A representation theorem for languages with generalized quantifiers through back-and-forth methods

We obtain in this paper a representation of the formulae of extensions ofL by generalized quantifiers through functors between categories of first-order structures and partial isomorphisms. The main tool in the proofs is the back-and-forth technique. As a corollary we obtain the Caicedo's version of Fraïssés theorem characterizing elementary equivalence for such languages. We also discuss informally some geometrical interpretations of our results.     

Propositional quantifiers

Summary In discussing propositional quantifiers we have considered two kinds of variables: variables occupying the argument places of connectives, and variables occupying the argument places of predicates.We began with languages which contained the first kind of variable, i.e., variables taking sentences as substituends. Our first point was that there appear to be no sentences in English that serve as adequate readings of formulas containing propositional quantifiers. Then we showed how a certain natural and illuminating extension of English by prosentences did provide perspicuous readings. The point of introducing prosentences was to provide a way of making clear the grammar of propositional variables: propositional variables have a prosentential character — not a pronominal character. Given this information we were able to show, on the assumption that the grammar of propositional variables in philosopher's English should be determined by their grammar in formal languages (unless a separate account of their grammar is provided), that propositional variables can be used in a grammatically and philosophically acceptable way in philosophers' English. According to our criteria of well-formedness Carnap's semantic definition of truth does not lack an essential predicate - despite arguments to the contrary. It also followed from our account of the prosentential character of bound propositional variables that in explaining propositional quantification, sentences should not be construed as names.One matter we have not discussed is whether such quantification should be called propositional, sentential, or something else. As our variables do not range over (they are not terms) either propositions, or sentences, each name is inappropriate, given the usual picture of quantification. But we think the relevant question in this context is, are we obtaining generality with respect to propositions, sentences, or something else?Because people have argued that all bound variables must have a pronominal character, we presented and discussed in the third section languages in which the variables take propositional terms as substituends. In our case we included names of propositions, that-clauses, and names of sentences in the set of propositional terms. We made a few comparisons with the languages discussed in the second section. We showed among other things how a truth predicate could be used to obtain generality. In contrast, the languages of the second section, using propositional variables, obtain generality without the use of a truth predicate.Special thanks are due to Nuel D. Belnap, Jr., who has given me much valuable assistance with the preparation of this paper. I also thank Alan Ross Anderson, Joseph Camp, Jr., Steven Davis, and Wilfrid Sellars for suggestions and corrections.The preparation of this paper was partly supported by a NSF grant.     

Computation of Aristotle's and Gergonne's syllogisms

A connection between Aristotle's syllogistic and the calculus of relations is investigated. Aristotle's and Gergonne's syllogistics are considered as some algebraic structures. It is proved that Gergonne's syllogistic is isomorphic to closed elements algebra of a proper approximation relation algebra. This isomorphism permits to evaluate Gergonne's syllogisms and also Aristotle's syllogisms, laws of conversion and relations in the square of oppositions by means of regular computations with Boolean matrices.     

Aristotelian necessities

In his Parts of Animals, Aristotle distinguishes three modes of the necessary.However, it is not clear just what these three modes are.Nor is it clear how this passage fits with other texts where Aristotle distinguishes modalities in different ways.Here I present and explain Aristotle’s three modes of necessity, and claim that they are the only three recognized by Aristotle.I then explain how this passage agrees with other passages where Aristotle mentions formal and structural features of the modalities.I end by showing how having three modes of necessity does not make ‘necessary’ ambiguous.Rather, I claim, Aristotle has a single, central notion of necessity and hence a unified theory of modality     

Aristotelian contraries

The first version of this paper was delivered as part of Problems and Changes in the Concept of Predication, a conference at the UC Humanities Research Institute organized by Karel Lambert and Alan Code. I am indebted to the other participants for helpful discussion, and to HRI for providing a wonderful environment for the conference.     

Reasoning with quantifiers

In the semantics of natural language, quantification may have received more attention than any other subject, and one of the main topics in psychological studies on deductive reasoning is syllogistic inference, which is just a restricted form of reasoning with quantifiers. But thus far the semantical and psychological enterprises have remained disconnected. This paper aims to show how our understanding of syllogistic reasoning may benefit from semantical research on quantification. I present a very simple logic that pivots on the monotonicity properties of quantified statements--properties that are known to be crucial not only to quantification but to a much wider range of semantical phenomena. This logic is shown to account for the experimental evidence available in the literature as well as for the data from a new experiment with cardinal quantifiers ("at least n" and "at most n"), which cannot be explained by any other theory of syllogistic reasoning.     

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.     

Ideological belief bias with political syllogisms

Abstract

The belief bias in reasoning occurs when individuals are more willing to accept conclusions that are consistent with their beliefs than conclusions that are inconsistent. The present study examined a belief bias in syllogisms containing political content. In two experiments, participants judged whether conclusions were valid, completed political ideology measures, and completed a cognitive reflection test. The conclusions varied in validity and in their political ideology (conservative or liberal). Participants were sensitive to syllogisms’ validity and conservatism. Overall, they showed a liberal bias, accepting more liberal than conservative conclusions. Furthermore, conservative participants accepted more conservative conclusions than liberal conclusions, whereas liberal participants showed the opposite pattern. Cognitive reflection did not magnify this effect as predicted by a motivated system 2 reasoning account of motivated ideological reasoning. These results suggest that people with different ideologies may accept different conclusions from the same evidence.     

Solving categorical syllogisms with singular premises

We elaborate on the approach to syllogistic reasoning based on “case identification” (Stenning & Oberlander, 1995 Stenning, K. and Oberlander, J. 1995. A cognitive theory of graphical and linguistic reasoning: Logic and implementation. Cognitive Science, 19: 97–140. [Crossref], [Web of Science ®] , [Google Scholar]; Stenning & Yule, 1997 Stenning, K. and Yule, P. 1997. Image and language in human reasoning: A syllogistic illustration. Cognitive Psychology, 34: 109–159. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). 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 Störring, G. 1908. Experimentelle Untersuchungen über einfache Schlussprozesse [Experimental research on simple inferential processes]. Archiv für die Gesamte Psychologie, 11: 1–127.  [Google Scholar]) 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.     

Complexly fractionated syllogistic quantifiers

Consider syllogisms in which fraction (percentage) quantifiers are permitted in addition to universal and particular quantifiers, and then include further quantifiers which are modifications of such fractions (such as “almost 1/2 the S are P” and “Much more than 1/2 the S are P”). Could a syllogistic system containing such additional categorical forms be coherent? Thompson's attempt (1986) to give rules for determining validity of such syllogisms has failed; cf. Carnes &; Peterson (forthcoming) for proofs of the unsoundness and incompleteness of Thompson's rules. Building on Peterson (1985), the coherence of such a syllogistic can, however, be demonstrated with an algebra which provides its semantics; e.g., “almost 1/2 the S are P” is represented as “?(3(SP)?SP)”.