Logical Concept Map

Logical-Concept-Map
Definition
act of reasoning
correct relations between concepts and propositions
various ways of inference, deduction, and argumentation
based on logical relations
subject
predicate
conjuction
disjunction
a class
true/false
judged by the first principles
Principle of identity
non contradiction
no third option between being and non being
everything has a cause or sufficient reason to exist
divided in three sections
simple apprehension
judgement
reasoning or syllogisms
The Concept
Simple apprehension
the comprehension of the minimum unities of thought
(blue, man, etc.)
ends with the formation of concepts
the concept of rock is the result of a psychic
act in which is found the essence of the rock
the concept is expressed in words
sign of essence
Natural Sign
the connection between it and the thing
it signifies is determined by nature
smoke as the sign of fire
Artificial Sign
the connection between it and what it signifies
is the result of mans arbitrary imposition
a green light indicates legal passage
Formal Sign
signify without first being known themselves
natural formal sign of the quiddita
which it expresses.
Instrumental Sign
signify only after being known themselves
words, on the other hand, are
artificial instrumental signs.
Properties
extension
The sum of the subjects of which it can be said
comprehension
The content of the concept
synonyms
intentio
that toward which the intellect is in tension at
the time it is grasping the essence in question
verbum mentis
the interior word, as if intellect were expressing
to itself the quiddditas it has grasped.
ratio
the concept as an intelligible notion
species expressa
that in which the quidditas is grasped and known
The Judgement
Divisions
Affirmative Universal (A)
Every S is P
All Men are Just
Negative Universal (E)
No S is P
No Men are just
Affirmative Particular (I)
Some S is P
Some Men are just
Negative Particular (O)
Some S is not P
Some Men are not just
Rules
In every affirmative proposition, the predicate is taken particularly
In every negative proposition, the predicate is taken universally
Relations
Contradiction
cannot be both simultaneously true nor false
A vs O (All Men are just VS Some men are not just)
E vs I (No Men are just VS Some Men are just)
Contrary
both cannot be true simultaneously
but both can be false
(You can still have SOME even without ALL or NONE)
A vs E (All Men are just VS No Men are just)
Subcontrary
both cannot be false simultaneously but both can be true*
*(Not valid for you can have both false and have A or E)
Some Men are just VS Some Men are not just
Subalternation
if universal is true, particular is also, but not viceversa
if particular is false, universal is also, but not viceversa*
*(Not valid for particular can be false and have universal as true)
All Men are just SO Some Men are just
No Men are just SO Some Men are not just
The Syllogism
Structure
T = major term
M = middle term
t = minor term
Ejemplo
every animal (M) is an organism (T) major
every man (t) is an animal (M) minor
therefore every man (t) is an organism (T) conclusion
NonConditional
Rules
1. There may be only 3 terms
the major, the middle and the minor
the middle term is found in one premise
with the major and in one the minor
the major and minor term are found in the conclusion
example
every triangle is a rectilinear figure
no rectangle is a curvilinear figure
no triangle is a curvilinear figure
(4 terms: triangle, rectilinear figure,
rectangle, curvilinear figure)
2. the middle term must be universal at least once
there is no communication or connection
between the major term and the minor term
example
Africans are black
John is black
John is an African
Middle term is particular in both cases
3. the middle term cannot enter into the conclusion
cannot mediate the conclusion
and make part of the conclusion
example
Africans are black
John is black
John and Africans are black
Nothing is concluded
4. the extremes cannot have more universality
in the conclusion than in the premises
an effect (conclusion) cannot be
greater than its causes (premises)
example
Every triangle is a rectilinear figure
no rectangle is a triangle
no rectangle is a rectilinear figure
We cannot deduce this relation
5. the conclusion follows the weakest premise
(if some premise is particular, negative,
contingent, doubtful, then the conclusion must be
the same)
example
all men are rational
rocks are not rational
rocks are not men
The middle term has been
completely affirmed and denied
6. if both premises are affirmative,
the conclusion cannot be negative
any negative conclusion will be
invented or contradictory
example
every animal is an organism
every man is an animal
no man es an organism
Nothing has been denied
7. from two particular premises, nothing follows
one of the extremes must be universal
example
some men are just
some men are not just
all men are just
The conclusion is a contradiction
8. from two negative premises, nothing follows
if the middle term is predicable of neither extreme,
the extremes cannot be related to one another
example
No African is black
John is not black
John is not African
John could be African precisely
because he is not black
Moods
First Figure
the major premise must be universal and
the minor premise must be affirmative
SUBPRAE
MT > tM = tT
BARBARA
every man is intelligent
every rational animal is a man
every rational animal is intelligent
CELARENT
no man is a rock
every rational animal is a man
No rational animal is a rock
DARII
every man is intelligent
some rational beings are men
some rational beings are intelligent
FERIO
no man is a rock
some rational beings are men
some rational beings are not rocks
Second Figure
the major term must be universal and
one of the premises must be negative
PRAEPRAE
TM > tM = tT
CESARE
no logician is a bad reasoner
every sophist is a bad reasoner
no sophist is a logician
CAMESTRES
every sophist is a bad reasoner
no logician is a bad reasoner
no logician is a sophist
FESTINO
no logician is a bad reasoner
some people are bad reasoners
some people are not logicians
BAROCO
every sophist is a bad reasoner
some people are not bad reasoners
some people are not sophists
Third Figure
the minor term must always be affirmative,
and the conclusion particular
SUBSUB
MT > Mt = tT
DARAPTI
every man is an animal
every man is rational
some rational beings are animals
FELAPTON
no man is a dragon
every man is rational
some rational beings are not dragons
DISAMIS
some rational beings are animals
every rational being is intelligent
some intelligent beings are animals
DATISI
every rational being is intelligent
some rational beings are animals
some animals are intelligent beings
BOCARDO
some bad reasoners are not logicians
every bad reasoner is a sophist
some sophist are not logicians
FERISON
no man is an insect
some men are intelligent
some intelligent beings are not insects
Fourth Figure
If the major is affirmative then the minor must be universal
if the minor is affirmative, the conclusion must be particular
if some premise is negative, the
major must be universal
PRAESUB
TM > Mt = tT
BRAMANTIP
all men are mammals
all mammals are animals
some animals are men
CAMENES
all men are rational
no rational beings are insects
no insects are men
DIMARIS
some men are American
all Americans speak English
Some who speak English are men
FESAPO
no rational beings are insects
all insects are instinctive
some instinctive beings are not rational
FRESISO
no rational beings are insects
some insects are ugly
some ugly beings are not rational
Conditional
major is a conditional proposition, minor is a
categorical proposition which affirms or denies one of
the members of the conditional proposition
Rules
1. affirming the condition, the conditioned is affirmed
2. affirming the conditioned, the condition is not affirmed
3. removing the conditioned, the condition is removed
4. removing the condition, the conditioned is not removed
Moods
1. ponendoponens
affirming in the minor the condition enunciated in the major, the
conditioned enunciated in the major is affirmed in the conclusion
form 1
If a is b, x is y
now a is b
so x is y
form 2
If a is b, x is not y
now a is b
so x is not y
form 3
If a is not b, x is y
now a is not b
so x is y
form 4
If a is not b, x is not y
now a is not b
so x is not y
2. tollendotollens
precluding in the minor the conditioned enunciated in the major, the
condition enunciated in the major is precluded in the conclusion
form 1
If a is b, x is y
now x is not y
so a is not b
form 2
If a is b, x is not y
now x is y
so a is not b
form 3
If a is not b, x is y
now x is not y
so a is b
form 4
If a is not b, x is not y
now x is y
so a is b
Examples
condition and conditioned
have the same subject
conditional
if man is a mammal, then man is an animal
man is a mammal
so man is an animal
unconditional
all mammals are animals
man is a mammal
so man is an animal
condition and conditioned do
not have the same subject
conditional
if legs work, then man moves
legs work
so man moves
conditional divided
part 1
if man has legs, then man moves
man has legs
man moves
part 2
if legs work, then legs move man
legs work
legs move man
unconditional
part 1
all that has legs, moves
man has legs
man moves
part 2
all that works in man moves man
legs work
so legs move man
Disjunctive
major is properly a disjunctive proposition, minor is a
categorical proposition which affirms or denies one of
the members of the disjunctive proposition
Moods
1. ponendotollens
affirming the first member of the disjunction, the
second is precluded, the two being incompatible
The minor affirms one of the predicates
and the conclusion denies the other
form 1
either a is b, or x is y
now a is b
so x is not y
form 2
either a is b, or x is not y
now a is b
so x is y
form 3
either a is not b, or x is y
now a is not b
so x is not y
form 4
either a is not b, or x is not y
now a is not b
so x is y
2. tollendoponens
precluded the first member of the disjunction, the
second is affirmed, the two being incompatible
The minor denies one of the predicates
and the conclusion affirms the other
form 1
either a is b, or x is y
now a is not b
so x is y
form 2
either a is b, or x is not y
now a is not b
so x is not y
form 3
either a is not b, or x is y
now a is b
so x is y
form 4
either a is not b, or x is not y
now a is b
so x is not y
Examples
either communism is a true philosophy or
christianity is a true religion
now communism is a true philosophy
so christianity is not a true religion
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.
2 possibilities
the major either a is b, or x is y becomes:
if a is b then x is not y or else if x is y, then a is not b
Valid vs. Sound
Fallacies
Informal
Revelance
Appeal to Ignorance
Principle
statement p is unproved. notp is true.
statement notp is unproved. p is true.
Example
all unproven things are nonexistent
God is unproven
God is a nonexistent thing
Appeal to Authority
Principle
authority on subject x, L says accept statement p.
p is outside the scope of or not germane to the subject x.
Example
John is an athlete
John says that Apple is best
to be an athlete is to be an expert in technology
Appeal to the Circumstantial
Principle
Person L says argument A.
Person L's circumstance or character is not satisfactory.
Argument A is not a good argument.
Example
all who rob are not trustworthy
all burglars rob
all burglars are not trustworthy
Appeal to the Majority
Principle
snob appeal
Person L says statement p or argument A.
Person L is in the elite.
Statement p is true or argument A is good.
bandwagon
Most, many, or all persons believe statement p is true.
Statement p is true.
appeal to emotion
Emotions are used to express evidence for statement p
Statement p is true.
Example
a group decision is the valid choice
a groups says Pepsi is best
Pepsi is the valid choice
Appeal to Pity
Principle
Person L argues statement p or argument A.
L deserves pity because of circumstance y.
Circumstance y is irrelevant to p or A.
Statement p is true or argument A is good.
Example
Feeling bad for someone is more
important than princples
someone failed their exam and I feel
bad for them
I should pass the student so that I
do not feel bad
Appeal to Force
Principle
If statement p is accepted then bad event x will happen.
Statement p is accepted.
Event x is bad, dangerous, or threatening will happen
Example
If you oppose me you will fail
You opposed me
You will fail
Irrelevant Conclusion
Principle
Either premise A or premise B have
nothing to do with the conclusión.
Example
a hero is admired
someone who survives is a hero
someone who survives is a volunteer
Presumption
Accident
Principle
some principle that is true, then errs by
applying it to a specific case that is atypical.
Example
Women earn less than men earn for X work
Oprah Winfrey is a woman.
Oprah Winfrey earns less than men for X work
Converse Accident
Principle
a case that is atypical is presented, then
errs by deriving from it a general rule
Example
Dennis Rodman wears earrings and plays well
I wear earrings
I play well
False Cause
Principle
infers the presence of a cause because
events appear to occur in strict correlation
Example
The moon was full last night
Last night I overslept
The moon caused me to oversleep
Begging the Question
Principle
using the conclusion of an argument as one
of the premises offered in its own support
Example
All dogs are mammals and all mammals have hair.
Since animals with hair bear young, dogs bear young.
But all animals that bear live young are mammals.
Therefore, all dogs are mammals.
Complex Question
Principle
presupposes the truth of its own conclusion by including
it implicitly in the statement of the issue to be considered
Example
Have you tried to stop smoking?
If so, you admit to smoking.
If not, then you still smoke.
So, you smoke.
Formal
Syllogistic
Fallacy of the affirmative conclusion
Principle
nothing positive can be concluded from negative premises
Example
no fish are dogs
no dogs can fly
so all fish can fly
Fallacy of the negative conclusion
Principle
nothing negative can be concluded from positive premises
Example
every animal is an organism
every man is an animal
so no man es an organism
Fallacy of exclusive premises
Principle
invalid because both of its premises are negative
Example
no mammals are fish.
some fish are not whales.
so some whales are not mammals.
Fallacy of the undistributed middle
Principle
middle term in a categorical syllogism is not distributed
in either the minor premise or the major premise
Example
Africans are black
John is black
John is an African
Fallacy of four terms
Principle
the middle term must be the same idea in both premises
Example
every triangle is a rectilinear figure
no rectangle is a curvilinear figure
so no triangle is a curvilinear figure
Fallacy of the Illicit major
Principle
invalid because its major term is undistributed in the
major premise but distributed in the conclusion
Example
every triangle is a rectilinear figure
no rectangle is a triangle
so no rectangle is a rectilinear figure
Fallacy of the Illicit minor
Principle
invalid because its minor term is undistributed in the
minor premise but distributed in the conclusion
Example
all cats are felines.
all cats are mammals.
so all mammals are felines.
Symbolic
affirming the consequent
Principle
to infer the converse from the original statement
the conclusion can be false even when the premises are true.
Example
If this is a poodle then this is a dog
This is a dog
Therefore this a poodle
Denying the Antecedent
Principle
to infer the inverse from the original statement
the conclusion can be false even when the premises are true.
Example
If this is a poodle then this is a dog
This is not a poodle
Therefore this not a dog
Affirming a Disjunct
Principle
to conclude that one disjunct must be false because the other disjunct is true.
they may both be true because "or" is defined inclusively rather than exclusively.
Example
Max is a cat or Max is a mammal.
Max is a cat.
Therefore, Max is not a mammal.
Symbolic Logic
A statement letter of is defined as any uppercase
letter written with or without a numerical
subscript.
Lower case letters can also be used
A connective or operator of is any of
the signs ‘&’, ‘v‘, ‘→’, ‘↔’, and ‘¬’.
‘&’ or ‘∧’ or ‘•’ is equal to "and" (conjunction)
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.
‘v‘ or "+" is equal to "or" (disjunction)
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.
The sign ‘v‘ is used for disjunction in the inclusive sense.
‘→’ or ‘⊃’ is equal to "if...then..."
(material implication)
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).
The "if...then..." here is really a "if...
then might be"
If Marie is in Paris, then she is in
France. (True if both are true).
"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.)
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.)
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."
‘↔’ or ‘≡’ is equal to "if and only
if"...then..." (material equivalence)
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 truthvalues.
If Romney is President, then he won
Florida. True if both are true.)
If Romney is NOT President, then
he DID NOT WIN Florida.
"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.
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.
‘¬’ or ‘~’ or ‘–’ is equal to "not" or a negative (negation)
( ) parentheses are used in forming even more complex statements.
I ↔ (C & P)
(I ↔ C) & P
¨ ¨ quotations are to talk about words and
sentences in other languages.
“(I ↔ C) & P”
Greek letters ‘α’, ‘β’, etc., are used for any object
language expression of a certain designated form.
α v β is the complex statement
“(I ↔ C) v (P & C)”
"∴" is used instead of the horizontal line for the conclusion
A wellformed formula corresponds to the notion of a
grammatically correct or properly constructed
statement
“¬(Q v ¬R)” is grammatical because
it is a wellformed formula
The string of symbols, “)¬Q¬v(↔P&”, while consisting
entirely of symbols, is not grammatical because it is
not wellformed.
The ¬ operator has higher precedence than ∧; ∧ has higher
precedence than ∨; and ∨ has higher precedence than → or ↔
When an operand is surrounded by operators of equal precedence,
the operand associates to the left.
EXAMEPLES
((P & Q) → ¬R)
If John is in Paris and if he is eating pie
then he is not outside of France
[(p → q) ∧ p] → q
If he is 10 years old, then he can watch the movie
...and he really is 10 years old
so he can watch the movie
[(¬p v q) ∧ (q → r)] → (r v ¬p)
Either you are not a man or you are intelligent
... and if you are intelligent then you can think
Then... either you can think or you are not a man
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