Syllogistic reasoning with intermediate quantifiers Niki Pfeifer ([email protected]) Department of Psychology; Hellbrunnerstrasse 34 5020 Salzburg, Austria Gernot D. Kleiter ([email protected]) Department of Psychology; Hellbrunnerstrasse 34 5020 Salzburg, Austria Abstract The present paper tries to exploit a not so well known non-standard development in syllogistic infer- A system of intermediate quantifiers (\Most S are P", m ence, namely intermediate quantifiers, for psychologi- \ n S are P") is proposed for evaluating the rational- cal purposes. The classical universal and the existential ity of human syllogistic reasoning. Some relations be- quantifiers in syllogisms are either too strict or too weak, tween intermediate quantifiers and probabilistic inter- pretations are discussed. The paper concludes by the respectively. On the one hand, the universal quantifier is generalization of the atmosphere, matching and conver- too strict because it does not allow for exceptions. One sion hypothesis to syllogisms with intermediate quanti- simple counterexample falsifies an all-assertion. In ev- fiers. Since our experiments are currently still running, eryday contexts exceptions are the rule, we reason with most of the paper is theoretical and intended to stimu- late psychological studies. defaults or rules of thumb that hold normally or most of the time. Nonmonotonic reasoning formalizes reasoning with exceptions and withdrawing conclusions in the light Introduction of new evidence. Experiments in which nonmonotonic reasoning is investigated are reported in Pfeifer & Kleiter In 1908, St¨orring published a psychological study on sim- (2003, 2005, in press). On the other hand, the existen- ple inference processes. The last part of his paper was tial quantifier is too weak because it quantifies only over devoted to syllogistic reasoning (St¨orring, 1908; Politzer, at least one individual. Such quantifiers hardly ever oc- 2004). He presented syllogism tasks like the following cur in everyday life reasoning. Quantifiers that|at least one to his subjects: implicitly|actually occur in everyday life reasoning like All p belong to class a,1 \most . ", \almost-all . ", or \90 percent . " are All a belong to class d. not expressible in classical syllogistics. Such quantifiers Therefore . that lie \in-between" the existential and the universal quantifier are called intermediate quantifiers 2. The subjects had to complete the conclusion. In 1935, the well known study of Woodworth & Sells (1935) was published. Most theoretical and empirical research work on syllogistic reasoning, though, was published during Table 1: The universal (A) and particular (I) affirmative, the last decades (Johnson-Laird & Byrne, 1994; Bacon and the universal (E) and particular (O) non-affirmative et al., 2003; Newstead, 2003; Morley et al., 2004). Syllo- moods of classical syllogisms and their predicate-logical gisms build a prototypical task to investigate human rea- (PL) form. soning in the field of monadic predicate calculus. Com- Mood Read PL-Formula parable prototypical tasks are Wasons's selection task in the field of propositional calculus and, more recently, also A All S are P 8x(Sx ! P x) in modal logic. Similarly, classical modus-ponens-type I At least one S is P 9x(Sx ^ P x) tasks investigate parts of propositional calculus. Syllo- E All S are not P 8x(Sx ! :P x) gisms are well-defined in standard logic and were studied O At least one S is not P 9x(Sx ^ :P x) in philosophy for more than two millennia. No wonder that they are considered as one of the benchmarks of human rationality. 2Also known as \generalized quantifiers”. Historically, the During the last 50 years many new approaches to old first system was proposed by Sir William Hamilton, his dis- logical problems were developed in philosophy, artificial pute with De Morgan is reprinted in (De Morgan, 1847). intelligence, and linguistics. These non-standard ap- Generalized quantifiers have been developed in mathematics proaches are often of special interest to psychology as and logics (Mostowski, 1957; Lindstr¨om, 1966; V¨a¨an¨anen, they try to do more justice to practical reasoning than 2004), artificial intelligence and computer science (Schwartz, 1997; Liu & Kerre, 1998; Nov´ak, 2001), linguistics (Barwise & classical logic. Typical examples are nonmonotonic rea- Cooper, 1981; van Bentham & ter Meulen, 1985; G¨ardenfors, soning or the probabilistic treatment of conditionals. 1987; van der Does & van Eijck, 1991; Keenan & Westerst˚ahl, 1997). Peterson's work is an improvement of Finch (1957) 1\Alle p geh¨oren zur Gattung a" (St¨orring, 1908, p.78) and is situated in the philosophy of language tradition. Table 2: The four figures of syllogisms. S, M, and P are Table 3: Classical syllogisms that are predicate-logically the subject, middle, and predicate term, respectively. (PL) and not predicate-logically valid. PL-valid Not PL-valid Fig. I Fig. II Fig. III Fig. IV Figure I AAA Barbara AAI Barbari Major Prem. MP P M MP P M AII Darii EAO Celaront Minor Prem. SM SM MS MS EAE Celarent Concl. SP SP SP SP EIO Ferio Figure II AEE Camestres AEO Camestrop AOO Baroco EAO Cesaro In the present paper we propose intermediate quanti- EAE Cesare fiers as developed by Peterson (1985, 2000) as a promis- EIO Festino ing candidate for investigating and evaluating human Figure III AII Datisi AAI Darapti syllogistic reasoning. After a short introduction to classi- cal syllogisms, we sketch the formal system of intermedi- EIO Ferison EAO Felapton ate quantifiers and formulate for this system some (clas- IAI Disamis sical) hypothesis of human syllogistic reasoning. The OAO Bocardo discussion of some relations between intermediate quan- Figure IV AEE Camenes AAI Bramantip tifiers and probabilistic interpretations concludes the pa- EIO Fresison AEO Camenop per. Since we are presently running experiments, we IAI Dimaris EAO Fesapo cannot provide empirical data yet. Classical Syllogisms Syllogisms with intermediate quantifiers The classical syllogism is a two-premise-one-conclusion argument made by three of four sentence types, or moods Intermediate quantifiers are quantifiers \between" the all (Table 1). The order of the predicates involved is reg- quantifier and the existential quantifier. Examples of in- termediate quantifiers are Almost-all S are P, Most S are imented by the four figures (Table 2). This leads to m 3 P Many S are P fractions S are 256 possible syllogisms, of which 24 are syllogistically , or quantifiers with , n valid. From a predicate logical point of view, only 15 P. As stated in the introduction, the universal quanti- syllogisms are predicate-logically valid (Table 3). All 15 fier is too strict and the existential quantifier is too weak a pri- predicate logically valid syllogisms are also syllogistically and not appropriate to model human reasoning on valid. The reason is that in syllogistics All S are P im- ori grounds. We therefore suggest to prefer intermediate plies At least one S is P (Some S are P ), because it is im- quantifiers for modeling human syllogistic reasoning. plicitly assumed that the subject term Sx is not empty. Intermediate quantifiers have hardly been investigated This assumption is called \existential import". In pred- by psychologists. Exceptions are the logical rule-based icate logic, 8x(Sx ! P x) does not entail 9x(Sx ^ P x). approach by Guerts (2003) and Chater & Oaksford's The reason is well known: In predicate logic, formulae (1999) Probability Heuristics Model. We will not dis- like 8x(Sx ! P x) can be \vacuously true". This is the cuss these approaches here. Studies on probability judg- case when there is no x such that x has the property S. ment as well can be close to studies on quantifiers with Then, clearly 9x(Sx ^ P x) is false (since :9xSx is as- fractions. sumed). However, if the existential assumption is made Peterson (2000) provides algorithms to evaluate syl- explicit, 9x(Sx ^ P x) is a predicate-logically valid con- logisms with intermediate quantifiers. These algorithms are correct and complete with respect to arbitrarily many clusion, 1 2 intermediate quantifier syllogisms ( 5 S are P, 5 S are P, m 8x(Sx ! P x) ^ 9xSx ` 9x(Sx ^ P x) : n S are P, . ). For his interpretation of intermediate quantifiers consider the Venn diagram in Figure 1 and The valid syllogisms got names like \Barbara" for Table 4. Figures 2, 3, and Figure 4 list the valid syllo- mnemotechnic reasons.4 The vowels in these names indi- gisms with the intermediate quantifiers Almost-all S are cate the moods of the first and second premise, and the P (P), Most S are P (T), and Many S are P (K) and mood of the conclusion of the respective syllogism (in their non-affirmative versions Almost-all S are :P (B), the order just stated) (Hughes & Londey, 1965). A list Most S are :P (D), and Many S are :P (G), respec- of valid syllogisms with their traditional names is given tively. Syllogism ATK of Figure I, e.g., is, All M are P in Table 3. (major premise), Most S are M (minor premise), there- Many S are P 3 3 fore (conclusion). The validity of figures 4 = 64 ways of constituting a two-premise argument (2 I, II, and IV can be directly inspected, since the interme- for the premises, 1 for the conclusion) by four moods (A, I, diate quantifiers strengthen the premises or weaken the E, O). Multiply 64 by the four figures gives 64 × 4 = 256 possible syllogisms. conclusion (solid or dashed arrows, respectively). Valid 4The first mnemotechnic verses of valid syllogisms ap- syllogisms of Figure III in the shaded boxes of Figure 3 peared in William of Sherwood's Introductiones Logicam are not derived trivially by strengthening or weakening. (Summulae), 13th Century (Kneale & Kneale, 1984, p. 231f.) Consider, e.g., syllogism TTI of Figure III: Most M are .................................