Andrew Parisi

13 Feb, 2-3:30pm, LH 306

Various forms of inferentialism require a proof-theoretic account of modality. Many modal logics can be adequately captured with the sequent calculus. But there is no account yet discovered that can capture the full range of modal logics that might be of interest to inferentialists. In particular, no straightforward cut-free account of S5 has been discovered, and no account of logics much weaker than S4 can uniquely characterize the necessitation operator, i.e. a second necessitation operator with the same introduction rules is not everywhere intersubstitutible with the first necessitation operator. An alternative to sequents, hypersequents, is offered as a solution to some of these problems. Hypersequent accounts of various modal logics have been discovered that meet the above two criteria for an inferentialist account of the modal connectives. The various systems are obtained only by restrictions of the external structural rules of the hypersequent calculus.

Francesca Boccuni

6 Feb, 2-3:30pm, LH 306

An abstraction principle has the form §F = §G RE(F, G), where § is an abstraction operator mapping Fregean concepts into objects, and RE is an equivalence relation holding between concepts. Abstraction principles trace back to Frege. Well-known examples are Hume’s Principle and Basic Law V. These principles provide identity conditions for the individuation of abstracta, since they provide a means to identify the entities the identity statement of the left-hand side involves, by appealing to the identity criterion embodied by the equivalence relation on the right-hand side. By individuation, they also provide a way to attach referents to the abstract-terms on the left-hand side. Famously, though, abstraction principles fail to provide sufficient conditions for such an individuation, and thus, it may be argued, they fail to provide a way to fix the reference of the abstract-terms they govern. This is known as the Julius Caesar problem. By the notion of parametric reference in logico-mathematical reasoning, I propose to detach individuation of abstracta from the fixing of reference of abstract-terms. I argue for this view and I provide an appropriate semantics for parametric reference. Furthermore, I investigate a possible argument against this view, according to which the philosophical advantages of the (neo-)Fregean interpretation of abstraction principles as unveiling the nature of Fregean abstracta are lost by the approach via parametric reference, and make this latter approach philosophically unsubstantial. As a reply, I claim that, by using parametric reference, philosophical advantages other than those the (neo-)Fregeans envisage may be obtained. In this respect, I investigate the relation between parametric reference and the notion of invariance under permutations, recently put forward by Aldo Antonelli, in order to retrieve the logicality of abstraction principles on different grounds than the (neo-)Fregeans’.

Katalin Bimbo

20 Feb, 2-3:30pm, LH 306

The use of a binary accessibility relation in the semantics for normal modal logics that was invented by Kripke in the late 1950s is well motivated not only philosophically, but also mathematically. A similar semantics for relevance logics that was introduced by Routley and Meyer in the early 1970s uses a ternary accessibility relation. Gaggle theory that was invented by Dunn in the early 1990s generalizes both Kripke’s as well as Routley and Meyer’s semantics. Gaggle theory predicts that a standard relational semantics for the major relevance logics such as T, E and R should have a ternary accessibility relation.

In this talk, I explore how the Routley-Meyer semantics for T and R fare, respectively, for the implicational fragments of these logics (or for slight extensions thereof).

Justin Khoo

23 Jan, 4pm-5:30pm, LH 306

I discuss three observations about backtracking counterfactuals not predicted by existing theories of counterfactuals, and then motivate a theory of counterfactuals that does predict them. On my theory, counterfactuals quantify over a suitably restricted set of historical possibilities from some contextually relevant past time. I motivate each feature of the theory relevant to predicting our three observations about backtracking counterfactuals.

Joel David Hamkins

5 Dec, 2pm-3:30pm, LH 201

I shall give a general introduction to the theory of infinite games, with a focus on the theory of transfinite ordinal game values. These ordinal game values can be used to show that every open game—a game that, when won for a particular player, is won after finitely many moves—has a winning strategy for one of the players. By means of various example games, I hope to convey the extremely concrete game-theoretic meaning of these game values for various particular small infinite ordinals. Some of the examples will be drawn from infinite chess, which is chess played on a chessboard stretching infinitely without boundary in every direction, and the talk will include animations of infinite chess positions having large numbers of pieces (or infinitely many) with hundreds of pieces making coordinated attacks on the chessboard. Meanwhile, the exact value of the omega_one of chess is not currently known.

Ilaria Frana

7 Nov, 2pm-3:30pm, LH 201

(Joint work with Kyle Rawlins, Johns Hopkins University.)

In this work, we focus on epistemic biases induced by the Italian particle mica in negative polar questions. Simple negative polar questions in Italian, just like their English counterparts, convey a positive epistemic bias on the part of the speaker, i.e. by asking a negative polar question (Non fumi?/Don’t you smoke?) the speaker indicates that (s)he previously expected the positive answer to the question to be true. Negative polar questions with mica (Non fumi mica?), on the other hand, reverse the polarity of the bias, thus conveying that the speaker previously expected the negative answer to the question to be true. Following the literature on polar questions, we propose that mica is an N-word that introduces an epistemic conversational operator (along the lines of Romero and Han (2003)’s VERUM operator). The different epistemic biases are then derived on the basis of the relative scope of negation and VERUM operator(s), and pragmatic (gricean) reasoning.

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”.