120A Final Exam Study Guide
1. KNOWLEDGE REPRESENTATION
Exemplar vs Prototype theory
● Exemplar Theory (concrete): Idea of dog is represented by all the dogs I’ve ever seen
● Prototype theory (abstract): We forget all the dogs we’ve seen
Problem with Classical theory: What is in and what is out?
Problems with Probabilistic approaches (Prototype and Exemplar) ● Both rely heavily on idea of similarity, which is vague and variable ○ Problem: similarity is context dependent (categorization varies based on what you are told about the figure)
○ Requires understanding of the relevant dimensions of similarity. ○ Similarity requires some metric assumptions that are not always met ○ Similarity may capture some shared properties among concepts, but it does not explain why two concepts are in the same category!
If you want to learn more check out Which orbitals do not exist based on the rules of quantum mechanics?
Theory-based categorization - We have an intuitive/implicit theory about what goes in a category
● We understand and categorize concepts in terms of implicit/intuitive theories about the world
● Categorization is knowledge-based inference (rather than similarity-based judgment)
What is “language”?
● A set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements (Chomsky 1957) We also discuss several other topics like Can our minds simultaneously operate?
● A system of symbols and rules that enable us to communicate (Harley, 2008)
Computational complexity & cognitive constraints
A. Mary can ride a bike, but Jack cannot
B. Mary can ride a bike, but Jack cannot [ride a car]
C. Mary is able to ride a bike, but Jack is not allowed to
- You’d be surprised if B or C were what I meant
● Parallel interpretation is like computational complexity If you want to learn more check out What set of decisions is involved in strategic management?
● Somehow in our minds we have constraints that guide our interpretation of language (like vision)
Properties of language
1. Symbolic: makes use of arbitrary relation between sounds and meaning (e.g., “dog”) 2. Discrete infinity: a finite set of elements can generate a (potentially) infinite set of ‘meanings’ (cf. Descartes)
3. Structure dependence: meaning is conferred through a specific arrangement of symbols
4. Displacement: language allows referring to ideas/elements that are not “there” 5. Organized at multiple levels: Sounds, words, sentences, paragraphs and text ● Phonology, morphology, syntax
● Phonemes = smallest unit of sound
● Morphemes = smallest unit of meaning
● Syntax = combining words together to convey a certain intention/meaning
Phonemes: the smallest unit of speech that can be used to distinguish one utterance from another (in a given language) Don't forget about the age old question of What did napoleon find in egypt?
○ Children can acquire any language they are dropped into If you want to learn more check out How proteins and carbohydrates are digested and absorbed by the body?
○ But they develop a certain sensitivity threshold for the phonemes of their own language
Waveform & spectrogram
● Phoneme/word segmentation
○ How do babies know which sounds go with which words?
● Coarticulation: the pronunciation of a phoneme is changed by the following phoneme If you want to learn more check out What are gnostic units for?
○ Between phonemes in same word: “Pool” vs. “Place”
○ Between phonemes in different words: “You can’t handle the truth” ■ Space between words is an illusion: we actually just string our
words together in one continuous string
The McGurk effect - We use visual information to help us understand auditory information
● Video: We hear the sound “Ba” and think it is “Fa” if the man changes his mouth to make it look like he is making an “F” sound
Morpheme - smallest unit of meaning within a given language
● Not meaningful in and of themselves
● Divided into root words and affixes (prefix, suffix)
○ “Dog” is single morpheme
○ “Dogs” is 2 morphemes (“Dog” + “s”)
Syntax - combining words to create meaningful sentences
● Rules are for categories of words, not for single words
● Simple sentences - finite number of possible sentences
○ “Mark likes Jack”
● Complex sentences - recursion allows infinite number of sentences with finite number of elements
○ “Mark likes that [Jack likes that][Mary hugged Mark]”
● Phrase structure
○ We perceive language as a linear signal, but our mind creates a rich hierarchical representation
● Grammar/syntax can be detached from meaning
○ Gibberish sentence can have correct syntax, but a meaningful sentence written backwards has incorrect syntax
● Syntactic ambiguity
○ The girl looked at the boy with the telescope: who has the telescope? ○ Garden path sentences - the sentence leads you down the wrong hierarchical representation → you have to go back and recreate the tree ■ “The man returned to his home was happy” (the man was returned to his home by someone else)
How do we learn language?
● Hearing language in our environment
○ Behaviorism - association/reward
■ Wrong because children aren’t rewarded for correct grammar
■ Language was the death of behaviorism. No type of instrumental or classical conditioning could explain the ability of children to acquire language
○ Scientific induction - the idea that you are Google, and you acquire language through lots of data and patterns
Evidence against Scientific Induction (we are NOT like Google): ● Uniform stages of language acquisition
○ Children of different backgrounds have a similar timeline of language development
○ If it were purely through scientific induction, it wouldn’t be this uniform ● Socioeconomic status (SES) and language
○ Children of different SES:
■ Language, structure, meaning? → Same
■ Vocabulary → Different
● Children acquire language uniformly even in the face of (some) pathologies: ○ Blind children acquire color words or sight-related words just like sighted children
● Children acquire language uniformly and suddenly despite different cognitive abilities
○ We all have that one dumb friend who we wonder how they even survive through life, but their language ability is as good as ours
● Language learning is largely unguided (i.e., parents don’t give real feedback) ● Children raised in pidgin language environments creolize the languages ● Poverty of the stimulus: Most of language is not fully formed sentences, but children are able to develop fully formed, ideal language
● Non-human primates can only pick up a few words, don’t use language productively, don’t pass it down to their children – yet they perform other tasks at the level of a 5-year-old human child (at that age, human children have high level of language)
Plato’s Problem: How can children with different linguistic environments arrive at an accurate grammar relatively rapidly, and with finite input?
Chomsky: Universal Grammar (Experience → Language Faculty → Grammar) ● Language faculty is a genetically transmitted algorithm for developing grammar ○ The reason for sudden and uniform acquisition of language is that our brain has a biologically endowed mechanism for language
● Universal Grammar - a set of genetically endowed grammatical principles which determine the structure and range of possible grammatical operations in natural language
○ Must be general enough to pick up any language, but specific enough that you can separate different languages
● Principles & Parameters
○ A person's syntactic knowledge can be modeled with two mechanisms: ■ A finite set of fundamental principles that are common to all
languages (e.g., a sentence must always have a subject)
■ A finite set of parameters that determine syntactic variability
amongst languages (binary parameter that determines whether a
given principle is ON or OFF; e.g., Do you have to say the subject
of a sentence?).
● Null subject parameter: do finite verbs allow for null
○ English: Maria speaks French → Speaks French*
(can’t omit “Maria” because it is unclear)
○ Italian: Maria parla Francese → Parla Francese
● Wh- parameter: can wh- expressions be fronted?
○ English: What do you think he will say?
○ Chinese: 你相信他会说什么？ [You think he will say
● A child’s learning task is 2-fold:
○ Structural learning (parameter setting): Determining the appropriate value for each parameter → ON or OFF
○ Lexical learning: Learning the language specific symbolic association between meaning and sound
Language in the Brain
● Broca’s area
● Wernicke’s area
Broca’s aphasia - nonfluent aphasia
● Trouble with speech production (short, simple sentences)
● Reading and writing not as affected
Wernicke’s aphasia - fluent aphasia
● Impaired comprehension (to varying degree)
● Impaired production of meaningful sentences (yet retain grammatical structure)
3. LANGUAGE AND THOUGHT
If you spoke a different language, or didn’t learn how to speak any language, would you be a different person?
● Theoretical positions
○ Linguistic relativism
○ Core knowledge systems and natural language
● Studies of language and thought
○ Color perception, Arithmetic cognition, Theory of mind, Spatial cognition ○ Evidence from: healthy individuals from different cultures/languages, “pre-linguistic” infants, aphasic patients, non-linguistic animals
1. Universalism - Thought → Language
● Aristotle, Fodor, Pinker
● Words are just the signs of ideas
○ Physical reality → thoughts/concepts → Language
○ Color spectrum → wavelengths → color names (blue, green…) ● Fodor’s Language of Thought (LOT)
○ Language = expression of thoughts
■ Evidence: sometimes we say things wrong, not what we mean
2. Piaget - Thought → Language
● Language development is a result of cognitive development
● Four stages of development (children):
○ 1. Sensory-motor
○ 2. Pre-operational
○ 3. Concrete operational
○ 4. Formal operational
● Object permanence: Developed during sub-stages 5 and 6 of sensory motor stage
● Prerequisite model (Tommasello & Farrar 1986)
○ Visible object movement words (‘move’, ‘fall-down’, ‘up)
○ Non visible object movement words (‘gone’, ‘find’, ‘more’)
○ Stage 6 children could learn and produce both types of words, but stage 5 children could only do so with visible displacement
3. Vygotsky - Thought ←→ Language
● Language and thought have independent origins, but they become interdependent through development
● Two functions of language: External vs. Internal
○ 1. Language and thought are unrelated when we’re babies (Beginning Stage). Children can have non-verbal thoughts (thought without
language), use of first words & babble is pre-intellectual (language without thought)
○ 2. External Speech (Middle Stage). Loose correlation between actions and the meaning and timing of words
○ 3. Internal Speech (Final Stage). External speech turns into internal speech, making more complex thoughts possible
4. Linguistic Relativism & Determinism (aka Whorfian hypothesis) - Language → Thought
● von Humboldt, Sapir, Whorf
● Language is logically prior to thought, and determines the kinds of thoughts someone can have
○ Linguistic labels shape and organize our thoughts
Linguistic Determinism - effects of language on thought (how strong a hold language has on your thoughts)
● Forms of linguistic determinism
○ 1. Strong: language determines thought
○ 2. Weak: language influences thought (makes it easier or habitual to think one way or another)
○ 3. Weakest: language affects memory because length and frequency of the name for something makes it easier to remember
○ 4. Cognitive Linguistic Determinism (Hunt & Agnoli 1991): Different languages make some thoughts easier/harder, hence, some thoughts or lines of reasoning may be easier in one language than in another
Linguistic Relativism - how thought is affected by language
● Language variability across languages (different labels/grammar)
→ Different organization-perception of the world
→ Different cognitive structures
● Specific level: the specific words a language has to describe something affects how the speaker thinks about it (e.g. words for snow)
● General level: the general grammar make-up of a language will make some concepts possible or absent (e.g. time)
Early evidence of Whorfian hypothesis
● Eskimo Snow
○ Different words for snow
● “Timeless” Hopi
○ No words to distinguish past present future
Problems with the early evidence of Whorfian hypothesis (Disproving Whorf strengthens argument for Universalism)
● The Great Eskimo Vocabulary Hoax: English has lots of terms for snow too ● Non-Grammatical Time: Hopi may not have grammatical markers as English (or Italian, French, …) have, but they do express time in other ways
● Translations: Whorf used simplistic word-by-word translations
● Circularity: Differences in language are taken to imply differences in thought, and the evidence for the differences in thought is that there are differences in language …
● Culture: it is hard to tell whether differences in vocabulary have to do with thought versus cultural and environmental factors (e.g. words for snow).
Evidence for Whorf AND for Universalism
● Navaho Verb Endings (Carroll & Casagrande 1958)
○ Hypothesis: Navaho expected to pay more attention to object shape because of how their language works.
○ Task: Look at 2 objects (yellow rope and blue stick), pick which one they would pair with the yellow rope
■ Navaho-Navaho (70%) > English-Navaho (40%)
■ Supports Linguistic Determinism
○ English-English (80%) ≈ Navaho-Navaho (70%)
■ Evidence against Linguistic Determinism/ Evidence for
Universalism (if language influenced thought, English wouldn’t be
similar to Navaho)
A Colorful Debate - Support for Linguistic Determinism
● Brown & Lenneberg (1954) - English speakers only: Colors with a verbal label are better recalled.
● Lenneberg & Roberts (1953) compared English v Zuni speakers. In Zuni, same word for yellow and orange. Zuni tend to mis-recall yellow/orange ● Robertson et al (2000) English v New Guinea Berinmo.
○ English & Berinmo have different colour boundaries
○ Categorical Perception in recall:
■ Berinmo confuse Blue/Green
■ English confuse Wor/Nol
A Colorful Debate - Evidence against Linguistic Determinism ● Kay and Regier (2003) - Is colour naming arbitrary across culture (110 languages)?
○ Color terms cluster across languages
■ Focal Colors = colors that are most discriminated due to genetic makeup (most important colors)
○ Color vocabulary develops systematically across languages
■ If 2 colors in a language, will be black and white
■ If 3 colors, will be black, white, and red
● Rosh Heider and Olivier (1972)
○ English v New Guinea Dani (‘mola’ bright, ‘mili’ dark)
○ Color memory was the same despite dramatic differences in naming ● Bornstein, Kessen and Weiskopf (1972)
○ What is the “pre-linguistic” state of colour perception?
○ Habituation paradigm: Show infants one color until they habituate, then show a new color (either in the same category, or across category) ■ Infant loses interest (habituates) if keep showing same color
■ Infant regains interest if show new color
● Shows that the infant can tell the difference between
■ Despite not having color vocabulary, they can recognize the same color boundaries that we recognize
● Davies & Corbett (1997): Setswana v English v Russian on the blue-green boundary.
■ Setswana: 1 word (“botula: blue-green”)
■ English: 2 words ( “green”, “blue”)
■ Russian: 3 words (“zelenyj: green”, “sinij: dark-blue”, “goluboj: light blue”)
■ 1. Sorting patterns were similar in all 3 groups (evidence AGAINST Whorf)
■ 2. Setswana more likely to group blue and green (evidence FOR Whorf)
■ 3. Russians did NOT group light-dark blue differently (evidence AGAINST Whorf)
5. Core knowledge systems and natural language - Language → Thought ● Language brings modules together
○ Language is like an octopus, can reach into each of the modules ● Continuity/discontinuity of the human mind
○ Continuity: sensory-motor
○ Discontinuity: cognitive
● Old core knowledge systems and new knowledge (modules)
● Properties of modules
○ Domain specific
○ Task specific
○ Informationally encapsulated
● Examples of modules
○ Object mechanisms
○ Social interaction
○ Goal directed actions
○ Natural geometry (navigation)
○ Numerosity “Number sense”
■ Habituation paradigm: Infants can tell the difference between
different number of items.
● Recognize different quantities, to a certain extent
● Can differentiate between 8 and 16, but not 8 vs 12
● Approximate aural numerosity (different number of sounds)
■ Skinner box (instrumental learning): rat has number sense
● Rats can learn how many times they have to press the lever
in order to get food
■ Abstract, approximate representation
● Weber’s Law: as the number increases, the animal’s
performance gets worse
● Subitizing - the ability to perceive small # of items without counting (1-3) ○ Instantaneous
○ Small-numbered, exact system
○ Exact calculation in pre-linguistic infants
■ Experiment: Start with one Mickey Mouse. Add another
● Control: End up with 2 Mickeys (likely outcome,
shouldn’t be surprised)
● Manipulation: Take the original Mickey out, so there’s
only one Mickey, even though there should be 2
(impossible outcome, child should be surprised)
■ Results: Longer looking time for impossible outcomes
○ Habituation experiment: also shows subitizing in infants
○ Exact small number calculation in non-linguistic animals (rhesus monkeys)
Limitations to number cognition (subitizing & number sense)
● You can represent a few distinct/exact individuals and add/subtract one; but you’re limited to “one, another one, and another one”
● You can represent large numbers of objects/events as sets with cardinal values ● You can’t represent sets of individuals
What makes us smart?
● The compositional semantics of natural language “provides the medium for combining the representations”
Mathematics and Natural Language
(O’Keine and Spelke et al)
● If core knowledge hypothesis is correct, how do you expect people to do in the trained and untrained language for exact small number (subitizing)? ○ Should do equally well
● If the core knowledge hypothesis is correct, how do you expect people to do in trained and untrained language for approximate large number?
○ Equally well
● If the core knowledge hypothesis is correct, how do you expect people to do in trained and untrained language for exact large number?
○ Better in trained language
→ Language plays a role in getting exact large number
● If core knowledge hypothesis is right, should non-human primates be able to calculate exact large number?
Evidence against Core knowledge systems hypothesis
● Brannon and Terrace (1998)
○ Monkeys can put items in order for exact small number AND exact high number
● Monti et al (2012)
○ Language parts of brain are not important for algebra
● Varley et al: Math syntax and Lang syntax
○ Aphasia patients can do algebra even though language is impaired
Bottom line: People are NOT rational decision makers
Expected Utility Theory
● This theory says that people are rational decision-makers
○ Have well-specified preferences
○ Choose the option that gives you the best expected utility (highest probability that it will turn out well)
■ Utility: the value of an outcome
■ Probability: the likeliness of an outcome
■ Expected Utility = Probability x Utility
Value and subjective value
1. Which one do you prefer:
○ a. Win $100 for sure
○ b. 50/50 chance of winning $200 or nothing
● Everyone picked (a)!
○ The subjective value of $100 for sure is greater than that of $100 expected!
○ When dealing with gains, we are risk averse (don’t want to take risk)
2. Which one do you prefer:
○ a. Lose $100 for sure
○ b. 50/50 chance of losing $200 or nothing
● Everyone picked (b)!
○ The subjective value of -$100 for sure is smaller than that of -$100 expected!
○ When dealing with losses, we are risk prone (more likely to take risk)
The subjective value of an amount of money is not the same as its monetary value! ■ e.g., What is the value of $1,000 to you? What is the value of $1,000 to Bill Gates?
● According to Expected Utility theory, we should have picked 50/50 for both ● So are we irrational?
Subjective Expected Utility Theory
● People are rational decision-makers
○ Have well specified preferences
○ Choose the option that maximizes their subjective EU
■ Subjective Utility: the subjective value of an outcome
■ Subjective Probability: the perceived prob of an outcome
■ Subjective Expected Utility = SP x SU
Evidence against Expected Utility Theory
1. Framing - how a question is framed influences your decision
● Would you rather save 200 people or have a ⅓ probability of saving 600? ● Would you rather let 400 people die, or have a ⅓ probability that nobody will die? ● These examples violate the Independence rule of Expected Utility Theory:
○ Preferences among alternative outcomes depend on how they are described!
2. Emotion influences decision making
● Trolley dilemma: You have to pull a lever to let the train kill only one person instead of five people
● Bridge: You have to push a person off the bridge to stop the train ● Best friend: What if the one person is your best friend?
From a numerical point of view, they are the same, but they FEEL different → Violates Expected Utility theory
3. Context Effects - we make decisions based on how appealing an option is compared to other options
1. Closer restaurant with low quality, further restaurant with mediocre quality, farthest with best quality
2. Closer restaurant with low quality, farthest and mediocre quality, kind of far and best quality
- Any time you have a crummy version of something, it shifts you toward a slightly better choice
- Violates Independence rule of Expected Utility: Preferences among alternatives are sometimes affected by irrelevant (in fact, asymmetrically dominated) alternatives
Heuristics and biases - using intuition to make decisions
● Reduce cognitive load by employing intuitive strategies (rather than full analytical reasoning) and simplifications to guide our decisions
○ Often get us into sub-optimal options and expose us to fallacious thinking and systematic errors
● Heuristics: mental shortcuts, rule of the thumb
● Biases: “tendencies” that people exhibit in their decision making
Types of Biases
1. Illusory correlations - false associations
● Tendency to see cause-effect relationships between variables even when there is none (personal prejudices and stereotypes play a big role)
○ E.g., racial stereotypes
2. Confirmation bias
● Tendency to search for or interpret information in a way that confirms one's existing beliefs, expectations or hypotheses.
Types of Heuristics:
1. Satisficing (Good enough)
● Rather than “rational” as described by Subjective Utility Theory, individuals show bounded rationality – rational, but within limits.
● We have limited cognitive resources and limited time, we satisfice instead of maximize/optimize:
○ Instead of picking “the best option” (after having considered all of them), we go through options one by one and once we find a “good enough” one we pick that one.
● People expect that a sequence of events generated by a random process will be representative of a longer random sequence (e.g., we thought BBBB was not likely because all the other sequences were GBGB, but really it’s equally likely)
● In flipping a coin, people think H-T-H-T-T-H is more likely than H-H-H-T-T-T (which seems nonrandom) or H-H-H-HH-H (which seems like an unfair coin)
● Our behaviors are often dictated by misperceptions of probabilities ○ We are more afraid of sharks than lightning
● How easily and frequently examples come to mind affects our probability of judgments
● People are influenced by an initial anchor value
● Anchor value may be unreliable or irrelevant, and adjustment is often insufficient ○ E.g.: Participants asked to calculate in 5 secs the answer to one of the following problems
■ 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8
■ 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
■ People overestimate when the first number is higher, underestimate when first number is lower
● Application of heuristics often leads to fallacious thinking
● Gambler’s fallacy
○ Example of the Representativeness heuristic gone bad (The mistaken belief that the probability of a random event is influenced by previous random events)
■ E.g., the number, you are “due” to win the next one, while of course each gamble is independent
● Hot hand fallacy
○ The mistaken belief that a certain course of events will continue ■ Most people believe in “hot streaks” for basketball shooting
■ Gilovich et al. (1985)
● Found that the hot streak is a fallacy
● The chance of making a shot does not depend upon whether
the player has made the previous shot
● The Linda problem (Conjunction fallacy)
○ We focus on the irrelevant info we are given about Linda, and assume there is a high probability that she is a successful bank teller AND an active feminist (but it’s more likely to just be a successful bank teller) ○ People don’t think mathematically (we’re irrational haha)
● Bayes Rule: the Bomb problem
○ We tend to think that probability of bomb given positive test equals probability of positive test if there is a bomb (WRONG!)
● Even pros (experts) don’t get it right!
○ If you have positive mammogram, what is chance of having breast cancer? → doctors say 75%
■ Bayes Rule says 9%!
■ The doctors ignore the base rate (0.8%)
Subjective Expected Utility Theory is WRONG
● Are people are rational decision-makers? NOPE
1. Our preference set is not complete, transitive, or independent
2. Our decisions are influenced by all sorts of factors, and often make use of “heuristics” which lead to non-rational decisions
3. We are pretty bad at estimating probabilities
Two Systems (theory of decision making)
● System #1 rapidly generates intuitive answers, which can be
monitored/evaluated by System #2, though this latter stage may be rare ○ System 1: Intuitive, automatic, produces heuristics
○ System 2: Analytical, conscious
■ System 2 is rarely used
● Example: Bat and ball problem
○ A bat and ball cost $1.10 together, how much does the ball cost if the bat costs $1 more than the ball?
■ The ball actually costs 5 cents: $0.05 + $1.05
→ It’s not that people can’t do the math, it just takes more complex processing
What is problem solving?
- Problem solving is cognitive processing directed at transforming a given situation into a goal situation when no obvious method of solution is available
Well-defined vs ill-defined problems
● Well-defined (e.g., mazes):
○ Clear initial state
○ Specific criterion for goal state
○ Clear operators
○ Clear path constraints
E.g., Animal crossing (you have to bring fox, chicken, and corn across river without them eating each other)
● Ill-defined (e.g., finding the perfect mate)
○ Initial resources not obvious
○ Unclear when goal has been reached
○ What actions do I take?
○ Are there any rules?
Solving Well-structured Problems
● Algorithms: series of operations applied over the problem space that may be represented over and over again, and continue until a final state is reached (very common in computers)
○ Example: Long division
● Heuristics: informal, intuitive strategies that sometimes lead to a effective solutions (and sometimes do not!)
○ Sometimes you have to go away from the goal to get to the goal ■ Example: hill climbing (hill dips down and then goes up again)
■ Example: dog gets to bone (has to go around the fence because can’t go over it)
Means-End Analysis - A heuristic based on creating sub-goals to reduce the distance between current state and goal state.
● Example: Tower of Hanoi
○ Goal: Get all disks on last peg, ordered by size with largest on bottom ○ Solution: Set sub-goals → must be willing to temporarily move away from the goal
● Example: Lily pad problem
○ A lily pad doubles in size every day. If on the 90th day the pond is totally covered with lily pads, on what day is the pond half covered?
○ How to solve problem?
■ Working forwards: forwards step-by-step approach to problem
■ Working backwards: backwards step-by-step approach to problem solving
● You don’t have to map out all the days, just work backwards
from the last day → more efficient solution!
Bronze coin problem
● It’s fake because if it really were made in 544 B.C., it wouldn’t say B.C., because the people of that time didn’t even know who Christ was yet
Woman married 20 men
● She conducted their wedding ceremonies
Insight problems - problems in which you need to restructure the initial representation of the problem
Examples of poor representation
● You have loose black and blue socks in a drawer, mixed in a ratio of 5 black to 1 blue. How many socks do you have to pull out to make sure you will have a pair of the same color?
● Sometimes, information is not missing, it is just obscured: the numbers here are a distraction
○ You only need to take out three socks (if you take a black and blue one, just need one more to have another blue one)
2. supersonic bumblebee
● Bee flies between two trains → The bee travels 1000 miles
How well do people predict how close they are to the solution when they solve insight vs non-insight problems? (Metcalfe & Wiebe, 1987)
● Non-insight: seemed like subjects were gradually getting closer to the solution ● Insight: seemed like subjects had no idea at all for the whole time, then suddenly figured it out
Insight vs non-insight problems
● Insight: problem-solvers are poor at predicting success or ‘closeness’ to solution ○ Sudden understanding of what is needed for the solution
○ Combining information in new ways
● Non-insight (routine): problem-solvers good at predicting their success and at monitoring how close they are to a solution
Gestalt Approach - Different modes of thinking
○ Productive thinking: involves insights that go beyond the bounds of existing associations
○ Reproductive thinking: based on existing associations involving what is already known (doesn’t require adding anything new)
● Insight occurs during productive thinking when the problem is suddenly restructured and the solution becomes clear.
● Non-human primates (Kohler 1927)
○ Monkey has to stack the boxes to get to the fruit
● Humans (Maier 1931)
○ Goal: hold one string in each hand
○ Problem: the strings are too far apart to grab one while holding the other ○ Solution: There are scissors on the chair. Person has to tie the scissors to one string and swing the string.
Does insight really exist?
● Evidence for insight
○ Metcalfe & Weibe (1987): warmth ratings → seems like insight exists ● Evidence against insight
○ Novick & Sherman (2003): Anagram task (unscramble letters → words) ■ Although insight appears sudden, in fact much processing goes on before insight is reached, and before we are conscious of it (parallel vs serial processing?)
Obstacles in problem solving
● Mental set - a frame of mind involving a certain representation of a problem, its context or a procedure for solving it
○ Mental set can mess up how you solve a problem. You are stuck in a mental set but you need to approach the problem in a new way
Mental Sets: Examples
1. Matchsticks equations
● Task: Move one matchstick so the equations are balanced
2. Water Jar
● The last problem requires a different problem-solving approach, but people try to use the solutions to the previous problems to help them solve the last problem 3. Nine Dots
● Task: draw lines through dots without lifting pencil
● Literally have to think outside the box
Functional fixedness - The inability to realize that something that has a certain use might also be used for performing other functions
● E.g., you have candle, matches, thumbtacks. You need to tack the candle to the wall but the thumbtacks aren’t long enough to stab through the candle ○ Solution: tack the matchbox to the wall
● Participants fixate on the box as having only one function: keep matches. ● When the match-box was shown empty, with matches on the table, Ss were more likely to solve the problem.
Transfer - Influence of previous problem solving experience on present problem solving ● Positive transfer: when our experience improves problem solving ● Negative transfer: when our experience disrupts problem solving (e.g., functional fixedness, mental set)
● Near transfer: beneficial effect of previous problem solving on problems in a similar context
● Far transfer: beneficial effects of previous problem solving on different contexts
Factors affecting transfer
● Task similarity: similarities between problems in superficial & structural features ● Context similarity: similarities in physical and social context
● Time interval: the period of time between the past and the present problem ○ Time decreases negative transfer (because over time, mental set is refreshed)
Elephant statue, gold, river problem
● Chinese students have higher solving rate than American students because they have heard a an ancient Chinese story like this
● Use story about radiation to find solution to story about the general with small army (apply what you learn in one problem to another problem)
○ Both involve combining multiple small forces, adds together to make big effect
● ~40% solved the radiation problem with no further help
● ~40% solved it after they were told it was relevant to solving the radiation problem
Incubation - Putting a problem aside for a while without consciously thinking about it ● Often decreases negative transfer and other problems
Example: Incubation and subconscious processing
● Subjects are asked “How many uses are there for a chair?”
● Given a math distractor
● Afterwards, tested to see how many uses for chair they can think of ○ Aware vs. unaware: Some subjects were told they would be tested afterwards (these guys did better than people who didn’t know)
Expertise - Experts have extensive knowledge about a particular domain, and they use it to organize, represent, and interpret information
● Thus expertise affects their abilities to remember, reason, and solve problems Chi, Feltovich, and Glaser (1981) - Experts see the world differently ● Task: categorize simple physics problems
● Subjects: novices vs. Ph.D. physics students
● Results: Novices grouped problems based on surface features (having an inclined plane, using a spring), experts sorted according to the physical principles relevant to the problems.
Expertise and Brain changes
● Ex. London taxi drivers “The Knowledge” (knowledge of London’s streets) ○ Posterior Hippocampus grows
○ Anterior Hippocampus shrinks
Reasoning - process of drawing conclusions from principles and from evidence
Inductive vs Deductive reasoning
● Inductive: Particular → General
○ Productive inference
○ Deriving novel information (you produce new information)
○ It’s about probability (you evaluate the inference based on how likely it is to be correct)
● Deductive: General → Particular
○ You already knew the general principle, you’re just applying your previous knowledge
○ Non-productive inference
○ It’s about certainty (50/50 chance to be correct, it’s either right or wrong) Examples of inductive and deductive reasoning
● Inductive reasoning example
○ 85% of known killers who use severe blunt force trauma to the faces of their victims live with their mother
○ 75% of known killers who tie up their victims during a crime are between the ages of 25-31, drive a 4x4 truck, are white, and are highly intelligent ○ Therefore, the offender may be a white male, age 25-31, who lives with his mother, and drives a 4x4 truck (it’s likely that an offender has these traits, but not necessarily)
● Deductive reasoning example
○ Fingerprint in blood is on knife
○ Fingerprint belongs to D
○ Therefore, D was in contact with knife after victim began to bleed (there is no other option)
The problem with induction
● We would like to form universal generalizations about the world ● All X’s are Y
○ e.g., All swans are white
○ These are based on our previous experience (Swan #1 was white … Swan #3265 was white → therefore, all swans are white)
● BUT: Induction is only guaranteed if we have experienced all possible instances ○ e.g., there are black swans in Australia
Benefits of inductive reasoning
● Computational complexity: It helps reduce uncertainty and predict possible events (at the cost of sometimes being wrong)
○ Helps us make sense of the world
● It’s about drawing secure conclusions given some ‘already known’ information. ● Some vocabulary:
○ Argument: A set of statements
■ Each argument must have at least 1 statement. If more statements, then the last one is referred to as the “conclusion,” all the other
ones are “premises”
■ Arguments are evaluated for validity (and not probability!)
Truth vs. Validity
● Truth - depends on relationship between assertion and the actual world. ○ Snow is white (T)
● Validity - depends on relationship between form of argument and conclusion. ○ Validity is a property of an argument, not a property of a given statement ○ Validity is a formal property: It depends on the logic structure of the argument, not on its semantic contents
● Modus Ponens (Valid, intuitive)
● Modus Tollens (Valid, not intuitive)
● Denying the Antecedent (invalid)
● Affirming the Consequent (invalid)
P → Q
(If P then Q. P. Therefore Q.)
P → Q
(If P then Q. Not Q. Therefore not P.)
Denying the antecedent (Invalid):
P → Q
(If P then Q. Not P. Therefore not Q.)
If I am from California, then I am from the USA
I am not from California.
I am not from the USA? → Wrong! I might be from Texas, so I am still from USA
Affirming the consequent (Invalid)
P → Q
(If P then Q. Q. Therefore P.)
If I am from California, then I am from the USA
I am from the USA.
I am from CA? → Wrong! I might be from Alabama
How do we solve deductions?
1. Mental rules (Osherson 1974, Rips 1994)
○ Reasoning makes use of mental representations that resemble propositional (“language like”) representations
○ Reasoners manipulate these representations by applying syntactic rules of inference
○ Different inferences might be of different complexity according to whether there is a rule for it or not
■ Modus ponens has a rule → easy
■ Modus tollens does not have a rule → difficult
● Mental Rules: Modus Ponens
○ (A v B) → C
(If A or B then C. B. Therefore C.)
● Mental rules: Modus tollens
○ (A v B) → C
○ ~(A v B)
(If A or B then C. Not C. Therefore not A or B)
2. Mental models (Johnson Laird, 1983)
○ Deductive reasoning involves diagrams rather than language-like representations (e.g., Venn diagrams)
○ Use general knowledge + meanings of assertions → construct mental models of the possibilities
● Evaluating the validity of arguments: a conclusion is valid if it holds in all the models generated by the premises
● Principle of truth - individuals by default do not represent what is false ○ When you hear “If P then Q”, what comes to mind is P and Q.
○ You don’t immediately think “What if it doesn’t rain?”
■ If the initial representation does not allow you to solve the
argument, then you have to “flesh out” all possible models (incl.
representing what is false)
Mental rules (Osherson 1974, Rips 1994)
● Is about deducibility: is the conclusion provable from the premises? ● Proceeds by proving validity (or failing to)
Mental models (Johnson-Laird 1983)
● Is about semantic entailment: is the conclusion true in all states of affairs in which the premises are true?
● Proceeds by uncovering invalidity (or failing to)
Atmosphere Heuristic - The logical terms (“some, no, all, some-not”) used in the premises of an argument create an atmosphere that makes participants accept conclusions that have the same terms.
● Part 1: people tend to accept particular conclusions (e.g., “some…”) to particular premises and general conclusions (e.g., “all…”) to general premises. When mixed, go with particular.
○ Some As are Bs, Some Bs are Cs, Some As are Cs → Invalid! (Part 1) ○ All As are Bs, Some Bs are Cs, Some As are Cs → Invalid! (Part 1)
● Part 2: people tend to accept positive conclusions (e.g., “some/all…”) to positive premises and negative conclusions (e.g., “no…”) to negative premises. When mixed, go with negative.
○ No As are Bs, All Bs are Cs, No As are Cs → Invalid! (Part 2) ● All As are Bs, All Bs are Cs, Some As are Cs → Valid!
Belief bias involves a tendency to:
1. Accept invalid conclusions if they are believable
2. Reject valid conclusions when they are unbelievable
The problem is that people seem to ‘confuse’ the truth value of the conclusion with the validity
All dogs are animals, Some animals are pets, Some dogs are pets? Invalid! All sharks are animals, Some animals are pets, Some sharks are pets? Invalid!
Wason Selection Task
● Each card has a number on one side and a letter on the other
● Proposed rule: If there is a G on one side of the card, then there is a 3 on the other side
● Which cards should you turn to test this rule? → 8
○ If there is another letter on the other side of the “8” card, then you can disprove the rule
● Each card has an action on one side and an age on the other
● Proposed rule: If a person is drinking beer then she must be 21 (or older) ● Which cards should you turn to test this rule? → 19
How to explain the WST effect?
● Mental rules and mental models are “formal” theories of deductive reasoning, so they don’t seem to address this specific case (not to mean one can’t impose semantic effects on top of deductive routines)