Early modern humans developed mental capabilities that were immeasurably greater than those of nonhuman primates. We see this in the rapid innovation in tool making, the development of complex language, and the creation of sophisticated art forms, none of which we find in our closest relatives. While we can readily observe the results of this high-order cognitive capacity, it is difficult to see how it could have developed. We take up the topic of cave art and archeoacoustics, particularly the discovery (...) that cave art is often closely connected to the acoustic properties of the cave chambers in which it is found. Apparently, early modern humans were able to detect the way sound reverberated in these chambers, and they painted artwork on surfaces that were acoustic “hot spots,” i.e., suitable for generating echoes. We argue that cave art is a form of cross-modality information transfer, in which acoustic signals are transformed into symbolic visual representations. This form of information transfer across modalities is an instance of how the symbolic mind of early modern humans was taking shape into concrete, externalized language. We also suggest that the earliest rock art found in Africa may constitute one of the first fossilized proxies for the expression of full-fledged human linguistic behavior. (shrink)

This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in (...) much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included. (shrink)

I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A.

Abduction is or subsumes a process of inference. It entertains possible hypotheses and it chooses hypotheses for further scrutiny. There is a large literature on various aspects of non-symbolic, subconscious abduction. There is also a very active research community working on the symbolic (logical) characterisation of abduction, which typically treats it as a form of hypothetico-deductive reasoning. In this paper we start to bridge the gap between the symbolic and sub-symbolic approaches to abduction. We are interested in benefiting from developments (...) made by each community. In particular, we are interested in the ability of non-symbolic systems (neural networks) to learn from experience using efficient algorithms and to perform massively parallel computations of alternative abductive explanations. At the same time, we would like to benefit from the rigour and semantic clarity of symbolic logic. We present two approaches to dealing with abduction in neural networks. One of them uses Connectionist Modal Logic and a translation of Horn clauses into modal clauses to come up with a neural network ensemble that computes abductive explanations in a top-down fashion. The other combines neural-symbolic systems and abductive logic programming and proposes a neural architecture which performs a more systematic, bottom-up computation of alternative abductive explanations. Both approaches employ standard neural network architectures which are already known to be highly effective in practical learning applications. Differently from previous work in the area, our aim is to promote the integration of reasoning and learning in a way that the neural network provides the machinery for cognitive computation, inductive learning and hypothetical reasoning, while logic provides the rigour and explanation capability to the systems, facilitating the interaction with the outside world. Although it is left as future work to determine whether the structure of one of the proposed approaches is more amenable to learning than the other, we hope to have contributed to the development of the area by approaching it from the perspective of symbolic and sub-symbolic integration. (shrink)

In this paper thecanonicalmodal logics, a kind of complete modal logics introduced in K. Fine [4] and R. I. Goldblatt [5], will be characterized semantically using the concept of anultrafilter extension, an operation on frames inspired by the algebraic theory of modal logic. Theorem 8 of R. I. Goldblatt and S. K. Thomason [6] characterizing the modally definable Σ⊿-elementary classes of frames will follow as a corollary. A second corollary is Theorem 2 of [4] which states that any (...) complete modal logic defining a Σ⊿-elementary class of frames is canonical.The main tool in obtaining these results is the duality between modal algebras and general frames developed in R. I. Goldblatt [5]. The relevant notions and results from this theory will be stated in §2. The concept of a canonical modal logic is introduced and motivated in §3, which also contains the above-mentioned theorems. In §4, a kind of appendix to the preceding discussion, preservation of first-order sentences under ultrafilter extensions is discussed.The modal language to be considered here has an infinite supply of proposition letters, a propositional constant ⊥, the usual Boolean operators ¬, ∨, ∨, →, and ↔ —with ¬ and ∨ regarded as primitives—and the two unary modal operators ◇ and □ — ◇ being regarded as primitive. Modal formulas will be denoted by lower case Greek letters, sets of formulas by Greek capitals. (shrink)

Reprinted in Quine, W. V. O. 1966. The Ways of Paradox. (New York: Random House.).

heories of embodied cognition (e.g., Perceptual Symbol Systems Theory; Barsalou, 1999, 2009) suggest that modality specific simulations underlie the representation of concepts. Supporting evidence comes from modality switch costs: participants are slower to verify a property in one modality (e.g., auditory, BLENDER-loud) after verifying a property in a different modality (e.g., gustatory, CRANBERRIES-tart) compared to the same modality (e.g., LEAVES-rustling, Pecher et al., 2003). Similarly, modality switching costs lead to a modulation of the N400 effect in event-related potentials (ERPs; Collins (...) et al., 2011; Hald et al., 2011). This effect of modality switching has also been shown to interact with the veracity of the sentence (Hald et al., 2011). The current ERP study further explores the role of modality match/mismatch on the processing of veracity as well as negation (sentences containing “not”). Our results indicate a modulation in the ERP based on modality and veracity, plus an interaction. The evidence supports the idea that modality specific simula- tions occur during language processing, and furthermore suggest that these simulations alter the processing of negation. (shrink)

This paper addresses the problem of intermodal perception, that of how warranted perception arises of objects having characteristics in multiple sense modalities. It first shows the inadequacy of the currently popular explanations of such perception in terms of special, innate mechanisms. It proposes instead a phenomenological account in terms of an infant's general capacities for observation and thought. To this end it prepares the terrain with brief investigations into four topics: spontaneity, non-symbolic thinking, the role of spontaneity in perception, and (...) the lived body. It then traces how an infant may, through spontaneous observations of invariable co-presence and sequence, integrate tactile objects, then tastes, smells, and sounds, and finally, with the help from topological and dynamic congruence, visual objects into a single world with the lived body as centre of orientation. (shrink)

Intensional logic is the technical study of such intensional phenomena in human reasoning as modality, knowledge, or flow of time. These all require a richer semantic picture than standard truth values in one static environment. Such a picture is provided by so-called possible worlds semantics, a paradigm which is surveyed in this book, both as to its external sources of motivation and as to the internal dynamics of the resulting program. In particular, Manual of Intensional Logic presents the major classical (...) topics, including modal logic, tense logic, and conditional logic, all of which illustrate motivations coming from philosophy and linguistics. The Book also discusses recent computational applications in computer science and AI. Finally, Manual of Intensional Logic takes up recent developments in the study of language and information making themselves felt in the area. The Book examines the role of partial information--with illustrations drawn from different branches of Intensional Logic--and various influences stemming from current theories of the semantics of natural language, involving generalized quantifiers and theories of types. (shrink)

Originally published in 1930, this well-known text by the late British philosopher Abraham Wolf offers the student a practical, consistent, and comprehensive approach to logic which remains unique in its field. Dr. Wolf here deals systematically with the two main types of reasoning - formal logic and inductive logic - and their various applications. All the main elements of logic - such as inference, syllogism, dilemmas, evidence, deductive and inductive methods, and probability - are subsumed under these general headings. Professor (...) Wolf strongly emphasizes the fact that logic cannot be mastered without some practical application; at the end of this volume, therefore, he includes a section of exercises based on each chapter. His unusally interesting appendix examines such matters as symbolic as logic, fallacies, the law of contradiction, modal propositions, the existential import of categorical propositions, predictables, and categories. (shrink)