Sequent-systems and groupoid models. I

The purpose of this paper is to connect the proof theory and the model theory of a family of propositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related toBCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hubert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.The first part of the paper, which follows, contains the first two sections, which deal with sequent-systems and Hubert-formulations. The second part, due to appear in the next issue of this journal, will contain the third section, which deals with groupoid models.     

Sequent-systems and groupoid models. II

The purpose of this paper is to connect the proof theory and the model theory of a family of prepositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related to BCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hilbert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.Below is the sequel to the first part of the paper, which appeared in the previous issue of this journal (vol. 47 (1988), pp. 353–386). The first part contained sections on sequent-systems and Hilbert-formulations, and here is the third section on groupoid models. This second part is meant to be read in conjunction with the first part.     

A note on sequent calculi intermediate between LJ and LK

We prove that every finitely axiomatizable extension of Heyting's intuitionistic logic has a corresponding cut-free Gentzen-type formulation. It is shown how one can use this result to find the corresponding normalizable natural deduction system and to give a criterion for separability of considered logic. Obviously, the question how to obtain an effective definition of a sequent calculus which corresponds to a concrete logic remains a separate problem for every logic.     

A cut-free gentzen-type system for the logic of the weak law of excluded middle

The logic of the weak law of excluded middleKC _p is obtained by adding the formula A A as an axiom scheme to Heyting's intuitionistic logicH _p . A cut-free sequent calculus for this logic is given. As the consequences of the cut-elimination theorem, we get the decidability of the propositional part of this calculus, its separability, equality of the negationless fragments ofKC _p andH _p , interpolation theorems and so on. From the proof-theoretical point of view, the formulation presented in this paper makes clearer the relations betweenKC _p ,H _p , and the classical logic. In the end, an interpretation of classical propositional logic in the propositional part ofKC _p is given.     

Like a bee on a windowpane: Heyting's reflections on solipsism

This paper presents the content of the unpublished notes that the Dutch mathematician Arend Heyting wrote in different periods of his life on solipsism and that are preserved in Heyting's archive at the University of Amsterdam. Most of the notes are quoted here and translated into English. Their study shows the originality of Heyting's reflections on a subject that was typical of his master, L. E. J. Brouwer, the father of intuitionism.     

A Negationless Interpretation of Intuitionistic Theories. II

This work is a sequel to our [16]. It is shown how Theorem 4 of [16], dealing with the translatability of HA(Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types.     

Finite Sets and Natural Numbers in Intuitionistic TT Without Extensionality

In this paper, we prove that Heyting's arithmetic can be interpreted in an intuitionistic version of Russell's Simple Theory of Types without extensionality.     

Models for normal intuitionistic modal logics

Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic.     

Quantum logical calculi and lattice structures

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus T _eff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the value-definiteness of propositions is not postulated, the calculus T _eff represents a calculus of effective (intuitionistic) quantum logic.Beginning with the tableaux-calculus the equivalence of T _eff to calculi which use more familiar figures such as sequents and implications can be investigated. In this paper we present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to T _eff. The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space.     

On Reduction Systems Equivalent to The Lambek Calculus with the Empty String

The paper continues a series of results on cut-rule axiomatizability of the Lambek calculus. It provides a complete solution of a problem which was solved partially in one of the author's earlier papers. It is proved that the product-free Lambek Calculus with the empty string (L ₀) is not finitely axiomatizable if the only rule of inference admitted is Lambek's cut rule. The proof makes use of the (infinitely) cut-rule axiomatized calculus C designed by the author exactly for this purpose.     

Symmetric Categorial Grammar

The Lambek-Grishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of type-forming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the Curry-Howard derivational semantics, and structure-preservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands of natural language form and meaning.     

Making context memories independent of the hippocampus

We present evidence that certain learning parameters can make a memory, even a very recent one, become independent of the hippocampus. We confirm earlier findings that damage to the hippocampus causes severe retrograde amnesia for context memories, but we show that repeated learning sessions create a context memory that is not vulnerable to the damage. The findings demonstrate that memories normally dependent on the hippocampus are incrementally strengthened in other memory networks with additional learning. The latter provides a new account for patterns of hippocampal retrograde amnesia and how memories may become independent of the hippocampus.Contextual fear conditioning can be supported by two neural systems, one that contains the hippocampus (HPC), and one that does not. Evidence for this assertion comes from studies in which the HPC, in rats, is damaged either before or after the contextual fear conditioning. Extensive damage to the HPC before conditioning has little effect on contextual fear conditioning (Maren et al. 1997; Frankland et al. 1998; Wiltgen et al. 2006). This result can only mean that there is a non-HPC memory system that can support fear of context. In contrast, there is unequivocal evidence that moderate to extensive damage to the HPC soon after learning severely impairs the ability of the conditioning context to evoke fear, suggesting that the HPC normally makes a major contribution to this type of memory (Kim and Fanselow 1992; Maren et al. 1997; Frankland et al. 1998; Anagnostaras et al. 1999; Debiec et al. 2002; Lehmann et al. 2007b; Sutherland et al. 2008; Wang et al. 2009).The dissociable effects of pre- and post-training HPC damage on contextual fear conditioning have been interpreted as suggesting that: (1) When the HPC is intact during learning it interferes with other systems and prevents them from acquiring an independent contextual fear conditioning memory, and (2) when the HPC is absent, these other systems are released from this interference and are able to rapidly acquire an independent memory (Maren et al. 1997; Frankland et al. 1998; Fanselow and Poulos 2004; Driscoll et al. 2005; Lehmann et al. 2006; Sutherland et al. 2006). The latter interference from the HPC on the other memory systems has been termed overshadowing. Supplemental Figure S1 depicts data from our laboratory demonstrating the overshadowing phenomenon and the dissociable effects of HPC damage induced before and after contextual fear conditioning.Very little, however, is known about the parameters determining the extent to which the HPC system interferes with the non-HPC system for control over contextual fear. The purpose of the current study is to provide some insight into this issue. Typically, contextual fear conditioning in rats is conducted in a single conditioning session in which a configuration of static background cues is paired with several footshocks. When returned to the conditioning context, rats display several species-specific defensive responses including freezing (i.e., absence of movement except for breathing). Several theorists have proposed that non-HPC systems are more likely to be recruited when there are multiple experiences with similar events, which, in turn, would mitigate the necessity of the HPC for memory expression (O''Keefe and Nadel 1978; Sherry and Schacter 1987; McClelland et al. 1995; O''Reilly and Rudy 2001; White and McDonald 2002). Accordingly, we hypothesized that repeated contextual fear conditioning sessions separated by hours and days would overcome the HPC interference or overshadowing effect. In other words, with repeated learning sessions, enough information would be incrementally captured by the non-HPC system to support a contextual fear memory that would survive complete damage to the HPC.Adult male rats received 11 fear-conditioning sessions across 6 d. In each session, they were placed in a context and received mild footshocks (Shock Context). Concurrently, the rats were exposed 10 times to another context in which they never received shock (No-Shock Context). The No-Shock Context served as a control condition to measure whether the rats simply showed generalized fear or could show context-specific memory. Within 72 h following the last conditioning session, rats either received sham surgery or complete lesions of the HPC using the neurotoxin N-methyl-d-aspartic acid (NMDA) (Lehmann et al. 2007a). Rats were then tested for retention in both the Shock and No-Shock Contexts in a counterbalanced order. In addition, in a single learning episode, another group of rats received a matching number of shocks (i.e., 12 shocks) and context exposure (i.e., 17 min), and then received surgery 7–10 d after conditioning. The latter interval is identical to the interval between the initial conditioning session and surgery in the repeated learning condition. Figure 1 illustrates and describes the design of the experiments.Open in a separate windowFigure 1.Illustration of the experimental design used in (A) the single conditioning session and (B) repeated conditioning session experiments. In A the rats were initially placed in the conditioning chamber for 17 min and received the first of 12 footshocks (1 mA/2 sec) at the 300-sec mark, and then one every following 58 sec after shock offset. Seven to 10 d later, the rats either received sham or HPC damage (Sx). Approximately 10 d after, the rats were returned to the chamber to assess freezing over a 5-min retention test. In B the rats were placed initially in the conditioning chamber for 1 min and received a shock at the 45-sec mark (Shock Context). Approximately 45 min later, the rats were placed in a different chamber for 1 min and did not receive shock (No-Shock Context). The procedure was repeated twice daily for five consecutive days, and the Shock and No-Shock chamber order was counterbalanced according to the principles of a Latin Square design. The rats then received sham or HPC damage 1–3 d later. The rats'' retention was assessed in both contexts ∼10 d after surgery in both the Shock and No-Shock Context in a counterbalanced order with a 24-h span between tests. Importantly, the number of shocks, context exposure time, and interval between initial learning and surgery were matched between both experiments.When all shocks were delivered in a single session, HPC damage caused profound retrograde amnesia. As illustrated in Figure 2A, the HPC rats displayed significantly less freezing than control rats during the retention test (t₍₈₎ = 23.895, P < 0.001). This result replicates all previous studies in which the HPC was damaged days after a single contextual fear conditioning training session (Kim and Fanselow 1992; Maren et al. 1997; Frankland et al. 1998; Anagnostaras et al. 1999; Debiec et al. 2002; Lehmann et al. 2007b; Sutherland et al. 2008).Open in a separate windowFigure 2.Mean (± SEM) percent time freezing by Sham and HPC rats during the retention test of the (A) single conditioning (12 shocks) experiment and (B) repeated conditioning session experiment. In A the HPC rats showed significantly less freezing (P < 0.001) than the Sham rats, suggesting that the damage caused profound retrograde amnesia for contextual fear conditioning learned in a single session 7–10 d before surgery. In B the performance of the HPC rats did not significantly differ from the Sham rats, and they exhibited significantly more freezing in the Shock Context than the No-Shock Context (P < 0.001). Consequently, repeated conditioning sessions prevented the retrograde amnesic effects normally observed in contextual fear conditioning following HPC damage, suggesting that other neural networks were now able to support the memory.In striking contrast, memory for contextual fear conditioning was spared when the HPC was damaged after repeated conditioning sessions. Figure 2B shows the percent time spent freezing during the retention test in the Shock and No-Shock Contexts. An ANOVA with between-group factor (Lesion: Sham and HPC) and within-group factor (Context: Shock and No-Shock) revealed a significant main effect of Context (F_(1,14) = 84.731, P < 0.001), indicating that the rats displayed higher levels of freezing in the Shock than in the No-Shock Context. The effect of Lesion (F_(1,14) = 4.280, P = 0.058) was not significant, nor was the Lesion × Context interaction (F_(1,14) = 0.877, P = 0.369), suggesting that extensive HPC damage did not impair memory. The tendency for an effect of Lesion is due to the HPC rats freezing less than the Sham rats in the No-Shock Context (P = 0.06) rather than freezing less in the Shock Context (P = 0.457).The repeated conditioning sessions clearly enabled a contextual fear representation to be established in non-HPC memory systems. However, it is surprising that the HPC damage did not impair the ability to discriminate between the Shock and No-Shock Context, because evidence suggests that context discrimination is dependent on the HPC (see Moscovitch et al. 2006). Indeed, studies of rats with HPC damage induced before learning have shown that contextual fear conditioning is acquired quickly by non-HPC systems in a single session, but the ability to discriminate between the training context and a new context is lost (Frankland et al. 1998; Antoniadis and McDonald 2000; Winocur et al. 2007). Hence, it is significant in the present study that the HPC damage did not impair context discrimination abilities in the rats that received repeated learning episodes. The latter appear to have established a context representation, outside of the HPC, that was not bereft of details. Yet, one should consider that the rats in the repeated sessions experiment received experience in both the Shock and No-Shock Contexts prior to surgery, and this discrimination training procedure may have established two different non-HPC representations. It remains possible that HPC damage would impair the ability to discriminate the Shock Context from a new context, which is what is found in anterograde amnesia studies (Frankland et al. 1998; Antoniadis and McDonald 2000; Winocur et al. 2007). To address this possibility, a new experiment examined whether HPC-damaged rats could discriminate the Shock Context from a Novel Context. Rats were trained with the same repeated learning protocol as described earlier, with the exception that the rats were never placed in the No-Shock Context prior to surgery. One to 3 d following learning, the rats either received Sham or complete HPC damage. They were then tested for retention in the Shock and the Novel (i.e., No-Shock) Context in a counterbalanced order. Figure 3 shows the percent time spent freezing during the retention test in the Shock and Novel Contexts. An ANOVA with between-group factor (Lesion: Sham and HPC) and within-group factor (Context: Shock and Novel) revealed that the rats froze significantly more in the Shock than the Novel Context (F_(1,10) = 57.393, P < 0.001). However, no significant difference was found between the HPC and Sham groups (F_(1,10) = 0.597, P = 0.458) and the Lesion × Context interaction did not reach significance (F_(1,10) = 0.123, P = 0.733). Thus, as in the previous repeated sessions experiment, the HPC damage did not cause retrograde amnesia for contextual fear conditioning and, more importantly, the HPC damage did not impair the ability to discriminate between the original context and new context.Open in a separate windowFigure 3.Mean (± SEM) percent time freezing by Sham and HPC rats in the Shock and Novel Contexts during the retention tests of the discrimination experiment. The rats exhibited significantly more freezing in the Shock than the Novel Context (P < 0.001), and the HPC rats did not significantly differ from the Sham rats, suggesting that the HPC-damaged rats remembered the specific meaning of the Shock Context as well as control rats. Hence, repeated conditioning sessions established a context-rich representation in non-HPC systems, which supports successful context discriminations.The absence of amnesia for contextual fear conditioning in the current study is not due to insufficient damage to the HPC. We calculated (see Lehmann et al. 2007b) that an average of 83% of the HPC was damaged across rats (smallest: 64%; largest: 90%) in the repeated learning experiments (see Supplemental material for more histological details). The amount of HPC damage is substantially more than that found in most studies reporting impairments for contextual fear conditioning following HPC damage (Kim and Fanselow 1992; Maren et al. 1997; Frankland et al. 1998; Anagnostaras et al. 1999; Debiec et al. 2002) and more than for the single-session experiment (average 76%) in which we currently report amnesia. Therefore, the amount of HPC damage inflicted in the rats in this study is certainly sufficient to disrupt HPC-dependent memories.Like others (Kim and Fanselow 1992; Maren et al. 1997; Frankland et al. 1998; Anagnostaras et al. 1999; Debiec et al. 2002; Lehmann et al. 2007b; Sutherland et al. 2008), we found that damage to the HPC after a single contextual fear-conditioning session involving multiple shocks produces profound retrograde amnesia for contextual fear conditioning. However, in two separate experiments, distributing shock across multiple conditioning sessions prevented this amnesia. In one case, the rats experienced Context–Shock pairings in one context and no shock in another context. Following this training, rats with damage to the HPC did not differ from control rats in the absolute amount of freezing in the training context nor in their ability to discriminate between the two contexts. In the second case, rats only received the multiple Context–Shock sessions. Rats with damage to the HPC could not be distinguished from control rats during the test in the training context or in their responses to a novel context. These findings provide new support for the general idea that contextual fear conditioning can be supported by both HPC and non-HPC systems. This conclusion is supported by (1) the finding that damage to the HPC following a single conditioning session virtually eliminates freezing during the test, implying the importance of the HPC system, and (2) that following multiple conditioning sessions, damage to the HPC has no effect on either contextual fear displayed in the training context or their ability to discriminate the training context from other contexts, suggesting the existence of non-HPC systems that can support contextual fear. The findings also reveal that the overshadowing or interference by the HPC over the non-HPC memory systems for control over contextual fear is not absolute. Following a single conditioning session, removal of the HPC produced a devastating retrograde amnesia, illustrating substantial overshadowing. However, distributing conditioning across several sessions completely attenuated the effects of damage to the HPC, revealing that non-HPC systems can support contextual fear conditioning despite the HPC, and revealed the importance of multiple sessions for this to occur.The overshadowing by the HPC is based on the familiar idea in associative learning at the behavioral level, where through a competitive process some of the cues that redundantly predict a reinforcer acquire the ability to generate strong conditioned responding, while other equally predictive, but less salient cues do not (Stout et al. 2003). Conditioning to the less potent cues proceeds more effectively if the more potent competitors are absent. Following the same principle, if the HPC representation is active, then learning in the non-HPC systems suffers strong interference. In contrast, in the absence of the HPC representation, learning in non-HPC systems is released from this interfering effect of the HPC. Thus, the learning rate in non-HPC networks is potently lowered by the activity of the HPC. However, with repeated learning, other structures, which are overshadowed by the HPC, may cumulatively build a representation that achieves HPC independence. The current findings clearly support this hypothesis, whereby repeated learning episodes incrementally established a contextual fear-conditioning representation outside of the HPC that mitigated the usual retrograde amnesic effects of HPC damage.One important question is where does the HPC interference occur? Biedenkapp and Rudy (2009) recently reported that the HPC competes with the basolateral region of the amygdala during fear conditioning. Previously, Guarraci et al. (1999) found that the amount of conditioned fear produced by training could be increased if the dopamine D1 receptor agonist {"type":"entrez-protein","attrs":{"text":"SKF82958","term_id":"1156217255","term_text":"SKF82958"}}SKF82958 was injected into the basolateral region. Biedenkapp and Rudy (2009) reasoned that if this is the area where the HPC interferes with non-HPC systems for the association with shock, then a local infusion of {"type":"entrez-protein","attrs":{"text":"SKF82958","term_id":"1156217255","term_text":"SKF82958"}}SKF82958 before a single session of contextual fear conditioning should attenuate the interference and allow the non-HPC system to gain more control over contextual fear. Their data supported this hypothesis, which leads to the possibility that with multiple conditioning sessions, the non-HPC system gradually gains association with these fear-supporting neurons in this region of the brain.Patients with bilateral damage to the HPC often exhibit temporally graded retrograde amnesia, such that recently acquired memories are lost, whereas remote memories, especially those acquired years before the damage, are more likely to be spared (Scoville and Milner 1957; Rempel-Clower et al. 1996). This pattern of amnesia is taken as evidence for temporally based systems consolidation, whereby over time the essential support for memories is “switched” from dependence on the HPC to neocortical networks (McClelland et al. 1995; Squire and Alvarez 1995; Anagnostaras et al. 2001; Meeter and Murre 2004; Squire et al. 2004; Wiltgen et al. 2004; Frankland and Bontempi 2005). Our research, however, points to another process for becoming independent of the HPC, a change in the strength of the representation in non-HPC systems during learning rather than a consolidation process linked to the passage of time since the learning episode. A study of a former London taxi driver with bilateral HPC damage alludes to this possibility (Maguire et al. 2006). This amnesic patient showed greater retrograde amnesia for roads that he used less commonly than the major arteries that he used regularly. Hence, greater exposure to the major arteries established memories in non-HPC systems, whereas roads with less exposure remained dependent on the HPC regardless of the age of the memory. Our findings add support to this view, because studies examining the effects of complete HPC damage after a single conditioning episode suggest that the HPC is permanently involved in contextual fear conditioning (Lehmann et al. 2007b; Sutherland et al. 2008); yet, with repeated learning episodes we clearly demonstrated that the memory rapidly becomes independent of the HPC. The latter is important because the process for memories becoming independent of the HPC need not require systems consolidation.In conclusion, this is the first example of intact contextual fear memories following complete HPC damage induced soon after learning. Importantly, repetition of the learning episode underlies the change in memory from HPC dependent to HPC independent. We argue that each learning episode incrementally establishes a representation in non-HPC memory systems—a representation that ultimately becomes sufficiently strong to support memory expression without the HPC. The current findings also demonstrate the critical need to consider learning parameters when discussing patterns of retrograde amnesia and the role of the HPC in memory.     

The general psychophysical differential equation: A comparison of three specifications

The general psychophysical differential equation, dy/dx = W₂(y)/W₁(x), with the solution y = f(x), where x and y are subjective variables and W₁ and W₂ their subjective Weber functions, is (a) compared with a corresponding functional equation, and (b) studied from a stochastic point of view by error calculus, Methods for evaluating and handling divergences are proposed and illustrated for a number of combinations of Weber functions. It is shown that either the differential: and the functional equations have the same solution or the difference between the solutions is negligible compared to empirical scatter. The error calculus gives the same result: either no error at all or a negligible one.     

Person‐Centered Emotional Support and Gender Attributions in Computer‐Mediated Communication

Without physical appearance, identification in computer‐mediated communication is relatively ambiguous and may depend on verbal cues such as usernames, content, and/or style. This is important when gender‐linked differences exist in the effects of messages, as in emotional support. This study examined gender attribution for online support providers with male, female, or ambiguous usernames, who provided highly person‐centered (HPC) or low person‐centered (LPC) messages. Participants attributed gender to helpers with gender‐ambiguous names based on HPC versus LPC messages. Female participants preferred HPC helpers over LPC helpers. Unexpectedly, men preferred HPC messages from male and gender‐ambiguous helpers more than they did when HPC messages came from females. Implications follow about computer‐mediated emotional support and theories of computer‐mediated communication and social influence.     

Hippocampal train stimulation modulates recall of fear extinction independently of prefrontal cortex synaptic plasticity and lesions

It has been shown that long-term potentiation (LTP) develops in the connection between the mediodorsal thalamus (MD) and the medial prefrontal cortex (mPFC) and between the hippocampus (HPC) and the mPFC following fear extinction, and correlates with extinction retention. However, recent lesion studies have shown that combined lesions of the MD and mPFC do not interfere with extinction learning and retention, while inactivation of the dorsal HPC disrupts fear extinction memory. Here we found in rats that immediate post-training HPC low-frequency stimulation (LFS) suppressed extinction-related LTP in the HPC-mPFC pathway and induced difficulties in extinction recall. HPC tetanus, applied several hours later, failed to re-establish mPFC LTP but facilitated recall of extinction. Delayed post-training HPC LFS also provoked mPFC depotentiation and difficulties with extinction recall. HPC tetanus abolished these two effects. We also found that damage to the mPFC induced fear return only in rats that received HPC LFS following extinction training. HPC tetanus also reversed this behavioral effect of HPC LFS in lesioned rats. These data suggest that the HPC interacts with the mPFC during fear extinction, but can modulate fear extinction independently of this interaction.     

Compressing movement information via principal components analysis (PCA): Contrasting outcomes from the time and frequency domains

PCA has become an increasingly used analysis technique in the movement domain to reveal patterns in data of various kinds (e.g., kinematics, kinetics, EEG, EMG) and to compress the dimension of the multivariate data set recorded. It appears that virtually all movement related PCA analyses have, however, been conducted in the time domain (PCA_t). This standard approach can be biased when there are lead-lag (phase-related) properties to the multivariate time series data. Here we show through theoretical derivation and analysis of simulated and experimental postural kinematics data sets that PCA_t and, PCA in the frequency domain (PCA_f), can lead to contrasting determinations of the dimension of a data set, with the tendency of PCA_t to overestimate the number of components. PCA_f also provides the possibility of obtaining amplitude and phase-difference spectra for each principal component that are uniquely suitable to reveal control mechanisms of the system. The bias in the PCA_t estimate of the number of components can have significant implications for the veracity of the interpretations drawn in regard to the dynamical degrees of freedom of the perceptual-motor system.     

A new formulation of discussive logic

S. Jakowski introduced the discussive prepositional calculus D ₂as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories (see, for example, N. C. A. da Costa and L. Dubikajtis, On Jakowski's discussive logic, in Non-Classical Logic, Model Theory and Computability, A. I. Arruda, N. C. A da Costa and R. Chuaqui edts., North-Holland, Amsterdam, 1977, 37–56). D ₂has afterwards been extended to a first-order predicate calculus and to a higher-order logic (cf. the quoted paper). In this paper we present a natural version of D ₂, in the sense of Jakowski and Gentzen; as a consequence, we suggest a new formulation of the discussive predicate calculus (with equality). A semantics for the new calculus is also presented.     

Hippocampal low-frequency stimulation and chronic mild stress similarly disrupt fear extinction memory in rats

Disruptions of fear extinction-related potentiation of synaptic efficacy in the connection between the hippocampus (HPC) and the medial prefrontal cortex (mPFC) have been shown to impair the recall of extinction memory. This study was undertaken to examine if chronic mild stress (CMS), which is known to alter induction of HPC–mPFC long-term potentiation, would also interfere with both extinction-related HPC–mPFC potentiation and extinction memory. Following fear conditioning (5 tone-shock pairings), rats were submitted to fear extinction (20 tone-alone presentations), which produced an increase in the amplitude of HPC–mPFC field potentials. HPC low-frequency stimulation (LFS), applied immediately after training, suppressed these changes and induced fear return during the retention test (5 tone-alone presentations). CMS, delivered before fear conditioning, did not interfere with fear extinction but blocked the development of extinction-related potentiation in the HPC–mPFC pathway and impaired the recall of extinction. These findings suggest that HPC LFS may provoke metaplastic changes in HPC outputs that may mimic alterations associated with a history of chronic stress.     

The abstract variable-binding calculus

Theabstract variable binding calculus (VB-calculus) provides a formal frame-work encompassing such diverse variable-binding phenomena as lambda abstraction, Riemann integration, existential and universal quantification (in both classical and nonclassical logic), and various notions of generalized quantification that have been studied in abstract model theory. All axioms of the VB-calculus are in the form of equations, but like the lambda calculus it is not a true equational theory since substitution of terms for variables is restricted. A similar problem with the standard formalism of the first-order predicate logic led to the development of the theory of cylindric and polyadic Boolean algebras. We take the same course here and introduce the variety of polyadic VB-algebras as a pure equational form of the VB-calculus. In one of the main results of the paper we show that every locally finite polyadic VB-algebra of infinite dimension is isomorphic to a functional polyadic VB-algebra that is obtained from a model of the VB-calculus by a natural coordinatization process. This theorem is a generalization of the functional representation theorem for polyadic Boolean algebras given by P. Halmos. As an application of this theorem we present a strong completeness theorem for the VB-calculus. More precisely, we prove that, for every VB-theory T that is obtained by adjoining new equations to the axioms of the VB-calculus, there exists a model D such that _T s=t iff _D s=t. This result specializes to a completeness theorem for a number of familiar systems that can be formalized as VB-calculi. For example, the lambda calculus, the classical first-order predicate calculus, the theory of the generalized quantifierexists uncountably many and a fragment of Riemann integration.The work of the first author was supported in part by National Science Foundation Grant #DMS 8805870.     

Distinct ventral and dorsal hippocampus AP5 anxiolytic effects revealed in the elevated plus-maze task in rats

The hippocampal formation (HPC) mediates processes associated with learning, memory, anxiety and fear. The glutamate N-methyl-d-aspartate (NMDA)-receptor subtype is involved in many HPC functional processes related to learning and memory. Although not tested for the HPC, NMDA-receptor antagonists reduced fear and anxiety related responses when applied to other brain regions mediating defensive behaviour. Consequently, this study evaluated the effects of ventral or dorsal HPC application of the NMDA-receptor antagonist, AP5, in rats submitted to the Trial 1/Trial 2 elevated plus-maze (EPM) task. Ventral, but not dorsal, infusions of AP5 (6 and 24 nmol) before EPM Trial 1 increased open arms exploration and reduced risk assessment behavior, suggesting an anxiolytic-like effect. Furthermore, no interference in the avoidance responses was detected during EPM Trial 2 after AP5 infusion into the ventral or dorsal HPC before Trial 1, post-trial 1, or before Trial 2. These data support the notion of differential involvement of ventral HPC, but not dorsal, in mechanisms associated with anxiety and suggest the participation of the glutamatergic transmission, through NMDA receptor, into the ventral HPC in the mediation of defensive behavior.