Peter Schroeder-Heister

A natural extension of natural deduction

One of the main ideas of calculi of natural deduction, as introduced by Jaskowski and Gentzen, is that assumptions may be discharged in the course of a derivation. As regards sentential logic, this conception will be extended in so far as not only formulas but also rules may serve as assumptions which can be discharged. The resulting calculi and derivations with rules of any finite level are informally introduced in ?1, while ??2 and 3 state formal definitions of the concepts involved and basic lemmata. Within this framework, a standard form for introduction and elimination rules for arbitrary n-ary sentential operators is motivated in ?4, understood as a contribution to the theory of meaning for logical signs. ?5 proves that the set {&, v, D, A } of standard intuitionistic connectives is complete, i.e. &, v a and A suffice to express each n-ary sentential operator having rules of the standard form given in ?4. ?6 makes some remarks on related approaches. For an extension of the conception presented here to quantifier logic, see [1 1].

Shift of functional cerebral asymmetry during the menstrual cycle.

This study investigated whether for females, who are said to be less strongly lateralized for cognitive functions than men, hemispheric superiority might depend on the phase of the menstrual cycle. The results show that while asymmetry in lexical decisions did not change throughout the menstrual cycle, asymmetry in face perception decreased linearly from a large right hemisphere superiority during menstruation to a small left hemisphere superiority during the premenstrual phase. This is seen as being relevant not only for the discussion of sex differences in cerebral asymmetry but also for the concept of cerebral organization in general.

Rules of definitional reflection

The author discusses two rules of definitional reflection: the logical version of definitional reflection, as used in the extended logic programming language GCLA, and the omega version of definitional reflection. The logical version is a left-introduction rule completely analogous to the left-introduction rules for logical operators in Gentzen-style sequent systems, whereas the omega version extends the logical version by a principle related to the omega rule in arithmetic. Correspondingly, the interpretation of free variables differs between the two approaches, resulting in different principles of closure of inference rules under substitution. This difference is crucial for the computational interpretation of definitional reflection.<<ETX>>

Extensions of Logic Programming

What do you do to start reading extensions of logic programming? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. Its not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this extensions of logic programming.

Spatial Coding and Spatio-Anatomical Mapping: Evidence for a Hierarchical Model of Spatial Stimulus-Response Compatibility

Publisher Summary This chapter examines evidence for a hierarchical model of spatial stimulus-response (S-R) compatibility. S-R compatibility usually denotes the reaction-time advantage for spatially compatible S-R pairings over incompatible pairings in choice-reaction tasks. The spatial S-R compatibility is related to the compatibility between perceived stimulus position and the position of the response key. In most designs, the apparent and actual stimulus positions, as well as the positions of the responding hands and response keys, are the same. It is found that with respect to the stage of cognitive processing in which S-R compatibility effects are generated, the major position is that the response-selection stage, rather than the stimulus-encoding stage, is relevant. The results again suggest the prevalence of the spatial coding factor in S-R compatibility, because the compatibility effect for fingers depends on the relative spatial positions of the responding fingers with respect to each other and not on their anatomical relation. The small hand-compatibility effect for the condition with the responding hand being fixed throughout an experimental session can be explained by a modified attentional hypothesis.

Validity Concepts in Proof-theoretic Semantics

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart.

A Proof-Theoretic Approach to Logic Programming. I. Clauses as Rules

In this paper definite Horn clause programs are investigated within a proof-theoreti c framework; program clauses being considered rules of a formal system. Based on this approach, the soundness and completeness of SLD-resolutio n is established by purely proof-theoretic methods. Extended Horn clauses are defined as rules of higher levels and related to an approach based on implication formulae in the bodies of clauses. In a further extension, which is treated in Part II of this series, program clauses are viewed as clauses in inductive definitions of atoms, justifying an additional inference schema: a reflection principle that roughly corresponds to interpreting the program clauses as introduction rules in the sense of natural deduction. The evaluation procedures for queries with respect to the defined extensions of definite Horn clause programs are shown to be sound and complete. The sequent calculus with the general elimination schema even permits the

Cut-elimination in logics with definitional reflection

Definitional Reflection is a principle for introducing atomic assumptions, given a set of definitional rules for atomic formulas. In this paper, proof-theoretic properties of first-order sequent systems with definitional reflection are proved. It is shown that the presence of contraction and the use of implication in the bodies of definitional clauses exclude each other, if cut-elimination is desired.

Spatial S-R compatibility effects with unimanual two-finger choice reactions for prone and supine hand positions

The influence of spatial stimulus-response (S-R) compatibility on unimanual two-finger choice reactions was investigated. Subjects had to react as fast as possible to a flash of light that was presented in the right or left visual field. They used the index or middle fingers of their left or right hands to press the spatially same (compatible) key or the spatially different (incompatible) key. In Condition 1 the subjects’ palms faced down (pronation); in Condition 2 their palms faced up (supination) so that the spatial order of the fingers was reversed. A strong compatibility effect between right and left field of stimulation and spatially right and left finger was found under both conditions. Compatible reaction times were shorter than incompatible reaction times by 52 msec in the palm-down condition and by 61 msec in the palm-up condition. The results are interpreted as supporting the coding hypothesis of spatial compatibility.

Spatial S-R compatibility with orthogonal stimulus-response relationship.

Spatial stimulus-response (S-R) compatibility with unimanual two-finger choice reactions was investigated under conditions in which the spatial orientation of response keys was either parallel to or perpendicular to the orientation of the stimuli. Subjects responded to green or red lights in the left or right visual field (irrelevant stimulus location). The response keys were oriented horizontally on the left or right side of the body midline parallel to the stimuli, and were pressed with the palms facing down (Condition A), or were oriented orthogonally to the stimuli in the midsaggital plane, either horizontally and pressed with palms facing down (B) or facing up (C), or vertically and pressed with palms facing the body (D). The results for Condition A demonstrate the usual spatial S-R compatibility effect between field of stimulation and spatial position of responding finger. For Conditions B and D, a strong reaction time advantage still obtained for those stimulus-finger pairings that are compatible under Condition A. Condition C revealed an RT advantage for the opposite pairings. This shift of the compatibility effect from Condition B to Condition C indicates that the left/right distinction of fingers does not follow a simple, fixed spatio-anatomical mapping rule. The results are discussed within the framework of a hierarchical model of spatial S-R compatibility, with spatial coding and spatio-anatomical mapping as factors.