Terry Horgan and John Tienson have suggested that connectionism might provide a framework within which to articulate a theory of cognition according to which there are mental representations without rules (RWR) (Horgan and Tienson 1988, 1989, 1991, 1992). In essence, RWR states that cognition involves representations in a language of thought, but that these representations are not manipulated by the sort of rules that have traditionally been posited. In the development of RWR, Horgan and Tienson attempt to forestall a particular (...) line of criticism, theSyntactic Argument, which would show RWR to be inconsistent with connectionism. In essence, the argument claims that the node-level rules of connectionist networks, along with the semantic interpretations assigned to patterns of activation, serve to determine a set of representation-level rules incompatible with the RWR conception of cognition. The present paper argues that the Syntactic Argument can be made to show that RWR is inconsistent with connectionism. (shrink)

A skeptical worry known as ‘the indeterminacy of computation’ animates much recent philosophical reflection on the computational identity of physical systems. On the one hand, computational explanation seems to require that physical computing systems fall under a single, unique computational description at a time. On the other, if a physical system falls under any computational description, it seems to fall under many simultaneously. Absent some principled reason to take just one of these descriptions in particular as relevant for computational explanation, (...) widespread failure of computational explanation would appear to follow. This paper advances a new solution to the indeterminacy of computation. Very roughly, I argue that the computational identity of a physical system is determinate relative to a contextually specified way of regarding that system computationally—known as a labelling scheme. When a system simultaneously implements multiple computations, it does so relative to different labelling schemes. But relative to a fixed labelling scheme, a physical system has a unique computational identity. I argue that this relativistic conception of computational identity vindicates computational explanation in the face of simultaneous implementation. (shrink)

One approach to achieving cooperation in the one-shot prisoner’s dilemma is Tennenholtz’s (Games Econ Behav 49(2):363–373, 2004) program equilibrium, in which the players of a game submit programs instead of strategies. These programs are then allowed to read each other’s source code to decide which action to take. As shown by Tennenholtz, cooperation is played in an equilibrium of this alternative game. In particular, he proposes that the two players submit the same version of the following program: cooperate if the (...) opponent is an exact copy of this program and defect otherwise. Neither of the two players can benefit from submitting a different program. Unfortunately, this equilibrium is fragile and unlikely to be realized in practice. We thus propose a new, simple program to achieve more robust cooperative program equilibria: cooperate with some small probability. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

A recent hypothesis in empirical brain research on language is that the fundamental difference between animal and human communication systems is captured by the distinction between finite-state and more complex phrase-structure grammars, such as context-free and context-sensitive grammars. However, the relevance of this distinction for the study of language as a neurobiological system has been questioned and it has been suggested that a more relevant and partly analogous distinction is that between non-adjacent and adjacent dependencies. Online memory resources are central (...) to the processing of non-adjacent dependencies as information has to be maintained across intervening material. One proposal is that an external memory device in the form of a limited push-down stack is used to process non-adjacent dependencies. We tested this hypothesis in an artificial grammar learning paradigm where subjects acquired non-adjacent dependencies implicitly. Generally, we found no qualitative differences between the acquisition of non-adjacent dependencies and adjacent dependencies. This suggests that although the acquisition of non-adjacent dependencies requires more exposure to the acquisition material, it utilizes the same mechanisms used for acquiring adjacent dependencies. We challenge the push-down stack model further by testing its processing predictions for nested and crossed multiple non-adjacent dependencies. The push-down stack model is partly supported by the results, and we suggest that stack-like properties are some among many natural properties characterizing the underlying neurophysiological mechanisms that implement the online memory resources used in language and structured sequence processing. (shrink)

According to the revision theory of truth, the binary sequences generated by the paradoxical sentences in revision sequence are always unstable. In this paper, we work backwards, trying to reconstruct the paradoxical sentences from some of their binary sequences. We give a general procedure of constructing paradoxes with specific binary sequences through some typical examples. Particularly, we construct what Herzberger called “unstable statements with unpredictably complicated variations in truth value.” Besides, we also construct those paradoxes with infinitely many finite primary (...) periods but without any infinite primary period, those with an infinite critical point but without any finite primary period, and so on. This is the first formal appearance of these paradoxes. Our construction demonstrates that the binary sequences generated by a paradoxical sentence are something like genes from which we can even rebuild the original sentence itself. (shrink)

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. The theme of this book lies at the interface of many disciplines. Logic is the main methodology, but the various chapters cross easily between mathematics, computer science, philosophy, linguistics, cognitive and social sciences, while also ranging from pure theory to empirical work. Accordingly, the authors of this book represent a wide variety of original thinkers from different research communities. And their interconnected themes challenge at the same time how we think of logic, philosophy and computation. Thus, very much in line with van Benthem's work over many decades, the volume shows how all these disciplines form a natural unity in the perspective of dynamic logicians (broadly conceived) exploring their new themes today. And at the same time, in doing so, it offers a broader conception of logic with a certain grandeur, moving its horizons beyond the traditional study of consequence relations. (shrink)

This paper investigates, in a simple risk-sharing framework, the extent to which the incompleteness of contracts could be attributed to the complexity costs associated with the writing and the implementation of contracts. We show that, given any measure of complexity in a very general class, it is possible to find simple contracting problems such that, when complexity costs are explicitly taken into account, the contracting parties optimally choose an incomplete contract which coincides with the ‘default’ division of surplus. Optimal contracts (...) with complexity costs are constrained efficient in our model. We therefore interpret our results as saying that, in the absence of a strategic role for complexity costs, their effect is entirely determined by their size relative to the size of payoffs. (shrink)

In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine (...) meaning-dependent mathematical characteristics that cannot be captured by formal calculi. ‘…there is a conflict between mathematical practice and the formalist doctrine.’ [Kreisel, 1969, p. 39]. (shrink)

An introduction to the special issue on epistemic logic and the foundations of game theory edited by Michael Bacharach and Philippe Mongin. Contributors are Michael Bacharach, Robert Stalnaker, Salvatore Modica and Aldo Rustichini, Luc Lismont and Philippe Mongin, and Hyun-Song Shin and Timothy Williamson.

We consider structures of the form, where Φ and Ψ are non-empty sets and is a relation whose domain is Ψ. In particular, by using a special kind of a diagonal argument, we prove that if Φ is a denumerable recursive set, Ψ is a denumerable r.e. set, and R is an r.e. relation, then there exists an infinite family of infinite recursive subsets of Φ which are not R -images of elements of Ψ. The proof is a very elementary (...) one, without any reference even to e.g. the -theorem. Some consequences of the main result are also discussed. (shrink)

Machines were introduced as calculating devices to simulate operations carried out by human computers following fixed algorithms. The mathematical study of (paper) machines is the topic of our essay. The first three sections provide necessary logical background, examine the analyses of effective calculability given in the thirties, and describe results that are central to recursion theory, reinforcing the conceptual analyses. In the final section we pursue our investigation in a quite different way and focus on principles that govern the operations (...) of physically realizable machines. (shrink)

The following inference is valid: There are exactly 101 dalmatians, There are exactly 100 food bowls, Each dalmatian uses exactly one food bowl Hence, at least two dalmatians use the same food bowl. Here, “there are at least 101 dalmatians” is nominalized as, "x1"x2…."x100$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x100) and “there are exactly 101 dalmatians” is nominalized as, "x1"x2…."x100$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ (...) x100) & Ø"x1"x2…."x101$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x101). This is abbreviated $101xDx. The validity of the above inference corresponds to the valid formula, PHP(100): [$101xDx & $100xFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). More generally, for variable n, the formula PHP(n) is PHP(n): [$n+1xDx & $nxFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). A mathematical proof that PHP(n) is valid, for all n > 0, is quite short (less than a page), but refers to numbers, functions and sets. It uses the Pigeonhole Principle. This explains why PHP(n) is valid, for all n>0. However, I estimate that a predicate calculus derivation of PHP(100), using natural deduction, say, would require around 107 symbols. Unfeasibility Problem: nominalism is the radical anti-realist view that there are no numbers, functions or sets. So, how could a nominalist know that PHP(100) is valid, without directly performing the rather long derivation? Can the nominalist “ride piggyback” on the standard mathematical proof? If so, how is this justified? (shrink)

In this paper, I evaluate the claim that the equivalence of multiple intensionally distinct definitions of random sequence provides evidence for the claim that these definitions capture the intuitive conception of randomness, concluding that the former claim is false. I then develop an alternative account of the significance of randomness-theoretic equivalence results, arguing that they are instances of a phenomenon I refer to as schematic equivalence. On my account, this alternative approach has the virtue of providing the plurality of definitions (...) of randomness with conceptual unity and a rationale for certain investigations that are carried out in the field. (shrink)