What’s the probability that in the next 500 years, the Earth’s polar ice caps will melt completely? How did you arrive at a specific number? Some theorists argue that your response was meaningless, because nothing similar to the event has ever occurred. But our studies show that individuals’ responses are systematic and predictable.

Those theorists who describe themselves as frequentists argue that whatever estimate you came up with for the melting of the Earth’s ice caps is meaningless: probabilities should be interpreted as the limits of repeated observations, and there is no event that is comparable to the the ice caps melting. Bayesian theorists, however, argue that the estimate you gave reflects your degree of belief in the event. With Phil Johnson-Laird and Max Lotstein, I developed a theory to explain how humans derive numerical estimates of subjective probabilities (Khemlani, Lotstein, & Johnson-Laird, 2012). This theory distinguishes between an intuitive prenumerical system and a deliberative system capable of arithmetic. A computer implementation of the theory uses mental models of evidence to construct iconic representations of degrees of belief.

The system works by constructing simulations of evidence in the form of quantified expressions, e.g., Some arctic glaciers have melted (Khemlani et al., under review). It then infers a bounded analog magnitude representation of belief by sampling from the simulations of evidence. The analog magnitude representation is similar to representations found in infants (Xu & Spelke, 2000), animals (Meck & Church, 1983), and adults in non-numerate cultures (Gordon, 2004). The procedures that operate over the analog belief representations are constrained: reasoners can combine and update beliefs, but they do so without relying on memory storage. The representations accordingly impose little cognitive strain – but their simplicity predicts systematic errors.

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For instance, consider the following problem: suppose you’ve estimated the probability of two unique events, e.g., the probability that the ice caps will melt is 10% and the probability that sea lions will go extinct is 2%. What is the probability that both events occur, i.e., the ice caps will melt and sea lions will go extinct? If the two events are independent, then individuals should multiply the probabilities, e.g, P(ice caps melt & sea lions die) = P(ice caps melt) * P(sea lions die) = .05 * .02 = .001. That general strategy is reflected in Figure A above, where P(A) = P(ice caps melt) and P(B) = P(sea lions die). However, our system is limited, and it generally does not multiply probabilities in such a manner. It has to resort to a simpler computation to combine the two probabilities, e.g., one that splits the difference between them (reflected in Figure B). Experimental data (Figure C) corroborates the second strategy and the computational model we developed.

• Barth, H. et al. (2006). Nonsymbolic arithmetic in adults and young children. Cognition, 98, 199–222.
• Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306, 496-499.
• Meck, W.H., & Church, R.M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psycholology: Animal Behavioral Processes, 9, 320-334.
• Khemlani, S., Lotstein, M., & Johnson-Laird, P.N. (2012). The probabilities of unique events. PLoS ONE, 7, e45975.
• Xu, F., & Spelke, E.S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74, B1-11.

We present a system for building mental models of quantified assertions. This system implements constraints on the size of model, the sorts of individual it represents, and on the likelihood of a search for alternative models. It yields quantitative predictions at a fine-grained level, and they fit data from experiments on quantificational reasoning better than previous accounts.
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Generic statements are common ways of expressing everyday generalizations. Most people agree with generalizations such as, “dogs have four legs”, “sharks bite swimmers”, “shoes have laces”, and “ducks lay eggs”, all of which are generics. Those generalizations are based on an underlying network of concepts, and a new set of studies investigates the structure of that network.
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We describe a new model-based theory of what negation means, how it is mentally represented, and how it is understood.

Negation is so commonplace is everyday speech that individuals are rarely aware of its problematic nature. For instance, negations in English include words, prefixes, and suffixes such as not, no, un-, im-, ab-, -less, and so forth. Often, individuals can handle multiple negations with ease, such as in Varzi’s negative biography, which is excerpted here. There are nine separate negations in that quote alone. Yet consider this shorter description:

It is wrong to deny that it is not the case that Franklin didn’t call for the police.

Given the sentence above, did Franklin call the police or not? (Answer: he did.) There are four negations in this sentence, and the nature of the negations makes the sentence extraordinarily difficult to understand. Linguists, philosophers, and psychologists have studied many aspects of negation, including its syntax, its pragmatics, its meaning, and how people verify a negative sentence as true or false. But there’s no general theory of the meaning of negation, its mental representation, or its use.

In a recent paper with Phil Johnson-Laird and Isabel Orenes (2012a), I have proposed such a theory and discovered the novel phenomena that it predicts. The theory postulates that individuals interpret negation as having the smallest feasible grammatical scope, e.g., one proposition rather than two, and that they do not know what “negates” a conjunction or other compound sentences, but have to enumerate these possibilities. We have implemented the theory computationally and corroborated it experimentally. Individuals do reduce the scope of negations, and so fail to envisage all the possibilities consistent with denials of such statements as, “the forks, knives, and spoons are on the table”. Likewise, the accuracy of their formulation and understanding of denials differs from one sort of assertion to another, e.g., denials of disjunctions are easier for them than denials of conjunctions (Khemlani, Orenes, & Johnson-Laird, 2012b).
The theory of negation extends the mental models theory of reasoning (Johnson-Laird, 2006) with four additional principles that concern the meaning, representation, interpretation, and verification:

1. The principle of negative meaning: negation is a function that takes a single argument, which is a set of fully explicit models of possibilities, and in its core meaning this function returns the complement of the set.
2. The principle of symbolic negation: negation is mentally represented by a symbol, which can occur in models, and the symbol is associated with a representation of the meaning of negation. When possible, as in the case of binary predicates, individuals can eliminate negation in terms of a representation of its positive contrast set.
3. The principle of verification: a proposition expressed by a sentence is evaluated as true perceptually if its mental model maps into the model based on perception of the relevant state of affairs. All of the referents, properties, and relations, in the model of the proposition map into the perceptual model. A corollary is that a successful mapping, a match, is faster than an unsuccessful mapping, a mismatch, because the latter calls for a more exhaustive search.
4. The principle of enumerative negation: Individuals formulate and interpret the negation of a statement containing a sentential connective, such as and, if, or or, and two main clauses, A and B, by constructing a series of models of conjunctive possibilities.  They first conjoin the negations of the main clauses: not-A and not-B, and check whether the resulting possibility is consistent with the unnegated statement. They then negate each clause, and check whether each of the resulting possibilities is consistent with the unnegated statement. Finally, they affirm both main clauses, A and B, which is a possibility consistent, say, with the negation of an exclusive disjunction.  In each case, if a model is consistent with the unnegated statement, they reject it; otherwise, they accept it as consistent with the negation.

The four principles outlined above, along with the broader theory of mental models, yield several novel predictions:

• The parsing of negative assertions should return a small scope for negation, where possible, because such a scope minimizes the number of mental models of assertions, and thereby reduces the processing load on working memory.
• The meaning of negated clauses, their referents, and knowledge of the topic or context, can modulate the interpretation of negation. It can block at least one possibility in which the negative would otherwise hold, and the result only a contrary to the corresponding affirmative proposition.
• When individuals have in mind a representation of a proposition – especially the set of possibilities to which it refers, its negation should be easier for them to understand than in other contexts, i.e., it should take less time. A corollary is that explicit negations using “not” should be easier to grasp as denials than implicit negations, such as the use of a complementary predicate (e.g., “open”) to deny its antonym (e.g., “closed”).
• Those affirmative assertions with only one mental model should be easier to understand than those with multiple mental models. Their respective negations should switch in difficulty, because the complement of one model is a set of multiple models, whereas the complement of multiple models is a set of one or two mental models.

Recent empirical findings support many of the theory’s postulations. Here’s a small sample:

• Individuals interpret the negation of a conditional, e.g., If A then B, to support both If A then not-B as well as If not-A then B (Espino & Byrne, 2012). This finding is a “small scope” effect, in line with the first prediction outlined above.
• Children misremember negatively quantified assertions (e.g., No As are Bs) as negative generics (e.g., As are not Bs; see Leslie & Gelman, 2010). This too corroborates the small scope prediction.
• Humans find it more difficult to comprehend and produce denials of disjunctions than denials of conjunctions (Khemlani, Orenes, & Johnson-Laird, 2012b;  Macbeth et al., under review). This finding supports the principle of enumerative negation.
• Eye-tracking data suggests that in specific cases, individuals are willing to fixate on objects that represent negated situations (Orenes, Beltrán, Santamaría, 2012). This supports the assumption that listeners represent negative assertions as combinations of iconic and symbolic representations, in line with the principle of symbolic negation.

• Espino, O., & Byrne, R.M.J. (2012). It is not the case that if you understand a conditional then you know how to negate it. Journal of Cognitive Psychology, 24, 329-334.
• Johnson-Laird, P. N. (2006). How we reason. Oxford, UK: Oxford University Press.
• Khemlani, S., Orenes, I., & Johnson-Laird, P. N. (2012a). Negation: A theory of its meaning, representation, and use. Journal of Cognitive Psychology, 24, 541–559.
• Khemlani, S., Orenes, I., & Johnson-Laird, P.N. (2012b). Negating compound sentences. In N. Miyake, D. Peebles, & R. Cooper (Eds.), Proceedings of the 34th Annual Conference of the Cognitive Science Society. Austin, TX: Cognitive Science Society.
• Leslie, S.J., & Gelman, S. (2010). Quantified statements are recalled as generics. Cognitive Psychology, 64, 186-214.
• Macbeth, G., Razumiejczyk, E., Crivello, M., Bolzán, C., Pereyra Girardi, C., & Campitelli, G. (under review). Mental models for the negation of conjunctions and disjunctions. Manuscript under review.
• Orenes, I., Beltrán, D., & Santamaría, C. (2012). On the need of simulating what is unspecified: negations in a visual world paradigm. Abstract presented at the 7th International Conference on Thinking, London, UK.