This book presents the first study of the development of the theory of modal syllogistic in the Middle Ages.

This article enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: Allxareyand Somexarey, There are at least as manyxasy, and There are morexthany. Herexandyrange over subsets (not elements) of a giveninfiniteset. Moreover,xandymay appear complemented (i.e., as$\bar{x}$and$\bar{y}$), with the natural meaning. We formulate a logic for our language that is based on the classical syllogistic. The main result is a soundness/completeness theorem. (...) There are efficient algorithms for proof search and model construction. (shrink)

Extensions of traditional syllogistics have been increasingly researched in philosophy, linguistics, and areas such as artificial intelligence and computer science in recent decades. This is mainly due to the fact that syllogistics is seen as a logic that comes very close to natural language abilities. Various forms of extended syllogistics have become established. This paper deals with the question to what extent a syllogistic representation in CL diagrams can be seen as a form of extended syllogistics. It will be shown (...) that the ontology of CL enables numerically exact assertions and inferences. (shrink)

This book uncovers and examines the confusion in antiquity between Aristotle's hypothetical syllogistic and Stoic logic, and offers a fresh perspective on the ...

The syllogism has often been criticized. Yet the theory of the syllogism cannot be omitted from logic. Even if it were not for its historical significance, its nature as a chapter of class logic assigns to it a place in any presentation of logic.The usual exposition of the theory of the syllogism, however, whether given by the use of the familiar rules of the syllogism, or by the help of diagrams, appears clumsy and lacks the lucidity of modern chapters of (...) logic. The reason seems to be given in the inefficient notation, taken over from ancient and medieval logic. In the following I should like to present an improved notation, which combines some of the traditional features with modern ones, and which is based on the criticism of the syllogistic theory which I have given elsewhere. It will be seen that in the revised form the theory of the syllogism is apt to meet the standards of modern logic. (shrink)

ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules which (...) establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out. (shrink)

This paper adds comparative adjectives to two systems of syllogistic logic. The comparatives are interpreted by transitive and irreflexive relations on the underlying domain. The main point is to obtain sound and complete axiomatizations of the valid formulas in the logics.

This paper reviews current psychological theories of syllogistic inference and establishes that despite their various merits they all contain deficiencies as theories of performance. It presents the results of two experiments, one using syllogisms and the other using three-term series problems, designed to elucidate how the arrangement of terms within the premises affects performance. These data are used in the construction of a theory based on the hypothesis that reasoners construct mental models of the premises, formulate informative conclusions about the (...) relations in the model, and search for alternative models that are counterexamples to these conclusions. This theory, which has been implemented in several computer programs, predicts that two principal factors should affect performance: the figure of the premises, and the number of models that they call for. These predictions were confirmed by a third experiment. Dans cet article sont passées en revue les théories psychologiques du traitement des syllogismes. On établit qu'en dépit de leurs mérites variés toutes sont en défaut en tant que théories de la performance. On présente les résultats de deux expériences, une utilisant des syllogismes et l'autre des problémes avec des séries de trois termes conçues pour élucider comment l'arrangement des termes dans les premises affecte la performance. Ces données sont utilisées pour construire une théorie fondée sur l'hypothése que les gens construisent des modèles mentaux des premises, formulent des conclusions explicites sur les relations dans le modèle et cherchent des modèles qui seraient des contre-exemples pour leurs conclusions. Cette théorie, utilisée dans plusieurs programmes d'ordinateur, prédit que deux principaux facteurs affectent la performance: la figure des premises et le nombre de; modèles qu'ils mettent en jeu. Cette prédiction est confirmée dans les trois expériences. (shrink)

This article describes a specific pedagogical context for an advanced logic course and presents a strategy that might facilitate students’ transition from the object-theoretical to the metatheoretical perspective on logic. The pedagogical context consists of philosophy students who in general have had little training in logic, except for a thorough introduction to syllogistics. The teaching strategy tries to exploit this knowledge of syllogistics, by emphasizing the analogies between ideas from metalogic and ideas from syllogistics, such as existential import, the distinction (...) between contradictories and contraries, and the square of opposition. This strategy helps to improve students’ understanding of metalogic, because it allows the students to integrate these new ideas with their previously acquired knowledge of syllogistics. (shrink)

Traditionally, syllogisms are arguments with two premises and one conclusion which are constructed by propositions of the form “All… are…” and “At least one… is…” and their respective negated versions. Unfortunately, the practical use of traditional syllogisms is quite restricted. On the one hand, the “All…” propositions are too strict, since a single counterexample suffices for falsification. On the other hand, the “At least one …” propositions are too weak, since a single example suffices for verification. The present contribution studies (...) algebraic interpretations of syllogisms with comparative quantifiers (e.g., “Most… are…”) and quantitative quantifiers (e.g., “n/m… are…”, “all, except n… are…”). This modern version of syllogistics is intended to be a more adequate framework for argumentation theory than traditional syllogistics. (shrink)

This paper presents a restructured set of axioms for categorical logic. In virtue of it, the syllogistic with indefinite terms is deduced and proved, within the categorical logic boundaries. As a result, the number of all the conclusive syllogisms is deduced through a simple and axiomatic methodology. Moreover, the distinction between immediate and mediate inferences disappears, which reinstitutes the unity of Aristotelian logic.

In elementary logic textbooks, Venn diagrams are used to analyze and evaluate the validity of syllogistic arguments. Although the method of Venn diagrams is shown to be a powerful analytical tool in these textbooks, it still has limitations. On the one hand, such method fails to represent singular statements of the form, “a is F.” On other hand, it also fails to represent identity statements of the form, “a is b.” Because of this, it also fails to give an account (...) of the validity of some obviously valid arguments that contain these types of statements as constituents. In this paper, owing to the developments in the literature on Venn diagrams, we offer a way of supplementing the rules of the Venn diagram found in textbooks, and show how this retooled Venn diagram technique could account for the problem cases. (shrink)

A novel theoretical formulation of Categorical Logic based on two properties of categorical propositions and three simple axioms has been introduced recently. This formulation allowed for the suppression of the distinction between immediate and mediate inferences, and also provided a theoretical framework to study opposition relations, thus restoring the theoretical unity of traditional Aristotelian logic. By using this approach, it has been reported that a total of 3072 conclusive syllogistic moods can be found when including indefinite terms in classical syllogistic, (...) but this result has yet to be proven. This paper presents an overview of the recently proposed theoretical formulation of Categorical Logic, along with the derivation of the 48 canonical syllogistic moods that are capable of generating the 3072 conclusive moods previously reported. (shrink)

Comparative syllogism is a type of scientific reasoning widely used, explicitly or implicitly, for inferences from observations to conclusions about effectiveness, but its philosophical significance has not been fully elaborated or appreciated. In its simplest form, the comparative syllogism derives a conclusion about the effectiveness of a factor (e.g. a treatment or an exposure) on a certain property via an experiment design using a test (experimental) group and a comparison (control) group. Our objective is to show that the comparative syllogism (...) can be understood as encoding a simulation view of counterfactuals, in that counterfactual situations are conceptual constructs that can be correctly simulated by homogeneous comparison groups. In this simulation view, the empirical data from the comparison groups play an evidential role in the evaluation of counterfactuals and in obtaining counterfactual knowledge. We further indicate how successful experimental designs can help us to obtain correct simulations, and thus to bring us to scientifically-empirically based counterfactual knowledge. (shrink)

Aristotle’s categorical syllogistic is the first formal deductive system in the history of formal sciences. Most parts or elements of the system are validated by modern (first-order) mathematical logic, but the system is quite limited in scope, as it is incapable of analyzing inferences other than the ‘figure syllogisms’ consisting of a couple of_ a-e-i-o_ premises and an _a-e-i-o_ conclusion, containing three ‘moderately’ universal terms – terms that express neither a highest genus nor a lowest species – each of which (...) is common to a different couple of propositions out of the three. Logicians in the following ages dealt with various questions concerning the addition of various (novel) logical forms into this limited system, among which are singular predications, i.e. categorical propositions with a singular term as subject, and the most common choice has been interpreting singular predications as _universal _predications. This paper tries to take a strictly formal perspective on the system of categorical syllogisms, and argues in detail for a much simpler and fairly effective assimilation or translation scheme for singular predications. The key to the proposed scheme is the preservation of the supposed inferential relation between opposite singular predications, namely _contradiction_. It is argued that the resulting _singularized _syllogistic moods, which are also validated by first-order logic (under its standard renderings of _a-e-i-o_ and singular predications), promote in turn the mentioned type of treatment, since they call for the employment of two of the four subaltern moods of the system, disregarded by Aristotle, which are _formally _there. (shrink)

A system of intermediate quantifiers (“Most S are P”, “m/n S are P”) is proposed for evaluating the rationality of human syllogistic reasoning. Some relations between intermediate quantifiers and probabilistic interpretations are discussed. The paper concludes by the generalization of the atmosphere, matching and conversion hypothesis to syllogisms with intermediate quantifiers. Since our experiments are currently still running, most of the paper is theoretical and intended to stimulate psychological studies.

The relation between ex falso and disjunctive syllogism, or even the justification of ex falso based on disjunctive syllogism, is an old topic in the history of logic. This old topic reappears in contemporary logic since the introduction of minimal logic by Johansson. The disjunctive syllogism seems to be part of our general non-problematic inferential practices and superficially it does not seem to be related to or to depend on our acceptance of the frequently disputable ex falso rule. We know (...) that the acceptance of the ex falso is a sufficient condition for the acceptance of the disjunctive syllogism, but the interesting question is: is the ex falso a necessary condition for the acceptance of the disjunctive syllogism? The aim of the present paper is to discuss some possible ways to define systems that combines the preservation of the disjunctive syllogism with the rejection of the ex falso. (shrink)

In 1883, while a student of C. S. Peirce at Johns Hopkins University, Christine Ladd-Franklin published a paper titled On the Algebra of Logic, in which she develops an elegant and powerful test for the validity of syllogisms that constitutes the most significant advance in syllogistic logic in two thousand years. Sadly, her work has been all but forgotten by logicians and historians of logic. Ladd-Franklin's achievement has been overlooked, partly because it has been overshadowed by the work of other (...) logicians of the nineteenth century renaissance in logic, but probably also because she was a woman. Though neglected, the significance of her contribution to the field of symbolic logic has not been diminished by subsequent achievements of others.In this paper, I bring to light the important work of Ladd-Franklin so that she is justly credited with having solved a problem over two millennia old. First, I give a brief survey of the history of syllogistic logic. In the second section, I discuss the logical systems called “algebras of logic”. I then outline Ladd-Franklin's algebra of logic, discussing how it differs from others, and explain her test for the validity of the syllogism, both in her symbolic language and the more familiar language of modern logic. Finally I present a rigorous proof of her theorem. Ladd-Franklin developed her algebra of logic before the methods necessary for a rigorous proof were available to her. Thus, I do now what she could not have done then. (shrink)

In the first book of the Prior Analytics, Aristotle sets out, for the first time in Greek philosophy, a logical system. It consists of a deductive system (I.4-22), meta-logical results (I.23-26), and a method for finding and giving deductions (I.27-29) that can apply in “any art or science whatsoever” (I.30). After this, Aristotle compares this method with Plato’s method of division, a procedure designed to find essences of natural kinds through systematic classification. This critical comparison in APr I.31 raises an (...) interpretive puzzle: how can Aristotle reasonably juxtapose two methods that differ so much in their aims and approach? What can be gained by doing so? Previous interpreters have failed to show how this comparison is legitimate or what important point Aristotle is making. The goal of this paper is to resolve the puzzle. In resolving this puzzle we not only learn more about the relationship be- tween division and the syllogistic in Aristotle. We will also learn something about the motivation for the syllogistic itself, by seeing the role that it plays in his philosophy of science. (shrink)

Prevailing interpretations of Aristotle’s use of syllogistic language outside the Organon hold that he offers a single, comprehensive theory of the practical syllogism spanning his ethical and biological works. These comprehensive theories of the practical syllogism are plausible neither philosophically nor as interpretations of Aristotle. I argue for a multivocal account of the practical syllogism that distinguishes (1) Aristotle’s use of syllogistic language to explain aspects of his account of animal motion in MA from (2) his use of syllogistic language (...) to explain aspects of his account of the distinctive practical cognition of the phronimos in EN. I offer a novel account of the role of syllogistic language in ethics, arguing that it elucidates a nuanced account of universals and particulars in ethics according to which acting virtuously requires an understanding of underlying universal values and a capacity to relate them to concrete, particular features of our circumstances. (shrink)

An investigation of Proclus' logic of the syllogistic and of negations in the Elements of Theology, On the Parmenides, and Platonic Theology. It is shown that Proclus employs interpretations over a linear semantic structure with operators for scalar negations (hypemegationlalpha-intensivum and privative negation). A natural deduction system for scalar negations and the classical syllogistic (as reconstructed by Corcoran and Smiley) is shown to be sound and complete for the non-Boolean linear structures. It is explained how Proclus' syllogistic presupposes converting the (...) tree of genera and species from Plato's diairesis into the Neoplatonic linear hierarchy of Being by use of scalar hyper and privative negations. (shrink)

Does cognition sometimes literally extend into the extra-organismic environment (Clark, 2003), or is it always “merely” environmentally embedded (Rupert, 2004)? Underlying this current border dispute is the question about how to individuate cognitive processes on principled grounds. Based on recent evidence about the active role of representation selection and construction in learning how to reason (Stenning, 2002), I raise the question: what makes two distinct, modality-specific pen-and-paper manipulations of external representations – diagrams versus sentences – cognitive processes of the same (...) kind, e.g. episodes of syllogistic reasoning? In response, I defend a “division of labor” hypothesis, according to which external representations are dependent on perceptually grounded neural representations and mechanisms to guide our behavior; these internal mechanisms, however, are dependent on external representations to have their syllogistic content fixed. Only their joint contributions qualify the extended computational process as an episode of syllogistic reasoning in good standing. (shrink)

We extend the language of the classical syllogisms with the sentence-forms “At most 1 p is a q” and “More than 1 p is a q”. We show that the resulting logic does not admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed.

The question as to what makes a perfect Aristotelian syllogism a perfect one has long been discussed by Aristotelian scholars. G. Patzig was the first to point the way to a correct answer: it is the evidence of the logical necessity that is the special feature of perfect syllogisms. Patzig moreover claimed that the evidence of a perfect syllogism can be seen for Barbara in the transitivity of the a-relation. However, this explanation would give Barbara a different status over the (...) other three first figure syllogisms. I argue that, taking into account the role of the being-contained-as-in-a-whole formulation, transitivity can be seen to be present in all four first figure syllogisms. Using this wording will put the negation sign with the predicate, similar to the notation in modern predicate calculus. (shrink)

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: $$\begin{array}{ll}\mathbf{Some}\, a \,{\rm are} \,R-{\rm related}\, {\rm to}\, \mathbf{some} \,b;\\ \mathbf{Some}\, a \,{\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{all}\, b;\\ \mathbf{All}\, a\, {\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{some}\, b;\\ \mathbf{All}\, a\, {\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{all} \,b.\end{array}$$ Such primitives formalize sentences from natural language like ‘ All students read some textbooks’. Here a, b denote arbitrary sets (of objects), and R denotes an (...) arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem. (shrink)

After 100+ years of studying syllogistic reasoning, what have we learned? Well, Riesterer and colleagues suggest that we have learned to throw away most of the data! If that seems like a bad idea to you then, be assured, that the authors agree with you. The sad fact is that the conclusion of “No Valid Conclusion” (NVC) is one of the most frequently selected responses in syllogistic reasoning but these “majority data” have been ignored by most researchers. Riesterer and colleagues (...) introduce heuristic rules for predicting NVC and provide directions to modelers on how to up their game. (shrink)

This paper explores the question of what makes diagrammatic representations effective for human logical reasoning, focusing on how Euler diagrams support syllogistic reasoning. It is widely held that diagrammatic representations aid intuitive understanding of logical reasoning. In the psychological literature, however, it is still controversial whether and how Euler diagrams can aid untrained people to successfully conduct logical reasoning such as set-theoretic and syllogistic reasoning. To challenge the negative view, we build on the findings of modern diagrammatic logic and introduce (...) an Euler-style diagrammatic representation system that is designed to avoid problems inherent to a traditional version of Euler diagrams. It is hypothesized that Euler diagrams are effective not only in interpreting sentential premises but also in reasoning about semantic structures implicit in given sentences. To test the hypothesis, we compared Euler diagrams with other types of diagrams having different syntactic or semantic properties. Experiment compared the difference in performance between syllogistic reasoning with Euler diagrams and Venn diagrams. Additional analysis examined the case of a linear variant of Euler diagrams, in which set-relationships are represented by one-dimensional lines. The experimental results provide evidence supporting our hypothesis. It is argued that the efficacy of diagrams in supporting syllogistic reasoning crucially depends on the way they represent the relational information contained in categorical sentences. (shrink)

European Journal of Philosophy, Volume 29, Issue 4, Page 864-886, December 2021.

Does syllogistic logic have the resources to capture mathematical proof? This volume provides the first unified account of the history of attempts to answer this question, the reasoning behind the different positions taken, and their far-reaching implications. Aristotle had claimed that scientific knowledge, which includes mathematics, is provided by syllogisms of a special sort: 'scientific' ('demonstrative') syllogisms. In ancient Greece and in the Middle Ages, the claim that Euclid's theorems could be recast syllogistically was accepted without further scrutiny. Nevertheless, as (...) early as Galen, the importance of relational reasoning for mathematics had already been recognized. Further critical voices emerged in the Renaissance and the question of whether mathematical proofs could be recast syllogistically attracted more sustained attention over the following three centuries. Supported by more detailed analyses of Euclidean theorems, this led to attempts to extend logical theory to include relational reasoning, and to arguments purporting to reduce relational reasoning to a syllogistic form. Philosophical proposals to the effect that mathematical reasoning is heterogenous with respect to logical proofs were famously defended by Kant, and the implications of the debate about the adequacy of syllogistic logic for mathematics are at the very core of Kant's account of synthetic a priori judgments. While it is now widely accepted that syllogistic logic is not sufficient to account for the logic of mathematical proof, the history and the analysis of this debate, running from Aristotle to de Morgan and beyond, is a fascinating and crucial insight into the relationship between philosophy and mathematics. (shrink)

Moti Mizrahi (2013) presents some novel counterexamples to Hypothetical Syllogism (HS) for indicative conditionals. I show that they are not compelling as they neglect the complicated ways in which conditionals and modals interact. I then briefly outline why HS should nevertheless be rejected.

Syllogistics reduces to only two rules of inference: monotonicity and symmetry, plus a third if one wants to take existential import into account. We give an implementation that uses only the monotonicity and symmetry rules, with an addendum for the treatment of existential import. Soundness follows from the monotonicity properties and symmetry properties of the Aristotelean quantifiers, while completeness for syllogistic theory is proved by direct inspection of the valid syllogisms. Next, the valid syllogisms are decomposed in terms of the (...) rules they involve. The implementation uses Haskell [8], and is given in ‘literate programming’ style [9]. (shrink)

Nicholas Rescher, by examining and reproducing sources in Arabic philosophy, seeks to definitively settle the debate over whether Galen originated the fourth ...

ABSTRACTIn two experiments, we examined the resolution of confidence judgments in syllogistic reasoning and their heuristic bases. Based on the assumptions of Koriat's Self-Consistency Model of confidence, we expected the confidence judgments to be related to conclusion consensuality, reflecting the role of consistency as a heuristic cue to confidence. In Experiment 1, the participants evaluated 24 syllogisms with conclusions that varied with respect to validity and consensuality. In Experiment 2, the participants produced conclusions to 64 pairs of premises. The correlation (...) between confidence and reasoning accuracy was low. In both experiments confidence was related to the consensuality of the responses. For consensually correct items, correlation between confidence and accuracy was positive; however, for consensually incorrect items it was negative. In Experiment 2, confidence was lower for syllogisms with higher response cardinality, or syllogisms that elicited a greater variety of conclusions. (shrink)

In this article, I present a schema for generating counterexamples to the argument form known as Hypothetical Syllogism with indicative conditionals. If my schema for generating counterexamples to HS works as I think it does, then HS is invalid for indicative conditionals.

In the paper, a new syllogistic system is built up. This system simulates a massive-parallel behavior in the propagation of collectives of parasites. In particular, this system simulates the behavior of collectives of trematode larvae.

The follow-up research of Aristotle’s syllogism has different approaches. The traditional syllogism follows Aristotle’s conceptual system and hopes to make improvements within Aristotle’s theory. Mathematical logic proposes a new conceptual system to accurately interpret Aristotle’s syllogism. Lei Ma puts forward an extended syllogism whose conceptual system is different from Aristotelian logic and mathematical logic. He thinks that Aristotle’s syllogism and traditional syllogism have tedious figures, moods, and reasoning rules, which are difficult for us to memorize. It is a theoretical conclusion (...) of the human reasoning process but does not conform to the actual human thinking process. Ma’s syllogism is called substitution logic, which mainly concerns the substitution characteristics of a human thinking process, and summarizes the substitution rules in the reasoning process. Substitution logic appropriately describes the actual human reasoning process, thus inspiring us to establish a unified scientific theory of thinking and carry out normative research on the unity of abstract thinking and imaginative thinking. Substitution logic may be applied to the field of artificial intelligence, making artificial intelligence closer to the reality of human thinking. It seems that the research direction of substitution logic will give birth to human-like AI systems and promote the revolutionary transformation of AI research. (shrink)

In [1] J. Perzanowski formulated, among others, an ontology expressed in the relational language. He presented some interesting connections which hold between these relations. In the present paper we focus on further analysis of these relations.

Syllogism is a form of logical argument allowing one to deduce a consistent conclusion based on a pair of premises having a common term. Although Aristotle was the first to conceive and develop this way of reasoning, he left open a lot of conceptual space for further modifications, improvements and systematizations with regards to his original syllogistic theory. From its creation until modern times, syllogism has remained a powerful and compelling device of deduction and argument, used by a variety of (...) figures and assuming a variety of forms throughout history. The Aftermath of Syllogism investigates the key developments in the history of this peculiar pattern of inference, from Avicenna to Hegel. Taking as its focus the longue durée of development between the Middle Ages and the nineteenth century, this book looks at the huge reworking scientific syllogism underwent over the centuries, as some of the finest philosophical minds brought it to an unprecedented height of logical sharpness and sophistication. Bringing together a group of major international experts in the Aristotelian tradition, The Aftermath of Syllogism provides a detailed, up to date and critical evaluation of the history of syllogistic deduction. (shrink)

In this paper we investigate the locus of believability effects in syllogistic reasoning. We identify three points in the reasoning process at which such effects could occur: the initial interpretation of premises, the examination of alternative representations of them (in all of which any valid conclusion must be true), and the “filtering” of putative conclusions. The effect of beliefs at the first of these loci is well established. In this paper we report three experiments that examine whether beliefs have an (...) effect at the other two loci. In experiments 1 and 2 subjects drew their own conclusions from syllogisms that suggested believable or unbelievable ones. In the third experiment they evaluated conclusions that were presented to them. The data show that beliefs both affect the examination of alternative models and act as a filter on putative conclusions. We conclude by showing how some types of problem and some problem contents make the existence of alternative models more obvious than others. (shrink)