Search

There has been debate between those who maintain that indexical expressions are not essential and those who maintain that such indexicals cannot be dispensed with without an important loss of content. This version of the essentialist view holds that thoughts must also have indexical elements. Indexical thoughts appear to be in tension with the computational theory of mind (CTM). In this case we have the following inconsistent triad:\ud \ud (i) De se thoughts are essential.\ud (ii) De se thoughts are indexical, they have a (Kaplanian) character.\ud (iii) Computations can only take the syntactic type into account, they cannot take tokens into account.\ud If (iii) is correct, then it seems we cannot make sense of a thought which uses a character such that its referent could vary from tokening to tokening. I argue that (iii) need not cause a problem, while maintaining the CTM. I claim that computations need not be sensitive to the features of a tokened symbol in the way that character demands. This job may be performed by a non‐modular part of the mind. Resolving the triad in this way provides a reason to accept that indexicals in thought are possible.

An incoherent visualization is when aspects of different senses of a word (e.g., the biological "mouse" vs. the computer "mouse") are present in the same visualization (e.g., a visualization of a biological mouse in the same image with a computer tower). We describe and implement a new model of creating contextual coherence in the visual imagination called Coherencer, based on the SOILIE model of imagination. We show that Coherencer is able to generate scene descriptions that are more coherent than SOILIE's original approach as well as a parallel connectionist algorithm that is considered competitive in the literature on general coherence. We also show that co-occurrence probabilities are a better association representation than holographic vectors and that better models of coherence improve the resulting output independent of the association type that is used. Theoretically, we show that Coherencer is consistent with other models of cognitive generation. In particular, Coherencer is a similar, but more cognitively plausible model than the C3 model of concept combination created by Costello and Keane (2000). We show that Coherencer is also consistent with both the modal schematic indices of perceptual symbol systems theory (Barsalou, 1999) and the amodal contextual constraints of Thagard's (2002) theory of coherence. Finally, we describe how Coherencer is consistent with contemporary research on the hippocampus, and we show evidence that the process of making a visualization coherent is serial.

Under some mild assumptions on the Levy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure-jump isotropic unimodal Levy processes in $R^d$., Minor changes

The main result of this paper states that if $k$ is a field of characteristic $p\gt 0$ and $A/ k$ is a central simple algebra of index $d= {p}^{n} $ and exponent ${p}^{e} $, then $A$ is split by a purely inseparable extension of $k$ of the form $k( \sqrt[{p}^{e} ]{{a}_{i} }, i= 1, \ldots , d- 1)$. Combining this result with a theorem of Albert (for which we include a new proof), we get that any such algebra is Brauer equivalent to the tensor product of at most $d- 1$ cyclic algebras of degree ${p}^{e} $. This gives a drastic improvement upon previously known upper bounds.

Token exchange inherently introduces an element of delay between behavior and reward and so token studies may help us better understand delay of gratification and self-control. To examine this possibility, we presented three language-trained chimpanzees with repeated choices involving different foods that could be eaten immediately or lexigram (graphic symbol) tokens that represented (and could be traded for) foods later. When both options were foods, chimpanzees always chose more preferred foods over less preferred foods. When both options were lexigram tokens representing those same foods, performance remained the same as chimpanzees selected the higher value token and then traded it for food. Then, when faced with choosing a token that could be traded later or choosing a food item that could be eaten immediately, most chimpanzees learned to make whatever response led to the more preferred food. They did this even when that meant selecting a high value lexigram token that could be traded only two to three minutes later instead of a medium value, but immediately available, food item. Thus, chimpanzees flexibly selected tokens even though such selections necessarily delayed gratification and required forgoing immediately available food. This finding illustrates the utility of symbolic token exchange for assessing self-control in nonhuman animals.

Previous research shows a correlation between individual differences in people’s school math abilities and the accuracy with which they rapidly and nonverbally approximate how many items are in a scene. This finding is surprising because the Approximate Number System (ANS) underlying numerical estimation is shared with infants and with non-human animals who never acquire formal mathematics. However, it remains unclear whether the link between individual differences in math ability and the ANS depends on formal mathematics instruction. Earlier studies demonstrating this link tested participants only after they had received many years of mathematics education, or assessed participants’ ANS acuity using tasks that required additional symbolic or arithmetic processing similar to that required in standardized math tests. To ask whether the ANS and math ability are linked early in life, we measured the ANS acuity of 200 3- to 5-year-old children using a task that did not also require symbol use or arithmetic calculation. We also measured children’s math ability and vocabulary size prior to the onset of formal math instruction. We found that children’s ANS acuity correlated with their math ability, even when age and verbal skills were controlled for. These findings provide evidence for a relationship between the primitive sense of number and math ability starting early in life.

We present a new approach to motion rearrangement that preserves the syntactic structures of an input motion automatically by learning a context-free grammar from the motion data. For grammatical analysis, we reduce an input motion into a string of terminal symbols by segmenting the motion into a series of subsequences, and then associating a group of similar subsequences with the same symbol. In order to obtain the most repetitive and precise set of terminals, we search for an optimial segmentation such that a large number of subsequences can be clustered into groups with little error. Once the input motion has been encoded as a string, a grammar induction algorithm is employed to build up a context-free grammar so that the grammar can reconstruct the original string accurately as well as generate novel strings sharing their syntactic structures with the original string. Given any new strings from the learned grammar, it is straightforward to synthesize motion sequences by replacing each terminal symbol with its associated motion segment, and stitching every motion segment sequentially. We demonstrate the usefulness and flexibility of our approach by learning grammars from a large diversity of human motions, and reproducing their syntactic structures in new motion sequences.

For a large class of pseudo-differential operators with a negative definite symbol q(x, ξ) in the sense of Hoh and for a large family of x-dependent Bernstein functions f(x, ·) we prove that the pseudo-differential operator with symbol −f(x, q(x, ξ)) has an extension generating a Feller semigroup. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

There has been debate between those who maintain that indexical expressions are not essential and those who maintain that such indexicals cannot be dispensed with without an important loss of content. This version of the essentialist view holds that thoughts must also have indexical elements. Indexical thoughts appear to be in tension with the computational theory of mind (CTM). In this case we have the following inconsistent triad: (i) De se thoughts are essential. (ii) De se thoughts are indexical, they have a (Kaplanian) character. (iii) Computations can only take the syntactic type into account, they cannot take tokens into account. If (iii) is correct, then it seems we cannot make sense of a thought which uses a character such that its referent could vary from tokening to tokening. I argue that (iii) need not cause a problem, while maintaining the CTM. I claim that computations need not be sensitive to the features of a tokened symbol in the way that character demands. This job may be performed by a non‐modular part of the mind. Resolving the triad in this way provides a reason to accept that indexicals in thought are possible.

One of the most striking differences between Frege's Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted, that is, which are being used to express judgements. There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role of the judgement stroke in a way that renders it readily comprehensible why Frege insisted that this symbol was an essential part of his logical system. The key point here is that Frege viewed logic as the study of inference relations amongst acts of judgement, rather than – as in the typical contemporary view – of consequence relations amongst certain objects (propositions or well-formed formulae). The paper also explains why Frege's use of the judgement stroke is not in conflict with his avowed anti-psychologism, and why Wittgenstein's criticism of the judgement stroke as ‘logically quite meaningless’ is unfounded. The key point here is that while the judgement stroke has no content, its use in logic and mathematics is subject to a very stringent norm of assertion.

Code-switching research has focused on spontaneous conversation, and code-switching has often been seen as a consequence of bilinguals attending to and extending the "macro" status and functions of the two languages in society, attitudes towards these languages, and their cultural connotations, for instance, the "we-code" vs. "they-code" distinction. In the Hong Kong context, code-switching to English has been primarily considered as quoting "Western" concepts and ideas, conveying referential and connotative meanings absent in Cantonese. Investigating a corpus of planned discourse, namely, Cantonese popular songs (i.e. Cantopop) in Hong Kong, this paper finds that the status and functions of English in Cantopop are more variable and flexible, beyond a mere symbol of "Western" culture or identity. Nonetheless, these functions can be attributed to the properties of the pop song genre, namely, Cantopop as poetic text, media text, and a product of pop culture; in particular, code-switching fits into the rhyming scheme, marks text structure, indexes prior texts, and conveys alternative identities.

In this paper we investigate Hankel operators with anti-holomorphic L2-symbols on generalized Fock spaces Am2 in one complex dimension. The investigation of the mentioned operators was started in [4] and [3]. Here, we show that a Hankel operator with anti-holomorphic L2-symbol is in the Schatten-class Sp if and only if the symbol is a polynomial with degree N satisfying 2N . The result has been proved independently before in the recent work [2], which also considers the case of several complex variables. However, in addition to providing a different proof for the result the present work shows that the methodology developed in [4] and [3] can be adopted in order to work to characterize Schatten-class membership. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are TIMSS items (a linear inequality and an equation involving square roots). The other problem involves square roots. We found that the majority of student pairs used symbol manipulation when solving the problems, and while many students seemed to prefer symbolic over graphical and tabular representations in their first attempt at solving the problems, we found that it was common for student pairs to use more than one strategy throughout the course of their solving. Students' use of graphing calculators to solve the problems is discussed.

Anterior negativities obtained during sentence processing have never been unambiguously reported in the mathematical domain. The reason for this might be that the tasks explored in the mathematical domain have been far from resembling those typically yielding language-related anterior negativities. To test this hypothesis, we explored three mathematical aspects: Order-relevant information, a parenthesis indicating the onset of an embedded calculation, and violations of the type of symbol displayed. Results yielded parieto-occipital instead of frontal negativities. Late posterior positivities were also found, largely comparable to linguistic P600 in topography, but dissociable in functional terms. Our data suggest that language-related anterior negativities may indeed reflect language-specific resources of the human brain and support recent claims that language and mathematical domains are more independent than previously thought.

Following the analogy between primes and knots, we introduce the refined Milnor invariants for prime numbers and establish their connection with certain Massey products in Galois cohomology. This generalizes the well-known relation between the power residue symbol and cup product and gives a cohomological interpretation of L. Redei's triple symbol.

Thirty-four matched pairs of sixth- and seventh-grade students were selected from 358 participants in a comparison of an explicit concrete- to-representational-to-abstract (CRA) sequence of instruction with traditional instruction for teaching algebraic transformation equations. Each pair of students had been previously labeled with a spe- cific learning disability or as at risk for difficulties in algebra. Students were matched according to achievement score, age, pretest score, and class performance. The same math teacher taught both members of each matched pair, but in different classes. All students were taught in inclusive settings under the instruction of a middle school mathematics teacher. Results indicated that students who learned how to solve algebra transformation equations through CRA outperformed peers receiving traditional instruction on both postinstruction and follow-up tests. Additionally, error pattern analysis indicated that students who used the CRA sequence of instruction performed fewer procedural errors when solving for variables. Abstract thinking requires a person to work with infor- mation that is not readily represented at the concrete or pictorial level (Hawker & Cowley, 1997). To work with abstract information is to understand theoretical properties and think beyond what a person can touch or see. On a practical level, an ability to work with abstract concepts allows one to work with predictions of what may happen and expectations about what is happening elsewhere. Abstract knowledge may also be considered a conscious awareness that a symbol stands for some

A spoken dialogue system for retrieving academic documents was constructed and evaluated. In order to realize speech reply with its prosodic features properly controlled to express dialogue focus, a concept-to-speech conversion scheme was developed where the reply sentence concept was finally converted step-by-step to phone and prosodic symbol strings to be fed to a speech synthesizer, instead of simply using a text-to-speech conversion scheme as done in other conventional spoken dialogue systems. The main research focuses were on the following three points: (1) generating a reply sentence easily understood by users and unambiguous to users by controlling the amount of information included in it: deleting redundant information and adding necessary information; (2) realizing functions necessary for increasing system usability; (3) generating a speech reply with natural-sounding prosody, with the dialogue focuses correctly conveyed. Evaluations of the original system, formerly developed, gave us guidelines to realize points (1) and (2). Based on them, a revised system was constructed and evaluated, showing the validity of its new functions. It was also shown that speech replies with prosodic features representing dialogue focuses could bring about clearer information transmission. © 2002 Wiley Periodicals, Inc. Syst Comp Jpn, 33(4): 25–39, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.1117

We study the pseudolocality and microlocality of pseudodifferential operators with general symbols. We treat the problem of pseudo– and microlocality, along finite dimensional linear subspaces, of the Weyl quantization and also generalize some results of Parenti and Rodino to the case of the Weyl–Hormander calculus of pseudodifferential operators. We do not require that the metric, in the definition of the symbol, is decomposable, σ temperate or satisfy the uncertainty principle neither the weight function of the symbol has to be σ — g temperate. Instead, we assume weaker conditions on the metric and the weight function and the only extra condition we are assuming on the symbol is one of its essential support.

This paper describes a connectionist inference architecture which performs standard symbolic inference on a subclass of first-order predicate calculus. This work first involves constructing efficient connectionist mechanisms to represent basic symbol components, dynamic bindings and basic symbolic inference procedures, and devising a set of algorithms which automatically translates input descriptions to localist neural networks. These connectionist mechanisms are built by taking an existing phase-locking mechanism and extending it further to obtain desirable features to represent and manipulate basic symbol structures. The existing phase-locking mechanism represents dynamic bindings very efficiently using temporal synchronous activity between neuron elements but has fundamental limitations in supporting standard symbolic inference. The extension addresses these limitations. The extended system encodes a significant subset of a Horn clause language in a connectionist style.

Summary An elusive probability paradox is analysed. The fallacy is traced back to improper use of a symbol that denotes at the same time a random variable and two different values that it may assume.

A formula ϕ(x) in the one non-logical symbol ϵ with one free variable x is Russell if the sentence Vχ(χ ϵ τ ⟷ Φ (χ)) is logically valid. This note describes a pattern common to the classical examples of Russell formulas, adds a couple of new ones, and constructs many formulas that are near-Russell.

Turing's analysis of computation is a fundamental part of the background of cognitive science. In this paper it is argued that a re-interpretation of Turing's work is required to underpin theorizing about cognitive architecture. It is claimed that the symbol systems view of the mind, which is the conventional way of understanding how Turing's work impacts on cognitive science, is deeply flawed. There is an alternative interpretation that is more faithful to Turing's original insights, avoids the criticisms made of the symbol systems approach and is compatible with the growing interest in agent-environment interaction. It is argued that this interpretation should form the basis for theories of cognitive architecture.

Ethnomathematics can consider recursion in two senses of the word. Mathematically, recursion consists of iterated functions, a kind of discrete feedback loop. Anthropologically, recursion is used in reflexive ethnographic description in which modern analytic methods become part of their own investigation. By comparing the recursive structure of Bamana divination techniques to recursion in modern mathematics, we can investigate some of the complex relations that bind technology and culture in both modern and traditional contexts.

A recently developed molecular technique (amplified fragment length polymorphisms, AFLP) was used for characterizing genetic heterogeneity within and among populations of a critically endangered species of plant, Astragalus cremnophylax var. cremnophylax. Using AFLP, up to 50 polymorphic genetic markers per AFLP-PCR amplification were generated, and a total of 220 variable markers overall. This information was used first to assess genetic diversity within each of the three known populations of Astragalus cremnophylax var. cremnophylax from Grand Canyon National Park in Arizona, USA: North Rim (NR; n = 970), South Rim Site 1 (SR1; n = 500), and South Rim Site 2 (SR2; n = 2). Diversity in the form of average heterozygosity [symbol: see text] H [symbol: see text] and the proportion of polymorphic genes [symbol: see text] P [symbol: see text] was greatest in the NR population ([symbol: see text] H [symbol: see text] = 0.13 and [symbol: see text] P [symbol: see text] = 0.38) and least in the SR2 population ([symbol: see text] H [symbol: see text] = 0.02 and [symbol: see text] P [symbol: see text] = 0.04). Diversity was also quite low for the SR1 population ([symbol: see text] H [symbol: see text] = 0.04 and [symbol: see text] P [symbol: see text] = 0.10). In addition, substantial genetic differentiation among populations was indicated by both phenetic (AMOVA) and genetic analyses (overall corrected FST = 0.41). This finding was corroborated by the results of several multivariate analyses which utilized the genetic data, including a UPGMA cluster analysis and a principal coordinate analysis which revealed the existence of discrete groups corresponding to the populations. Population structure was further revealed within the NR population which was known to consist of four spatially separated groups of plants. Several recommendations for the future management of the species are discussed.

The present invention teaches a source-level debugger that defines symbol tables with a grammar. A grammar interface component is used by the compiler to automate portions of the debugger to construct and write the symbol table in a grammar thus simplifying the compiler. The grammar interface component is used by the debugger during execution of the target to read and decode the symbol table, thus simplifying the debugger. Using a grammar to specify the symbol table also documents the symbol table concisely and emphasizes that symbol tables are data structures, not file formats. Many of the pitfalls of working with low-level file formats can be avoided by focusing instead on grammar-based high-level data structures and automating the implementation details. Management of breakpoints is divided by splitting the nub into a client nub that is local to the target, a server nub that is local to the debugger. The management of breakpoints is localized to the client nub, thus saving communication time between the client nub and the server nub in support of breakpoint processing.

We consider the following generalization of the notion of a structure recursive relative to a set X. A relational structure A is said to be a Γ(X)-structure if for each relation symbol R, the interpretation of R in A is ∑ relative to X, where β = Γ(R). We show that a certain, fairly obvious, description of classes ∑ of recursive infinitary formulas has the property that if A is a Γ(O)-structure and S is a further relation on A, then the following are equivalent: (i) For every isomorphism F from A to a Γ(X)-structure, F(S) is ∑ relative to X, (ii) The relation is defined in A by a ∑ formula with parameters. Mathematics Subject Classification: 03D45, 03C57, 03C75.

The glycan symbol nomenclature proposed by Harvey et al. in these pages has relative advantages and disadvantages. The use of symbols to depict glycans originated from Kornfeld in 1978, was systematized in the First Edition of "Essentials of Glycobiology" and updated for the second edition, with input from relevant organizations such as the Consortium for Functional Glycomics. We also note that >200 illustrations in the second edition have already been published using our nomenclature and are available for download at PubMed.

This paper argues that educational software should be written in such a way that teachers have access to the program's educational content. One such program, a computer-mediated reading and language development scheme, is discussed. The program allows each word of traditional orthography to be replaced by a graphic symbol if the teacher wishes.

The literature on pencil and paper tests and the effects of psychotropic drugs is reviewed. Performance at digit symbol substitution, symbol copying, letter cancellation, arithmetic, logic and cognitive processing with a relatively constant level of triazolam was compared with placebo. Analysis of variance, covariance analysis and principal component analysis were applied to the data. Pencil and paper tests which measure various skills are useful for detecting drug effects. Letter cancellation, arithmetic and DSST appear to be most sensitive, although logic and symbol copying may be useful.

Two series of three symbol balanced arrays of strength two are constructed. Using special classes of BIB designs, two classes of two symbol orthogonal arrays of strength three are constructed.

Two non-verbal tasks of logical symbol use (SU) and of symbol discovery (SD) were devised. SU differentiated adults of higher and lower intelligence. Most pupils of chronological age (CA) 12 and older, but not younger, discovered the significance of the logical symbols in learning the paired associate task of SD. SD and SU were given to deaf subjects and to hearing subjects from a rural, lower-class milieu. These experimental subjects (CA around 18) performed similarly to controls on SU, but did not attain the success of 12-year-old controls on SD. On the basis of these results it is proposed that stimulating past experience may be required for success on certain conceptual tasks which are of the discovery type and depend on the initiative of the subject, but that more structured tasks of logical reasoning are not adversely affected by restricted experience or linguistic deficiency.

This paper will discuss an alteration in the phonological components of Alemannic and Bavarian within the framework of generative phonology. First, an apparent discrepancy in the earliest documents concerning the relative chronology of two early rules will be accounted for. The rules are the second sound shift and the rule lowering au to ao. Internal evidence will also be presented for the diphthongal status of the orthographic symbol ao (and likewise ae). Second, the changes discussed here represent further evidence for the simplification principle recently discussed by Kiparsky. It is important to note that documented examples of simplification of synchronically existing rules are empirical evidence for the generative model of natural language.

StrOmnes, F. J. A semiotic theory of imagery processes with experiments on an Indo-European and a Ural-Altaic language: Do speakers of different languages experience different cognitive worlds?Scand. J. Psychol., 1973, 14, 291–304.-A theory of information transmission by symbol systems is presented. The theory is related to results on imagery and verbal behaviour found in the literature, and a new experiment is briefly reported. It is argued that information transmission can exclusively take place by the transmission of geometric isomorphs. Some of the conditions which must be fulfilled by a sign system which can transmit geometric isomorphs are pointed out. A supporting experiment, which purports to show that the underlying geometric systems of Ural-Altaic (represented by Finnish) and Indo-European (represented by Swedish) are strikingly different, is reported.

A typical chain of symbol-processors transforms events into data (inquiring), data into messages (communicating), messages into actions (deciding). A symbol-processor can be a man or a machine. This paper takes some examples from legal practice with criminal evidence as “data,” the “benefit” of condemning an innocent being even less than that of acquitting the guilty. In addition to the costs of inquiring and deciding, the costs of communication are considered. Related examples from the information industry are examined.

latter. If nothing is known about the cause of the apparent intergradation it is better not to attribute it to hybridization. It is suggested that X be used only when there is definite evidence for hybridization, and that the dash. -, be chosen to signify morphological intergradation. A suitable symbol needs to be standardized for international use, just as X is standardized. The existing symbols > and and < in the same formula could denote intermediacy. Other considerations relating to hybrid and non-hybrid formulae are the following: (1) one of the symbols, either X or -, is always present in a formula, and (2) the order of taxa in a formula is always alphabetical except perhaps in artificial crosses. Clarification of terms for hybrids and intergradient plants is needed and a standardized set of terms is desirable. To designate intergradient plants (not necessarily due to hybridization) the noun intergradant, an etymological equivalent of introgressant, is suggested. An attempt is made to find reasonable, approximate meanings for certain other terms.

Psychological diagnosis is sometimes based on clinical judgment. This article illustrates an attempt to make these judgments more objective. It describes a relatively new diagnostic technique in which the scoring is objective and does not require extensive psychological training. It may also save time by being a possible substitute for lengthier test batteries.

THE shift of emphasis from reading ability to conversational ability in foreign languages is bringing about changes in our methods of teaching. Particularly is this apparent in the initial presentation of a foreign language to a class of beginners. Most teachers now avoid the use of text or readers as far as possible in this first presentation. The development of good pronunciation is the primary purpose, and the teacher's voiced model of the expressions is the "text." Good pronunciation is an important objective throughout the learning of a language, but it is particularly important when the new sounds make their way to the brain for the first time to be associated with old concepts. Through aural comprehension, the foundation is laid for good pronunciation which must be taught from the start. To meet the aural-oral approach, a vocabulary must therefore be selected for the initial presentation which will be interesting and stimulating, and within the experience of the entire class. In Pittaro and Green's "El Primer Curso de Espafiol" we find numbers in the very first lesson. Herman Barnstorffl and Ruth Jackson2 in two separate articles describe the first lesson in a beginning class using the same material. And indeed it seems logical to present numbers first, for they are almost universally understood in their symbol form. If one decides to present numbers in the first lesson, the use of flash cards provides at once a motivation for the activity and the opportunity for a variety of drill. We must keep in mind that the numbers are not to be written out in the foreign language. They appear upon the flash cards only in their symbol form. They are pronounced by the class after the vocal model of the teacher until they are mastered by most of the students. Then simple problems of arithmetic are pronounced by the teacher,first in addition, later in subtraction, multiplication, and division. For example, he says, "Cuintos son dos y cinco?" He holds up the flash card for all to see and gives the model of the answer in a complete sentence. "Dos y cinco son siete."

This project was an empirical investigation of predictions derived from a theory “by Piaget concerning the development of intelligence. In particular, three content areas were to be studied, making use of group testing procedures and special statistical techniques for analysis. Piaget defines a mental operation as an internalized act, i.e., the external object is represented symbolically in the mind of the subject who instigates behavior with respect to the symbol. Concrete operations are operations on the symbols of objects that can be manipulated through the senses. Formal operations are operations on concrete operations. His theory implies a steplike development of intelligence and the possiblility of demonstrating that there are children who can carry out concrete operations but not formal and that all children who can carry out formal operations can also carry out concrete operations. An empirical finding reported by Piaget is that children in general change from the concrete to the formal operations between their twelfth and fifteenth birthdays. Of course, some children make the change before this period and some afterwards, and others again probably never do. Conditions which facilitate or inhibit this change are not explicitly stated by the theory. In the area of arithmetic, for example, Piaget's theory of intelligence is interpreted as representing the development of intelligence as a process of building up a set of elements and an operation with respect to this set, such that the conditions for a semigroup are satisfied. In this way the stage of concrete operations is established. Later in the developments of the individual the condition of an inverse is added and this with the unit changes the semigroup into a group. This is the stage of formal operations. The theory was examined with respect to three content areas, arithmetic, probability and inequalities (also referred to as “comparisons”). In the area of arithmetic, pairs of problems were prepared, one of which required reversibility of operations in the sense that an operation combined with its inverse leaves the original number unchanged, whereas the other could be solved by normal computational operations. In the area of probability theory formal operations were examined through the notion of independence. The children in effect were given in one problem that a certain number of object A and a certain number of object B were placed in a box and were asked which object, A or B, was most likely to come out by random drawing. In another problem it is given that one more of either object is in the box but that it has been removed by chance drawing before they have to decide which is most likely to come out. Formal operations in this case lead to the notion of independence which is an important condition for operations on probabilities. In the case of the comparisons items the items differed solely on the basis of generality. Such items do not examine the theory as stated, but were included for other reasons related to Piaget's work and to evidence found in the literature. The effect of generality was examined also in the other content areas. Data were obtained by administering 74 items, some of which had several parts, to children from Grades IV, VI, VII, VIII, and X. Approximately two hundred from each of the first two grades and three hundred from each of the upper three grades were tested. These data were punched onto IBM cards and analyzed in terms of the relationship between matched pairs of items. The results of this analysis indicated that six of the seven arithmetic item pairs were related in a way suggested by the theory. The seventh item pair required operations with fractions; further study of such items is suggested. For the probability items effects which could be attributed to lack of appreciation of the notion of independence were obtained with Grade IV and Grade VI children, Children in Grades VII, VIII, and X behaved somewhat differently, but this could have been due in part to the effect of the influence of a variable not anticipated in the planning of these items. Further study of this type of item is suggested. No effects were obtained with the inequalities items. Specific suggestions are made as to how items, in this area could be constructed that require reversibility of operations. In general, it was concluded that the group testing situation could be used experimentally with systematic variation of the content of items. Piaget's theory was used as the basis of this variation and certain predicted results were obtained. Some specific suggestions for further studies arose from the results of this study.