmons

18 Jan 2013, 4pm-6pm, FS 216

There are paradoxes of reference (Richard, Berry, KÓ§nig), paradoxes of predication (Russell), and paradoxes of truth (the Liar). I argue that there is a phenomenon that cuts across all of these semantic paradoxes: given a paradoxical expression, we can produce an expression composed of the very same words that is not defective, but instead has a definite semantic value. Call this phenomenon Repetition. I critically examine Kripke’s and Field’s responses to Repetition in the case of truth. I suggest an alternative response – a contextual account according to which our semantic concepts apply everywhere except for certain ‘singular points’. Now any account of the semantic paradoxes is vulnerable to ‘revenge’: new forms of the Liar that the account cannot handle. Repetition is one manifestation of revenge, and I go on to consider revenge more broadly. I argue that even dialetheism — according to which there are true contradictions — is subject to revenge. And I suggest a way we might approach revenge along singularity lines.

Jan Reimann

13 Apr 2012, 2pm-4pm, Class of 1947 Room

In mathematical logic, one tries to classify objects by their descriptive complexity, for example, how many quantifier changes are needed to define a given subset of the natural numbers. On the other hand, concepts like entropy allow for a measure-theoretic classification of complex, i.e. random, behavior.

Both approaches can be combined to define a notion of randomness for individual objects such as infinite binary sequences. I will discuss the resulting interplay between probability and definability. I will argue that the view from logic opens up new and perhaps unexpected perspectives on the concept of randomness, for example, concerning the role of infinity.