The Measurement of Inductive Abilities According to the Schemata of Inductive Logic United States mCivil Service Commission Bureau of Policies and Standards SUI\l'AT',I! T i ~ "'' .)I '-, [ ~ / "f Technical Study 78-2 THE MEASUREMENT OF INDUCTIVE ABILITIES ACCORDING TO THE SCHEMATA OF INDUCTIVE LOGIC Magda Munoz-Colberg Mary Anne Nester Test Services Section Personnel Research and Development Center United States Civil Service Commission Washington, D.C. 20415 October 1978 Abstract The logic of induction is described in its two stages: law-generative judgments and judgments of probability. The latter have been neglected in the psychometric approach to induction, which has mainly been limited to nonlinguistic law-generative questions. In order to correct this deficiency, thirty multiple choice items based on schemata for judgments of probability were written in the linguistic medium and administered to adult job applicants in three different tests. Statistical characteristics of the tests (means and reliabilities) and items (E-values and item-test point-biserial correlations) were very satisfactory when compared to operational subtests. Test correlations with inductive (law-generative nonlinguistic) and deductive subtests were substantial and higher with the deductive subtest. It is argued that the new test improves the predictive value sought in measurement because judgments of probability and reasoning from linguistic data prevail in every situation for which the instrument is intended to select. THE MEASUREMENT OF INDUCTIVE ABILITIES ACCORDING TO THE SCHEMATA OF INDUCTIVE LOGIC If the testing of inductive abilities is analyzed from the standpoint of inductive logic, it is evident ~ priori that a pure and comprehensive approach to testing would necessarily include both forms of induction: law-generative inductive judgments and inductive judgments of prob ability. In other words, in terms of reasoning process, the testing situation reflects with precision the situation for which it selects only when there is an identity between both situations. Regrettably, in the psychometric tradition, tests of inductive abilities have not extended beyond law-generative judgments. Inductive judgments of probability thus far have been ignored in the attempt to measure inductive abilities. The seriousness of this omission in terms of predictive value hardly needs to be emphasized. Inductive judgments of probability are prevalent in every "real" situation, on the job as well as in the academic milieu. In fact, they are present in the everyday existential situation and if one were testing for existence as such, judgments of probability would constitute a contextual sine qua non. Additionally, in attention to all circumstances presented by "real" situations, inductive tests should include questions presented in linguistic form. In the psychometric tradition the nonlinguistic medium has been generally utilized to test inductive reasoning (Ekstrom, French, and Harman, 1976). The research and development effort presented in this study had, therefore, a dual objective: First, to develop a test designed to measure inductive abilities in the form of judgments of probability, and second, to develop an inductive test in linguistic form. A third objective will be pursued in the immediate future, namely, to ascertain whether or not an inductive test in linguistic form is appropriate to attempt to measure inductive abilities in blind persons. Figural inductive items presented in embossed form have been found to be inadequate for universal testing since they rely to some extent on tactual abilities. This leaves serial items, among among those traditionally used, as the sole inductive item type appropriate to test the blind. The inadequacy of this situation need not be perused. Additionally, the issue mentioned above applies to the blind as well as to the sighted, namely, that it is necessary to approach the measurement of inductive abilities from a comprehensive standpoint and to include inductive judgments of probability alongside of law-generative judgments. LOGICAL ANALYSIS In contradistinction to deductionl and as a form of argument, induction is nondemonstrative. In other words, it is impossible for a deductive conclusion, Qa, to be false if the premises, (x) (Px ) Qx) h Pa, are true2. However, even if the-premises are A logic-based approach to the testing of deductive abilities is described in Munoz-Colberg and Nester, 1978 (Note 1) The symbol) is a connective which indicates implication. The symbol h indicates conjunction. The first premise asserts that for all x, if x has the the property P, then x also has the property Q. The term~ at the beginning of the formula represents a universal quantifier and indicates that reference is being made to all cases of ~ (all deductive schemata include a universal premise). The second premise asserts that a particular object ~has the property P. From these premises the conclusion that object a has the property Q follows with logical necessity. 1 true, it is possible for the conclusion ~to be false if it is inductively established. In Black's (1970) words: [The name induction may be used to] cover all cases of nondemonstrative argument, in which the truth of the premises, while not entailing the truth of the conclusion, purports to be a good reason for belief in it. (p. 57) Induction stands here for any kind of nondemonstrative argument whose conclusion is not intended to follow from the premises by sheer logical necessity. The negation of an inductive conclusion is compatible with the amalgamated premise (the conjunction of all the reasons offered in support of the conclusion). (p. 137) There are two classes or levels of inductive reasoning. One, ~ priori probability in itself, which deals with "the incidence of certain features in the total of all possible mathematical combinations of a given sort" (Ayer, 1972, p. 30), will not be explored as such in this paper. Because of its ultimately psychometric orientation, this paper is essentially conce1Qed with inductive reasoning as it functi(ns within the empirical context. ! priori probability if and when applied to the empirical necessitates certain assumptions and added empirical meanings which go beyond its definition. As Ayer (1972) has put it: Mathematically, the alternatives 1-6, in the case of the dice, are equally probable in the entirely trivial sense that in the series 1-6 each of them occurs just once. But clearly this triviality cannot be applied to any actual game, unless one makes suitable assumptions about actual frequencies, and then introduces a new notion of probability, or at least gives the old notion a new application which will need to be defined. (p. 30) The second class or level is constituted by what we might call statistical/logical judgments of probability. It is important to take note of the use of the word "logical" here. Although we have distinguished this type of inductive judgment from the ~ priori calculus of chances itself, this does not mean that there are no elements of a priorism in what we call statistical/ logical inductions. This point will be discussed below. The basic schema of these judgments may be stated simply as: g (Q,P) .x Pa ~(with a probability of .x) (Carnap, 1974, p. 37) The first premise states that the relative frequency of Q with respect to ! is ·~· The second premise states that a certain particular a has the property P, and the third statement asserts that this particular ~ has the property Q with a probability of .x. Essentially the point is that the conclusion about ~ regarding Q, as all inductive conclusions, must be derived in advance of possession of all relevant knowledge about the conjunctive incidence of Q and !• The inescapability of deriving these conclusions arises from the epistemic confrontation with the empirical manifold which cannot be reduced to the universal laws which operate in deductive judgments. As regards the conclusion Qa, the probability of .x is affirmed of the statement itself~ In other words, what is affirmed is that statement ~. with regard to. the evidence presented in the premises, has a probability of ·~· This point is extremely important: the derivation of statement Qa on the basis of the premises represents-a logical statement of probability, i.e., what is expressed in affirming it is a logical relation between the evidence and the conclusion. As such this affirmation represents an analytic statement or a statement of logical probability (i.e., degree of confirmation), not an empirical statement. In Carnap's (1974) words, "It is analytic because no empirical investigation is demanded. It expresses a logical relation between a sentence that states the evidence and a sentence that states the hypothesis" (p. 35). By contrast, the premise !f~= ·~ represents a statistical law, an empirical statement. The formula taken as a whole, therefore, exhibits both concepts of proba bility, statistical or empirical and logical \ 2 or analytic: the first premise is empirical, but the derivation of the conclusion, which is to say the inductive judgment itself, is analytic or logical. The derivation of the statistical law is, needless to say, a matter of crucial importance, although as such it is independent from the issue of the inductive judgment expressed by the formula. Certain obvious, basic questions regarding the derivation of the law would be: Was it derived on the basis of the frequency of Q,P in an observed sample? Or was it derived on the basis of the frequency of Q,P in the total population? If the former is the case, then, as Carnap (1974) states: Only the value of the frequency in the sample is known. The value of the fre quency in the population is not known. The best we can do is make an estimate of the frequency in the population. This estimate must not be confused with the value of the frequency in the sample. In general such estimates should deviate in a certain direction from the observed relative frequency in a sample. (p. 38) Carnap discusses the question and presents a number of techniques for making these estimates in The Continuum of Inductive Methods (1952). Beyond these basic and essentially answerable questions there are fundamental issues regarding the general cadre of induction which have literally plagued the philosophy of induction and which are actually regarded as unresolvable. It is essential for anyone concerned with induction, at any level, to be fairly well acquainted with these issues. A relatively brief dis'cussion will be presented here. More extensive presentations may be found in, e.g., Aye'r (1972), Black (1970), or Barker (1967). Hume's famous analysis presented in! Treatise ~Human Nature (1740/1955) and in An Enquiry Concerning Human Understanding (1748/1955) is at the root of the inductive impasse. Hume maintained that through empirical observation no epistemic effort can succeed in discovering more than contiguity and an internal habit of association. One of the most concise expositions of Hume's impact is presented by Ayer (1972). As he puts it: There is no such thing as a synthetic necessary connection between events. These are not, of course, the terms in which Hume -puts it, but this is what it comes to. No matter what events A and B are, if A is presented to us in some-spatia-temporal relation to B, there is nothing in this situation from which we could validly infer, without the help of other premisses, that events of the same type as ! and ! are connected in the same way on any other occasion. There is no such thing as seeing that A must be attended by B ••• [and] •••clearly the inference from the premiss 'Events of the type A and B have invariably been found in conjunction,' or to put it more shortly, 'All hitherto observed As bear the relation R to Bs,' to the conclusion 'All As bear the relation R to Bs,' or even to the conclusion 'This A will have the relation R to some B,' is not formally valid. There-is what we may call an inductive jump. (p. 4) One might recall here Bertrand Russell's (1950) celebrated witticism on induction: "The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken" (p. 115). It must be observed that the inductive impasse affects deduction as well as induction. The empirical possibility of a universal law, which constitutes, of necessity, one of the premises in deductive judgments, is negated by this argument. The universal law can only exist as a law of logic or of pure mathematics, or else it must be inductively established (in order to exist as an empirical law). Thus, deductive conclusions are logically axiomatic but would be epistemically unassailable if and only if the truth of the universal evidence were firm and unassailable. Analogously, as we have seen, the derivation of an inductive conclusion is in itself analytic and is only dismantled epistemically if it seeks to subsume the empirical, that is to say, if it intends to refer to particular events and to be empirically predictive. From the logical standpoint, event Qa cannot be said to have 3 probability in itself. What the inductive judgment affirms logically is that the probability value of proposition~ is affected and defined as .x by the amalgamated premise rf (~) = •!./\ Pa. It does not thereby affirm that particular event ~ in itself possesses a .x probability of having the property g_. The difference between deduction and induction is hence that if universal propositions were empirically possible, deductive judgments would constitute reliable epistemic tools; inductive judgments, by contrast, can never constitute a reliable epistemic tool. In other words, they do not reliably tell us anything about particulars. Ironically, however, we do in fact rely upon induction. Indeed, in its very limitations and in spite of its inherent unreliability, induction constitutes our only possibility of making contact with the empirical manifold. Epistemically, the amalgamated inductive premise, perhaps against our superior logical sense, constitutes grounds for tentative belief. As Black (1970) has put it, "Standard induction is preferable to soothsaying because we know that it will work (will approach limiting values in the long run) if anything will" (p. 79). Or, in Barker's (1967) words: •••given the evidence that we have, we must draw whatever nondemonstrative conclusions we can from it. To contend that given evidence cannot be employed in nondemonstrative argument unless there is further positive evidence that the given evidence is not misleading is to embark on a vicious infinite regress, a regress which would destroy the possibility of there being any nondemonstrative arguments at all. (pp. 23-24) In a sense the inductive problem is a pseudoproblem, for it is impossible even to think about perceivable phenomena without being caught into probabilities and uncertainties. The empirical manif,old, far from being susceptible to definite .enclosure, exceeds and overflows all formulae. Miguel de Unamuno's (1913/1954) ingenious words on Hegel seem particularly suitable: "Hegel, a great framer of definitions, attempted with definitions to reconstruct the universe, like that artillery sergeant who said that cannon were .made by taking a hole and enclosing it with steel" (p. 5). Thus, whereas inductive evidences are in themselves summative, they are, as Ayer (1972) declares, " •••marginally predictive••• [and] for the most part it is their predictive element which gives them their point" (p. SO). In recognizing the pseudoproblematic character of the inductive impasse, Black (1970) states: If induction is by definition nondeductive and if the demand for justification is, at bottom, that induction be shown to satisfy conditions of correctness appropriate only to deduction, then the task is certainly hopeless. But to conclude, for this reason, that induction is basically invalid or that a belief based upon inductive grounds can never be reasonable is to transfer, in a manner all too enticing, criteria of evaluation from one domain to another domain, in which they are inappropriate. Sound inductive conclusions do not follow (in the deductive sense of "follow") from even the best and strongest set of premises (in the inductive sense of "strongest"); there is no good reason why they should. (p. 83) In what is perhaps an attitude of compromise one is forced to affirm that the ultimate purpose of induction is to establish the tentative reliability of hypotheses upon which we are forcibly called to act. We might conclude with Russell (1950) that "•••our instincts certainly cause us to believe that the sun will rise tomorrow, but we may be in no better a position than the chicken which unexpectedly has its neck wrung" (p. 115). THE MEASUREMENT OF INDUCTIVE ABILITIES The primary purpose of the testing of reasoning abilities is to connect, to the point of essential convergence, two situations: the testing situation and 4 the "real" situation for which the test is intended to serve as a predictive instrument. In other words, in terms of reasoning process, the testing situation must reflect the "real" situation with precision. It follows that the issue of whether ability is acquired or innate3 does not substantially affect the concept of testing. In fact, whether or not the ability exists as such, predictive access to the specific performance of a person in a "real" situation is relevant. This predictive access is possible only when, as stated above, the measuring instrument adjusts with precision to the demands of the "real" situation. These comments may appear redundant in a discussion of testing, but they certainly are not: there are degrees of connectedness, in terms of reasoning pro cess, between performance in a "real" situation and performance on the test. What we are advocating here is that unless the degree of connectedness or convergence is essential, the test will bifurcate from the "real" situation. This is the very basis of a logical approach to the testing of inductive abilities. The inductive process conforms to the laws of logic. Hence, if inductive reasoning forms part of the definition of a job or academic situation, the measuring instrument for inductive abilities must include and conform to inductive schemata, i.e., in the "real" situation as well as on the test, correct inductions are crucial. On the basis of this criterion, three inductive tests were constructed in the form of judgments of probability. One, Test A, was a 30-item test. Test B and Test C were 20-item tests composed of two different combinations of items from Test A. All questions were developed in linguistic form and were designed according to three principal schemata. These are: 3 I. rf (~) = .x Pa ~(with a probability of .x) (Carnap, 1974, p. 37) II. Most P's each bear R to some~ or other. Some ~'s are ~'s (and no ~'s to which they bear Rare observed). Therefore (probably) there are some g's (not observed) to which theseS's bear R. -(Barker,-1967, p. 97) III. Of all the things that are ~. m are P. n a is an M. Therefore (with a probability of ~) ~ is a P. n (Barker, 1967, p. 70) These schemata express the basic logic of inductive argument. In all respects the questions were designed with attention to this basic logic. Thus, the statement "with respect to this evidence" was included in lead form before five alternative statements of probability, which represent the answer and four incorrect alternatives (so-called "distracters"). If this qualifying statement is not included, statements of probability exceed the limits of logic and confuse prediction with truth-value. Additionally, all distracters were constructed according to a logical plan whereby in every case the test taker is asked to discern, on the basis of the evidence, a correct statement of probability from incorrect statements of probability. In terms of predictive value, this construction criterion is significant: being able to discern a valid conclusion of prob ability from invalid conclusions of prob ability is crucial in any "real" situation for which the test serves as predictor. It is pertinent to reiterate at this point that these tests of inductive abilities deviate from traditional psychometrics and represent a refinement thereof in that they are constructed in the form of pure The term innate in any case may be interpreted in the Platonic sense (e.g., in the Meno) in which the innateness of the trait does not negate the relevance of the learning process. In fact, the learning process is the sine qua ~for the actualization of the trait. 5 judgments of probability. In the history of psychometrics to date, judgments of probability are found only in the WatsonGlaser Inference Test although this test does not exhibit (and does not intend to exhibit) a purely inductive form (Watson and Glaser, 1964). That is to say, the test-taker must discern, in some cases, the probability established by a certain body of evidence and choose an alternative which expresses this probability from among a set of conclusions which offer alternative statements of falsehood, truth, or insufficient evidence. Thus, as Ross (Note 2) correctly indicates, "The items which require T [True] and F [False] responses are apparently deductive items, while the items requiring PT [Probably True] and PF [Probably False] responses are probably inductive" (p. 12). Ross goes on to suggest that " •••it is possible that the inductive-deductive dichotomy is a poor way to conceptualize reasoning abilities. Perhaps the two are inseparable" (p. 25). In logic they are indeed inseparable and have never been dichotomized. In fact the schema for the deductive Modus Ponens (presented earlier as the generalized schema for deduction) and the generalized schema for an inductive judgment of probability are almost the same: (.!.) (Px ) Qx) rf (Q,P) .x Pa Pa Qa ~ (with a probability of •.!.) The two schemata differ only insofar as the first premise may represent universality or relative frequency and therefore the de rived conclusion differs in terms of truthvalue rather than in terms of the reasoning process through which it is derived. From the epistemic standpoint one could indeed conceive of inductive judgments of probability as being the deductive intent as it becomes transformed by the epistemic needs which arise upon confrontation with the resistance of the empirical manifold to deductive subsumption. Psychometrically, therefore, the measuring approa·ches to induction (in the form of judgments of probability) and deduction should be regarded as adjunctive measurements and correlated results should be expected, especially if the tests are constructed in the same medium (linguistic or nonlinguistic). As discussed before, the three inductive tests in the form of judgments of probability described in this paper were constructed in linguistic form. It has also been mentioned that in the psychometric tradition the nonlinguistic medium has been exclusively utilized to test inductive reasoning, which, as we have often reiterated, has been tested thus far only in the form of law-generative judgments. In this context it is relevant to note that although the nonlinguistic medium has traditionally been utilized to test law-generative inductive abilities, this medium does not constitute a corollary of the testing of these abilities. Conceivably these abilities could be tested through the linguistic medium. The schema would be one of the form: pn Therefore (probably) K (Black, 1970, p. 147) As Black (1970) points out, "Here, the qualifier 'probably' may be conceived to be attached, as shown, to the 'Therefore' (the sign of illation) ..... (p. 147). The choice of an inductive test depends directly on which form of argument (lawgenerative inductive reasoning or inductive reasoning in the form of judgments of probability) the "real" situation will primarily necessitate. However, it is likely that in most domains both forms will be called for equally. One could safely venture to say that although we, as reasoning subjects, sometimes carry out one form of argument without ,attention to its complementary form, it is extremely unlikely that this would be the case universally. Thus most testing situations, if at all intended for specific predictive purposes, would attend to both needs. In consonance with this criterion the research discussed in this paper included the experimental administration of 6 a 30-question inductive test in the form of law-generative judgments and in the nonlinguistic medium, in conjunction with the experimental administration of each of the three inductive tests, described previously, in the form of judgments of probability and in the linguistic medium. The 30 law-generative questions were of the serial and figural types described in the inductive section of the Ekstrom et al. manual (1976). The results of these experimental administrations will be reported below. EXPERIMENTAL ADMINISTRATIONS Method The three experimental subtests, the construction of which was described in the preceding section, were administered as part of the Federal civil service test battery known as the Professional and Administrative Career Examination; the individuals who took the tests were actual job applicants. Since jobs filled by this examination require a college degree or three years of professional or administrative experience, it probably can be assumed that most of the people who took the tests met this requirement (although there is no way to verify this assumption). The three experimental subtests were given to large random samples of the nationwide applicant pool in one of the months when the test was administered. The 30-question test was administered in May, 1976, and the two 20-question tests were given in May, 1977. The applicants were not routinely informed that there were experimental questions on the test. The instructions immediately preceding the questions consisted of a definition of inductive judgments of probability and an example constructed according to the schema: rf (Q,P) = .x Pa ~ (with a probability of .~) The time limit was 35 minutes for both the 30-question test and the 20-question tests. Results As mentioned before, the two 20-question tests consisted of two different combinations of the questions from the 30-question test. The 20-question tests, therefore, had 10 items in common with each other. This reassembly into two shorter tests was necessitated by the finding that many applicants had insufficient time to finish the 30-question test (perhaps because of the length of the individual items). In fact, 40% of the applicants failed to mark an answer for the last question on that test. Table 1 presents the means, standard deviations, and Kuder-Richardson formula 20 reliability coefficients (KR-20) for the three experimental tests. In addition, the Spearman-Brown formula was used to estimate from the reliabilities of the 20-question tests what the KR-20 reliabilities would be for corresponding 30-question tests. These estimates are also presented in Table 1. Table 1 Descriptive statistics for tests of inductive judgments of probability N Mean Standard KR-20 KR-20 Deviation Obtained Estimated for 30-item test A 30-item 1,473 14.24 5.18 .80 B 20-item 998 10.21 3.93 .74 .81 c 20-item 986 10.15 3.53 .68 .76 7 All three tests have desirable statistical characteristics. The average difficulty level of the questions (which in terms of the proportion of applicants answering a question correctly would be .47 for the 30-question test and .51 for the 20-question tests) is at the level which is generally sought for maximum differentiation among applicants (Adkins, 1947). The reliabilities are within the range of reliabilities which are found for the operational subtests (Wing, 1977), although they are all slightly lower both 20-item tests (they were given in the same position in both tests). The 30 items had a very desirable range of ~-values and point-biserial correlations. All but two of the items had ~-values in the range of .20 to .80 from which most operational test questions are chosen. All but one of the items had point-biserial correlations above .20 (the minimum value considered acceptable in our test construction program) and 60% had values of .40 or higher. than the average reliability of .814 which is found for the inductive test part which consists of nonlinguistic questions of the law-generative form. (It should be noted, however, that the operational subtests were assembled by selecting the best from pre tested questions, while the data for the experimental questions represent their initial trial.) The available item analysis data included ~-values (proportion of applicants answering the question correctly) and itemtest point-biserial correlations. Data from the 30-question test were not studied, since they were affected by high omit rates in items toward the end of the test. The ~-values and point-biserial correlations were averaged for the 10 items that were given in Table 2 presents the intercorrelations of the operational deductive subtest, the operational inductive subtest (nonlinguistic questions of the law-generative form) and the two 20-question subtests of inductive judgments of probability. All of these intercorrelations are substantial, and there is consistency between the results for the Test B and C groups. (This latter finding is to be expected, since the operational subtests were identical for the two groups and the two experimental tests had one-half of their questions in common.) Both Tests Band C have a significantly higher correlation with the deductive subtest than with the inductive subtest (for Test B, ~ = 5.77, ~ <•0001; for Test C, z = 4.28, ~ <.0001). Table 2 Subtest intercorrelations for operational deductive and nonlinguistic law-generative inductive subtests with two 20-question tests of inductive judgments of probability. Deduction Induction Test B .62 .64 Induction .60 .52 Test C .58 .49 Deduction Note: Above diagonal, n = 998 Below diagonal, n = 986 8 The significance of this study in terms DISCUSSION of the predictive value sought in ability The data presented in the preceding measurement must be reiterated and empha sized. Inductive judgments of probabilitysection demonstrate that the objective of are present, indeed prevalent, in every real this study, namely, the development of an inductive test which is in the linguistic situation for which an inductive measuring medium and in the form of judgments of instrument is intended to select. Thus, the probability, was fulfilled successfully. measurement of inductive abilities should be approached from a comprehensive standpoint The means and reliabilities of the deand include judgments of probability as wellveloped tests, as well as individual item as law-generative judgments. statistics, are all in the range which might be expected of operational tests Similarly, both judgments of probabilityassembled from selected, pre-tested items. and law-generative judgments are carried outTaken together, these findings indicate in the linguistic medium, with the admixturethat the developed tests constitute of numerical data which are necessitated byhomogeneous tests that discriminate well both inductive forms. Therefore, it is desiramong applicants on the basis of their able, if not preferable, to approach the ability to follow through the inductive measurement of these abilities through this process. medium. Figural and serial items do not reflect real situations since they exclude both In terms of subtest intercorrelations, linguistic and numerical data. the experimental tests corroborated their Induction, in the form logical groundwork. In view of these considerations, the of judgments of probability, is inseparable authors plan to conduct further research from deduction, as this paper has indiin the area of law-generative judgments, In fact, as stated before, the cated. which will result in the development andgeneralized schemata are almost identical. experimental administration of a measuring In a testing situation, therefore, it is instrument consisting of these judgments desirable to obtain correlated results presented in the linguistic medium. between inductive and deductive subtests (especially if the inductive subtest con Additionally, the experimental sists of judgments of probability drawn administration of inductive judgments of from linguistic data). Accordingly, as probability in linguistic form to blind shown in Table 2, the correlations between persons is planned for the immediate future the experimental tests of linguistic inducin order to ascertain whether or not such a tive judgments of probability and the test is appropriate to attempt to measure deductive subtest were higher than the inductive abilities in blind persons.correlations between the experimental Figural inductive items presented intests and the inductive subtest consisting embossed form have been found to be of nonlinguistic law-generative judgments. inappropriate for universal testing because they rely to an extent on tactual abilities. On the other hand, the substantial Serial items therefore appear to be the correlation obtained between the deductive only existing approach to the measurement subtest and the nonlinguistic law-generative of inductive abilities in blind persons- inductive subtest should be interpreted in a situation which is obviously deficient. light of the fact that the deductive subtest includes 15 questions of numerical deduction. Furthermore, as is the case with sighted persons, it is desirable--and this study This finding raises the possibility that the would argue, necessary--to approach the testing medium (linguistic versus nonlin measurement of inductive abilities in guistic) contributed to the obtained correblind persons from a comprehensive stand lation between the two tests. point and include judgments of probability alongside of law-generative judgments. 9 Reference Notes 1. Munoz-Colberg, M. and Nester, M. A. 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