Categorical Syllogisms Research Articles

The Analytical Perspective of Aristotle’s Categorical and Modal Syllogisms

What is meant under the genuine title of Aristotle’s ta Analytika is rarely properly understood. Presumably, his analytics was inspired by the method of geometric analysis. For Aristotle, this was a regressive or heuristic procedure, departing from a proposed conclusion (or prob­lem) and asking which premises could be found in order to syllogize, demonstrate or explain it. The terms that form categorical and modal propositions play a fundamental role in analytics. Aristotle introduces letters in lieu of the triples of terms (major – middle – minor) constitut­ing the propositions and the three syllogistic figures that schematize them. His formulation of the three syllogistic figures refers to a syntacti­cal and predicative order and position of the triples of terms, arranged in some diagrammed schemata, which, regrettably, are missing from the extant text of the Prior Analytics. Considering planar and graphic arrangements, both vertical and horizontal orders as well as the posi­tion of the three terms involved, we propose a reconstruction, at least to some extent, of these probable lettered diagrams. In such reconstructed diagrams, we can appropriately capture the definition of syllogism as a predicative connexion of terms, and easier survey a synoptic account of all valid predicative relations and transpositions, and also reduce the imperfect syllogisms into the moods of the first figure. Aristotle’s syllogistic is an analytical calculation of terms, understood as predicates and subjects within the categorical propositions, and more precisely of three terms schematized in three figures in predicative links such that, by means of a middle, follows from necessity a conclusion of the extreme terms. The necessity of the consequence is not based on the implication or inference of the propositions, but on a predictive transi­tivity through the middle term within the syllogistic figures. Syllogism must draw its conclusion through the way its terms are predicated of one another. Aristotle in his Prior Analytics (I 3, 8–22) developed also a complex account of modal syllogisms within necessity and possibility of belonging (predicating). This account involves also such an analyti­cal reduction to the syllogistic figures. In this analytical perspective, we try to throw some light on his modal syllogisms, although this difficult and nowadays thoroughly discussed topic would require a much wider treatment.

TWO SYLLOGISMS IN THE MOZI: CHINESE LOGIC AND LANGUAGE

AbstractThis article examines two syllogistic arguments contrasted in an ancient Chinese book, the Mozi, which expounds doctrines of the Mohist school of philosophers. While the arguments seem to have the same form, one of them (the one-horse argument) is valid but the other (the two-horse argument) is not. To explain this difference, the article uses English plural constructions to formulate the arguments. Then it shows that the one-horse argument is valid because it has a valid argument form, the plural cousin of a standard form of valid categorical syllogisms (Plural Barbara), and argues that the two-horse argument involves equivocal uses of a key predicate (the Chinese counterpart of ‘have four feet’) that has the distributive/nondistributive ambiguity. In doing so, the article discusses linguistic differences between Chinese and English and explains why the logic of plural constructions is applicable to Chinese arguments that involve no plural constructions.

Proofs as Cognitive or Computational: Ibn Sı̄nā’s Innovations

We record the advances made by the eleventh century Persian logician Ibn Sina—known in the West as Avicenna—away from a purely cognitive view of proofs and towards a more computational view, and the kinds of consideration that led him to these advances. Some of Ibn Sina’s new logics, which stand somewhere between Aristotle’s categorical syllogisms and modern first-order logic, can serve as a kind of laboratory for testing what are the differences between Aristotelian and modern logic, and where these differences come from.

Spatial representation of coherence.

Four experiments examined spatial correlates of the experience of coherence, that is, the extent to which propositions "fit together." Experiment 1 demonstrates for Heiderian triads (i.e., sets of liking/disliking relations between 3 fictitious persons) that name pairs from balanced triads, such as 2 friends commonly disliking a third person (high coherence) are seen as closer to each other in physical space as compared to name pairs from unbalanced triads, such as 2 persons disliking each other and having a common friend (low coherence). This pattern of results is conceptually replicated in 2 further experiments for categorical syllogisms. Two terms in conclusions from valid syllogisms (high coherence) were seen as spatially closer to each other than when 2 terms came from invalid syllogisms (low coherence). In the final 2 experiments, similar closeness effects are demonstrated for word pairs from scenarios that "made sense" in terms of causal connectedness (latent causality) as opposed to word pairs from scenarios perceived as causally unconnected. These findings are discussed in the context of spatial binding theories, applied psychology, and embodied cognition in general, and their methodological implications are highlighted. (PsycINFO Database Record

Preadolescents Solve Natural Syllogisms Proficiently.

"Natural syllogisms" are arguments formally identifiable with categorical syllogisms that have an implicit universal affirmative premise retrieved from semantic memory rather than explicitly stated. Previous studies with adult participants (Politzer, 2011) have shown that the rate of success is remarkably high. Because their resolution requires only the use of a simple strategy (known as ecthesis in classic logic) and an operational use of the concept of inclusion (the recognition that an element that belongs to a subset must belong to the set but not vice versa), it was hypothesized that these syllogisms would be within the grasp of non-adult participants, provided they have acquired the notion of deductive validity. Here, 11-year-old children were presented with natural syllogisms embedded in short dialogs. The first experiment showed that their performance was equivalent to adults' highest level of performance in standard experiments on syllogisms. The second experiment, while confirming children's proficiency in solving natural syllogisms, showed that they outperformed children who solved non-natural matched syllogisms in the same experimental setting. The results are also in agreement with the argumentation theory of reasoning.

Transitivity in coherence-based probability logic

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Moreover, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases. Finally, we apply our results to study selected probabilistic versions of classical categorical syllogisms and construct a new version of the square of opposition in terms of defaults and negated defaults.

Conflict, metacognition, and analytic thinking

One hundred and three participants solved conflict and non-conflict versions of four reasoning tasks using a two-response procedure: a base rate task, a causal reasoning task, a denominator neglect task, and a categorical syllogisms task. Participants were asked to give their first, intuitive answer, to make a Feeling of Rightness (FOR) judgment, and then were given as much time as needed to rethink their answer. They also completed a standardized measure of IQ and the actively open-minded thinking questionnaire. The FORs of both high- and low-capacity reasoners were responsive to conflict, such that FORs were lower for conflict relative to non-conflict problems. Consistent with the quantity hypothesis, high-capacity reasoners made a greater distinction between conflict and non-conflict items on measures of Type 2 thinking, namely, rethinking time and probability of changing answers. In contrast to the quality hypothesis, however, this rethinking time did not advantage the ability of the high-capacity group to produce normative answers, except for the base rate task. Indeed, we observed that the correlation between capacity and the probability of normative answers emerged at the initial response, rather than after rethinking.

Towards explaining the cognitive efficacy of Euler diagrams in syllogistic reasoning: A relational perspective

Although diagrams have been widely used as methods for introducing students to elementary logical reasoning, it is still open to debate in cognitive psychology whether logic diagrams can aid untrained people to successfully conduct deductive reasoning. In our previous work, some empirical evidence was provided for the effectiveness of Euler diagrams in the process of solving categorical syllogisms. In this paper, we discuss the question of why Euler diagrams have such inferential efficacy in the light of a logical and proof-theoretical analysis of categorical syllogisms and diagrammatic reasoning. As a step towards an explanatory theory of reasoning with Euler diagrams, we argue that the effectiveness of Euler diagrams in supporting syllogistic reasoning derives from the fact that they are effective ways of representing and reasoning about relational structures that are implicit in categorical sentences. A special attention is paid to how Euler diagrams can facilitate the task of checking the invalidity of an inference, a task that is known to be particularly difficult for untrained reasoners. The distinctive features of our conception of diagrammatic reasoning are made clear by comparing it with the model-theoretic conception of ordinary reasoning developed in the mental model theory.

An investigation of belief-bias and logicality in reasoning with emotional contents

The effects of emotion and belief on logicality have been examined independently in the literature. The aim of this paper is to explore the possible effect of emotion on both logicality and belief-bias. We also examine the nature of the belief-bias associated with emotion in terms of Type 1 versus Type 2 processes. In two studies we employed a categorical syllogisms task to compare reasoning about emotional and neutral contents. Study 1 was conducted with women from a university population and Study 2 included victims of sexual abuse in addition to control participants. The categorical syllogisms were manipulated in terms of logical validity (valid, invalid), believability (believable, unbelievable), and content type (generally emotional, sexual abuse, neutral). The results from control participants and victims were in line with the literature, indicating that reasoning about emotional contents is associated with decreased logicality compared to neutral contents. We also found some evidence that emotion leads to an increase in belief-bias, at least for sexual abuse contents. Overall the findings are consistent with increased Type 1 and decreased Type 2 processing with emotional contents, most strongly for contents related to sexual abuse.

FIGURES OF PROSLEPTIC SYLLOGISMS IN PRIOR ANALYTICS 2.7

In chapters 2.5–7 of the Prior Analytics Aristotle is concerned with what he calls circular proof. He gives an account of circular proofs within the framework of his syllogistic theory, and discusses how they come about in the three figures of categorical syllogisms. The results of this discussion are summarized at the end of chapter 2.7, at 59a32–41. The summary contains several statements to the effect that certain circular proofs come about in the third figure. Some of these statements are problematic because the circular proofs in question are actually not in the third figure of categorical syllogisms; in fact, these circular proofs are not categorical syllogisms at all, but what Theophrastus called prosleptic syllogisms. Hence, the statements are incorrect if they are understood to refer to the third figure of Aristotle’s categorical syllogisms. Since it seems natural to understand them in this way, Ross and others conclude that the passage at 59a32–41 is spurious and should be excised, although it is found in all MSS. 1 By contrast, this paper aims to show that the passage is not spurious. Following Pacius, I argue that the problematic statements in it refer not to the third figure of categorical syllogisms, but to the third figure of prosleptic syllogisms. On this interpretation, the statements are correct and can be regarded as genuine. Given that they are genuine, they show that Aristotle was aware of a classification of prosleptic syllogisms into three figures, even though such a classification does not occur elsewhere in his writings. Thus, the passage at 59a32–41 appears to be the earliest evidence we have of figures of prosleptic syllogisms. I begin with an overview of Aristotle’s treatment of circular proofs in Prior Analytics 2.5–7, focussing on his use of prosleptic syllogisms (§1). Readers familiar with the contents of Prior Analytics 2.5–7 may wish to skip this overview. Next we consider the problematic statements in 59a32–41 (§2). I will argue that these statements refer to the third figure of prosleptic syllogisms, and that there is no reason to doubt Aristotle’s authorship of the passage (§3).

Aristotle’s Theory of Deduction and Paraconsistency DOI:10.5007/1808-1711.2010v14n1p71

In the Organon Aristotle describes some deductive schemata in which inconsistencies do not entail the trivialization of the logical theory involved. This thesis is corroborated by three different theoretical topics by him discussed, which are presented in this paper. We analyse inference schema used by Aristotle in the Protrepticus and the method of indirect demonstration for categorical syllogisms. Both methods exemplify as Aristotle employs classical reductio ad absurdum strategies. Following, we discuss valid syllogisms from opposite premises (contrary and contradictory) studied by the Stagerian in the Analytica Priora (B15). According to him, the following syllogisms are valid from opposite premises, in which small Latin letters stand for terms such as subject and predicate, and capital Latin letters stand for the categorical propositions such as in the traditional notation: (i) in the second figure, Eba,Aba ` Eaa (Cesare), Aba, Eba ` Eaa (Camestres), Eba, I ba ` Oaa (Festino), and Aba,Oba ` Oaa (Baroco); (ii) in the third one, Eab,Aab ` Oaa (Felapton), Oab,Aab ` Oaa (Bocardo) and Eab, Iab ` Oaa (Ferison). Finally, we discuss the passage from the Analytica Posteriora (A11) in which Aristotle states that the Principle of Non-Contradiction is not generally presupposed in all demonstrations (scientific syllogisms), but only in those in which the conclusion must be proved from the Principle; the Stagerian states that if a syllogism of the first figure has the major term consistent, the other terms of the demonstration can be each one separately inconsistent. These results allow us to propose an interpretation of his deductive theory as a broad sense paraconsistent theory. Firstly, we proceed to a hermeneutical analysis, evaluating its logical significance and the interplay of the results with some other points of Aristotle’s philosophy. Secondly, we point to a logical interpretation of the Aristotelian syllogisms from opposite premises in the antilogisms method proposed by Christine Ladd-Franklin in 1883, and we also present a logical treatment of the Aristotelian demonstration with inconsistent material in the paraconsistent logics Cn, 1 _ n _ !, introduced by da Costa in 1963. These two issues seem having not yet been analysed in detail in the literature.

Revolutions of geometry

Preface. Acknowledgments. PART I FOUNDATIONS. 1 The First Geometers. 1.1 Egypt. 1.2 Babylon. 1.3 China. 2 Thales. 2.1 The Axiomatic System. 2.2 Deductive Logic. 2.3 Proof Writing. 3 Plato and Aristotle. 3.1 Form. 3.2 Categorical Propositions. 3.3 Categorical Syllogisms. 3.4 Figures. PART II THE GOLDEN AGE. 4 Pythagoras. 4.1 Number Theory. 4.2 The Pythagorean Theorem. 4.3 Archytas. 4.4 The Golden Ratio. 5 Euclid. 5.1 The Elements. 5.2 Constructions. 5.3 Triangles. 5.4 Parallel Lines. 5.5 Circles. 5.6 The Pythagorean Theorem Revisited. 6 Archimedes. 6.1 The Archimedean Library. 6.2 The Method of Exhaustion. 6.3 The Method. 6.4 Preliminaries to the Proof. 6.5 The Volume of a Sphere. PART III ENLIGHTENMENT. 7 Francois Viete. 7.1 The Analytic Art. 7.2 Three Problems. 7.3 Conic Sections. 7.4 The Analytic Art in Two Variables. 8 Rene Descartes. 8.1 Compasses. 8.2 Method. 8.3 Analytic Geometry. 9 Gerard Desargues. 9.1 Projections. 9.2 Points at Infinity. 9.3 Theorems of Desargues and Menelaus. 9.4 Involutions. PART IV A STRANGE NEW WORLD. 10 Giovanni Saccheri. 10.1 The Question of Parallels. 10.2 The Three Hypotheses. 10.3 Conclusions for Two Hypotheses. 10.4 Properties of Parallel Lines. 10.5 Parallelism Redefined. 11 Johann Lambert. 11.1 The Three Hypotheses Revisited. 11.2 Polygons. 11.3 Omega Triangles. 11.4 Pure Reason. 12 Nicolai Lobachevski and Janos Bolyai. 12.1 Parallel Fundamentals. 12.2 Horocycles. 12.3 The Surface of a Sphere. 12.4 Horospheres. 12.5 Evaluating the Pi Function. PART V NEW DIRECTIONS. 13 Bernhard Riemann. 13.1 Metric Spaces. 13.2 Topological Spaces. 13.3 Stereographic Projection. 13.4 Consistency of Non-Euclidean Geometry. 14 Jean-Victor Poncelet. 14.1 The Projective Plane. 14.2 Duality. 14.3 Perspectivity. 14.4 Homogeneous Coordinates. 15 Felix Klein. 15.1 Group Theory. 15.2 Transformation Groups. 15.3 The Principal Group. 15.4 Isometries of the Plane. 15.5 Consistency of Euclidean Geometry. References. Index.

Complexity of categorical syllogisms: An integration of two metrics

The complexity of categorical syllogisms was assessed using the relational complexity metric, which is based on the number of entities that are related in a single cognitive representation. This was compared with number of mental models in an experiment in which adult participants solved all 64 syllogisms. Both metrics accounted for similarly large proportions of the variance, showing that complexity depends on the number of categories that are related in a representation of the combined premises, whether represented in multiple mental models, or by a single model. This obviates the difficulty with mental models theory due to equivocal evidence for construction of more than one mental model. The “no valid conclusion” response was used for complex syllogisms that had valid conclusions. The results are interpreted as showing that the relational complexity metric can be applied to syllogistic reasoning, and can be integrated with mental models theory, which together account for a wide range of cognitive performances.

Phonological and visual distinctiveness effects in syllogistic reasoning: Implications for mental models theory

Two experiments are reported in which the representational distinctiveness of terms within categorical syllogisms was manipulated in order to examine the assumption of mental models theory that abstract, spatially based representations underpin deduction. In Experiment 1, participants evaluated conclusion validity for syllogisms containing either phonologically distinctive terms (e.g., harks, paps, and fids) or phonologically nondistinctive terms (e.g., fuds, fods, and feds). Logical performance was enhanced with the distinctive contents, suggesting that the phonological properties of syllogism terms can play an important role in deduction. In Experiment 2, participants received either the phonological materials from Experiment 1 or syllogisms involving distinctive or nondistinctive visual contents. Logical inference was again enhanced for the distinctive contents, whether phonological or visual in nature. Our findings suggest a broad involvement of multimodal information in syllogistic reasoning and question the assumed primacy of abstract, spatially organized representations in deduction, as is claimed by mental models theorists.

Solving categorical syllogisms with singular premises

We elaborate on the approach to syllogistic reasoning based on “case identification” (Stenning & Oberlander, 1995; Stenning & Yule, 1997). It is shown that this can be viewed as the formalisation of a method of proof that dates back to Aristotle, namely proof by exposition (ecthesis), and that there are traces of this method in the strategies described by a number of psychologists, from Störring (1908) to the present day. We hypothesised that by rendering individual cases explicit in the premises, the chance that reasoners would engage in a proof by exposition would be enhanced, and thus performance improved. To do so, we used syllogisms with singular premises (e.g., this X is Y). This resulted in a uniform increase in performance as compared to performance on the associated standard syllogisms. These results cannot be explained by the main theories of syllogistic reasoning in their current state.

Theories of categorical reasoning and extended syllogisms

The aim of this study was to examine the predictions of three theories of human logical reasoning, (a) mental model theory, (b) formal rules theory (e.g., PSYCOP), and (c) the probability heuristics model, regarding the inferences people make for extended categorical syllogisms. Most research with extended syllogisms has been restricted to the quantifier “All” and to an asymmetrical presentation. This study used three-premise syllogisms with the additional quantifiers that are used for traditional categorical syllogisms as well as additional syllogistic figures. The predictions of the theories were examined using overall accuracy as well as a multinomial tree modelling technique. The results demonstrated that all three theories were able to predict response selections at high levels. However, the modelling analyses showed that the probability heuristics model did the best in both Experiments 1 and 2.

Expanding the universe of categorical syllogisms: A challenge for reasoning researchers

Syllogistic reasoning, in which people identify conclusions from quantified premise pairs, remains a benchmark task whose patterns of data must be accounted for by general theories of deductive reasoning. However, psychologists have confined themselves to administering only the 64 premise pairs historically identified by Aristotle. By utilizing all combinations of negations, the present article identifies an expanded set of 576 premise pairs and gives the valid conclusions that they support. Many of these have interesting properties, and the identification of predictions and their verification will be an important next step for all proponents of such theories.

Early and late processes in syllogistic reasoning: Evidence from eye-movements

An eye-movement monitoring experiment was carried out to examine the effects of the difficulty of the problem (simple versus complex problems) and the type of figure (figure 1 or figure 4) on the time course of processing categorical syllogisms. The results showed that the course of influence for these two factors is different. We found early processing effects for the figure but not for the difficulty of the syllogism and later processing effects for both the figure and the difficulty. These results lend support to the Model Theory (Johnson-Laird, P. N., Byrne, R. M. J. (1991). Deduction. Hillsdale, New Jersey: LEA.) as opposed to other theories of reasoning (Chater, N., Oaksford, M. (1999). The probability heuristics model of syllogistic reasoning. Cognitive Psychology, 38, 191–258; Rips, L. J. (1994). The psychology of proof. Cambridge, MA: MIT Press; Rips, L. J. (1994). The psychology of proof. Cambridge, MA: MIT Press).

THE FACILITATING EFFECT OF THE HIERARCHICAL ORGANISATION OF CONCEPTS TO THE SYLLOGISTIC REASONING

1. Introduction The traditional tasks of logical reasoning can be both strikingly easy and extremely difficult for adults as well as children. From the four possible conditional reasoning problems, modus ponens (if p then q, p, therefore q) seldom invokes errors starting from six-year old children (Staudenmayer, Bourne 1977, Wildman, Fletcher 1977). However, even adults are reluctant to draw a perfectly valid conclusion from modus tollens (if p then q, not q, therefore not p), or are prone to make invalid conclusions with the conditional problems of the form denial of the antecedent (if p then q, not p, therefore not q) and affirmation of the consequent (q, therefore p). The amount of correct answers to the latter three tasks gradually increases with age, although adults' performance is far from perfect (see Overton 1990 for a review). Another group of tasks of traditional logic, categorical-syllogisms, appear to be more difficult. Similarly to the conditional problems, some of the possible 64 categorical syllogisms are solved correctly by preschoolers (e.g. All A are B, All B are C, therefore all C are A) (Dias, Harris 1988, Hawkins et al. 1984), however many other syllogisms are frequently solved incorrectly. The developmental data provided for the easier subset of 28 syllogisms regarding 9-year-old children and adults indicates that children solve correctly approximately one-third and adults one-half of the syllogisms (Bara et al. 1995). Thus, both in case of conditional syllogisms and categorical syllogisms, some development seems to occur. Several ideas about the origins of the development of logical reasoning have been suggested. According to the theory of mental models, children's deficiencies in syllogistic reasoning stem from their yet limited working memory abilities and skills in searching for counterexamples (in finding several alternative mental models for a particular syllogism) (Bara et al. 1995). Thus, according to this view, the growth of logical reasoning has nothing inherently logical about it (these are not the inferential rules that matter), instead the development is rooted in the increase of working memory abilities and knowledge base to interpret particular premises (Johnson-Laird et al. 1986). The theory of mental logic developed by Braine (1978, 1990) takes a different route. It divides logical skills into primary and secondary skills. Primary skills consist of inferential rules that are either innate or develop very early and are available to all of us. An example of one primary inferential rule is the above mentioned modus ponens; it is automatic and universally obvious to all people. Secondary skills, although based on the primary skills, are considered to be the result of education and have more academic nature. The correct answers to modus tollens would require some secondary skills; here it is necessary to construct a longer chain of reasoning where the primary rule of modus ponens and the reductio ad absurdum rule are applied. Therefore, fallacies with modus tollens are frequent and they gradually diminish with the cognitive development of a child. These longer reasoning chains impose greater demands on memory and often the use of written language is necessary. Another characteristic feature of secondary skills is the requirement to accept the strategy of minimal commitment--focus is not on what the speaker means, however imprecisely and elliptically it might have been expressed (as is the case with everyday reasoning), but on what the sentence is committed to (Braine 1978). Additionally, in non-academic everyday reasoning situations people use all the information available, whereas in case of more advanced logical tasks they should compartmentalise information, i.e. use only the information contained in the premises. Hence, Braine's theory, in contrast to the theory of mental models, proposes that specific skills that are introduced in school may be the constituent factors for the development logical reasoning. …

Equivalence of Syllogisms

We consider two categorical syllogisms, valid or invalid, to be equivalent if they can be transformed into each other by certain transformations, going back to Aristotle, that preserve validity. It is shown that two syllogisms are equivalent if and only if they have the same models. Counts are obtained for the number of syllogisms in each equivalence class. For a more natural development, using group-theoretic methods, the space of syllogisms is enlarged to include nonstandard syllogisms, and various groups of transformations on that space are studied.