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  • 1

    Non-constructive inference

    and conditionals

    David Over

    Psychology Department

    Durham University

    Thanks to:

    • The organizers

    • Nagoya University

    • Japan Society for the Promotion of Science

    Non-constructive reasoning

    Modified example from Toplak &

    Stanovich (2002):

    Jack is looking at Ann, and Ann is

    looking at George. Jack is a cheater, but

    George is not. Is a cheater looking at a

    non-cheater?

    A) Yes B) No C) Cannot tell

    The non-constructive aspect

    • Jack is looking at Ann but Ann is looking

    at George. Jack is a cheater but George is

    not. Is a cheater looking at non-cheater?

    • Ann is either a cheater or not. If she is,

    then a cheater (Ann) is looking at a non-

    cheater (George). If she is not, then a

    cheater (Jack) is looking at a non-cheater

    (Ann). Therefore, the answer is “Yes”.

    A constructive approach

    • Jack is looking at Ann but Ann is looking

    at George. Jack is a cheater but George is

    not. Is a cheater looking at non-cheater?

    • We get hold of Ann and try to cooperate

    with her in reciprocal altruism. She does

    not cooperate. Our “cheater detection

    mechanism” fires, and we conclude she is

    a cheater. Therefore, the answer is “Yes”.

    The distinction and dual

    process theory

    • Non-constructive inference is the purest

    example of a type 2 analytic process. It

    is an inference from “above”, using logic.

    • Constructive inference is from “below”:

    it is grounded in type 1 heuristic

    processes, such as those of perception.

  • 2

    Jonathan Evans’s list of dual process theories

    Perception & attention: Egeth & Yanis (1997), stimulus &

    goal driven attention.

    Skilled performance: Anderson (1983), procedural &

    declarative knowledge.

    Learning & memory: Reber (1993), implicit & explicit

    learning.

    Social cognition: Strack & Deustch (2004), impulsive &

    reflective.

    Reasoning & decision making: Evans (2006) and Evans &

    Over (1996), heuristic & analytic; Barbey & Sloman (in

    press), associative & rule based; Stanovich (1999) and

    Kahneman & Frederick (2002), System 1 & System 2.

    System 1 mental processes

    (Stanovich, 2004)

    • Associative

    • Holistic

    • Parallel

    • Automatic

    • Undemanding of cognitive capacity

    • Fast

    • Highly contextualized

    • “Old” in evolutionary terms

    System 2 mental processes

    (Stanovich, 2004)

    • Rule based

    • Analytical

    • Serial

    • Controlled

    • Demanding of cognitive capacity

    • Slow

    • Decontextualized

    • “New” in evolutionary terms

    Jonathan Evans’s characterization

    Type 1 processes: Fast and automatic, with

    high capacity and low effort.

    Type 2 processes: Slow and controlled, with

    limited capacity and high effort. These

    processes make use of working memory.

    Dual process theory is opposed to

    the massive modularity hypothesis

    • Many leading evolutionary psychologists

    have argued for what has been called the

    massive modularity hypothesis.

    • Some leading evolutionary psychologists

    do not accept this hypothesis, but the

    following support it: Cosmides & Tooby,

    Buss, Pinker, and Gigerenzer.

    Massive modularity implies:

    • There is no mental logic: no formal

    system for performing valid inferences.

  • 3

    Massive modularity implies:

    • There are no content independent

    mechanisms for inference or learning.

    Massive modularity implies:

    • There are only content specific or domain

    specific mechanisms – the modules - for

    solving adaptive problems.

    Massive modularity metaphor:

    • The mind is a Swiss army knife - it has

    many special blades for solving adaptive

    problems but no general purpose blade.

    Dual process theory implies:

    • Type 1 processes result from content

    specific mechanisms for perception,

    memory, and heuristic inference - the

    modules.

    Dual process theory implies:

    • Type 2 processes result from general

    purpose mechanisms, including a means

    of logical inference.

  • 4

    Dual process theory implies:

    • That the mind has two systems, System 1

    and System 2, or at least has two kinds of

    processes, type 1 and type 2. The Swiss

    army knife metaphor could also be used

    for dual process theory.

    My claim:

    • The best example of a type 2 process is

    non-constructive reasoning. A good

    example of this reasoning is inferring a

    disjunction, “p or q”, from “above”.

    Validly inferring a disjunction

    from “above”

    • We may infer, “Ann is a cheater or not a

    cheater”, from “above” using pure logic –

    in this case we cannot say which disjunct

    is true. We do not know which property

    Ann has: being a cheater or not one.

    Justifiably inferring a disjunction

    from “above”

    • We may infer, “Ann is a cheater or Jack

    is a cheater”, from “above” using

    probabilistic inference. Resources are

    missing. Someone is taking more than

    their share. Other general considerations

    point to Ann or Jack, but we do not know

    which is the cheater.

    What use is non-constructive

    reasoning?

    • We may infer, “Ann is a cheater or Jack

    is a cheater”, from “above”. This enables

    us to infer, “If Ann is not the cheater then

    Jack is.” That is a useful conditional to

    infer. For when we later get evidence that

    Ann is not the cheater, we may infer that

    Jack is.

  • 5

    Constructive reasoning is not

    useful in this way

    • Suppose we infer, “Ann is a cheater or

    Jack is a cheater”, from “below”, from

    Ann is the cheater. We now cannot infer,

    “If Ann is not the cheater then Jack is.”

    If it does turn out that our information

    from “below” was wrong, we do not have

    any reason to suspect Jack.

    Is non-constructive inference “old”?

    • It is often said that type 2 processes are “new”

    in evolutionary terms. Is this true of non-

    constructive inference? Do any other animals

    show signs of this kind of reasoning? The Stoic

    logician Chrysippus claimed that a dog could

    know that an animal went down one of three

    roads and infer that, if it did not go down the

    first two, then it went down the third. Is there

    any scientific evidence of such inference?

    The logical form of the inference

    The form is that of inferring “if not-p then q”

    from “p or q” or, equivalently inferring “if p

    then q” from “not-p or q”. My claim is that

    such inferences are justified only when the

    disjunction is inferred non-constructively. But

    in elementary logic, “if p then q” just means

    “not-p or q” and so such inferences are always

    justified, that is also so in the main

    psychological theory of conditional reasoning.

    The material conditional

    In elementary extensional logic, “if p then

    q” just means “not-p or q”, and so “if

    not-p then q” means “p or q”. This kind

    of conditional is the material conditional.

    The mental model theory of Johnson-

    Laird & Byrne (2002) implies that the

    ordinary conditional of natural language

    is the material conditional.

    What the mental model theory of

    Johnson-Laird & Byrne implies

    • Johnson-Laird & Byrne (2002, p. 650) hold

    that the following inference is valid:

    • In a hand of cards, there is an ace or a king or

    both. So if there isn’t an ace in the hand, then

    there is a king.

    • Now if the above were valid, then the ordinary

    conditional would be the material conditional,

    which Johnson-Laird & Byrne deny in places,

    but that is logically implied in what they write

    down as their theory.

    The form of the inference

    referred to by Johnson-Laird

    & Byrne (2002)

    • Inferring “if not-p then q” from “p or q”.

    • From “not-p or q”, we get “if not-not-p

    then q” by the form, from which we infer

    by double negation “if p then q”.

    • So one could equally well study inferring

    “if p then q” from “not-p or q”.

  • 6

    More logical points

    For all conditionals we must have that

    “if p then q”

    logically implies

    “not-p or q”

    But only for the material conditional,

    can the converse hold, as the material

    conditional just means “not-p or q”.

    If Johnson-Laird & Byrne

    (2002) are right

    • Then “if p then q” is logically equivalent

    to the truth function material conditional,

    “not-