Second-order logic

In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence \forall P,\forall x (Px \lor \neg Px) says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. Examples First-order logic can quantify over individuals, but

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Regularity of solutions for a fourth-order elliptic system via Conservation law

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Oscillatory patterns of activity in various frequency ranges are ubiquitously expressed in cortical circuits. While recent studies in humans emphasized rhythmic modulations of neuronal oscillations ("second-order" rhythms), their potential involvement in information coding remains an open question. Here, we show that a rhythmic (similar to 0.7 Hz) modulation of hippocampal theta power, unraveled by second-order spectral analysis, supports encoding of spatial and behavioral information. The phase preference of neuronal discharge within this slow rhythm significantly increases the amount of information carried by action potentials in various motor/cognitive behaviors by (1) distinguishing between the spikes fired within versus outside the place field of hippocampal place cells, (2) disambiguating place firing of neurons having multiple place fields, and (3) predicting between alternative future spatial trajectories. This finding demonstrates the relevance of second-order spectral components of brain rhythms for decoding neuronal information.

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