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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