Non-standard Modes Research Articles

Nonaromatic benzenoid hydrocarbons

Abstract

A simulation study of electron hole distributions

Narrowband electrostatic noise (NEN) is frequently observed in the Earth‧s distant magnetotail. NEN exhibits a spectral peak between 100 and 316 Hz and is very impulsive, having a peak amplitude in the range 10−4−10−3V/m√Hz. It has been suggested that NEN represents a plasma wave mode which results from a hole in the low energy electron velocity distribution. In this paper, it is shown by means of numerical simulation that a wave mode is excited in the presence of a modest sized electron hole. In the simulations the mode saturates at low amplitudes, in agreement with observations, without destroying the hole. It is suggested that the burst‐like behavior of NEN is due to the persistence of these nonstandard modes and their ability to be excited by a variety of weak free energy sources.

Presumption and prejudice in logical inference

Two nonstandard modes of inference, confirmation and denial, have been shown by Bandler and Kohout to be valid in fuzzy propositional and predicate logics. If denial is used in combination with modus ponens, the resulting inference mode (“augmented modus ponens”) yields more precise bounds on the consequent of an implication than are usually called for in approximate reasoning. Similar results hold for augmented modus tollens constructed from confirmation and conventional fuzzy modus tollens. Two simpler modes of inference, presumption and prejudice, are also valid under the same assumptions as confirmation and denial. Prejudice imposes an upper bound on the truth value of the consequent of a fuzzy implication regardless of the truth value of the antecedent; presumption imposes a lower bound on the truth value of the antecedent regardless of that of the consequent. Some of the consequences of presumption and prejudice cast doubt on the suitability of fuzzy propositional and predicate logics for use in expert systems that are designed to process real-world data. A logic based directly on fuzzy sets is explored as an alternative. Fuzzy set logic supports fuzzy modus ponens and modus tollens but does not entail the more problematic modes of confirmation, denial, presumption, and prejudice. However, some of the expressive power derivable from the diversity of fuzzy propositional logics and their derivative fuzzy predicate logics is lost.

Through the mincing machine with a Boolean layer cake

The purpose of the present paper is threefold. Firstly, it introduces an abstract theory of straight-line (circuit) Boolean computations and complexity based on a strict separation of the syntax and semantics of Boolean circuits; the usual (standard) way to compute a circuit becomes thus only one of many other (nonstandard) operation modes over the circuit. Secondly, we construct and investigate three particular nonstandard operation modes, of which one, the mincing (M-)mode, is the superposition of two others, the injection and ejection modes respectively. The M-mode uses the circuit as a pipe-lining device, channeling simultaneously through each layer of the circuit (a layer comprises all nodes of a given individual depth) information stored in previous layers and renewing the input and storing the output information after every such parallel step; the number of steps is equal to the double depth of the circuit. The function computed by the M-mode is richer as a functional device than the standard functional (S-)representation of a circuit; in particular, it not only detects paths connecting individual inputs and outputs of a circuit, as the S-representation does, but their lengths as well. Thirdly, based on the M-mode, a new (in fact, the first known nontrivial) general lower bound to the relative circuit-size complexity of a Boolean function is presented, in the form of the inequality $$s(f, C') \geqq m(f, C')$$ , which means that the circuit-size (CS-)complexity of a Boolean function f with respect to a fixed subset C′ ⫅ C of the set C of all Boolean circuits is at least as big as the mincing complexity of f with respect to the same subset; choosing appropriate subsets, one arrives at the complete, monotone, synchronous, locally synchronous (and in algebraic computations which can be naturally treated in the same vein, linear, bilinear) or other CS-complexities. The corresponding mincing complexity is defined as the minimum, taken over all circuits from C′ standardly computing f, of the Boolean dimension of the function computing by the M-mode; the Boolean dimension of a function is equal by definition to the 2-logarithm of the cardinality of the function's image. The advantage to work with the new lower bound is that it is defined as a minimum taken over a relatively well defined functional subspace associated with f and C′, and not, as in the CS-complexity case, over a subset of circuits, i.e., a mixed discrete-continuum structure of an obscure nature (the real reason why the lower bound problem is so difficult). To demonstrate how the above inequality works, we prove an Ω(m·log m) lower bound to the locally synchronous CS-complexity of the m-dimensional cyclic convolution. This particular result is important for several reasons: (i) it improves the more than ten-years old non-linear lower bound to the synchronous CS-complexity of the same function, (ii) it can not be, apparently, proved by previously known methods, (iii) there exist functions (an example will be given in this paper) with the linear locally synchronous and a non-linear synchronous CS-complexities.

Propositional Attitudes in the Logic of Walter Burley and William Ockham

PROPOSITIONAL ATTITUDES IN THE LOGIC OF WALTER BURLEY AND WILLIAM OCKHAM [1] While there is no separate treatise in either Burley or Ockham dealing with propositional attitudes, there are a considerable number ofpassages in the logical writings ofeach ofthe two authors which do in fact deal with propositional attitudes. As may be expected, many of these passages are found in the discussions of modal logic in general , while some appear in other contexts, in particular in those places in which the question of supposition of terms is raised, or in the context ofdiscussion of general inferential relations involving the so-called non-standard (i.e. epistemic and doxastic) modes, or in the theory of formal disputation. [2] By 'propositional attitude' I mean locutions such as 'a knows that p,' 'a believes that p,' 'a doubts that p,' and so on, and their counterparts in the passive voice, presumably equivalent in their logical force to those in the active voice; e.g., 'p is known,' 'p is believed,' 'p is doubted,' and so on. [3] These latter ways offorming such attitudinal claims are much more frequent in both Burley and Ockham than their counterparts in the active voice. Both authors, at least occasionally formulate such claims in an expanded form, e.g., 'p is known by you' or 'p is known by me,' and I shall understand this to be a sufficent indication that Burley and Ockham at least half-consciously assumed an epistemic subject , even though their tendency to use an impersonal passive mode may have prevented them from fully realizing the sui generis character of the epistemic/doxastic logic which they were in fact helping to develop. [4] The study of propositional attitudes constitutes what has be- 32IVAN ??? come known as epistemic or epistemic/doxastic logic.1 In order to be logic, such an enterprise must consider relations among propositions such as compatibility or consistency, inconsistency, implication or entailment , and the like; in order for it to be epistemic/doxastic logic, such an enterprise must consider the logical relations precisely as these are present in virtue of propositional-attitude-functors such as 'is known by a,' 'is believed by a,' etc. [5] Although nowadays epistemic/doxastic logics are sometimes presented as deductive calculi, the late-medievals (as well as the rediscoverers of the discipline in our own century, e.g., J. Hintikka, R. Chisholm and a great many philosophers of recent decades), thought of this enterprise primarily as a critical examination of our ordinary, natural-language uses.2 Thus, if there are cases of natural uses wherein we seem to allow locutions such as 'p, but I don't believe it,' we have to take such cases seriously and then decide whether our current statements of necessary and sufficient condition for knowledge should continue to include the claim that if I know that p, then I must believe that p. [6] The full-fledged developments of epistemic logic seem to have taken place in England during the second quarter of the fourteenth century. As far as we know at this stage of research, the leaders in those development appear to have been William Heytesbury (d.1372/3) [and other Oxford Fellows Richard Kilvington (d. 1361), John Dumbleton (fl. 1340s), Richard Swineshead (fl.1340-55), Ralph Strode (d.1387)] and the distinguished "followers" of the English tradition in 1 In our own century, epistemic/doxastic logic was rediscovered and systematically investigated by Jaakko Hintikka in his Knowledge and Belief (Ithaca &t London: Cornell Univ. Press, 1962). Within the framework of this logic, an especially influential article was that by Edmund L.Gettier, "Is Justified True Belief Knowledge?" Analysis, 23 (1963): 121-23. 2 Cf. Wolfgang Lenzen, Recent Work in Epistemic Logic, Acta Philosophica Fennica, vol. 30, issue 1 (Amsterdam: North-Holland Publishing Co., 1978). In this comprehensive study Professor Lenzen examines many arguments the structure and components of which amply illustrate the reliance of the proponents on our natural language uses. For medievals, the Latin language was of course not natural in exactly the same sense, but even though it was a learnt "scientific" language, it was not an artificial language in the sense in which our schematic and axiomatic languages...