Reasoning

Reasoning

What is reasoning? The world typically does not give us complete

information Reasoning is the set of processes that

enables us to go beyond the information given

What types of reasoning are there?

Validity vs. Truth Valid argument: true premises guarantee a true conclusion It does not necessarily correspond to the truth in the world

Deductive reasoning Allows us to draw conclusions that must hold given a set of

facts (premises) Inductive reasoning

Allows us to expand on conclusions Conclusions need not be true given premises Category-based induction Analogical reasoning Mental models

The logic of the situation You have tickets to the football game. Go Mean Green! You agree to meet Bill and Mary at the corner of Fry and

Hickory or at the seats If you see Mary on the corner of Fry and Hickory, you expect

to see Bill as well. If you do not see either of them at the corner, you expect to

see them at the seats when you get to the stadium. The agreement has a logical form

(Bill AND Mary) will be located at corner OR (Bill AND Mary) will be located at seats

AND and OR are logical operators They have truth tables

The logic of the situation

Simple logical arguments If you see Mary Bill AND Mary You expect to see Bill

Limits of logical reasoning

We are good at this kind of reasoning We do it all the time We can do it in novel situations

Are we good at all kinds of logical reasoning? What are our limitations?

Conditional Reasoning

Modus Ponens Modus Tollens

Conditional Reasoning

Each card has a letter on one side, and a number on the other

Which Cards must you turn over to test the rule: If there is a vowel on one side of the card,

then there is an odd number on the other side

Conditional Reasoning

Who do you have to check? If you have a beer, then you must be 21 or

older?

Conditional Reasoning

These cases are logically the same Valid Arguments: If premises are true, conclusion

must be true Affirming the Antecedent

P Q P

Q (Modus Ponens) Denying the Consequent

P Q NOT Q

NOT P (Modus Tollens)

Conditional Reasoning

Invalid Arguments: Conclusion need not be true, even if premises are true.

Affirming the Consequent P Q Q

P Denying the Antecedent

P Q NOT P

NOT Q

The ambiguity of if. In everyday language, sometimes implies a bidirectional relationship between P and Q (i.e. if and only if)

Logical thinking

Pure logic says that we should be able to reason about any content The Ps and Qs in the argument could be anything

However, we are more likely to accept an argument when the conclusion is true (in the real world) whether it is valid or not All professors are educators Some educators are smart

Some professors are smart

This conclusion may be true The argument is not valid It is possible that the smart educators are not professors

Logical thinking We are good with simple logical operators

AND, OR, NOT Earlier we saw content effects

Wason selection task With neutral content it is more difficult With familiar content it is easier

Social schemas are easy to reason about and may be context dependent rather that

Cheng & Holyoak; Tooby & Cosmides E.g. Permission: Some precondition must be filled in order to carry out some

action

More complex argument forms can be difficult, especially in unfamiliar contexts Why do we see these content effects? Valid deductive arguments ensure that a conclusion is true if the premises are

true Truth cannot be determined with certainty, thus we must generally reason about

content We will look at how people reason about content later

Inductive Reasoning

Luci’s presentation!

Abductive Reasoning

Say what? Another form of reasoning is provided by the philosopher C.S.

Peirce It essentially provides a means for coming up with rules based

on new instances experiences One way you might think of it is coming up with hypotheses

based on new findings (whereas deduction would deal with outlining the consequences of a hypothesis and induction in testing the hypothesis)

Observation: the grass is wet Explanation: it rained

The explanation is consistent with the domain of the problem

Abuduction

Deduction Necessary inferences (if A leads to B and B leads to C, then A

leads to C)

All balls in this urn are red All balls in this particular random sample are taken from this urn Therefore All balls in this particular random sample are red

Peirce regarded the major premise here as being the Rule, the minor premise as being the particular Case, and the conclusion as being the Result of the argument.

The argument is a piece of deduction (necessary inference): an argument from population to random sample.

Abuduction

Induction Interchange the conclusion (the Result) with the major

premise (the Rule). Argument becomes:

All balls in this particular random sample are red All balls in this particular random sample are taken

from this urn Therefore, All balls in this urn are red

Here is an argument from sample to population, and this is what Peirce understood to be the core meaning of induction: argument from random sample to population

Abuduction Abduction

New argument: Interchange the conclusion (the Result) with the minor premise (the Case)

Argument becomes: All balls in this urn are red All balls in this particular random sample are red Therefore, All balls in this particular random sample are taken

from this urn.

This is nothing at all like an argument from population to sample or an argument from sample to population: it is a form of probable argument different from both deduction and induction

Would later see these as three aspects of the scientific method

Scientific reasoning

Scientific reasoning Combination of reasoning abilities

Hypothesis testing Generate an explanation for some phenomenon Develop an experiment to test the hypothesis Seek disconfirming evidence

How good are people at this type of reasoning? How good are scientists at living up to this ideal?

Hypothesis Testing

Deductive side (conditional reasoning) If the null hypothesis is true, this data would not occur The data has occurred The null hypothesis is false

This is true by denying the consequent (modus tollens) Unfortunately this is not how hypothesis testing takes place

If the null hypothesis is true, this data would be unlikely The data has occurred The null hypothesis is false

The problem is that we make the first statement probabilistic, and that changes everything

Hypothesis Testing

If a person is an American, then he is not a member of Congress FALSE

This person is a member of Congress

Therefore, he is not an American

This is a valid argument but untrue as the first premise is false

If a person is an American, then he is probably not a member of Congress TRUE

This person is a member of Congress

Therefore, he is not an American

This is the form of hypothesis testing we undertake, and is logically incorrect

Hypothesis testing

Induction Take a sample, calculate a statistic Generalize to the population

Problem: often no real reason to believe the population statistic is a constant Example: though the transformed score is of course a

mean of 100 IQ, IQ raw scores have been improving over the past couple decades

Begs the question, to what are we generalizing? Just this population at this time?

Hypothesis testing

People tend to have a confirmation bias We seek confirming evidence

Scientists also show a confirmation bias They tend to be more critical of evidence that is

inconsistent with their beliefs. This always may not be a bad thing (Koehler)

Wason 246 task You are told to find a rule that generates “correct” three

number sequences. You are told that 2-4-6 is a “correct” sequence You search for the rule by testing as many sequences

as you want until you are confident you know the rule

Hypothesis testing

Confirmation bias Many people initially assume the rule is

“Sequences increasing by 2” They try sequences like “4-6-8” and “13-15-17”

These are sequences that would confirm their hypothesis

Few people try sequences that would disconfirm their hypothesis (e.g., “1-2-3” or “3-2-1”)

The actual rule is “Any increasing sequence” Few people find the correct rule

Hypothesis testing

Scientists ignore base rates (prior research) Bayes theorem allows for incorporating prior

probabilities to give a (posterior) probability about a hypothesis Yet most of social science does not use Bayesian

methods

Some do not realize that the end of their scientific efforts is a probability about data, not a hypothesis Not p(H|D) But p(D|H)

MC’s experience at Research and Statistical Support People (students and faculty) come in with: No clear hypothesis to test Lack of knowledge regarding the methods

that would allow a hypothesis to be tested Heavy reliance on prescribed ‘rules’ which do

little to aid their reasoning about the problem Vague notion as to which population they are

generalizing to And a host of other issues…

MC’s Suggestions for Having Fun with Science Have clear ideas

Regarding concepts (operational definitions), their implications, and the coherency of hypotheses regarding them

Sounds easy but is probably the hardest part and the source of most problems

Do not ignore prior efforts Sorry to break it to you, but much has been done in your area of

research Don’t be afraid to explore

Engage your natural curiosity (try new methods and really investigate your data)

Think causally Every method is an investigation of a causal model, what’s yours?

Remember the big picture Your research should speak well beyond its specific results (esoterism

≠ progress)

Importance of Content

Analogy and Similarity How do we use past experience? What are analogies? Structural alignment Similarity

What to do... How do you decide what to buy?

Use your past experience

How do you figure out which experience is relevant? Using prior knowledge

Use of prior knowledge is guided by similarity

How can we study this process? Studying pairs of items Study perceptions of similarity when all information is

available

Contrast model

Tversky (1977) Had people list features of concepts Had other people rate the similarity of concepts Compared the feature lists

Similarity increases with common features, similarity decreases with distinctive features Similarity ratings were positively related to the number

of common features Similarity ratings were negatively related to the number

of distinctive features

Analogy Often, things being compared are not very similar.

Atom vs. Solar system

Analogies preserve relations The Atom and the Solar System have similar relations among their parts.

The Atom cause( greater(charge(nucleus) charge(electron)), revolve(electron,nucleus))

The Solar System cause( greater(mass(Sun) mass(planet)), revolve(planet,Sun))

The attributes of the objects are not similar. The nucleus is not hot, the planets are not small etc.

Structure mapping

Structured representations

Relations connect the objects

Items are placed in correspondence when they play the same role in a matching relational system

Analogical Inference

Can make inferences about target domain

Inferences based on correspondences between the base and target

Allows us to learn from experience

Types of similarity

Focus on alignable differences Gentner & Markman (1994) Ss given 40 word pairs

20 highly similar, 20 highly dissimilar Hotel-Motel Magazine-Kitten

List one difference for as many pairs as possible in 5 minutes

More differences listed for similar pairs than dissimilar pairs Reflects that alignable differences are easier to find for

similar pairs than for dissimilar pairs

Similarity and cognition

Similarity enables us to use background knowledge Recognize how a new case is like an old one

Structure mapping/structural alignment Relations are important in similarity comparisons Commonalities and Alignable differences are key

Nonalignable differences are less important Differences are easy to find for similar things

Structural alignment affects cognitive processing

Reasoning and Mental Models

Mental models Intuitive Theories and Naïve physics

Mental Models and Intuitive Theories

Mental models allow us to reason about devices Kind of like scripts and schemas discussed earlier

People often have causal information about the way things work Used to allow us to get through the world Information may be flawed

Three types of mental models Logical mental models Analogical mental models Causal models

Logical and Analogical Models Logical mental models Used to solve logic problems

Johnson-Laird Contain “empty” symbols that are manipulated

All Archers are Bankers No Bankers are Chemists ?

Useful primarily for logic puzzles

Analogical mental models Sometimes we understand one device by analogy to another

Electricity and water flow Voltage <--> Water pressure Current <--> Flow rate Resistance <--> Width of pipe

Causal Models

Causal models allow us to explain and understand the world around us

Note that it is not exactly clear what may be determined what a cause is, and that is a separate question from how we come to a determination of a causal relationship White 1990, Ideas about Causation in Philosophy and

Psychology Nevertheless our (often flawed) notions of causal

relationships can have profound effects on our ability to reason and understand

Intuitive Theories

Naïve physics What would happen to a ball shot through this

pipe? People often respond by assuming curvilinear

momentum McCloskey and Proffitt Even happens if they carry out an action

Intuitive Theories Why do we err? Our naïve physics matches our

observations The world has friction, and so

there are unseen forces that act in opposition to seen forces

Our naïve physics is often accurate for things we can do with our bodies

Only when we create larger machines do the differences become important

Should not be a surprise Newtonian physics is only a

few hundred years old Aristotelian mechanics is

closer to our daily experience

How deep are our models?

Shallowness of explanation Keil

People believe they understand more than they do Asked college students about devices

Toilet, Car ignition, Bicycle derailleur Said they understood devices, but could not actually explain

them Why does this happen? When we know how to use an object and it is familiar, we

believe we know how it works

Summary

Mental models Logical mental models Analogical mental models Causal mental models

Naïve physics Physical beliefs sometimes diverge from truth Sufficient to get us around the world

Scientific reasoning People generate pretty good tests Often show a confirmation bias