Feature Review
Logic, probability, and human reasoning

https://doi.org/10.1016/j.tics.2015.02.006Get rights and content

Highlights

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    We describe a way to integrate probability with deductive reasoning.

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    Theories based on logic do not withdraw conclusions that conflict with the facts.

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    A ‘new paradigm’ based on probabilities solves this problem, but introduces others.

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    Mental model theory resolves the issues and new studies corroborate its predictions.

This review addresses the long-standing puzzle of how logic and probability fit together in human reasoning. Many cognitive scientists argue that conventional logic cannot underlie deductions, because it never requires valid conclusions to be withdrawn – not even if they are false; it treats conditional assertions implausibly; and it yields many vapid, although valid, conclusions. A new paradigm of probability logic allows conclusions to be withdrawn and treats conditionals more plausibly, although it does not address the problem of vapidity. The theory of mental models solves all of these problems. It explains how people reason about probabilities and postulates that the machinery for reasoning is itself probabilistic. Recent investigations accordingly suggest a way to integrate probability and deduction.

Section snippets

The nature of deductive reasoning

To be rational is to be able to make deductions – to draw valid conclusions from premises. A valid conclusion is one that is true in any case in which the premises are true [1]. In daily life, deductions yield the consequences of rules, laws, and moral principles [2]. They are part of problem solving, reverse engineering, and computer programming 3, 4, 5, 6 and they underlie mathematics, science, and technology 7, 8, 9, 10. Plato claimed that emotions upset reasoning. However, individuals in

Problems for logic as a theory of deductive reasoning

An ancient proposal is that deduction depends on logic (see also 16, 17, 18, 19, 20). Sentential logic concerns inferences from premises such as conjunctions (‘and’) and disjunctions (‘or’). Like most logics, it has two parts: proof theory and model theory [35]. Proof theory contains formal rules of inference for proofs. One major rule of inference in most formalizations is:

A → C

A

therefore, C

where A and C can be any sentences whatsoever, such as:

‘2 is greater than 1’ → ‘1 is less than 2’.

Proof

Probability logic

As a consequence of the preceding arguments, some cognitive scientists propose that probability should replace logic. Their theories differ in detail but overlap enough to have a label in common – the new paradigm 25, 26, 27, 28, 29. We refer to the paradigm as ‘probability logic’ or ‘p-logic’ for short. It presupposes that degrees of belief correspond to subjective probabilities 45, 46, 47, 48, 49, an idea that not all psychologists accept 50, 51. It focuses on conditionals, and one p-logician

The theory of mental models

During World War II, Kenneth Craik proposed that individuals simulate the world in mental models to make predictions, but that reasoning depends on verbal rules [59]. A more recent theory – the mental model theory – postulates that simulation underlies reasoning too 30, 31, 32, 33. It is a simple idea – people simulate possibilities – and most of its ramifications are integrated in a computer program, mReasoner [60], which is in the public domain at http://mentalmodels.princeton.edu/models/.

An evaluation of p-logic and mental models

P-logic has the great merit of allowing that reasoning can be tentative, uncertain, and probabilistic. It has inspired much ingenious research. However, what is the standing of its four key hypotheses?

Ramsey's test assesses the probability of conditionals. Perhaps reasoners use the test [45], but their estimates of conditional probabilities tend to violate the complete joint probability distribution (see below).

Conditionals have a defective truth table. Some experiments corroborate its

Do probabilities enter into pure deductions?

By ‘pure’ deductions, we mean those that make no reference to probabilities. P-logic, however, holds that probabilities are ubiquitous and that they unconsciously enter into pure deductions 21, 22, 23, 25, 26, 27, 28, 29. By contrast, the model theory implies that probabilities enter into the contents of reasoning only if invoked explicitly. We examine this proposal for conditional, syllogistic, and causal reasoning.

According to p-logic, a conditional such as:

If the FDA approves a drug then it

The probabilistic machinery of reasoning

At this point, readers may suspect that the model theory is deterministic through and through. In fact, it postulates that the machinery underlying reasoning – even for pure deductions – is probabilistic [30]. One illustration concerns evaluations of consistency – an important task because inconsistent beliefs can lead to disaster 109, 110. The only general way to use formal rules of inference [18] to establish the consistency of a set of assertions is to show that the negation of one assertion

Reasoning about probabilities

The model theory can explain how people reason about probabilities of various sorts. Consider the following inference:

A sign of a particular viral infection – a peculiar rash – occurs only in patients who are infected, but some patients with the infection do not have the rash. Is the infection more likely than the rash? (Yes.)

This deduction follows from the mental model of the possible individuals. Here is a numerical example:

There is a box in which there is at least a red marble, or else there

Concluding remarks

We began with two questions: does logic underlie human deductions and how do probabilities fit together with them? Despite the importance of logic to mathematics and the theory of computability, unconscious logical rules do not appear to be the basis of everyday reasoning. Arguments for this claim motivated the new paradigm in which reasoners rely instead on probability logic. It focuses on conditional assertions and postulates that individuals assess them by imagining that their if-clauses are

Acknowledgments

The authors thank Ruth Byrne, Rebecca Schwarzlose (TiCS editor), and two anonymous reviewers for their constructive criticisms of an earlier draft. They are also grateful to Igor Douven, Niki Pfeifer, Gernot Kleiter, and Klaus Oberauer for helping them to clarify their views about the four key hypotheses of the new paradigm. This research was supported by a Jerome and Isabella Karle Fellowship from the Naval Research Laboratory to S.S.K.

Glossary

Bayesian net
a directed graph in which each node represents a variable and arrows from one node to another represent conditional dependencies. It captures the complete joint probability distribution in a parsimonious way.
Consistency
a set of assertions is consistent if they can all be true at the same time.
Counterexample
in an inference, a possibility to which the premises refer but which is inconsistent with the conclusion.
Deductive reasoning
a process designed to draw a conclusion that follows

References (160)

  • M. Bucciarelli et al.

    Strategies in syllogistic reasoning

    Cogn. Sci.

    (1999)
  • J.K. Kroger

    Distinct neural substrates for deductive and mathematical processing

    Brain Res.

    (2008)
  • R. Jeffrey

    Formal Logic: Its Scope and Limits

    (1981)
  • M. Bucciarelli

    The psychology of moral reasoning

    Judgm. Decis. Mak.

    (2008)
  • N.Y.L. Lee

    The psychological problem of Sudoku

    Think. Reason.

    (2008)
  • N.Y.L. Lee et al.

    Strategic changes in problem solving

    J. Cogn. Psychol.

    (2013)
  • S.S. Khemlani

    Kinematic mental simulations in abduction and deduction

    Proc. Natl. Acad. Sci. U.S.A.

    (2013)
  • N.Y.L. Lee et al.

    A theory of reverse engineering and its application to Boolean systems

    J. Cogn. Psychol.

    (2013)
  • E.W. Beth et al.

    Mathematical Epistemology and Psychology

    (1966)
  • P.N. Johnson-Laird

    How We Reason

    (2006)
  • J. Baron

    Thinking and Deciding

    (2008)
  • R. Nickerson

    Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs

    (2011)
  • E. Blanchette et al.

    The influence of affect on higher level cognition: a review of research on interpretation, judgement, decision-making and reasoning

    Cogn. Emot.

    (2010)
  • A. Gangemi

    Models and cognitive change in psychopathology

    J. Cogn. Psychol.

    (2013)
  • K.C. Klauer

    Working memory involvement in propositional and spatial reasoning

    Think. Reason.

    (1997)
  • D.N. Osherson
    (1974–1976)
  • J. Macnamara

    A Border Dispute: The Place of Logic in Psychology

    (1986)
  • L.J. Rips

    The Psychology of Proof

    (1994)
  • L.J. Rips

    Reasoning

  • M. Oaksford et al.

    Bayesian Rationality

    (2007)
  • M. Oaksford et al.

    Précis of Bayesian Rationality: the probabilistic approach to human reasoning

    Behav. Brain Sci.

    (2009)
  • J.M. Keynes

    A Treatise on Probability

    (1921)
  • D.E. Over

    New paradigm psychology of reasoning

    Think. Reason.

    (2009)
  • J.St.B.T. Evans

    Questions and challenges for the new psychology of reasoning

    Think. Reason.

    (2012)
  • S. Elqayam

    Rationality in the new paradigm: strict versus soft Bayesian approaches

    Think. Reason.

    (2013)
  • N. Pfeifer

    The new psychology of reasoning: a mental probability logical perspective

    Think. Reason.

    (2013)
  • M. Oaksford et al.

    Dynamic inference and everyday conditional reasoning in the new paradigm

    Think. Reason.

    (2013)
  • P.N. Johnson-Laird

    Mental Models

    (1983)
  • P.N. Johnson-Laird et al.

    Deduction

    (1991)
  • P. Koralus et al.

    The erotetic theory of reasoning: bridges between formal semantics and the psychology of deductive inference

    Philos. Perspect.

    (2013)
  • G. Boolos et al.

    Computability and Logic

    (1989)
  • A.M. Turing

    On computable numbers, with an application to the Entscheidungsproblem

    Proc. Lond. Math. Soc. Ser.

    (1937)
  • M. Davis

    Engines of Logic: Mathematicians and the Origin of the Computer

    (2000)
  • G. Antoniou

    Nonmonotonic Reasoning

    (1997)
  • J. Pijnacker

    Reasoning with exceptions: an event-related brain potentials study

    J. Cogn. Neurosci.

    (2011)
  • G. Baggio

    Logic as Marr's computational level: four case studies

    Top. Cogn. Sci.

    (2014)
  • M. Oaksford et al.

    Probabilistic single function dual process theory and logic programming as approaches to non-monotonicity in human vs artificial reasoning

    Think. Reason.

    (2014)
  • T. Achourioti

    The empirical study of norms is just what we are missing

    Front. Psychol.

    (2014)
  • I. Orenes et al.

    Logic, models, and paradoxical inferences

    Mind Lang.

    (2012)

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