logic & reasoning what is reasoning? reasoning is arriving at new beliefs based on old beliefs....

Logic & Reasoning

What is reasoning?Reasoning is arriving at new beliefs based on old beliefs.

That is: Using premises (old beliefs) to reach conclusions (new beliefs)Using beliefs about experiences to form beliefs about the world.

How do we create new beliefs from old beliefs?

Two main ways: Deductive Logic: Reasoning where the conclusion is guaranteed to be true, given the premises. Inductive Logic: Reasoning where the conclusion is likely to be true, given the premises.

Deductive Logic / Reasoning

A valid deductive argument:

B1: If you have had 5 beers in the last half hour, then you are drunk.B2: If you are drunk and you are driving, then you are breaking the law.B3: You have had 5 beers in the last half hour.B4: You are driving.

B*: You are breaking the law.

If you know B1-B4 to be true, then you can adopt new beliefB* with certainty: it must be true!


A valid deductive argument:

B1: If I say the word “Balumbo”, then I will get a million dollars

B2: I say the word “Balumbo”

B*: I get a million dollars.

Whether or not an argument is TRUE is completely separate

from whether the argument is VALID.


An invalid deductive argument:

B1: If my car uses gasoline, then it pollutes the environment.B2: My car pollutes the environment.

B*: My car uses gasoline.

Whether or not an argument is TRUE is completely separatefrom whether the argument is VALID.


General form of an implication:

B1: If X , then Y.B2: X

B*: Y.modus ponens

B1: If X , then Y.B2: not Y

B*: not X.

modus tollens


General form of an implication:

B1: If X , then Y.B2: X

B*: Y.modus ponens

B1: If it’s a nice day, then it is sunny.B2: It’s not sunny

B*: It’s not a nice day.

modus tollens

Wason’s Card Selection Task

A D 4 7

RULE: If a card has a vowel on one side, then it has an even number on the other side


beer coke 22 18

RULE: If a person is drinking beer, then that person must be 21 or over.


A D 4 7



A D 4 7


modus ponens: If vowel, then check for even...modus tollens: If not even, then check for not vowel...


A D 4 7


Mistake: affirming the consequent


A D 4 7


Mistake: denying the antecedent


beer coke 22 18

Mistake: affirming the consequent

RULE: If a person is drinking beer, then that person must be 21 or over.


Invalid deductive argument (affirming the consequent)

B1: If the switch is turned on then the light will go on.B2: The light is on.

B*: The switch was turned on

B1: If X , then Y.B2: Y

B*: X.


A ________ syllogism:

B1: All cat are mammals.B2: Some mammals are furry

B*: Some cats are furry


A invalid syllogism:

B1: All cat are mammals.B2: Some mammals are furry

B*: Some cats are furry

mammals

cats furry


A ________ syllogism:

B1: No addictive things are inexpensive.B2: Some cigarettes are inexpensive.

B*: Some cigarettes are not addictive.


A valid syllogism:

B1: No addictive things are inexpensive.B2: Some cigarettes are inexpensive.

B*: Some cigarettes are not addictive.

addictive things inexpensive things

cigarettes


Invalid deductive argument

B1: If the switch is turned on then the light will go on.B2: The light is on.

B*: The switch was turned on

Knowing B2 to be true increases the likelihood of B*.

Logic & Reasoning

What is reasoning?Reasoning is arriving at new beliefs based on old beliefs.

That is: Using premises (old beliefs) to reach conclusions (new beliefs)Using beliefs about experiences to form beliefs about the world.

How do we create new beliefs from old beliefs?

Two main ways: Deductive Logic: Reasoning where the conclusion is guaranteed to be true, given the premises. Inductive Logic: Reasoning where the conclusion is likely to be true, given the premises.

Inductive Logic / Reasoning

A valid inductive argument:

S1: Today, Psych 85 is exciting and fun!S2: Last class, Psych 85 was exciting and fun!S3: The class before, Psych 85 was exciting and fun!

Therefore, next class, Psych 85 will be exciting and fun!

This is enumerative induction: Every F seen so far has been a G, therefore all F’s are G’s, or at least the next F will be a G.

Enumerative Induction

• Vision: Non-Accidental Properties Each retinal pattern of type X I’ve seen so far has been caused by objects in the world of type Y, therefore, all retinal patterns of type X are caused by objects of type Y.• Memory: Categorization Each thing with property X I’ve seen so far has been a Y, therefore, all things with property X are Y’s.• Language: Learning of Grammatical Morphemes Each object-word I’ve seen so far can be made plural by adding -s, therefore, all object-words can be made plural by adding -s.


• Classical Conditioning Each time Mr. Doggie has heard the bell ring, food has appeared, therefore, every time the bell rings, food will appear!• The Hebb Rule

raven unit blackness unit


Maybe all induction is is association and pattern matching?

It’s not that simple.

There are limits to what can and can’t be associated…There are limits to what can be considered evidence.

clicking bottle

sweet water

pain nausea

aversion

aversion

no aversion

no aversion


We are biased in what we consider “evidence” for beliefs.This is good.

Consider the belief:

All ravens are black. = All non-black things are non-ravens.

If Raven then Blacknot Black

not Raven

Evidence that all ravens are black:• Yellow cars• Gray Chihuahuas• Green toads...


Maybe all induction is is association and pattern matching,… with some biases about what can and can’t be learned.

What about rules? What about non-enumerative induction?

Other forms of induction: Forming the belief about thebest possible explanation for a set of observations.

Give the next number in the series:

1, 1, 2, 3, 5, 8, 13, ___


Maybe all induction is is association and pattern matching,… with some biases about what can and can’t be learned.

What about rules? What about non-enumerative induction?

Other forms of induction: Forming the belief about thebest possible explanation for a set of observations.

Give the next number in the series:

1, 1, 2, 3, 5, 8, 13, 0, 0, 0, ...

Problem of Induction:How do we ever know whatthe “best explanation” isfor a given set of data?

Problem Solving

Well-defined problems:• A set of possible states that define a problem space• A clearly defined initial state and goal state • A set of operators that move you between states • And any path constraints that tell you what kinds of solutions are acceptable

• The states are the stops• The operators are the trains • The path constraint is you want to get there as quickly as possible

Solving the problem involves a search.

Problem Solving

Well-defined problems:• A set of possible states that define a problem space• A clearly defined initial state and goal state • A set of operators that move you between states • And any path constraints that tell you what kinds of solutions are acceptable

Problems can be then solved by searching fordifferent paths from the initial state to the goal state.

…but we don’t have time for all that silliness!

Searching Problem Space

Short-cuts in searching through problem space:

Heuristic Search: Don’t search all pathways, just ones likely to be the shortest.

Satisficing: Don’t look for the shortest, just look for one that is short enough.

Means-Ends Analysis: Divide the problem up into sub-goals, that direct your search toward the goal.

Means-Ends Analysis

1) Compare the current state to the goal state (If they are the same, you’re done)

2) Find some operator that will make the difference between them smaller

3) If you can apply the operator, apply it! Otherwise, set up a sub-goal to make it so you can apply it

4) Go back to 1)

Tower of Hanoi Puzzle

1234

A B C

Stack[1]


1234

A B C

Goal: Move stack[1] to peg CProduction: disk[1] is not on peg C, and is not free subgoal: move stack[2] to peg B

Stack[1]


1234

A B C

Goal: Move stack[2] to peg BProduction: disk[2] is not on peg B, and is not free subgoal: move stack[3] to peg C

Stack[2]


1234

A B C

Goal: Move stack[3] to peg CProduction: disk[3] is not on peg C, and is not free subgoal: move stack[4] to peg B

Stack[3]


1234

A B C

Goal: Move stack[4] to peg BProduction: disk[4] is not on peg B, but is free! so move disk[4] to peg B

Stack[4]


123

4

A B C

Goal: Move stack[4] to peg BProduction: disk[4] is not on peg B, but is free! so move disk[4] to peg B

Searching Problem Space

Short-cuts in searching through problem space:

Heuristic Search: Don’t search all pathways, just ones likely to be the shortest.

Satisficing: Don’t look for the shortest, just look for one that is short enough.

Means-Ends Analysis: Divide the problem up into sub-goals, that direct your search toward the goal.

Improving The Search

What gets better with expertise:

Chunking: States are represented in terms of meaningful arrangements, rather than individual pieces.

Composition: Several separate steps in a process can be consolidated into a single step.

Schemas: Experience with similar situations in the past can be used to guide new searches.

Problem Representation

Ill-Defined Problems

“Insight Problems”



One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two across, spiraled around the mountain to a glittering temple at the summit. The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting, he began his journey back along the same path, again beginning exactly at sunrise, walking at variable speeds and resting along the way.

Show that there is a time when the monk was at exactly the samespot at exactly the same time of day on each journey.


“Insight Problems” involve a sudden change in themental representation of the problem space, ratherthan a search of paths within one problem space.

How do we discover a new representation?

Creativity: How do we find a totally new way of looking at something?

Scientific Creativity often seems to make sure of abstract similarities between otherwise unrelated domains.

Analogy

Analogy

“Light is to lamps as heat is to furnaces.”

Light: helps vision electromagnetic waves […]

Heat: can be felt speed of molecular movement […]

(Light, Lamps): A comes out of B B is designed to increase the amount of A in a room […]

(Heat, Furnaces): A comes out of B B is designed to increase the amount of A in a room […]

Analogy


“Atoms are like the Solar System.”

Nucleus: positive charge too small to be seen between 1 and 100 AMU […]

Sun: hot yellow millions of tons […]

Electrons: negative charge too small to be seen have wavelike properties […]

Planets: cold solid, with atmosphere sometimes have moons […]

Analogy


“Atoms are like the Solar System.”

(Nucleus, Electrons): B revolves around A A is bigger than B A is more massive than B […]

(Sun, Planets): B revolves around A A is bigger than B A is more massive than B […]

The mapping aligns nucleus with sun, electrons with planets(even though individually these items are not similar)

Discovering New Representations

• Many problems can be solved simply by chosing a representation space and searching through it (with heuristics and other shortcuts)• When the correct problem space is not obvious, or the chosen problem space isn’t working, we somehow have to generate a new representation of the problem space.• One factor that can help determine what new representation should be used is analogy: finding a mapping between the problem and another domain.• Whereas searching a space is step-by-step, gradual, and conscious, changing from one representation to another is sudden, and probably the result of parallel constraint satisfaction.

logic & reasoning what is reasoning? reasoning is arriving at new beliefs based on old beliefs....

Documents

inductive logic

logic reasoningwhat

premises old beliefs

modus ponensb1

new beliefb

nice day

half hour

word balumbob