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LOGIC SUMMARY by Mind Map: LOGIC SUMMARY

1. 1. Definition

1.1. act of reasoning

1.2. correct relations between concepts and propositions

1.3. various ways of inference, deduction, and argumentation

1.4. based on logical relations

1.4.1. subject

1.4.2. predicate

1.4.3. conjuction

1.4.4. disjunction

1.4.5. a class

1.4.6. true/false

1.5. judged by the first principles

1.5.1. Principle of identity

1.5.2. non contradiction

1.5.3. no third option between being and non being

1.5.4. everything has a cause or sufficient reason to exist

1.6. divided in three sections

1.6.1. simple apprehension

1.6.2. judgement

1.6.3. reasoning or syllogisms

2. 2. The Concept

2.1. Simple apprehension

2.1.1. the comprehension of the minimum unities of thought

2.1.2. (blue, man, etc.)

2.1.3. ends with the formation of concepts

2.1.4. the concept of rock is the result of a psychic act in which is found the essence of the rock

2.1.5. the concept is expressed in words

2.2. sign of essence

2.2.1. Natural Sign

2.2.1.1. the connection between it and the thing it signifies is determined by nature

2.2.1.2. smoke as the sign of fire

2.2.2. Artificial Sign

2.2.2.1. the connection between it and what it signifies is the result of mans arbitrary imposition

2.2.2.2. a green light indicates legal passage

2.2.3. Formal Sign

2.2.3.1. signify without first being known themselves

2.2.3.2. natural formal sign of the quiddita which it expresses.

2.2.4. Instrumental Sign

2.2.4.1. signify only after being known themselves

2.2.4.2. words, on the other hand, are artificial instrumental signs.

2.3. Properties

2.3.1. extension

2.3.1.1. The sum of the subjects of which it can be said

2.3.2. comprehension

2.3.2.1. The content of the concept

2.4. synonyms

2.4.1. intentio

2.4.1.1. that toward which the intellect is in tension at the time it is grasping the essence in question

2.4.2. verbum mentis

2.4.2.1. the interior word, as if intellect were expressing to itself the quiddditas it has grasped.

2.4.3. ratio

2.4.3.1. the concept as an intelligible notion

2.4.4. species expressa

2.4.4.1. that in which the quidditas is grasped and known

3. 3. The Judgement

3.1. Divisions

3.1.1. Affirmative Universal (A)

3.1.1.1. Every S is P

3.1.1.2. All Men are Just

3.1.2. Negative Universal (E)

3.1.2.1. No S is P

3.1.2.2. No Men are just

3.1.3. Affirmative Particular (I)

3.1.3.1. Some S is P

3.1.3.2. Some Men are just

3.1.4. Negative Particular (O)

3.1.4.1. Some S is not P

3.1.4.2. Some Men are not just

3.2. Rules

3.2.1. In every affirmative proposition, the predicate is taken particularly

3.2.2. In every negative proposition, the predicate is taken universally

3.3. Relations

3.3.1. Contradiction

3.3.1.1. cannot be both simultaneously true nor false

3.3.1.2. A vs O (All Men are just VS Some men are not just)

3.3.1.3. E vs I (No Men are just VS Some Men are just)

3.3.2. Contrary

3.3.2.1. both cannot be true simultaneously but both can be false

3.3.2.2. (You can still have SOME even without ALL or NONE)

3.3.2.3. A vs E (All Men are just VS No Men are just)

3.3.3. Subcontrary

3.3.3.1. both cannot be false simultaneously but both can be true*

3.3.3.2. *(Not valid for you can have both false and have A or E)

3.3.3.3. Some Men are just VS Some Men are not just

3.3.4. Subalternation

3.3.4.1. if universal is true, particular is also, but not vice-versa

3.3.4.2. if particular is false, universal is also, but not vice-versa*

3.3.4.3. *(Not valid for particular can be false and have universal as true)

3.3.4.4. All Men are just SO Some Men are just

3.3.4.5. No Men are just SO Some Men are not just

4. 4. The Syllogism

4.1. Structure

4.1.1. T = major term

4.1.2. M = middle term

4.1.3. t = minor term

4.1.4. Ejemplo

4.1.4.1. every animal (M) is an organism (T) major

4.1.4.2. every man (t) is an animal (M) minor

4.1.4.3. therefore every man (t) is an organism (T) conclusion

4.2. Non-Conditional

4.2.1. Rules

4.2.1.1. 1. There may be only 3 terms

4.2.1.1.1. the major, the middle and the minor

4.2.1.1.2. the middle term is found in one premise with the major and in one the minor

4.2.1.1.3. the major and minor term are found in the conclusion

4.2.1.1.4. example

4.2.1.2. 2. the middle term must be universal at least once

4.2.1.2.1. there is no communication or connection between the major term and the minor term

4.2.1.2.2. example

4.2.1.3. 3. the middle term cannot enter into the conclusion

4.2.1.3.1. cannot mediate the conclusion and make part of the conclusion

4.2.1.3.2. example

4.2.1.4. 4. the extremes cannot have more universality in the conclusion than in the premises

4.2.1.4.1. an effect (conclusion) cannot be greater than its causes (premises)

4.2.1.4.2. example

4.2.1.5. 5. the conclusion follows the weakest premise

4.2.1.5.1. (if some premise is particular, negative, contingent, doubtful, then the conclusion must be the same)

4.2.1.5.2. example

4.2.1.6. 6. if both premises are affirmative, the conclusion cannot be negative

4.2.1.6.1. any negative conclusion will be invented or contradictory

4.2.1.6.2. example

4.2.1.7. 7. from two particular premises, nothing follows

4.2.1.7.1. one of the extremes must be universal

4.2.1.7.2. example

4.2.1.8. 8. from two negative premises, nothing follows

4.2.1.8.1. if the middle term is predicable of neither extreme, the extremes cannot be related to one another

4.2.1.8.2. example

4.2.2. Moods

4.2.2.1. First Figure

4.2.2.1.1. the major premise must be universal and the minor premise must be affirmative

4.2.2.1.2. SUB-PRAE

4.2.2.1.3. M-T > t-M = t-T

4.2.2.1.4. BARBARA

4.2.2.1.5. CELARENT

4.2.2.1.6. DARII

4.2.2.1.7. FERIO

4.2.2.2. Second Figure

4.2.2.2.1. the major term must be universal and one of the premises must be negative

4.2.2.2.2. PRAE-PRAE

4.2.2.2.3. T-M > t-M = t-T

4.2.2.2.4. CESARE

4.2.2.2.5. CAMESTRES

4.2.2.2.6. FESTINO

4.2.2.2.7. BAROCO

4.2.2.3. Third Figure

4.2.2.3.1. the minor term must always be affirmative, and the conclusion particular

4.2.2.3.2. SUB-SUB

4.2.2.3.3. M-T > M-t = t-T

4.2.2.3.4. DARAPTI

4.2.2.3.5. FELAPTON

4.2.2.3.6. DISAMIS

4.2.2.3.7. DATISI

4.2.2.3.8. BOCARDO

4.2.2.3.9. FERISON

4.2.2.4. Fourth Figure

4.2.2.4.1. If the major is affirmative then the minor must be universal

4.2.2.4.2. if the minor is affirmative, the conclusion must be particular

4.2.2.4.3. if some premise is negative, the major must be universal

4.2.2.4.4. PRAE-SUB

4.2.2.4.5. T-M > M-t = t-T

4.2.2.4.6. BRAMANTIP

4.2.2.4.7. CAMENES

4.2.2.4.8. DIMARIS

4.2.2.4.9. FESAPO

4.2.2.4.10. FRESISO

4.3. Conditional

4.3.1. major is a conditional proposition, minor is a categorical proposition which affirms or denies one of the members of the conditional proposition

4.3.2. Rules

4.3.2.1. 1. affirming the condition, the conditioned is affirmed

4.3.2.2. 2. affirming the conditioned, the condition is not affirmed

4.3.2.3. 3. removing the conditioned, the condition is removed

4.3.2.4. 4. removing the condition, the conditioned is not removed

4.3.3. Moods

4.3.3.1. 1. ponendo-ponens

4.3.3.1.1. affirming in the minor the condition enunciated in the major, the conditioned enunciated in the major is affirmed in the conclusion

4.3.3.1.2. form 1

4.3.3.1.3. form 2

4.3.3.1.4. form 3

4.3.3.1.5. form 4

4.3.3.2. 2. tollendo-tollens

4.3.3.2.1. precluding in the minor the conditioned enunciated in the major, the condition enunciated in the major is precluded in the conclusion

4.3.3.2.2. form 1

4.3.3.2.3. form 2

4.3.3.2.4. form 3

4.3.3.2.5. form 4

4.3.4. Examples

4.3.4.1. condition and conditioned have the same subject

4.3.4.1.1. conditional

4.3.4.1.2. unconditional

4.3.4.2. condition and conditioned do not have the same subject

4.3.4.2.1. conditional

4.3.4.2.2. conditional divided

4.3.4.2.3. unconditional

4.4. Disjunctive

4.4.1. major is properly a disjunctive proposition, minor is a categorical proposition which affirms or denies one of the members of the disjunctive proposition

4.4.2. Moods

4.4.2.1. 1. ponendo-tollens

4.4.2.1.1. affirming the first member of the disjunction, the second is precluded, the two being incompatible

4.4.2.1.2. The minor affirms one of the predicates and the conclusion denies the other

4.4.2.1.3. form 1

4.4.2.1.4. form 2

4.4.2.1.5. form 3

4.4.2.1.6. form 4

4.4.2.2. 2. tollendo-ponens

4.4.2.2.1. precluded the first member of the disjunction, the second is affirmed, the two being incompatible

4.4.2.2.2. The minor denies one of the predicates and the conclusion affirms the other

4.4.2.2.3. form 1

4.4.2.2.4. form 2

4.4.2.2.5. form 3

4.4.2.2.6. form 4

4.4.3. Examples

4.4.3.1. either communism is a true philosophy or christianity is a true religion

4.4.3.2. now communism is a true philosophy

4.4.3.3. so christianity is not a true religion

4.4.3.4. The disjunctive is reduced to the conditional by means of the negation of one of the members of the disjunction and its conversion into conditioned.

4.4.3.5. 2 possibilities

4.4.3.5.1. the major either a is b, or x is y becomes:

4.4.3.5.2. if a is b then x is not y or else if x is y, then a is not b

4.5. Valid vs. Sound

5. 5. Fallacies

5.1. Informal

5.1.1. Revelance

5.1.1.1. Appeal to Ignorance

5.1.1.1.1. Principle

5.1.1.1.2. Example

5.1.1.2. Appeal to Authority

5.1.1.2.1. Principle

5.1.1.2.2. Example

5.1.1.3. Appeal to the Circumstantial

5.1.1.3.1. Principle

5.1.1.3.2. Example

5.1.1.4. Appeal to the Majority

5.1.1.4.1. Principle

5.1.1.4.2. Example

5.1.1.5. Appeal to Pity

5.1.1.5.1. Principle

5.1.1.5.2. Example

5.1.1.6. Appeal to Force

5.1.1.6.1. Principle

5.1.1.6.2. Example

5.1.1.7. Irrelevant Conclusion

5.1.1.7.1. Principle

5.1.1.7.2. Example

5.1.2. Presumption

5.1.2.1. Accident

5.1.2.1.1. Principle

5.1.2.1.2. Example

5.1.2.2. Converse Accident

5.1.2.2.1. Principle

5.1.2.2.2. Example

5.1.2.3. False Cause

5.1.2.3.1. Principle

5.1.2.3.2. Example

5.1.2.4. Begging the Question

5.1.2.4.1. Principle

5.1.2.4.2. Example

5.1.2.5. Complex Question

5.1.2.5.1. Principle

5.1.2.5.2. Example

5.2. Formal

5.2.1. Syllogistic

5.2.1.1. Fallacy of the affirmative conclusion

5.2.1.1.1. Principle

5.2.1.1.2. Example

5.2.1.2. Fallacy of the negative conclusion

5.2.1.2.1. Principle

5.2.1.2.2. Example

5.2.1.3. Fallacy of exclusive premises

5.2.1.3.1. Principle

5.2.1.3.2. Example

5.2.1.4. Fallacy of the undistributed middle

5.2.1.4.1. Principle

5.2.1.4.2. Example

5.2.1.5. Fallacy of four terms

5.2.1.5.1. Principle

5.2.1.5.2. Example

5.2.1.6. Fallacy of the Illicit major

5.2.1.6.1. Principle

5.2.1.6.2. Example

5.2.1.7. Fallacy of the Illicit minor

5.2.1.7.1. Principle

5.2.1.7.2. Example

5.2.2. Symbolic

5.2.2.1. affirming the consequent

5.2.2.1.1. Principle

5.2.2.1.2. Example

5.2.2.2. Denying the Antecedent

5.2.2.2.1. Principle

5.2.2.2.2. Example

5.2.2.3. Affirming a Disjunct

5.2.2.3.1. Principle

5.2.2.3.2. Example

6. 6. Symbolic Logic

6.1. A statement letter of is defined as any uppercase letter written with or without a numerical subscript.

6.1.1. Lower case letters can also be used

6.2. A connective or operator of is any of the signs ‘&’, ‘v‘, ‘→’, ‘↔’, and ‘¬’.

6.2.1. ‘&’ or ‘∧’ or ‘•’ is equal to "and" (conjunction)

6.2.1.1. The conjunction of two statements "P & Q"is true if both are true, and is false if either P is false or Q is false or both are false.

6.2.2. ‘v‘ or "+" is equal to "or" (disjunction)

6.2.2.1. The disjunction of two statements P v Q, is true if either P is true or Q is true, or both P and Q are true, and is false only if both P and Q are false.

6.2.2.2. The sign ‘v‘ is used for disjunction in the inclusive sense.

6.2.3. ‘→’ or ‘⊃’ is equal to "if...then..." (material implication)

6.2.3.1. A statement of the form P → Q, is false if P is true and Q is false, and is true if either P is false or Q is true (or both).

6.2.3.1.1. The "if...then..." here is really a "if... then might be"

6.2.3.1.2. If Marie is in Paris, then she is in France. (True if both are true).

6.2.3.1.3. "If Marie is in Paris" is false, then she "is" (could still be) in France. (she could still be in France somewhere even though she is not in Paris.)

6.2.3.1.4. If Marie is in Paris, then she is in France. (If she is not in France, then there is no way she could be in Paris.)

6.2.3.1.5. If Marie is in Paris, then she is in France. (if both are false then it would be true: "If Marie is NOT in Paris, then she is NOT (might not be) in France."

6.2.4. ‘↔’ or ‘≡’ is equal to "if and only if"...then..." (material equivalence)

6.2.4.1. A statement of the form P ↔ Q is regarded as true if P and Q are either both true or both false, and is regarded as false if they have different truth-values.

6.2.4.1.1. If Romney is President, then he won Florida. True if both are true.)

6.2.4.1.2. If Romney is NOT President, then he DID NOT WIN Florida.

6.2.4.1.3. "If Romney is President" is false, then "then he won Florida" cannot be true and the whole sentence false because we presupposed that by winning Florida he would be president.

6.2.4.1.4. BUT, if we say that "If Romney is President" is true, and that "then he won Florida" is false, then all would be false because we presupposed it was a necessary conditional.

6.2.5. ‘¬’ or ‘~’ or ‘–’ is equal to "not" or a negative (negation)

6.3. ( ) parentheses are used in forming even more complex statements.

6.3.1. I ↔ (C & P)

6.3.2. (I ↔ C) & P

6.4. ¨ ¨ quotations are to talk about words and sentences in other languages.

6.4.1. “(I ↔ C) & P”

6.5. Greek letters ‘α’, ‘β’, etc., are used for any object language expression of a certain designated form.

6.5.1. α v β is the complex statement “(I ↔ C) v (P & C)”

6.6. "∴" is used instead of the horizontal line for the conclusion

6.7. A well-formed formula corresponds to the notion of a grammatically correct or properly constructed statement

6.7.1. “¬(Q v ¬R)” is grammatical because it is a well-formed formula

6.7.2. The string of symbols, “)¬Q¬v(↔P&”, while consisting entirely of symbols, is not grammatical because it is not well-formed.

6.8. The ¬ operator has higher precedence than ∧; ∧ has higher precedence than ∨; and ∨ has higher precedence than → or ↔

6.9. When an operand is surrounded by operators of equal precedence, the operand associates to the left.

6.10. EXAMEPLES

6.10.1. ((P & Q) → ¬R)

6.10.1.1. If John is in Paris and if he is eating pie

6.10.1.2. then he is not outside of France

6.10.2. [(p → q) ∧ p] → q

6.10.2.1. If he is 10 years old, then he can watch the movie

6.10.2.2. ...and he really is 10 years old

6.10.2.3. so he can watch the movie

6.10.3. [(¬p v q) ∧ (q → r)] → (r v ¬p)

6.10.3.1. Either you are not a man or you are intelligent

6.10.3.2. ... and if you are intelligent then you can think

6.10.3.3. Then... either you can think or you are not a man