Elsevier

Cognition

Volume 193, December 2019, 103950
Cognition

Original Articles
Possibilities as the foundation of reasoning

https://doi.org/10.1016/j.cognition.2019.04.019Get rights and content

Highlights

  • We present a theory of the meaning of ‘possible’ and its cognates based on finite alternatives.

  • The theory distinguishes alethic, deontic, and epistemic: the believability of propositions.

  • Compound assertions based on ‘or’ and ‘if’ refer to conjunctions of possibilities.

  • These interpretations occur in system 1 for intuitive inferences, and in system 2 for deliberations.

  • The theory diverges from all normal modal logics.

Abstract

Reasoning about possibilities is fundamental in daily life. Yet, it has been little studied in psychology. We present a psychological theory in which it is the foundation of human reasoning. The theory explains how possibilities have distinct interpretations (deontic, epistemic, and alethic), how people represent them in models, and how these models yield inferences. Key principles are that the semantics of possibilities are the same finitary alternatives underlying probabilities, that speech acts can create obligations inexpressible as probabilities, that compound assertions – conditionals and disjunctions – refer to conjunctions of possibilities holding in default of knowledge to the contrary, and that mental models condense multiple consistent possibilities into one. The theory is incompatible with all normal modal logics and with probabilistic logic. Yet, experiments have corroborated its predictions. The article discusses its precursors, rivals, and potentials.

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.

These modalities have been studied in modal logic (e.g., Kripke, 1963), linguistics (e.g., Palmer, 2001), psycholinguistics (e.g., Miller & Johnson-Laird, 1976, Sec. 7.3.1), linguistic philosophy (e.g., White, 1975), formal semantics (e.g., Kratzer, 2012, Portner, 2009), and computational linguistics (e.g., Isard, 1975, Moens and Steedman, 1988). The new theory takes into account these studies. It is based neither on normal modal logics nor on their semantics of ‘possible worlds,’ but on a finitary semantics for possibility and its cognates. Reasoning, in turn, depends on models of possibilities. So, we refer to the new theory as the ‘model’ theory. It has a long-standing account of what reasoners are trying to compute in drawing their own conclusions (Johnson-Laird, 1983, Johnson-Laird and Byrne, 1991, Ch. 2). They aim for conclusions that are new, parsimonious, and encapsulations of the semantic information in the premises. If no such conclusion exists, they respond that nothing follows. If they are given a conclusion to evaluate, they check only that the conclusion maintains the premises’ semantic information. So, unlike other accounts of reasoning, the theory distinguishes between conclusions that individuals accept and the subset of them that they draw for themselves. It treats all inferences in daily life as depending on tacit assumptions, and so the caveat that we mentioned earlier is that conclusions hold in default of knowledge to the contrary. Given such knowledge, they can be withdrawn. Reasoning is therefore defeasible (or nonmonotonic). When individuals withdraw a conclusion, they amend its premises, and try to explain the provenance of the inconsistency (Johnson-Laird et al., 2004, Khemlani and Johnson-Laird, 2012).

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.

Rips suggested that such inferences are based on formal rules of

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.

In this section, we show that these three interpretations are distinct, and we establish linguistic cues to them (cf. Kratzer, 2012). These components may vary

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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