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, 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)

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.

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)

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)

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)

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)

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.

Doctors’ strikes are legally permissible in the UK, with the situation differing in other countries. But are they morally permissible? Doug McConnell and Darren Mann have systematically attempted to dismiss the arguments for the moral impermissibility of doctors’ strikes and creatively attempted to provide further moral justification for them. Unfortunately for striking doctors, they fail to achieve this. Meanwhile, junior doctors’ strikes have continued in the UK through 2023 and have now extended into 2024. In this response, which focuses on (...) the UK situation and specifically junior doctors’ strikes in the National Health Service (NHS) in England, I will demonstrate a central problem with their arguments—namely that they underplay the harms caused by prolonged doctors’ strikes by ignoring the harms to patients with cancer. This weakens their conclusion that strikes are morally permissible in terms of the conditions and thresholds they set. I then provide a psychological critique of their justification for strikes in terms of the interests of the public. It follows that invoking the controversial concept of supererogatory action is ungrounded but also absurd when you consider time-critical cancer care. If those representing striking doctors wish to maintain a modicum of moral respectability, they should mitigate for patients with cancer and negotiate reasonably and with urgency. (shrink)

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 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)

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)

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)

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)

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.

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.

When I say ‘Hesperus is Phosphorus’, I seem to express a proposition. And when I say ‘Joan believes that Hesperus is Phosphorus’, I seem to ascribe to Joan an attitude to the same proposition. But what are propositions? And what is involved in ascribing propositional attitudes?

What is involved in the meaning of our expressions? Frege suggested that there is an aspect of an expression.

I argue that virtual reality is a sort of genuine reality. In particular, I argue for virtual digitalism, on which virtual objects are real digital objects, and against virtual fictionalism, on which virtual objects are fictional objects. I also argue that perception in virtual reality need not be illusory, and that life in virtual worlds can have roughly the same sort of value as life in non-virtual worlds.

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.

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 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)

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)

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)

The physician-patient relationship has changed over the last several decades, requiring a systematic reevaluation of the competing demands of patients, physicians, and families. In the era of genetic testing, using a model of patient care known as the family covenant may prove effective in accounting for these demands. The family covenant articulates the roles of the physician, patient, and the family prior to genetic testing, as the participants consensually define them. The initial agreement defines the boundaries of autonomy and benefit (...) for all participating family members. The physician may then serve as a facilitator in the relationship, working with all parties in resolving potential conflicts regarding genetic information. The family covenant promotes a fuller discussion of the competing ethical claims that may come to bear after genetic test results are received. (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. ]].

To clarify the notion of computation and its role in cognitive science, we need an account of implementation, the nexus between abstract computations and physical systems. I provide such an account, based on the idea that a physical system implements a computation if the causal structure of the system mirrors the formal structure of the computation. The account is developed for the class of combinatorial-state automata, but is sufficiently general to cover all other discrete computational formalisms. The implementation relation is (...) non-vacuous, so that criticisms by Searle and others fail. This account of computation can be extended to justify the foundational role of computation in artificial intelligence and cognitive science. (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.

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.

One of the enduring concerns of moral philosophy is deciding who or what is deserving of ethical consideration. Much recent attention has been devoted to the "animal question" -- consideration of the moral status of nonhuman animals. In this book, David Gunkel takes up the "machine question": whether and to what extent intelligent and autonomous machines of our own making can be considered to have legitimate moral responsibilities and any legitimate claim to moral consideration. The machine question poses a (...) fundamental challenge to moral thinking, questioning the traditional philosophical conceptualization of technology as a tool or instrument to be used by human agents. Gunkel begins by addressing the question of machine moral agency: whether a machine might be considered a legitimate moral agent that could be held responsible for decisions and actions. He then approaches the machine question from the other side, considering whether a machine might be a moral patient due legitimate moral consideration. Finally, Gunkel considers some recent innovations in moral philosophy and critical theory that complicate the machine question, deconstructing the binary agent--patient opposition itself. Technological advances may prompt us to wonder if the science fiction of computers and robots whose actions affect their human companions could become science fact. Gunkel's argument promises to influence future considerations of ethics, ourselves, and the other entities who inhabit this world. (shrink)

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)

The philosophical interest of verbal disputes is twofold. First, they play a key role in philosophical method. Many philosophical disagreements are at least partly verbal, and almost every philosophical dispute has been diagnosed as verbal at some point. Here we can see the diagnosis of verbal disputes as a tool for philosophical progress. Second, they are interesting as a subject matter for first-order philosophy. Reflection on the existence and nature of verbal disputes can reveal something about the nature of concepts, (...) language, and meaning. In this article I first characterize verbal disputes, spell out a method for isolating and resolving them, and draw out conclusions for philosophical methodology. I then use the framework to draw out consequences in first-order philosophy. In particular, I argue that the analysis of verbal disputes can be used to support the existence of a distinctive sort of primitive concept and that it can be used to reconstruct a version of an analytic/synthetic distinction, where both are characterized in dialectical terms alone. (shrink)

It is widely accepted that conscious experience has a physical basis. That is, the properties of experience (phenomenal properties, or qualia) systematically depend on physical properties according to some lawful relation. There are two key questions about this relation. The first concerns the strength of the laws: are they logically or metaphysically necessary, so that consciousness is nothing "over and above" the underlying physical process, or are they merely contingent laws like the law of gravity? This question about the strength (...) of the psychophysical link is the basis for debates over physicalism and property dualism. The second question concerns the shape of the laws: precisely how do phenomenal properties depend on physical properties? What sort of physical properties enter into the laws' antecedents, for instance; consequently, what sort of physical systems can give rise to conscious experience? It is this second question that I address in this paper. (shrink)

Is conceptual analysis required for reductive explanation? If there is no a priori entailment from microphysical truths to phenomenal truths, does reductive explanation of the phenomenal fail? We say yes . Ned Block and Robert Stalnaker say no.