# Contextual analysis, epistemic probabilities, and paradoxes

### Ehtibar Dzhafarov

Contextual analysis deals with systems of random variables. Each random variable within a system is labeled in two ways: by its content (that which the variable measures or responds to) and by its context (conditions under which it is recorded). Dependence of random variables on contexts is classified into (1) direct (causal) cross-influences and (2) purely contextual (non-causal) influences. The two can be conceptually separated from each other and measured in a principled way. The theory has numerous applications in quantum mechanics, and also in such areas as decision making and computer databases. A system of deterministic variables (as a special case of random variables) is always void of purely contextual influences. There are, however, situations when we know that a system is one of a set of deterministic systems, but we cannot know which one. In such situations we can assign epistemic (Bayesian) probabilities to possible deterministic systems, create thereby a system of epistemic random variables, and subject it to contextual analysis. In this way one can treat, in particular, such logical antinomies as the Liar paradox. The simplest systems of epistemic random variables describing the latter have no direct cross-influences and the maximal possible degree of purely contextual influences.

# Brouwer, Plato, and classification

### Sam Sanders

Classification is an essential part of all the exact sciences, including mathematical logic.
The program Reverse Mathematics classifies theorems of ordinary mathematics according
to the minimal axioms needed for a proof. We show that the current scale, based on
comprehension and discontinuous functions, is not satisfactory as it classifies many
intuitively weak statements, like the uncountability of \$\mathbb{R}\$ or properties of
the Riemann integral, in the same rather strong class. We introduce an alternative/
complimentary scale with better properties based on (classically valid) continuity
axioms from Brouwer’s intuitionistic mathematics. We discuss how these new
results provide empirical support for Platonism.

# What Can Theoretical Computer Science Contribute to the Discussion of Consciousness?

### Lenore Blum

We propose a mathematical model, which we call the Conscious Turing Machine (CTM), as a formalization of neuroscientist Bernard Baars’ Theater of Consciousness. The CTM is proposed for the express purpose of understanding consciousness. In settling on this model, we look not for complexity but simplicity, not for a complex model of the brain or cognition but a simple mathematical model sufficient to explain consciousness. Our approach, in the spirit of mathematics and theoretical computer science, proposes formal definitions to fix informal notions and deduce consequences. We are inspired by Alan Turing’s extremely simple formal model of computation that is a fundamental first step in the mathematical understanding of computation. This mathematical formalization includes a precise definition of chunk, a precise description of the competition that Long Term Memory (LTM) processors enter to gain access to Short Term Memory (STM)), and a precise definition of conscious awareness in the model. Feedback enables LTM processors to learn from their mistakes and successes and emerging links enable conscious processing to become unconscious. The reasonableness of the formalization lies in the breadth of concepts that the model explains easily and naturally. The model provides some understanding of the Hard Problem of consciousness, which we explore in the particular case of pain and pleasure. The understanding depends on the dynamics of the CTM, not on chemicals like serotonin, dopamine, and so on. We set ourselves the problem of explaining the feeling of consciousness in ways that apply as well to machines made of silicon and gold as to animals made of flesh and blood. With regard to suggestions for AI, the CTM is well suited to giving succinct explanations for whatever high level decisions it makes. This is because the chunk in STM either articulates an explanation or points to chunks that do.

# Logic Group’s Statement on Black Lives Matter

The UConn Logic Group, as a founding and principal member of the Logic Supergroup, is a co-signatory on the Supergroup’s recent Statement on Black Lives Matter. The full statement appears below.

Statement from the Logic Supergroup organizers on Black Lives Matter

The killings of George Floyd and Breonna Taylor by police have resulted in deep sadness and outrage worldwide. This sadness and outrage has swept through academic communities as well. We members of the Logic Supergroup join our colleagues throughout academia in denouncing anti-Black violence in North America specifically, and systemic racism affecting Black, Indigenous, and People of Color generally. We are committed to working with our BIPOC colleagues in logic and with our colleagues in other fields to speak out against injustice and to work to make logic an inclusive community. Black Lives Matter.

In early 2020, logic groups around the world began to combine their meetings into an online Logic Supergroup as a reaction to the coronavirus pandemic. As part of #ShutDownSTEM, the Supergroup held 24 hours of Inclusive Logic on June 10, 2020 to build concrete actions against racism in logic.

Here are four concrete actions that we identified as important and have committed ourselves to immediately, and which we encourage our member groups to commit themselves to as well:

We commit to doing our small part to increase the diversity in logic, as a discipline, by first increasing the diversity represented in our speaker series. We acknowledge that we have, to this point, failed in this regard. We acknowledge, in fact, that we have failed to have a speaker series that even manages to be representative of the diversity that is present in logic, let alone one that contributes to increasing this diversity.Second, we commit to seeking funding from a broad range of sources in order to overcome barriers to participation in the supergroup, and in logic meetings and seminars more broadly. Such funding could be used, for example, in order to allow for graduate student travel stipends if the supergroup holds in-person events in the future, or to aid researchers who need equipment in order to participate in our virtual events now. This will remove some of the barriers faced by BIPOC students to entering and participating in the logic community.

Third, we commit to hosting periodic events to discuss, evaluate, and develop resources for logic education that are inclusive and that encourage participation in logic from a diverse audience. This includes setting regular Logic Supergroup meeting times to discuss further actions in support of inclusion. We recognize that addressing issues of inclusiveness and working in support of anti-racism within individuals and within academia require continued commitment and ongoing, sustained effort.

Fourth, we commit to developing, promoting, and enunciating guidelines and codes of conduct for our conferences, workshops and other events, as well as other events associated with our constituent organizations.

# 2019 Workshop: “If” by any other name

### “If” by any other name

It is a relatively recent development that research on conditionals is taking a deep and sustained interest in the full range of linguistic markers, their interactions with each other and with other linguistic categories, and the ways in which they drive and constrain the interpretation of the sentences they occur in. Tense and aspect is an area where such attention has already borne fruit; to a lesser extent, we may mention conditional connectives and pro-forms (especially thanks to works like Iatridou 2000 and Iatridou & Embick 1993). More recently, there seems to be a growing interest in two things: on the one hand, more varied aspects of formal marking of conditionals and the ways in which different grammatical categories may be recruited to encode conditional meaning (including aspect, different types of connectives, conjunctions, etc.); on the other hand, the appearance of these markers in other linguistic contexts (like optatives, complement clauses, temporal clauses, interrogatives, etc.).

Program

Saturday, April 6

12:00-2:00: Kai von Fintel & Sabine Iatridou (MIT) “Prolegomena to a Theory of X-Marking”

2:30-3:15: Muyi Yang (UConn) “Explaining negative counterfactuals”
3:15-4:00: Teruyuki Mizuno (UConn) “The structure of might-counterfactuals: a view from Japanese”

4:30-5:30: Paolo Santorio (UC San Diego) “The Double Life of Antecedent Strengthening”

Sunday, April 7

10:00-11:00: Una Stojnić (Columbia): t.b.a.

11:15-12:00: Hiromune Oda (UConn): t.b.a.

1:30-2:30: Will Starr (Cornell) “Indicative Conditionals, Strictly”
After 2:30: coffee & discussion as desired

# Recursive counterexamples and the foundational standpoint

### Benedict Eastaugh

Recursive counterexamples are classical mathematical theorems that are made false by restricting their quantifiers to computable objects. Historically, they have been important for analysing the mathematical limitations of foundational programs such as constructivism or predicativism. For example, the least upper bound principle is recursively false, and thus unprovable by constructivists. In this talk I will argue that while recursive counterexamples are valuable for analysing foundational positions from an external, set-theoretic point of view, the approach is limited in its applicability because the results themselves are not accessible to the foundational standpoints under consideration. This limitation can be overcome, to some extent, by internalising the recursive counterexamples within a theory acceptable to a proponent of a given foundation; this is, essentially, the method of reverse mathematics. I will examine to what extent the full import of reverse mathematical results can be appreciated from a given foundational standpoint, and propose an analysis based on an analogy with Brouwer’s weak and strong counterexamples. Finally, I will argue that, at least where the reverse mathematical analysis of foundations is concerned, the epistemic considerations above show that reverse mathematics should be carried out within a weak base theory such as RCA0, rather than by studying ω-models from a set-theoretic perspective.

# De Re and De Dicto Knowledge of Mathematical Statements

### Marianna Antonutti

In this talk, I will apply the de re/de dicto distinction to the analysis of mathematical statements and knowledge claims in mathematics. A proof will be said to provide de dicto knowledge of a mathematical statement if it provides knowledge of a purely existential statement, and to provide de re knowledge when it carries additional information concerning the identity criteria for the objects that are proven to exist. I will examine two case studies, one from abstract algebra and one from discrete mathematics, and I will suggest that reverse mathematics can help measuring the ‘de re content’ of two different proofs of the same theorem, and that the de re/de dicto distinction introduced here lines up with certain model theoretic properties of subsystems of second order arithmetic, such as the existence of certain kinds of minimal model. Furthermore, I will argue that there are good reasons not to identify the de re content of a proof with its constructive content nor with its predicative content.

# How to model the probabilities of conditionals

### Branden Fitelson

David Lewis (and others) have famously argued against Adams’s Thesis (that the probability of a conditional is the conditional probability of its consequent, given it antecedent) by proving various “triviality results.” In this paper, I argue for two theses — one negative and one positive. The negative thesis is that the “triviality results” do not support the rejection of Adams’s Thesis, because Lewisian “triviality based” arguments against Adams’s Thesis rest on an implausibly strong understanding of what it takes for some credal constraint to be a rational requirement (an understanding which Lewis himself later abandoned in other contexts). The positive thesis is that there is a simple (and plausible) way of modeling the probabilities of conditionals, which (a) obeys Adams’s Thesis, and (b) avoids all of the existing triviality results.

# At-issueness in direct quotation: the case of Mayan quotatives

### Scott AnderBois

In addition to verba dicendi, languages have a bunch of different other grammatical devices for encoding reported speech. While not common in Indo-European languages, two of the most common such elements cross-linguistically are reportative evidentials and quotatives. Quotatives have been much less discussed then either verba dicendi or reportatives, both in descriptive/typological literature and especially in formal semantic work. While quotatives haven’t been formally analyzed in detail previously to my knowledge, several recent works on reported speech constructions in general have suggested in passing that they pattern either with verba dicendi or with reportatives. Drawing on data from Yucatec Maya, I argue that they differ from both since they present direct quotation (like verba dicendi) but make a conventional at-issueness distinction (like reportatives). To account for these facts, I develop an account of quotatives by combining an extended Farkas & Bruce 2010-style discourse scoreboard with bicontextualism (building on Eckardt 2014’s work on Free Indirect Discourse).