Abstracts

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.

What Problem Did Ladd-Franklin (Think She) Solve(d)?

Sara Uckelman

Christine Ladd-Franklin is often hailed as a guiding star in the history of women in logic—not only did she study under C.S. Peirce and was one of the first women to receive a PhD from Johns Hopkins, she also, according to many modern commentators, solved a logical problem which had plagued the field of syllogisms since Aristotle. In this paper, we revisit this claim, posing and answering two distinct questions: Which
logical problem did Ladd-Franklin solve in her thesis, and which problem did she think she solved? We show that in neither case is the answer “a long-standing problem due to Aristotle”. Instead, what
Ladd-Franklin solved was a problem due to Jevons that was first articulated in the 19th century.

Logical Nihilism

Éno Agolli

Logical nihilism is the view that there is no logic, or more precisely that no single, universal consequence relation governs natural language reasoning. Here, I present three different arguments for logical nihilism from philosophically palatable premises. The first argument comes from a combination of pluralism with the desideratum that logical consequence should be universal, properly understood. The second argument is a slippery slope argument against monists who support weak logical systems on account of their power to characterize a vast range of true theories. The third argument is a general strategy of generating counterexamples to any inference rule, including purportedly fundamental ones such as disjunction introduction. I close by discussing why a truth-conditional approach to the meaning of the logical connectives not only does not force us to reject such counterexamples but also reveals that right truth-conditions are far more general than the classical ones, at the price of nihilism.

Ordering Anything: Rejiggering Linnebo’s Ordinal Abstraction

Eileen Nutting

Øystein Linnebo develops an abstractionist account of the natural numbers as ordinals. On this account, the natural numbers are abstracted from orderings of concrete numerals. But Linnebo also gestures towards an alternative version of his account, on which the restriction to concrete numerals is lifted. I develop something like this alternative account, show how it avoids the Burali-Forti paradox, and show how it guarantees that every number has a successor. Given these and other good features, I claim that Linnebo should prefer this alternative account to the one he develops.

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.

References:

Kujala, J.V., Dzhafarov, E.N., & Larsson, J.-A. (2015). Necessary and sufficient conditions for extended noncontextuality in a broad class of quantum mechanical systems. Physical Review Letters 115, 150401 (available as arXiv:1407.2886.).

Dzhafarov, E.N., Cervantes, V.H., Kujla, J.V. (2017). Contextuality in canonical systems of random variables. Philosophical Transactions of the Royal Society A 375: 20160389 (available as arXiv:1703.01252).

Cervantes, V.H., & Dzhafarov, E.N. (2018). Snow Queen is evil and beautiful: Experimental evidence for probabilistic contextuality in human choices. Decision 5, 193-204 (available as arXiv:1711.00418).

Core type theory

David Ripley

The Curry-Howard correspondence between intuitionistic logic and the simply-typed lambda calculus forms an important bridge between logical and computational research. This talk develops a variant typed lambda calculus, called “core type theory”, that stands in a similar correspondence to Neil Tennant’s “core logic” (fka “intuitionistic relevant logic”), and shows some basic (and some surprising!) results about this calculus.

Variable Free Semantics: Putting competition effects where they belong

Pauline Jacobsen

This talk will have two parts. First I will discuss the approach to semantics making no use of variable names, indices, or assignment functions that I have advocated in a series of papers (see especially Jacobson, 1999, Linguistics and Philosophy and 2000, Natural Language Semantics, also exposited in Jacobson 2014 textbook Compositional Semantics, OUP). There are a number of theoretical and empirical advantages to this approach, which will be just briefly reviewed. To mention the most obvious theoretical advantage: the standard use of variable names and indices in semantics requires meanings to be relativized to assignment functions (assignments of values to the variable names), adding a layer to the semantic machinery. This program eliminates this and treats all meanings as ‘healthy’ model theoretic objects (the meaning of a pronoun, for example, is simply the identity function on individuals, not a function from assignments to individuals). I will then show a new empirical payoff, which concerns competition effects found in ellipsis constructions. These competition effects have gone under the rubric of MaxElide in the linguistics literature. One example centers on the contrast in (1) (on the reading where each candidates hope is about their own success):

  1. a. Harris is hoping that South Carolina will seal the nomination for her, and Warren is too. (= ‘hoping that it will seal nomination for her (Warren)’)
    b. ?*Harris is hoping that South Carolina will seal the nomination for her, and Warren is also hoping that it will. (= ‘seal the nomination for her (Warren)’)

The ‘standard’ wisdom is there is a constraint in the grammar that when material is ‘missing’ (or, ‘elided’) if a bigger constituent can be elided, the bigger ellipsis is required. Why the grammar should contain such a constraint is a total mystery; moreover I and others have argued elsewhere that grammatical competition constraints represent a real complication in the grammar. When there are competition effects they should be located in speakers and hearers (we know that speakers and hearers do compute alternatives – Gricean reasoning, for example, is based on that assumption). Under the variable free account, the missing material in (1a) is of a different type than that in (1b). In (1a), the listener need only supply the property ‘be an x such that x hopes that SC will seal nomination for x‘ which is the meaning of the VP in the first clause. In (1b) what must be supplied is the 2-place relation ‘seal the nomination for’ (note that this is in part because the pronoun her in the first clause is not a variable, and so [[seal the nomination for her]] is the function from an individual x to the property of sealing the nomination for x, which in turn is the two place relation named above. The competition effect is thus about types not size, and can be given a plausible explanation in terms of communicative pressures. Assuming that meanings of more complex types are more difficult to access than those of simpler types, there is a pressure for speakers to choose the simpler type ellipsis. The type competition story crucially relies on the claim that expressions containing pronouns unbound within them denote functions from individuals to something rather than functions from assignment functions.

How to deal with semantic paradox (Hint: accept defeat)

Steven Dalglish

I contend that semantic paradox shows we should regard the rules of inference for semantic notions as defeasible. Truth is a prominent semantic notion for which semantic paradox poses a problem, and so a first step in solving semantic paradox is handling a semantic notion of truth. This talk investigates how defeasible rules of inference for a semantic notion of truth can form the basis for a successful truth-conditional theory of meaning. I start by using Default Logic for representing the rules of inference for a truth predicate as defeasible, before adding preferences over rules and modifying the criteria for defeat.

A Dynamic Solution to the Liar Paradox

Martin Pleitz

The Liar paradox arises when we combine the assumption that a sentence can refer to itself with our naïve notion of truth and apply our unrevised logic. Most current approaches to the Liar paradox focus on revising our notion of truth and logic because nowadays almost everyone is convinced that there are self-referential sentences. I will argue against this conviction. My argument starts from observations about the metaphysics of expressions: A meaningful expression is based in a syntactic expression which in turn is based in a non-semiotic object, and these are pairwise distinct. As all objects of this three-fold ontology exist only relative to contexts, we can import ideas from tense logic about how existence and reference can interact in a contextualist metaphysics. Semantico-metaphysical reasoning then shows that in this dynamic setting, an object can be referred to only after it has started to exist. Hence the self-reference needed in the Liar paradox cannot occur, after all. As this solution is contextualist, it evades the expressibility problems of other proposals.

Proof-theoretic Harmony

Intensional aspects from a second-order perspective

Luca Tranchini

The talk will focus on harmony, the widely discussed condition that two collections of introduction and of elimination rules for a certain logical constant should satisfy in order to perfectly match each other. Attention will be restricted to rules for propositional connectives in single-conclusion natural deduction format.

I will first argue that a decent criterion of harmony should be (hyper-)intensional, in the sense that taken two equivalent collections of elimination rules (i.e. such that each rule of the first collection is derivable from the rules in the second collection, and vice versa), it may well be that only one of the two collections is in harmony with a given collection of introduction rules.

To precisely formulate an intensional criterion for harmony we propose to modify the (extensional) criterion for harmony recently proposed by Schroeder-Heister (for short SH-harmony) and based on the idea that to any collection of introduction and of elimination rules one can associate a formula of intuitionistic propositional logic with propositional quantification (IPC2). Two collections of introduction and elimination rules for a certain connective are in SH-harmony if the two formulas associated to them are interderivable in IPC2.

An intensional criterion is achieved by imposing on the two formulas a condition stricter than intederivability: isomorphim. We will present the notion of isomorphism (well-known from category theory and computational study of proof systems), stress its significance for proof-theoretic semantics, and finally apply it to clarify the relationship between different ways of generating collections of elimination rules from a given collection of introduction rules known in the literature.