This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in (...) theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search including proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences.

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in (...) theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism, Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

The lambdaPi-calculus, a theory of first-order dependent function types in Curry-Howard-de Bruijn correspondence with a fragment of minimal first-order logic, is defined as a system of (linearized) natural deduction. In this paper, we present a Gentzen-style sequent calculus for the lambdaPi-calculus and prove the cut-elimination theorem. The cut-elimination result builds upon the existence of normal forms for the natural deduction system and can be considered to be analogous to a proof provided by Prawitz for first-order logic. The type-theoretic setting considered (...) here elegantly illustrates the distinction between the processes of normalization in a natural deduction system and cut-elimination in a Gentzen-style sequent calculus. (shrink)

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use (...) the operational view of them as provided by uniform proof-search—the proof-theoretic foundation of logic programming (LP)—to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS. (shrink)

In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems |$K$|⁠, |$KT$|⁠, |$K4$| and (...) |$S4$|⁠, with |$\square $| as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between |$\square $| and a natural presentation of |$\lozenge $|⁠. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

We give a novel approach to proving soundness and completeness for a logic (henceforth: the object-logic) that bypasses truth-in-a-model to work directly with validity. Instead of working with specific worlds in specific models, we reason with eigenworlds (i.e., generic representatives of worlds) in an arbitrary model. This reasoning is captured by a sequent calculus for a _meta_-logic (in this case, first-order classical logic) expressive enough to capture the semantics of the object-logic. Essentially, one has a calculus of validity for the (...) object-logic. The method proceeds through the perspective of reductive logic (as opposed to the more traditional paradigm of deductive logic), using the space of reductions as a medium for showing the behavioural equivalence of reduction in the sequent calculus for the object-logic and in the validity calculus. Rather than study the technique in general, we illustrate it for the logic of Bunched Implications (BI), thus IPL and MILL (without negation) are also treated. Intuitively, BI is the free combination of intuitionistic propositional logic and multiplicative intuitionistic linear logic, which renders its meta-theory is quite complex. The literature on BI contains many similar, but ultimately different, algebraic structures and satisfaction relations that either capture only fragments of the logic (albeit large ones) or have complex clauses for certain connectives (e.g., Beth’s clause for disjunction instead of Kripke’s). It is this complexity that motivates us to use BI as a case-study for this approach to semantics. (shrink)

We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic (...) logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers ∀ new and ∃ new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels. (shrink)

The reductive, as opposed to deductive, view of logic is the form of logic that is, perhaps, most widely employed in practical reasoning. In particular, it is the basis of logic programming. Here, building on the idea of uniform proof in reductive logic, we give a treatment of logic programming for BI, the logic of bunched implications, giving both operational and denotational semantics, together with soundness and completeness theorems, all couched in terms of the resource interpretation of BI’s semantics. We (...) use this set-up as a basis for exploring how coalgebraic semantics can, in contrast to the basic denotational semantics, be used to describe the concrete operational choices that are an essential part of proof-search. The overall aim, toward which this paper can be seen as an initial step, is to develop a uniform, generic, mathematical framework for understanding the relationship between the deductive structure of logics and the control structures of the corresponding reductive paradigm. (shrink)

[This is an edited version of a keynote talk at the conference on Neural Information Processing Systems (NeurIPS) on November 28, 2022, with some minor additions and subtractions.] -/- There has recently been widespread discussion of whether large language models might be sentient or conscious. Should we take this idea seriously? I will break down the strongest reasons for and against. Given mainstream assumptions in the science of consciousness, there are significant obstacles to consciousness in current models: for example, their (...) lack of recurrent processing, a global workspace, and unified agency. At the same time, it is quite possible that these obstacles will be overcome in the next decade or so. I conclude that while it is somewhat unlikely that current large language models are conscious, we should take seriously the possibility that successors to large language models may be conscious in the not-too-distant future. (shrink)

*[[This paper appears in _Toward a Science of Consciousness II: The Second Tucson Discussions and Debates_ (S. Hameroff, A. Kaszniak, and A.Scott, eds), published with MIT Press in 1998. It is a transcript of my talk at the second Tucson conference in April 1996, lightly edited to include the contents of overheads and to exclude some diversions with a consciousness meter. A more in-depth argument for some of the claims in this paper can be found in Chapter 6 of my (...) book _The Conscious Mind_ (Chalmers, 1996). ]]. (shrink)

Confronted with the apparent explanatory gap between physical processes and consciousness, there are many possible reactions. Some deny that any explanatory gap exists at all. Some hold that there is an explanatory gap for now, but that it will eventually be closed. Some hold that the explanatory gap corresponds to an ontological gap in nature.

The book is an extended study of the problem of consciousness. After setting up the problem, I argue that reductive explanation of consciousness is impossible , and that if one takes consciousness seriously, one has to go beyond a strict materialist framework. In the second half of the book, I move toward a positive theory of consciousness with fundamental laws linking the physical and the experiential in a systematic way. Finally, I use the ideas and arguments developed earlier to defend (...) a form of strong artificial intelligence and to analyze some problems in the foundations of quantum mechanics. (shrink)

[[This was written as a commentary on Ned Block 's paper "On A Confusion about a Function of Consciousness" . It appeared in _Behavioral_ _and Brain Sciences_ 20:148-9, 1997, and also in the collection _The Nature of Consciousness: Philosophical Debates_ edited by Block, Flanagan, and Guzeldere. ]].

A month ago, I bought an iPhone. The iPhone has already taken over some of the central functions of my brain. It has replaced part of my memory, storing phone numbers and addresses that I once would have taxed my brain with. It harbors my desires: I call up a memo with the names of my favorite dishes when I need to order at a local restaurant. I use it to calculate, when I need to figure out bills and tips. (...) It is a tremendous resource in an argument, with Google ever present to help settle disputes. I make plans with it, using its calendar to help determine what I can and can’t do in the coming months. I even daydream on the iPhone, idly calling up words and images when my concentration slips. Friends joke that I should get the iPhone implanted into my brain. But if Andy Clark is right, all this would do is speed up the processing, and free up my hands. The iPhone is part of my mind already. Clark is a connoisseur of the myriad ways in which the mind relies on the world to get its work done. The first part of this marvelous book explores some of these ways: the extension of our bodies, the extension of our senses, and crucially, the use of language as a tool to extend our thought. The second part of the book defends the thesis that in at least some of these cases, the world is not serving as a mere instrument for the mind. Rather, the relevant parts of the world have become parts of my mind. My iPhone is not my tool, or at least it is not wholly my tool. Parts of it have become parts of me. This is the thesis of the extended mind: when parts of the environment are coupled to the brain in the right way, they become parts of the mind. The thesis has a long history: I am told that there are hints of it in Dewey, Heidegger, and Wittgenstein. But no-one has done as much to give life to the idea as Andy Clark. In a series of important books and articles—Being There, Natural-Born Cyborgs, “Magic words: How language augments human computation”, and many others—he has explored the many ways in which the boundaries between mind and world are far more flexible than one might have thought.. (shrink)

There is a long tradition in philosophy of using a priori methods to draw conclusions about what is possible and what is necessary, and often in turn to draw conclusions about matters of substantive metaphysics. Arguments like this typically have three steps: first an epistemic claim , from there to a modal claim , and from there to a metaphysical claim.

Does consciousness collapse the quantum wave function? This idea was taken seriously by John von Neumann and Eugene Wigner but is now widely dismissed. We develop the idea by combining a mathematical theory of consciousness (integrated information theory) with an account of quantum collapse dynamics (continuous spontaneous localization). Simple versions of the theory are falsified by the quantum Zeno effect, but more complex versions remain compatible with empirical evidence. In principle, versions of the theory can be tested by experiments with (...) quantum computers. The upshot is not that consciousness-collapse interpretations are clearly correct, but that there is a research program here worth exploring. (shrink)

This chapter analyzes aspects of the relationship between consciousness and intentionality. It focuses on the phenomenal character and the intentional content of perceptual states, canvassing various possible relations among them. It argues that there is a good case for a sort of representationalism, although this may not take the form that its advocates often suggest. By mapping out some of the landscape, the chapter tries to open up territory for different and promising forms of representationalism to be explored in the (...) future. In particular, it argues for a nonreductive, narrow, and Fregean variety of representationalism, which contrasts strongly with more widely explored varieties. It concludes with some words about the fundamental relationship between consciousness and intentionality. (shrink)

In the Garden of Eden, we had unmediated contact with the world. We were directly acquainted with objects in the world and with their properties. Objects were simply presented to us without causal mediation, and properties were revealed to us in their true intrinsic glory.

Why is two-dimensional semantics important? One can think of it as the most recent act in a drama involving three of the central concepts of philosophy: meaning, reason, and modality. First, Kant linked reason and modality, by suggesting that what is necessary is knowable a priori, and vice versa. Second, Frege linked reason and meaning, by proposing an aspect of meaning (sense) that is constitutively tied to cognitive signi?cance. Third, Carnap linked meaning and modality, by proposing an aspect of meaning (...) (intension) that is constitutively tied to possibility and necessity. (shrink)

What happens when machines become more intelligent than humans? One view is that this event will be followed by an explosion to ever-greater levels of intelligence, as each generation of machines creates more intelligent machines in turn. This intelligence explosion is now often known as the “singularity”. The basic argument here was set out by the statistician I.J. Good in his 1965 article “Speculations Concerning the First Ultraintelligent Machine”: Let an ultraintelligent machine be defined as a machine that can far (...) surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an “intelligence explosion”, and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make. The key idea is that a machine that is more intelligent than humans will be better than humans at designing machines. So it will be capable of designing a machine more intelligent than the most intelligent machine that humans can design. So if it is itself designed by humans, it will be capable of designing a machine more intelligent than itself. By similar reasoning, this next machine will also be capable of designing a machine more intelligent than itself. If every machine in turn does what it is capable of, we should expect a sequence of ever more intelligent machines. This intelligence explosion is sometimes combined with another idea, which we might call the “speed explosion”. The argument for a speed explosion starts from the familiar observation that computer processing speed doubles at regular intervals. Suppose that speed doubles every two years and will do so indefinitely. Now suppose that we have human-level artificial intelligence 1 designing new processors. Then faster processing will lead to faster designers and an ever-faster design cycle, leading to a limit point soon afterwards. The argument for a speed explosion was set out by the artificial intelligence researcher Ray Solomonoff in his 1985 article “The Time Scale of Artificial Intelligence”.1 Eliezer Yudkowsky gives a succinct version of the argument in his 1996 article “Staring at the Singularity”: “Computing speed doubles every two subjective years of work.. (shrink)

The basic question of ontology is “What exists?”. The basic question of metaontology is: are there objective answers to the basic question of ontology? Here ontological realists say yes, and ontological anti-realists say no. (Compare: The basic question of ethics is “What is right?”. The basic question of metaethics is: are there objective answers to the basic question of ethics? Here moral realists say yes, and moral anti-realists say no.) For example, the ontologist may ask: Do numbers exist? The Platonist (...) says yes, and the nominalist says no. The metaontologist may ask: is there an objective fact of the matter about whether numbers exist? The ontological realist says yes, and the ontological anti-realist says no. Likewise, the ontologist may ask: Given two distinct entities, when does a mereological sum of those entities exist? The universalist says always, while the nihilist says never. The metaontologist may ask: is there an objective fact of the matter about whether the mereological sum of two distinct entities exists? The ontological realist says yes, and the ontological anti-realist says no. Ontological realism is often traced to Quine (1948), who held that we can determine what exists by seeing which entities are endorsed by our best scientific theory of the world. In recent years, the practice of ontology has often presupposed an ever-stronger ontological realism, and strong versions of ontological realism have received explicit statements by Fine (this volume), Sider (2001; this volume), van Inwagen (1998; this volume), and others. (shrink)

The term ‘emergence’ often causes confusion in science and philosophy, as it is used to express at least two quite different concepts. We can label these concepts _strong_ _emergence_ and _weak emergence_. Both of these concepts are important, but it is vital to keep them separate.

The search for neural correlates of consciousness (or NCCs) is arguably the cornerstone in the recent resurgence of the science of consciousness. The search poses many difficult empirical problems, but it seems to be tractable in principle, and some ingenious studies in recent years have led to considerable progress. A number of proposals have been put forward concerning the nature and location of neural correlates of consciousness. A few of these include.

The Matrix presents a version of an old philosophical fable: the brain in a vat. A disembodied brain is floating in a vat, inside a scientist’s laboratory. The scientist has arranged that the brain will be stimulated with the same sort of inputs that a normal embodied brain receives. To do this, the brain is connected to a giant computer simulation of a world. The simulation determines which inputs the brain receives. When the brain produces outputs, these are fed back (...) into the simulation. The internal state of the brain is just like that of a normal brain, despite the fact that it lacks a body. From the brain’s point of view, things seem very much as they seem to you and me. (shrink)

Hilary Putnam has argued that computational functionalism cannot serve as a foundation for the study of the mind, as every ordinary open physical system implements every finite-state automaton. I argue that Putnam's argument fails, but that it points out the need for a better understanding of the bridge between the theory of computation and the theory of physical systems: the relation of implementation. It also raises questions about the class of automata that can serve as a basis for understanding the (...) mind. I develop an account of implementation, linked to an appropriate class of automata, such that the requirement that a system implement a given automaton places a very strong constraint on the system. This clears the way for computation to play a central role in the analysis of mind. (shrink)

Addresses the issues at the heart of international medicine and social responsibility. A number of international declarations have proclaimed that health care is a fundamental human right. But if we accept this broad commitment, how should we concretely define the state’s responsibility for the health of its citizens? Although there is growing debate over this issue, there are few books for general readers that provide engaging accounts of critical incidents, practices, and ideas in the field of human rights, health care, (...) and medicine. Included in the book are case studies of such issues as AIDS among orphans in Romania, organ trafficking, prison conditions, health care rationing, medical research in the third world, and South Africa’s constitutionally guaranteed right of access to health care. It uses these topics to address themes of protection of vulnerable populations, equity and fairness in delivering competent medical care, informed consent and the free flow of information, and state responsibility for ensuring physical, mental, and social well-being. (shrink)

In Philosophy Without Intuitions, Herman Cappelen focuses on the metaphilosophical thesis he calls Centrality: contemporary analytic philosophers rely on intuitions as evidence for philosophical theories. Using linguistic and textual analysis, he argues that Centrality is false. He also suggests that because most philosophers accept Centrality, they have mistaken beliefs about their own methods.To put my own views on the table: I do not have a large theoretical stake in the status of intuitions, but unreflectively I find it fairly obvious that (...) many philosophers, including myself, appeal to intuitions. Cappelen’s arguments make a provocative challenge to this unreflective background conception. So it is interesting to work through the arguments to see what they might and might not show.In what follows I aim to articulate a minimal notion of intuition that captures something of the core everyday philosophical usage of the term, and that captures the sense .. (shrink)

Is there progress in philosophy? A glass-half-full view is that there is some progress in philosophy. A glass-half-empty view is that there is not as much as we would like. I articulate a version of the glass-half-empty view, argue for it, and then address the crucial question of what explains it.

In the carefully argued central chapters of Adapting Minds, Buller scrutinizes several of evolutionary psychology's most highly publicized "...

Does the capacity to think require the capacity to sense? A lively debate on this topic runs throughout the history of philosophy and now animates discussions of artificial intelligence. Many have argued that AI systems such as large language models cannot think and understand if they lack sensory grounding. I argue that thought does not require sensory grounding: there can be pure thinkers who can think without any sensory capacities. As a result, the absence of sensory grounding does not entail (...) that large language models cannot think or understand. I also consider to what extent quasi-sensory grounding can at least boost the performance of a language model. (shrink)

*[[This paper is largely based on material in other papers. The first three sections and the appendix are drawn with minor modifications from Chalmers 2002c . The main ideas of the last three sections are drawn from Chalmers 1996, 1999, and 2002a, although with considerable revision and elaboration. ]].

Two-dimensional approaches to semantics, broadly understood, recognize two "dimensions" of the meaning or content of linguistic items. On these approaches, expressions and their utterances are associated with two different sorts of semantic values, which play different explanatory roles. Typically, one semantic value is associated with reference and ordinary truth-conditions, while the other is associated with the way that reference and truth-conditions depend on the external world. The second sort of semantic value is often held to play a distinctive role in (...) analyzing matters of cognitive significance and/or context-dependence. (shrink)

Conscious experience is at once the most familiar thing in the world and the most mysterious. There is nothing we know about more directly than consciousness, but it is extraordinarily hard to reconcile it with everything else we know. Why does it exist? What does it do? How could it possibly arise from neural processes in the brain? These questions are among the most intriguing in all of science.

Cartesian arguments for global skepticism about the external world start from the premise that we cannot know that we are not in a Cartesian scenario such as an evil-demon scenario, and infer that because most of our empirical beliefs are false in such a scenario, these beliefs do not constitute knowledge. Veridicalist responses to global skepticism respond that arguments fail because in Cartesian scenarios, many or most of our empirical beliefs are true. Some veridicalist responses have been motivated using verificationism, (...) externalism, and coherentism. I argue that a more powerful veridicalist response to global skepticism can be motivated by structuralism, on which physical entities are understood as those that play a certain structural role. I develop the structuralist response and address objections. (shrink)

In recent years there has been an explosion of scientific work on consciousness in cognitive neuroscience, psychology, and other fields. It has become possible to think that we are moving toward a genuine scientific understanding of conscious experience. But what is the science of consciousness all about, and what form should such a science take? This chapter gives an overview of the agenda.

Was human nature designed by natural selection in the Pleistocene epoch? The dominant view in evolutionary psychology holds that it was -- that our psychological adaptations were designed tens of thousands of years ago to solve problems faced by our hunter-gatherer ancestors. In this provocative and lively book, David Buller examines in detail the major claims of evolutionary psychology -- the paradigm popularized by Steven Pinker in The Blank Slate and by David Buss in The Evolution of Desire (...) -- and rejects them all. This does not mean that we cannot apply evolutionary theory to human psychology, says Buller, but that the conventional wisdom in evolutionary psychology is misguided.Evolutionary psychology employs a kind of reverse engineering to explain the evolved design of the mind, figuring out the adaptive problems our ancestors faced and then inferring the psychological adaptations that evolved to solve them. In the carefully argued central chapters of Adapting Minds, Buller scrutinizes several of evolutionary psychology's most highly publicized "discoveries," including "discriminative parental solicitude". Drawing on a wide range of empirical research, including his own large-scale study of child abuse, he shows that none is actually supported by the evidence.Buller argues that our minds are not adapted to the Pleistocene, but, like the immune system, are continually adapting, over both evolutionary time and individual lifetimes. We must move beyond the reigning orthodoxy of evolutionary psychology to reach an accurate understanding of how human psychology is influenced by evolution. When we do, Buller claims, we will abandon not only the quest for human nature but the very idea of human nature itself. (shrink)

Although A-consciousness and P-consciousness are conceptually distinct, a refined notion of A-consciousness makes it plausible that the two are empirically inseparable. I suggest that the notion of direct availability for global control can play a central role here, and draw out some consequences.

There has been much interest in the possibility of connectionist models whose representations can be endowed with compositional structure, and a variety of such models have been proposed. These models typically use distributed representations that arise from the functional composition of constituent parts. Functional composition and decomposition alone, however, yield only an implementation of classical symbolic theories. This paper explores the possibility of moving beyond implementation by exploiting holistic structure-sensitive operations on distributed representations. An experiment is performed using Pollack’s Recursive (...) Auto-Associative Memory. RAAM is used to construct distributed representations of syntactically structured sentences. A feed-forward network is then trained to operate directly on these representations, modeling syn- tactic transformations of the represented sentences. Successful training and generalization is obtained, demonstrating that the implicit structure present in these representations can be used for a kind of structure-sensitive processing unique to the connectionist domain. (shrink)