Computer supported mathematics with Ωmega

Classical automated theorem proving of today is based on ingenious search techniques to find a proof for a given theorem in very large search spaces—often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited.The shift from search based methods to more abstract planning techniques however opened up a paradigm for mathematical reasoning on a computer and several systems of that kind now employ a mix of interactive, search based as well as proof planning techniques.The Ωmega system is at the core of several related and well-integrated research projects of the Ωmega research group, whose aim is to develop system support for a working mathematician as well as a software engineer when employing formal methods for quality assurance. In particular, Ωmega supports proof development at a human-oriented abstract level of proof granularity. It is a modular system with a central proof data structure and several supplementary subsystems including automated deduction and computer algebra systems. Ωmega has many characteristics in common with systems like NuPrL, CoQ, Hol, Pvs, and Isabelle. However, it differs from these systems with respect to its focus on proof planning and in that respect it is more similar to the proof planning systems Clam and λClam at Edinburgh.     

TPS: A hybrid automatic-interactive system for developing proofs

The theorem proving system Tps provides support for constructing proofs using a mix of automation and user interaction, and for manipulating and inspecting proofs. Its library facilities allow the user to store and organize work. Mathematical theorems can be expressed very naturally in Tps using higher-order logic. A number of proof representations are available in Tps, so proofs can be inspected from various perspectives.     

Purple sweet potato color repairs d-galactose-induced spatial learning and memory impairment by regulating the expression of synaptic proteins

Purple sweet potato color (PSPC), a class of naturally occurring anthocyanins used to color food (E163), has been reported to possess a variety of biological activities, including anti-oxidant, anti-tumor, and anti-inflammatory. The effect of PSPC on the spatial learning and memory of mice treated with d-galactose (d-gal) was evaluated by the Morris water maze; d-gal-treated mice had decreased performance compared with mice in the vehicle and PSPC groups, while the PSPC + d-gal group showed significantly shortened escape latency to platform, increased swimming speed, more target quadrant search time and more platform crossings as compared with the d-gal group. Brain functions, such as memory formation and recovery of function after injury, depend on proper regulation of the expression levels of the pre- and post-synaptic proteins. We investigated the expression of four pre-synaptic proteins (growth-associated protein-43, synapsin-I, synaptophysin, and synaptotagmin) and two post-synaptic proteins (post-synaptic density protein-95 and Ca²⁺/calmodulin-dependent protein kinase II) in the hippocampus and cerebral cortex, respectively, in response to different treatments. Western blotting analysis showed that there were significant decreases in the expression of these representative synaptic proteins in the hippocampus and cerebral cortex of d-gal-treated mice. Interestingly, these decreased expression levels of synaptic proteins could be reversed by PSPC. The levels of expression of these representative synaptic proteins in mice treated with PSPC alone were not significantly different from those in untreated mice. The results of this study suggested that memory impairment and synaptic protein loss in d-gal-treated mice may be improved by treatment with PSPC.     

Pure Logic of Many-Many Ground

A logic of grounding where what is grounded can be a collection of truths is a “many-many” logic of ground. The idea that grounding might be irreducibly many-many has recently been suggested by Dasgupta (2014). In this paper I present a range of novel philosophical and logical reasons for being interested in many-many logics of ground. I then show how Fine’s State-Space semantics for the Pure Logic of Ground (plg) can be extended to the many-many case, giving rise to the Pure Logic of Many-Many Ground (plmmg). In the second, more technical, part of the paper, I do two things. First, I present an alternative formalization of plg; this allows us to simplify Fine’s completeness proof for plg. Second, I formalize plmmg using an infinitary sequent calculus and prove that this formalization is sound and complete.     

Medial prefrontal cortex infusions of bupivacaine or AP-5 block extinction of amphetamine conditioned place preference

The present experiments used reversible lesion techniques and intra-mPFC infusions of the n-methyl d-aspartate (NMDA) receptor antagonist d,l-2-amino-5-phosphonovaleric acid (AP-5) to examine the role of the mPFC in extinction of an amphetamine conditioned place preference (CPP). Following initial training and testing for an amphetamine (2 mg/kg) CPP, adult male Long–Evans rats were given extinction trials that were identical to training, except in the absence of peripheral amphetamine injections. Immediately prior to each extinction trial, rats received intra-mPFC infusions of the anesthetic drug bupivacaine (0.75% solution/0.5 μl), AP-5 (1.25, 2.5, 5.0 μg/0.5 μl), or saline. Following extinction training, rats were given a second CPP test session. Rats receiving intra-mPFC infusions of saline displayed extinction of CPP behavior. In contrast, intra-mPFC infusions of bupivacaine or AP-5 (2.5, 5.0 μg) blocked CPP extinction. The findings indicate (1) the mPFC mediates extinction of approach behavior to drug-associated environmental contexts, and (2) NMDA receptor blockade within the mPFC is sufficient to block extinction of amphetamine CPP behavior.     

On Bourbaki’s axiomatic system for set theory

In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to Bourbaki’s system the axiom of universes for the purpose of considering the theory of categories. In this regard, we make some historical and epistemological remarks that could explain the conservative attitude of the Group.     

Information closure and the sceptical objection

In this article, I define and then defend the principle of information closure (pic) against a sceptical objection similar to the one discussed by Dretske in relation to the principle of epistemic closure. If I am successful, given that pic is equivalent to the axiom of distribution and that the latter is one of the conditions that discriminate between normal and non-normal modal logics, a main result of such a defence is that one potentially good reason to look for a formalization of the logic of “ $S$ is informed that $p$ ” among the non-normal modal logics, which reject the axiom, is also removed. This is not to argue that the logic of “ $S$ is informed that $p$ ” should be a normal modal logic, but that it could still be insofar as the objection that it could not be, based on the sceptical objection against pic, has been removed. In other word, I shall argue that the sceptical objection against pic fails, so such an objection provides no ground to abandon the normal modal logic B (also known as KTB) as a formalization of “ $S$ is informed that $p$ ”, which remains plausible insofar as this specific obstacle is concerned.     

Two Kinds of Consequential Implication

The first section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisfied by at least two different operators of decreasing strength (symbolized by \(\rightarrow \) and \(\Rightarrow \)). Only the former has been analyzed in recent literature, so the paper focuses essentially on the latter. Both operators may be axiomatized in systems which are shown to be translatable into standard systems of normal modal logic. The central result of the paper is that the minimal consequential system for \(\Rightarrow \), CI\(\Rightarrow \), is definitionally equivalent to the deontic system KD and is intertranslatable with the minimal consequential system for \(\rightarrow \), CI. The main drawback ot the weaker operator \(\Rightarrow \) is that it lacks unrestricted contraposition, but the final section of the paper argues that \(\Rightarrow \) has some properties which make it a valuable alternative to \(\rightarrow \), turning out especially plausible as a basis for the definition of operators representing synthetic (i.e. context-dependent) conditionals.     

Positive and negative mediation as a function of whether the absent cue was previously associated with the outcome

After presenting two cues, A and B, together, later pairings of one of the cues alone with an outcome can generate changes in the associative value of the absent cue. These changes can be in the same direction as the present cue (i.e., positive mediation) or in the opposite direction to the present cue (i.e., negative mediation). We found both mediational effects in a human contingency task. In addition, we found that the direction of the change was determined by the existence of a prior association between the absent cue and the outcome. When a prior association exists, the absent cue tends to change its value in the opposite direction to the present cue, whereas when there is no prior association, the absent cue tends to change its value in the same direction as the present cue. Recent associative models (Stout & Miller, 2007 Stout, S. C. and Miller, R. R. 2007. Sometimes-competing retrieval (SOCR): A formalization of the comparator hypothesis. Psychological Review, 114: 759–783.  [Google Scholar]) can explain our results.     

Probability logic and combining evidence

In Hailperin 1996 Hailperin, T. 1996. Sentential Probability Logic, Bethlehem and London: Lehigh University Press and Associated University Presses.  [Google Scholar], in addition to its formal development of Probability Logic, there are many sections devoted to historical origins, illustrative examples, and discussion of related work by other authors. Here selected portions of its formal treatment are summarized and then used as a basis for a probability logic treatment of combining evidence.     

Information dynamics and uniform substitution

The picture of information acquisition as the elimination of possibilities has proven fruitful in many domains, serving as a foundation for formal models in philosophy, linguistics, computer science, and economics. While the picture appears simple, its formalization in dynamic epistemic logic reveals subtleties: given a valid principle of information dynamics in the language of dynamic epistemic logic, substituting complex epistemic sentences for its atomic sentences may result in an invalid principle. In this article, we explore such failures of uniform substitution. First, we give epistemic examples inspired by Moore, Fitch, and Williamson. Second, we answer affirmatively a question posed by van Benthem: can we effectively decide when every substitution instance of a given dynamic epistemic principle is valid? In technical terms, we prove the decidability of this schematic validity problem for public announcement logic (PAL and PAL-RC) over models for finitely many fully introspective agents, as well as models for infinitely many arbitrary agents. The proof of this result illuminates the reasons for the failure of uniform substitution.     

Glymour and Quine on Theoretical Equivalence

Glymour (1970, 1977, 1980) and Quine (1975) propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.     

Editorial

Abstract

The concept of mentalization (Fonagy, Steele, Moran, Steele, &; Higgitt, 1991 Fonagy, P., Steele, H., Moran, G., Steele, M. and Higgitt, A. 1991. The capacity for understanding mental states: The reflective self in parent and child and its significance of security of attachment. Infant Mental Health Journal, 13: 201–218. [Crossref], [Web of Science ®] , [Google Scholar]) has been used to describe the way in which the child's ability to understand his own and others' minds crucially depends on the developmental opportunity to recognize himself as represented in the mind of another. Within the context of a case presented for supervision, this paper discusses the importance of mentalization or ‘reflective function’ in the development of the psychological self, drawing on recent attachment-related research. Peter Webber's film Girl with a Pearl Earring is also presented and discussed as illustration of the psychotherapeutic significance of searching for the individual's inner psychological potential.     

d-Serine facilitates the effects of extinction to reduce cocaine-primed reinstatement of drug-seeking behavior

Male Sprague Dawley rats were allowed to self-administer cocaine (0.5 mg/kg) during 90 min sessions for a period of 15 days. On day 16, rats were either held abstinent in their home cage environment or experienced an extinction session in which the active lever had no programmed consequences. Facilitating N-methyl-d-aspartate (NMDA) receptor activity with the coagonist d-serine (100 mg/kg i.p.) before or following the extinction session significantly reduced the subsequent cocaine-primed reinstatement of drug-seeking behavior tested on day 17. d-Serine significantly reduced drug-primed reinstatement only when combined with extinction, and its effectiveness when administered following the training session suggested that an enhancement of consolidation of extinction learning had occurred. In contrast, d-serine treatment did not reduce sucrose-primed reinstatement, indicating that the beneficial effects of this adjunct pharmacotherapy with extinction training were specific to an addictive substance (cocaine) and did not generalize to a natural reward (sucrose).     

Spinoza on the Essences of Modes

This paper examines some aspects of Spinoza's metaphysics of the essences of modes.2 ²I would like to thank John Carriero, Calvin Normore, Eliot Michaelson, Eileen Nutting, Paul Nichols, Alexi Patsaouras, Rachel Johnson and Sarah Jansen for reading and commenting on earlier versions of this paper. I situate Spinoza's use of the notion of essence as a response to traditional, Aristotelian, ways of thinking about essence. I argue that, although Spinoza rejects part of the Aristotelian conception of essence, according to which it is in virtue of its essence that a thing is a member of a kind, he nevertheless retains a different part of such a conception, according to which an essence is some structural feature of a thing which causally explains other, non-essential features. I go on to develop an account of Spinoza's metaphysics of essence, according to which essences, what he sometimes calls formal essences, are produced by the divine essence prior to and independent of the creation of finite modes, and according to which essences are the formal or exemplar causes of finite modes. I then argue that finite modes, in virtue of the formal essences which they actualize, are genuine causal relata. Finally, I offer some speculations about Spinoza's answer to the question, ‘Why, in a necessitarian cosmos filled with formal essences, should there be temporal finite modes at all?’     

Reasoning About Games

A mixture of propositional dynamic logic and epistemic logic that we call PDL + E is used to give a formalization of Artemov??s knowledge based reasoning approach to game theory, (KBR), [4, 5]. Epistemic states of players are represented explicitly and reasoned about formally. We give a detailed analysis of the Centipede game using both proof theoretic and semantic machinery. This helps make the case that PDL + E can be a useful basis for the logical investigation of game theory.     

The Doomsday Argument and the Simulation Argument

The Doomsday Argument and the Simulation Argument share certain structural features, and hence are often discussed together (Bostrom 2003, Are you living in a computer simulation, Philosophical Quarterly, 53:243–255; Aranyosi 2004, The Doomsday Simulation Argument. Or why isn’t the end nigh, and you’re not living in a simulation, http://philsci-archive.pitt.edu/190/; Richmond 2008, Doomsday, Bishop Ussher and simulated worlds, Ratio, 21:201–217; Bostrom and Kulczycki 2011 A patch for the Simulation Argument, Analysis, 71:54–61). Both are cases where reflecting on one’s location among a set of possibilities yields a counter-intuitive conclusion—in the first case that the end of humankind is closer than you initially thought, and in the second case that it is more likely than you initially thought that you are living in a computer simulation. Indeed, the two arguments do have some structural similarities. But there are also significant disanalogies between the two arguments, and I argue that these disanalogies mean that the Simulation Argument succeeds and the Doomsday Argument fails.     

The Madhyamaka Concept of Svabhāva: Ontological and Cognitive Aspects

This paper considers the philosophical interpretation of the concept of svabhāva, sometimes translated as ‘inherent existence’ or ‘own-being’, in the Madyamaka school of Buddhist philosophy. It is argued that svabhāva must be understood as having two different conceptual dimensions, an ontological and a cognitive one. The ontological dimension of svabhāva shows it to play a particular part in theories investigating the most fundamental constituents of the world. Three different understandings of svabhāva are discussed under this heading: svabhāva understood as essence, as substance, and as the true nature of phenomena (absolute svabhāva). The cognitive dimension shows svabhāva as playing an important rôle in our everyday conceptualization of phenomena. Svabhāva is here seen as a superimposition (samāropa) which the mind projects onto the world.