Original ArticlesPossibilities as the foundation of reasoning
Introduction
Possibility can be close to probability, as Sherlock Holmes’s remark bears out (see Conan Doyle, 1981, p. 339). Both introduce uncertainty into discourse, and so inferences about possibilities – modal reasoning – are ubiquitous in daily life. Here are three everyday examples, which we invite you to consider:
- 1.
Either Trump will be re-elected or he won’t be.
So, it’s possible that he won’t be.
- 2.
The probability of snow today is 90%.
So, it’s very possible that it will snow today.
- 3.
Pat may be married and Viv may not be.
So, maybe Pat is married and Viv isn’t.
Modal logics deal with possibilities, and so you might wonder which of their main versions underlie these three inferences. The answer as we show is: none. The inferences above do not follow in any normal modal logic. Yet, naive reasoners – those who know nothing of logic – make them. You might think that these individuals are either blundering or else introducing additional, perhaps pragmatic, factors that justify the inferences. This article takes a more radical stance. It presents a theory of modal reasoning in which the inferences are valid, that is: if their premises are true then so too are their conclusions (Jeffrey, 1981, p. 1), but with one caveat that we explain below.
The theory distinguishes three principal interpretations of possible (for examples, see Table 1 below):
- 4.
Alethic interpretations concern possible or necessary consequences.
Deontic interpretations concern permissible or obligatory actions (or inactions).
Epistemic interpretations concern degrees of belief in propositions.
The present article cannot describe the vast literature on modality, but it provides synopses of modal logic and modal linguistics (in Appendices A and B). Readers who suffer existential dread faced with logical symbols should read only the current state of the science (Section 2). The article next analyses the different interpretations of modal assertions, the cues to these interpretations, and their common underlying finitary semantics (Section 3). It shows how models can represent each interpretation (Section 4), and it illustrates how they yield different sorts of modal inference (Section 5). And it ends with a discussion of rival accounts, open questions for the model theory, and future lines of research (Section 6).
So, what do modals mean? How are they represented in the mind? And what mechanism uses these representations in modal reasoning? Our goal is to answer these three questions, relying in part on precursors. And, as we proceed, we enumerate the model theory’s predictions and their empirical corroborations. They include long-standing results, and new tests of novel predictions.
Section snippets
Modality: The state of the science
Psychologists have studied modality for over fifty years (e.g., Byrnes and Beilin, 1991, Inhelder and Piaget, 1958, Piéraut-Le Bonniec, 1980, Shtulman, 2009), but know much less about it than about other sorts of reasoning. One hypothesis (Rips, 1994, p. 322) is that individuals can make deontic inferences without being familiar with their contents, e.g.:
- 5.
It is obligatory that P given Q.
Therefore, it is permissible that P given Q.
Three interpretations of modals
Our first task is to distinguish among the different interpretations of modals in daily life, which have repercussions for reasoning. There are at least three main interpretations, which occur in this example in the order alethic, deontic, and epistemic:
- 8.
It follows of necessity that she is permitted to resign if it’s possible she wants to.
Mental models and fully explicit models
The model theory postulates that a mental model is a small finite representation that is iconic, i.e., insofar as possible it has the same structure as what it represents. So, each of the alternative outcomes of, say, the Democratic Iowa caucus in Fig. 1 can be represented in a model of a possibility. The theory postulates that a parser uses a grammar and a lexicon to compose the meanings of assertions. An intuitive system of reasoning, system 1, uses these meanings to construct a mental model
What mechanisms use models for modal reasoning
The model theory applies to all sorts of reasoning including induction, abduction, and deduction (Johnson-Laird, 2006, Khemlani and Johnson-Laird, 2013). It depends on simulating the world in models of possibilities, either static or kinematic (Johnson-Laird and Byrne, 1991, Khemlani et al., 2013). We have implemented the present theory of deduction in the mSentential program at http://mentalmodels.princeton.edu/models/. The program ensures that the model theory does not take too much for
General discussion
Ask a psychologist how people reason, and the answer you tend to get is that they use logical rules (e.g., Piaget, 1957, Rips, 1994). So, they should use modal logics to reason about possibilities. But, in contrast to all normal modal logics, people are happy to infer that nothing follows from certain premises, to withdraw conclusions that evidence contradicts, not to infer anything from a contradiction, and to deal with possibilities of different sorts within a single assertion (e.g., 8). But,
Conclusions
The model theory postulates that the meanings of possible and its cognates refer to small finite numbers of alternatives. Knowledge can interpret them as alethic relations, which include allowing and causing. It can interpret them as deontic permissions or obligations, which speech acts can create. And it can interpret them as epistemic possibilities or certainties, which are probabilities with or without numbers. The meaning of ‘or’, ‘if … then …’, and other sentential connectives, refers to
Acknowledgements
This research was made possible by a grant to the first author from the DFG in proposal RA 1934/3-1 and RA 1934/4-1. The article has had a long gestation, and we are indebted to many individuals. We thank Sunny Khemlani for advice and a critical reading of an earlier draft. We thank colleagues for their help and ideas: Monica Bucciarelli, Ruth Byrne, Geoff Goodwin, Thomas Hinterecker, Marie Jacobs, Markus Knauff, Philipp Koralus, Robert Mackiewicz, Gorka Navarrete, Isabelle Orenes, Cristina
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