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.
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.
(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.
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.
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.
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.
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”.
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.
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.
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.