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ToK - Logic

we're gonna pwn this. pwnage to the max.

Niki Shi

on 24 October 2011

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Transcript of ToK - Logic

LOGiC All men are mortal
All Greeks are men
a. All mortals are men
B. All Greeks are mortal
C. All mortals are Greek
D. All men are Greeks Question 2: I dream about monsters. My brother dreams about monsters.
A. Everyone dreams about monsters
B. Monsters is an image commonly provoked to appear in dreams
C. We will always dream about monsters.
D. My brother and I might be exposed to images of monsters, which may have caused the occurrence of dreams. Are you female?
A. True
B. False Question 4: Using the diagram on the right of a fuzzy representation of height labels. Person A is approximately 185cm tall. What would he be categorized as?
A. Tall
B. Average
C. Average to a degree of 0.2 and tall to a degree of 0.3
D. Average to a degree of 0.6 and tall to a degree of 0.8 B. All Greeks are mortal Answer: D. My brother and I might be exposed to images of monsters, which may have caused the occurrence of dreams. Answer: C.Average to a degree of 0.2 and tall to a degree of 0.3 Formal
Logic Informal
Logic Boolean
Logic Fuzzy Logic Definition:
study of the principles of valid inference and correct reasoning
used in philosophy, mathematics, semantics, and computer science
the formal and systematic tool used to examine different forms of arguments and their validity To what extent do different modes of logic influence our decisions and conclusions? 4 TYPES OF LOGIC

Fuzzy Jacob is justified in drinking the milk.
Chuck is justified in killing his father. INFORMAL LOGIC

Decent discipline
Precedents in 19th century works on Logic and Rhetoric
Aim to raise general standards of reasoning through public education
A child of the 1960s.

Equivalent to critical thinking
Attempt to develop a logic to assess, analyze and improve ordinary language
In contrast to formal logic, informal logic is the study of particular arguments in the CONTEXTS in which they occur.
Less generalizations and more critical analysis Knowledge Issue 4 Criteria when analyazing informal logic:

Whether all the premises are true.
Whether the conclusion is at least probable, given the truth of the premises.
Whether the premises are relevant to the conclusion.
Whether the conclusion is vulnerable to new evidence. I dream about monsters.
My brother dreams about monsters.

a. Everyone dreams about monsters
b. Monsters is an image commonly provoked to appear in dreams
c. We will always dream about monsters.
d. My brother and I might be exposed to images of monsters, which may have caused the occurrence of dreams. Question 4: Using the diagram on the right of a fuzzy representation of height labels. Person A is approximately 185cm tall. What would he be categorized as?
A. Tall
B. Average
C. Average to a degree of 0.2 and tall to a degree of 0.3
D. Average to a degree of 0.6 and tall to a degree of 0.8 An Example of Russell's Paradox:

There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved themselves or were shaved by the barber. And the barber only shaved the men who did not shave themselves. Did the barber shave himself? ? Fuzzy Logic Founders:

The first idea of multifaced truths found by an important person came through Buddha
Bertrand Russell discovered the Russell's Paradox dealing with fuzzy sets in 1901
Theory of fuzzy sets was developed by Lotfi Zadeh in 1965 Lotfi's Fuzzy Sets:

instead of 'x' found within only set A or not in set A, 'x' will be in the set of A to a degree within the range of [0,1] and not in the set of A to a degree within the range of [0,1] also embraces multifaced truths
where an object can be A and not A at once
a method in which quick, simple, and sufficiently good solutions are reached
models reality which is not found in only black and white but rather in shades of grey You tell me. Supposing no transvestites. Named after George Boole
Defined an algebraic system of logic in the mid 19th century
Many applications in electronics, computer hardware and software
The basis of all modern digital electronics.
In 1938, Claude Shannon showed how electric circuits with relays could be modelled with Boolean logic.
Enormously consequential with the emergence of the electronic computer. Binary logic:
Logic that must be used when you take a true or false test.
Clear definite answer; true or false, yes or no, on or off, 1 or 0

In a set, every element is either a member or a non-member. No in between.

Boolean logic:
A complete system for logical operations, used often since popularization of mathematical logic and discussions concerning the foundations of mathematics. Terms:
Element: a member in a set
Universe: all elements being considered
Empty/null set: set of no elements
Subset: Every element in set A is also in set B [A is subset of B]
Superset: Every element in set B is also in set A [A is superset of B]
Identity/equivalence: every element in A is in B and vice versa
Proper subset: subset but sets are NOT identical
Proper superset: superset but sets are NOT identical

Unary operator: operator applied to only ONE set
. NOT: takes the complement (opposite) with respect to the universe
Binary operators: Applies to TWO sets:
. Basic ones:
. OR: Intersection.
. AND: Union.
. Others:
. XOR: exlusive OR; "one or the other but not both"
. A-B: set difference Formal logic is defined as the branch of logic concerned
exclusively with the principles of deductive reasoning The works of Aristotle are the
earliest known works of formal logic 384 BC - 322 BC Known as the Organo Principal of identity: if a statement is true, then it is true.

Principle of excluded middle: a statement is either true or false
Principle of contradiction: no statement can be both true and false He preferred induction over deduction All men are mortal
All Greeks are men
a) All mortals are men
b) All Greeks are mortal
c) All mortals are Greek
d) All men are Greeks
All A are B
All C are A.
Therefore, all C are B.
Perception and Arts
- Many things in life cannot be sorted into precise categories
- These include music, art, poetry, and even feelings
Popular until the 1600s when it was rejected by Sir Francis Bacon knowledge issues and fallacies o o o Perception and Arts:
many things in life cannot be sorted into precise categories
these can include music, art, dance, and even feelings Mathmatics:
math is very much based on absolutes
however, fuzzy logic can be seen in some aspects of mathmatical sets
it can also be seen in percentages and probability
the mathematical model of fuzzy logic itself is complex Emotions:
we often have contrasting emotions
it is difficult to say exactly how we feel at any given point in time
emotions are often found in ranges rather than in exacts Language
Formal System
1. A set of symbols, such as the alphabet
2. A well-formed formula
3. A set of axioms
4. A set of rules for inferring
The Awkwardness of Formal Logic

All men are mortals
All Socrates are men
All Socrates are mortals

Not necessarily true if there is a woman called Socrates
Simple decision making

It is raining outside.
Rain can make you sick.
Going outside will make me sick.

This is a reasonable assumption to make given the circumstances.
Irrational Assumptions
Cutting people is a crime.
Surgeons cut people.
Therefore, surgeons are criminals.

A Sweeping Generalization: a dicto simpliciter ad dictum secundum quid Perception and Paradigms
The weather is bad.
Bad weather is caused by angry gods.
Therefore, the gods are angry.

Your location and culture can change your perception on a statement. Ethics

Law says we should not kill.
We should follow laws.
Therefore, we should not kill.

Given a terrorist threatening to kill 500 innocent people:

A deontologist would stick to the law and not kill the terrorist.
A consequentialist would take one life to spare 500 other lives. Ethics:
fuzzy logic contradicts deontological ethics usually
fuzzy logic can aide in consequentialist/utilitarian ethics
generally, fuzzy logic makes it difficult in making exact choices but gives more information in return Language:
Different meanings, implications and connations.
If I work hard, we will do well on the next test.

Difference between “good”, “well”, “not bad”, and “alright”?

To what extent can we trust logic in deriving our conclusions if such simple linguistic words can provoke such different interpretations?

To what extent does personal experience elude expression in language?
Feelings, Emotions and Intuition

If you add vodka to your life, your sleepy life will be transformed into a life of cosmopolitan excitement
A life of cosmopolitan excitement is desirable.
You desire vodka. Should emotion play a role in the evaluation of knowledge claims?
Are there circumstances under which, in order to evaluate a knowledge claim, one should ignore or pay attention to one’s emotions?
Do we follow emotions and intuition more than logic in making decisions? If so, should we attempt to encourage or avoid it?

In certain cases AND and OR can be used interchangeably.
"I always carry an umbrella for when it rains and snows" Language
Ambiguity of logical operators Religion

Informal fallacy:
An argument whose stated premises fail to support their proposed conclusions

Science is the only way to test claims
Science disproved God a long time ago
Hence, God does not exist. Christianity believes in free will.
You are a Christian.
You believe in free will. Begging the question, aka. Cicurla argument
Aristotle – when a proposition which require proof is assumed without proof.
Religious person and atheist? How do we apply logical thinking in religion? To what extent are we able to justify the truth of our premises in matters of religions? Reason
What we do when we make a transition from a preise to a conclusion.
True, justified premises, but incorrect reasoning.

Everytime I wash my car, it snows within an hour.
Washing my car brings snow.
What constitutes as a good argument?
What is the value of learning to distinguis between valid and invalid arguments? Boolean Logic AND/OR

The English words "and" and "or" have a meaning that usually applies more to sets of things than specific characteristics of items.
"Give me all the red and blue berries" Formal Logic Informal Logic Fuzzy Logic Boolean Logic Formal Logic Informal Logic Fuzzy Logic OR/XOR
"or" may correspond to either logical OR (and/or) or logical XOR (either/or)
. OR: "Would you like cream or sugar with your coffee?"
. -set taking precedence over item
. XOR: "Would you like soup or salad with your meal?"
. -item taking precedence over set Example
Computer programming in English:
-"The program should verify that the applicant has checked the male or female box."
-Should be XOR (can't be male AND female) Ambiguity of terms
"Are you in the IB program?"
Can be viewed as "Do you take any IB courses?" (Jamilyn; takes only some IB courses)
Do you take only IB courses at your school? (summer school excluded) A Fallacy of Boolean Logic Existential fallacy
Assuming that a set contains elements
Any argument whose conclusion implies that a class has at least one member, but whose premisses do not so imply. Example:
All trespassers will be prosecuted.
Therefore, some of those prosecuted will have trespassed.

Existential Import
There are trespassers. Dilemmas

Mapping more complex problems
In between yes or no, 1 or 0, true or false, etc
For this there is Fuzzy logic as discussed Jacob is hungry.
If one is hungry, one eats.
Therefore Jacob should eat. The law allows for killing in self defense.
Chuck killed in self defense.
Therefore Chuck did not break that law. Has Jacob eaten anything today?

No, he has not. There is no food.
However, milk is filling.
He should drink milk. He is not famished but is relatively hungry and thus does not need to drink all of the milk. He will drink some of it. Was Chuck's father pointing a gun at him?
Yes. All of the 500 innocent victims have family. Chuck wants to save their family members from suffering. Chuck should try his best to prevent the victims from getting killed. Though he did not know whether his dad would kill him for sure or not kill him at all, the percentage of his dad doing so was on the higher end of killing him. Counter-Example:
All unicorns are animals.
Therefore, some animals are unicorns.

No Existential Import
There are no unicorns. Despite the extremities of binary logic, is it possible to assess situations without binary logic? Final Decision will need to use it:
"Do I carry out the decision or not" Can boolean logic be used to arrive at knowledge claims on its own, independent from other types of logic? Mathematics
Only binary logic
Either Right or wrong Can NO binary logic be used to arrive to a claim? Almost anything can be rephrased as a question requiring binary logic.
Do you like school more than you dislike it?
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