The Theory of Negation

Negation is a fundamental element of natural language (^{Horn, 1989}). It appears in all human languages, although it does not have the same grammar structure in all of them (^{Khemlani et al., 2012}). Its importance is evidenced by the attention received from various disciplines, especially in the study of propositional reasoning (^{Macbeth et al., 2013}; ^{Macbeth et al., 2015}; ^{Orenes et al., 2016}).

According to ^{Khemlani et al. (2012}), while the goal of a Linguistic Theory of Negation is to explain its syntactic, semantic and pragmatic elements, the purpose of a Psychological Theory is to explain how such grammar, meaning, and knowledge are understood by people and how people represent them mentally. And here lies the central goal of the proposal: to decipher the meaning, representation, and use granted by people to negation (^{Khemlani et al., 2012}, ²⁰¹⁴; ^{Macbeth et al., 2013}, 2017a). Out of all these elements, meaning is the one which acquires fundamental value, due to its effect on negation (^{Khemlani et al., 2012}, ²⁰¹⁴). Thus, given any proposition, the MMT postulates that individuals use the meaning to represent it mentally. This meaning is a product of their knowledge of the world, and of the contextual factors that intervene in the moment (^{Johnson-Laird, 1983}, 2006; ^{Johnson-Laird et al., 2015}, 2018). The representation is produced using mental models (^{Johnson-Laird, 1983}; 2006; ^{Johnson-Laird et al., 2015}, 2018), the scale of which will be linked to the scope of the subject’s interpretation of the proposition (^{Khemlani et al., 2012}).

In this way, the Theory of Negation works with the assumptions of the MMT and defines it as a function that takes the set of mental models as a single argument, and produces the complement for that set as a result (^{Khemlani et al., 2012}, 2014; ^{Macbeth et al., 2013}; 2014b, 2017a). In this sense, the meaning of negation coincides with that proposed by logic (^{Khemlani et al., 2012}). For both, the relationship between an affirmation and its negation is of mutual contradiction, and the possibilities between both are complete and exhaustive (^{Khemlani et al., 2012}, 2014). Since inferences are based on mental models, the MMT also considers that people will only succeed in their reasoning if they start from properly represented premises (^{Khemlani et al., 2012}). In this way, the number of mental models takes on a crucial role. According to the model, as the number of possibilities that subjects must represent increases, so does the overload in their working memory (^{Khemlani, 2018}; ^{Khemlani et al., 2012}, ²⁰¹⁴, 2017, ²⁰¹⁸). Consequently, the task becomes more difficult, while both time to resolution and tendency to error increase (^{Khemlani, 2018}; ^{Khemlani et al., 2012}, ²⁰¹⁴, 2017, 2018).

Starting from these assumptions, five main predictions derive from the Theory of Negation (^{Khemlani et al., 2012}); however, we will only focus on three of them in this research. One of these predictions (prediction 1 of the Theory of Negation) is related to the scope of the subject’s interpretation of a proposition. According to the theory, given the negation of a compound proposition, that is, a proposition made up of two atomic sentences joined by a connector (and, or, if... then...), the individual tends to represent the negation by considering only one of the atomic propositions, and not the proposition composed in its entirety (^{Khemlani et al., 2012}; ^{Macbeth et al., 2013}). This form of representation is called small scope and corresponds to a processing bias that people use whenever they have the opportunity (^{Khemlani et al., 2012}). This interpretation also implies the representation of a smaller number of mental models (hence the name of the bias), and consequently a lower overload in working memory (^{Khemlani et al., 2012}; ^{Macbeth et al., 2013}).

A second prediction (prediction 3 of the Theory of Negation) establishes that the interpretation of a compound proposition is also modulated by the meaning, knowledge, and reference people assign to the connectives involved. This is fundamentally determined by the use people give them in everyday language (^{Khemlani et al., 2012}). According to this prediction, this modulation can interfere, or even block, the correct interpretation of a proposition, especially in the case of individuals who do not have training in logic (^{Khemlani et al., 2012}).

Finally, a third prediction (prediction 5 of the Theory of Negation) establishes that propositions that require the representation of a single mental model are easier to interpret than those that require more mental models. The interesting thing about this prediction is that it also postulates that difficulty is reversed in the case of a negation (^{Khemlani et al., 2012}). This characteristic arises as an effect of the complementarity condition described above. Thus, the complement of a single mental model requires the representation of several mental models, while the complement of a set of mental models requires only one or two (^{Khemlani et al., 2012}).

This last prediction is clearly expressed in De Morgan’s laws. As these laws formally establish, the negation of a conjunction is equivalent to an inclusive disjunction (Law 1), while the negation of an inclusive disjunction is equivalent to a conjunction (Law 2) (^{De Morgan, 1847}; ^{Suppes & Hill, 1992}). Since the representation of a conjunction requires a single mental model, and the representation of an inclusive disjunction involves three mental models, the theory predicts greater difficulty in processing Law 1 than Law 2 (^{Macbeth et al., 2014}b; 2017a).

In this way, the purpose of this study is to analyze the propositional reasoning of both laws in relation to the natural language of people. In particular, it aims to test the aforementioned predictions. It should be borne in mind, however, that such predictions will only make sense in a theory that considers the representation of mental models as a general mechanism of reasoning. This hardly happens in the case of theories based on logicist assumptions (^{Khemlani et al., 2012}).

Personality

The theories of personality have changed greatly and have accompanied the development of psychology since its inception. As from the 1970s, a proposal seems to gain momentum with each passing day: the Theory of Personality founded on the Five Factor Model - FFM (^{McCrae & Costa, 1987}; 1999; 2004; ^{Costa, McCrae, & Löckenhoff, 2019}; ^{McCrae & Mõttus, 2019}). This model proposes a pentafactorial structure of personality, according to which variations in behavior and overt acts are explained around five big personality traits: the Big Five (^{Sánchez & Ledesma, 2007}). These factors are called neuroticism, extraversion, openness, agreeableness, and conscientiousness (^{McCrae & Costa, 2004}; ^{McCrae & Mõttus, 2019}).

An important aspect of the proposal is that these dimensions are not the result of the free inventiveness of researchers but of numerous empirical studies (^{McCrae & Mõttus, 2019}; ^{Sánchez & Ledesma, 2007}). Concurrently, several studies support the existence of these five factors, in which genetic predisposition plays a fundamental role (^{Costa & McCrae, 1999}; ^{Laajaj et al., 2019}; ^{van der Linden, 2019}). Such structure has been proved independent of language, religion, and culture (^{Allik & McCrae, 2002}; ^{Laajaj et al., 2019}; McCrae & Terraciano, 2005; ^{van der Linden, 2019}). According to these authors, the fact that these regularities are manifested in diverse cultures and contexts suggests the presence of basic characteristics in humans. This conception, however, does not seek to deny the role of the environment. On the contrary, it provides the final conditions for people’s development (^{Sánchez & Ledesma, 2007}).

The FFM offers a general description of the factors and attributes second-order factors called facets to each of them (^{Sánchez & Ledesma, 2007}). Using these descriptors as a reference, it is possible to outline the distinctive characteristics of people according to their preponderant factor. Thus, an individual who scores high in agreeableness can probably be described as generous, considerate, supportive, altruistic, and conciliatory (Ledesma, Sánchez, & Díaz-Lázaro, 2011). The agreeableness factor can be understood as altruistic because it is concerned with the benefit of others in situations that might result in total loss of benefits for everyone. A person whose central feature is neuroticism will probably be anxious, unsteady, undecided, might worry a lot, and be prone to rumination (^{Introzzi, Andrés, Canet-Juric, Stelzer, & Richard’s, 2016}; ^{Pereira et al., 2012}). An individual characterized by extraversion is surely sociable, talkative, spontaneous, full of energy, and excitement seeking; on the other hand, if openness predominates, the person will be imaginative, adventurous, creative, original, and non-conservative (^{Ledesma et al., 2011}). Finally, a subject with the conscientiousness trait is probably persistent, organized, cautious, tidy, and productive (^{Ledesma et al., 2011}).

Another distinctive aspect of the FFM is the emphasis placed on evaluation (^{Sánchez & Ledesma, 2007}). Several instruments have been developed around the Big Five theory. In general, these are selfadministered instruments composed of adjectives or short phrases from which one seeks to infer basic personality trends. One of the most successful instruments is the NEO-FFI (^{McCrae & Costa, 2004}). This questionnaire has proven to be very useful, reliable, and valid in many contexts (McCrae & Costa, 2004; ^{McCrae & Mõttus, 2019}). Another instrument of interest here is the Adjective Checklist for Personality Assessment (AEP), developed by Ledesma and collaborators for the Argentine population (^{Ledesma et al., 2011}; ^{Sánchez & Ledesma, 2013}). The AEP replicated, at the local level, the findings of the FFM worldwide, proving to be of great regional value (^{Ledesma et al., 2011}; ^{Sánchez & Ledesma, 2013}).

If we understand the concept of trait as a relatively stable predisposition to react to the environment (^{Allport, 1974}), it is not strange to think that traits can influence the way we reason. Along this line, ^{Fumero et al. (2010}) argue that if reasoning is a semantically driven process according to which we imagine possible worlds, then the ability to consider one or another possibility can be linked to characteristics of our personality. Under these assumptions, the authors postulate that those individuals who score high in a certain personality trait perform better in tasks in which the contents are linked to that trait. For example, extrovert individuals would reason better in social situations, while those who tend to be neurotic perform better in anxiogenic situations (^{Fumero et al. (2010}). The general idea is that our personality has inferential consequences. This hypothesis has been partially confirmed by empirical evidence (^{Fumero et al. (2010}). ^{Sternberg et al. (2008}), for example, were interested in studying the relationship between personality and school performance from the perspective of cognitive styles. On the other hand, ^{Castro-Solano and Casullo (2001}) have warned that intellectual factors alone are poor predictors of academic performance, while pointing out that certain personality dimensions enable to distinguish between profiles of high and low school performance. More recently, other studies (^{Brandt, Lechner, Tetzner, & Rammstedt, 2019}; ^{Stajkovic, Bandura, Locke, Lee, & Sergent, 2018}) have also been interested in the influence of personality traits and cognitive ability on academic performance. Finally, ^{Macbeth, Razumiejczyk and Campitelli (2011}) and Macbeth, Razumiejczyk, & and Crivello (2014a) have presented evidence in favor of introspection as an intervening variable in logical reasoning.

The aim of this study was to describe the relationship between personality and reasoning. Our working hypothesis stated that neuroticism, extraversion, and agreeableness generate specific response patterns in compound reasoning tasks concerned with negation. Literature on reasoning in general and the MMT in particular confer great importance to difficulty and context as intervening factors in the reasoning process (Khemlani & Johnson-Laird, 2016; ^{Macbeth et al., 2013}; Markovits, Brisson, & Chantal, 2016; ^{Valiña, Martin, & Seoane, 2014}). In this study, although none of these phenomena are particularly considered in the statistical analysis, we nevertheless assume their existence as mediating factors of the process.

This paper continues as follows. First, we introduce our experimental paradigm to evaluate reasoning. Then, we present five experimental hypotheses derived from the working hypothesis included in the Method section. Afterwards, we continue with replication details related to participants, design, materials, and procedure. The statistical results are discussed in the context of previous evidence. Finally, the General Discussion section covers the scope and limitations of the contribution of this study in the context of the current state of the art.

Paradigm

When evaluating reasoning, two variables are of particular interest: response time and response type. In this study, we have focused on response type. Thus, we chose an experimental paradigm that was specifically designed to evaluate propositional reasoning regarding compound negations. It is a selection test used in previous studies (^{Macbeth et al., 2015}; 2017a), in which subjects are asked to select a response option that they consider appropriate. Given that the paradigm was also based on the MMT, it presupposes a close relationship between each specific response and specific mental models leading to such selection.

The task begins with a statement that describes an everyday situation. The statement is formulated in colloquial language but made in such a way that it contains one of De Morgan’s laws in its internal structure. In other words, the statement that represents Law 1 contains an internal structure of no type (p & q) (see Figure 1), while the statement of Law 2 is of no type (p or q) (see Figure 2), where the letters p and q symbolize atomic propositions. The instruction involves choosing from a set of options the one that best represents the situation described. In terms of reasoning, subjects have to find the logical equivalent of the proposition. Two statements are considered to be logically equivalent when they express the same idea (^{Macbeth et al., 2017}a).

The answer options are four: one corresponding to the normative response, and three corresponding to additional erroneous statements. The normative response is the one that captures the logical equivalent for each of ^{De Morgan’s laws (De Morgan, 1847}) in its internal structure. In the case of Law 1, the correct answer is of no p or no q, nor any type (see Figure 1, option a), while in the case of Law 2 it is of no p & no q type (see Figure 2, option d). As regards the number of mental models required in each case, it is clear that the task related to Law 1 is more difficult than the one related to Law 2, since the resolution of the first one requires the representation of three mental models, while for the second, only one is necessary (^{Macbeth et al., 2014}b; 2017a).

Figure 1 Sample of the task for the negation of a conjunction (Law 1).Note: The asterisk * signals the correct answer according to De Morgan’s Law 1. 

Figure 2 Sample of the task for the negation of a disjunction (Law 2).Note: The asterisk * signals the correct answer according to De Morgan’s Law 1. 

Regarding the other options, although they are erroneous, they do not lack theoretical relevance. On the contrary, each has different logical and psychological implications, as a result of differential processing. These responses were called transformation bias, scope bias and matching bias-like, according to the type of bias they referred to. Thus, the transformation bias describes the mental transformation operation performed by people as a result of an erroneous inferential effort. According to the laws established by logic, it is valid to consider the proposition «no p or no q» as the equivalent of the conditional «if p, then not q» (^{Suppes & Hill, 1992}). In this task, however, a different conditional was included (of «if not p, then not q» type), which is logically invalid (seeFigure 1, option d;Figure 2, option a). Considering the logical aspect, it is likely that those who chose this response option made an inferential processing effort intended to represent the fully explicit mental models. Despite the effort made, however, the difficulty of the task was such that they could not complete it successfully, and this led to a wrong transformation (^{Macbeth et al., 2017}a).

Another option offered was the so-called scope bias, associated with the processing bias proposed by ^{Khemlani et al. (2012}). This deviation is observed in the case of compound negation, according to which individuals tend to limit their interpretation to only one of the atomic propositions (small scope) and not to the compound proposition in its entirety (large scope) (^{Khemlani et al., 2012}; ^{Macbeth et al., 2013}). To illustrate it better, let’s check the example inFigure 1, where Laura realizes that she does not have enough money to buy a chocolate cake and a lemon cake, as intended. Finding the appropriate logical equivalent in this task requires to be able to deny both atomic propositions jointly: Laura does not buy the chocolate cake, or she does not buy the lemon cake, or she does not buy any of the two (inclusive disjunction:Figure 1, option a;Figure 2, option c). Despite this, most individuals tend to choose the option in which the propositions are denied but only separately; that is, Laura does not buy the chocolate cake, or does not buy the lemon cake; she chooses one but not both (exclusive disjunction: Figure 1, option c;Figure 2, option b).

From a logical point of view, the scope bias can be interpreted as an attempt by people to reduce the overload in their working memory and consequently the processing effort (^{Macbeth et al., 2017}a). From a psychological perspective, however, the implications are different. If the contextual nature of the statement is remembered, it is possible that this bias was mediated by a second phenomenon called loss aversion (^{Tversky & Kahneman, 1991}). This refers to a behavioral tendency according to which, when having to choose, the person would show greater propensity to reduce losses rather than to increase profits ((^{Tversky & Kahneman, 1991}). This phenomenon, originally studied in economic contexts, is nonetheless applicable whenever the person weighs losses and gains. In the case of our task, the bias is observed when the individual chooses the option that keeps some type of benefit for the agents involved. In the previous example, subjects cannot consider the idea of buying no cake at all when they intended to buy both, which explains the trend to choose the option that preserves some gain (buy at least one cake).

Finally, the third erroneous option is called matching-bias-like and is linked to superficial processing: in this case, the person chooses the answer option that contains the same propositional connector (seeFigure 1, option b;Figure 2, option c). This processing, in which the statement and response option are matched according to their appearance, was proposed by ^{Evans in the context of the Wason Selection Task (Evans, 1998}). It was also detected in recent studies with similar tasks (^{Macbeth et al., 2013}; 2017a). Accumulated evidence suggests that it is an economic process in inferential terms, since it seeks to minimize the overload in the working memory.

As noted, the used paradigm has certain characteristics that make it an optimal instrument for the study of propositional reasoning. First, the neutrality and simplicity of the format greatly facilitate the comparison between theories (^{Macbeth et al., 2017}a). Secondly, the use of a selection task makes it possible both to control the offered answers and to manage the level of difficulty more easily (Macbeth et al., 2014b).

Participants

A non-probabilistic sample of 150 students from the National University of Entre Ríos, a public university located in the city of Paraná, Argentina, was used in this study. All the participants were social sciences students with no formal training in mathematics or logic. No formal training in logic was the only exclusion criterion. This item was not directly asked to the participants, but was corroborated by the researchers in the Social Sciences program curricula. The mean age was 24.48 years old (SD= 6.64). Female participants (n= 77) accounted for 51.33% of the sample. All the participants signed an Informed Consent form. No monetary or academic rewards were given.

Materials and Procedure

Each participant received a booklet with two tasks: one aimed to evaluate compound reasoning with negation, and the other one aimed to evaluate the personality. The tasks were assigned using paper and pencil at the beginning of a regular class at the university. Each student had to complete it individually, without any time limit. Before starting to work, we explained the students that the instructions were given at the beginning of each task. We also indicated them that the study was not harmful or tricky, and that everyone could withdraw from the study at any moment. To evaluate reasoning, we used a sentenceequivalence task implemented in previous studies (^{Macbeth et al., 2014}b, 2015, 2017a), in which the psychometric results that support the task and other similar ones can be found (For more details, see ^{Campitelli, Macbeth, Ospina, & Marmolejo-Ramos, 2016}; ^{Khemlani el al., 2014}). Participants were asked to find a logical equivalence for a given compound negation. The task consisted of a total of 8 exercises: four evaluated reasoning with negation of a conjunction (De Morgan’s Law 1) and four evaluated reasoning with negation of a disjunction (De Morgan’s Law 2). We randomized the sequence of items and the sequence of options within each item for all the subjects. Four response options were given, that is, the normative response and three non-normative response options: scope bias, transformation bias, and matching-bias-like response. As we explained before, the first one matched De Morgan’s laws. The others were developed according to the number of mental models required to process them¯i.e., in line with the Mental Models Theory (Macbeth et al., 2014b).

To evaluate personality factors, we used the Adjective Checklist, an instrument developed in Argentina based on the Big Five theory (^{Ledesma et al., 2011}). The list consists of 67 adjectives covering the five personality factors. The authors of this checklist presented evidence that supports the good psychometric properties of the instrument (^{Ledesma et al., 2011}; ^{Sánchez & Ledesma, 2013}). Factor analysis results were consistent with the Five- Factor Model. The internal consistency of the five scales was very good (Cronbach’s á ranging from .74 to .85) and the convergent correlations with the Big Five Inventory were substantial (with Pearson’s correlation levels ranging from .60 to .78). Face validity, as evaluated by two independent raters, was good. Moreover, the instrument showed a convergent validation with an Argentine version of the Big Five Inventory (BFI), adapted by ^{Castro-Solano (2001)}, and ^{Castro Solano and Casullo, 2001} (For more details, see ^{Ledesma et al., 2011}; ^{Sánchez & Ledesma, 2013}). The inventory is structured as a Likert scale with five response options: from not characteristic of me to very characteristic of me. Participants were instructed to choose a single option for each adjective.

The completion of the experiment lasted roughly 20 minutes.

Design and Analyses

A correlational cross-sectional study was conducted to test five empirical hypotheses that relate personality factors to response types in a compound reasoning task. We correlated five personality factors (neuroticism, extraversion, agreeableness, openness, conscientiousness) with four response types (normative response, transformation bias, scope bias, matching-bias-like response) for both De Morgan’s laws (negation of a conjunction, negation of a disjunction). We developed a correlation matrix of 13x13 cells, that is, five (personality factors) plus eight reasoning conditions (four response types x two laws of logic). Since we are interested in correlates rather than manipulations, we used the 13x13 matrix instead of a 5x4x2 matrix (personality factors x response types x laws of logic) required for a factorial experiment focused on the effects of controlled manipulations, and the ANOVA family of statistics to produce evidence. To perform correlational analyses, we used the non-parametric Spearman’s rho and the set type I error at 0.05. The choice to apply robust statistics was because reasoning vectors were incompatible with the assumption of normality according to Kolmogorov- Smirnov test. In the case of both laws, all types of responses yield p values close to zero as per the aforementioned test.

Empirical Hypotheses

We derived five empirical hypotheses from our working hypothesis. That is, we conjectured that some specific cognitive patterns can be correlated with certain personality factors during reasoning with compound negations. We focused on neuroticism, extraversion, and agreeableness. Since the reasoning task we selected requires abstraction (^{Macbeth et al., 2014}b), we did not focus on openness to experience or conscientiousness because both are concrete experiences related rather than abstract thinking related. Moreover, due to the materials and procedure of our reasoning task, we focused on three response types: normative response, transformation bias, and scope bias. We did not consider the matching-biaslike response because it seems to be more a linguistic effect rather than a specific psychological phenomenon (Macbeth et al., 2014b). The empirical hypothesis H1 predicts a direct and significant correlation between neuroticism and transformation bias for the negation of a conjunction. H2 predicts a direct and significant correlation between neuroticism and transformation bias for the negation of a disjunction. H3 predicts a direct and significant correlation between agreeableness and scope bias for the negation of a conjunction. H4 predicts a direct and significant correlation between extraversion and scope bias for the negation of conjunction. H5 predicts an inverse and significant correlation between extraversion and normative response for the negation of a disjunction.

The relationship between neuroticism and transformation bias predicted in H1 and H2 can be justified as a psychological effect of erroneous overthinking. That is, the transformation bias response type requires more mental computation than other response types because it is formulated as a conditional. This transformation is biased both for the negation of conjunctions in H1, and for the negation of a disjunction in H2.

The scope bias response type is concerned with gains and losses, which are sensitive traits for both the extraversion and agreeableness personality factors. This kind of loss can be related to a lack of sociability skills (^{Ariely, 2008}). Therefore, H3 is justified as an altruistic concern in a social situation. The scope bias response in our reasoning task preserves some kind of gain because the total loss is cancelled by the exclusive disjunction. Following the same line of reasoning the prediction stated in H4 is concerned with the same social altruism criterion. In this case, extraversion might be understood as the source of such altruistic tendency.

On the other hand, the extraversion factor is also related to characteristics such as spontaneity and impulsivity, while the normative response does it with a greater capacity for reflection and attention. In this way, the inverse and significant correlation established in H5 between both variables can be justified by their opposite tendencies.